1 - Andrzej Materka

Transcription

1 - Andrzej Materka

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POLITECHNIKA L6DZKA
WYDZIAt ELEKTROTECHNIKI I ELEKTRONIKI
Andrzej Materka
Analog Electronics
Lecture notes
L6dz,1998
Recenzent / Reviewer
Prof, dr in?. Jersgi Ludnski
Sklad komputerowy, przygotowanie rysunkow i przelamanie tekstu /
Computer text-editing, figure artworks, text formatting
Andrgj Materka
Projekt graficzny okladki / Cover graphics design
Wojtek Materka
© Copyright by Wydzial Elektrotechniki i Elektroniki Politechniki Lodzkiej
ISBN 83-87202-06-1
Na okladce: Seria dziewi^ciu analogowych ukladow scalonych, z ktorych kazdy pelni
funkcje. sztucznej sieci neuronowej Kohonena. Uklady zaprojektowano w zespole
naukowo-badawczym kierowanym przez Autora w Instytucie Elektroniki Politechniki
Lodzkiej, we wspolpracy z prof. Alexisem De Vosem z Uniwersytetu w Gandawie.
Zostaly one wyprodukowane przez fabryke. ukladow scalonych 'Alcatel' w Belgii. Uklad
sieci Kohonena pozwala na praktyczna. realizacje. idei uczenia nienadzorowanego, m. in.
do celow rozpoznawania obrazow.
Front cover: A lot of 9 analog integrated circuits, each performing the function of
Kohonen-type artificial neural network. The research team led by the Author in the
Institute of Electronics, Technical University of Lodz - in cooperation with Professor
Alexis De Vos of Gent University - has designed the circuits. The 'Alcatel' silicon
foundry in Belgium fabricated the ICs. The Kohonen network implements the idea of
unsupervised learning for pattern recognition - among many other applications.
QM- 0041 ml&oif
Contents
1 Introduction, 9
1.1
1.2
1.3
1.4
1.5
1.6
Signals and Their Spectra, 9
Analog and Digital Signals, 12
Electronic-System Block Diagrams, 15
Information-Processing Electronics Versus Power Electronics, 16
Behavior of Circuit Components Versus Frequency, 16
Summary, 17
2 Amplifiers: Behavioral Description, 18
2.1
Basic Concepts, 18
2.2
Cascaded Amplifiers, 22
2.3
Power Supplies and Efficiency, 23
2.4
Decibel Notation, 25
2.5
Frequency Response, 26
2.6
The Miller Theorem, 34
2.7
Linear Distortion, 37
2.8
Pulse Response, 39
2.9
Nonlinear Distortion, 44
2.10 Summary, 46
3 D i o d e Circuits, 48
3.1
The Ideal Diode, 48
3.2
Terminal Characteristics of Semiconductor Diodes, 52
3.3
Analysis of Diode Circuits, 56
3.4
The Diode Small-Signal Model at Low Frequencies, 61
3.5
Rectifier Circuits, 66
3.6
Zener-Diode Voltage Regulator Circuits, 70
3.7
Wave-Shaping Circuits, 73
3.b
Switching and High-Frequency Behavior of thepn Junction, 76
3.9
Special Diodes, 84
3.10 Summary, 86
4 Field-Effect Transistor Circuits, 88
4.1
The »-channel Junction FET, 88
4.2
Metal-Oxide-Semiconductor FETs, 91
4.3
Load-Line Analysis of a Simple J F E T Amplifier, 94
4.4
The Self-Bias Circuit, 96
4.5
The Fixed- Plus Self-Bias Circuit, 98
4.6
The Small-Signal Equivalent Circuit, 100
4.7
Basic Small-Signal FET Amplifier Circuits, 103
4.8
The FET as a Voltage-Controlled Resistance, 113
4.9
CMOS Analog Switch, 115
4.10
CMOS Logic Circuits, 116
4.11
FET Dynamic Circuit Model, 121
4.12
Summary, 125
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5 Bipolar Transistor Circuits, 127
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
Load-Line Analysis of a Common-Emitter Amplifier, 130
The pnp Bipolar Junction Transistor, 133
Secondary Effects, 135
Large-Signal dc BJT Models, 139
Large-Signal dc Analysis of BJT Circuits, 141
Four-Resistor Bias Circuit, 145
Small-Signal Equivalent Circuits, 149
The Common-Emitter Amplifier, 150
The Emitter Follower, 153
Review of Small-Signal Equivalent Circuit Analysis, 156
The Common-Emitter Hybrid-Parameter Small-Signal Model, 160
The Hybrid-* Model, 161
Bipolar Transistor Behavior at High Frequencies, 164
Large-Signal Dynamic Model for die BJT, 173
Switching Behavior of die BJT, 177
Summary, 181
6 Feedback Circuits, 183
6.1
Effects on Sensitivity, Bandwidth and Distortion, 184
6.2
Feedback Types, 194
6.3
Effect of Feedback Types on Input and Output Impedance, 196
6.4
Summary of the Effect of Various Feedback Types, 198
6.5
Practical Feedback Networks, 199
6.6
Stability of Feedback Amplifiers, 202
6.7
Sinusoidal Oscillators, 205
6.8
Summary, 221
References, 223
Review Questions, 224
Problems, 230
Appendix
A.
B.
C.
D.
Nominal Values and the Color Code for Resistors, 256
Introduction to PSpice, 258
English-Polish Dictionary of Selected Terms, 266
Manufacturers' Date Sheets for 2N2222A transistor, 272
-4-
Przedmowa / Preface
Niniejszy skrypt powstal w wyniku zebiania i uporzajikowania notatek z wykladow
Analog Electronics oraz Electronic Devices and Systems, ktore mialem przyjemnosc
prowadzic w Politechnice Lodzkiej dla kilku grup studenckich IFE (International Facility
of Engineering) w latach 1995-1997. Opisuje. w nim zasady dzialania i wlasciwosci
podstawowych ukiadow elektronicznych - analogowych.
Zakres materialu opisanego w skrypcie w niewielkim jedynie stopniu obejmuje olbrzymia,
rozmaitosc znanych ukiadow elektronicznych. Staralem si? zawrzec w nim informacje
podstawowe, zdefiniowac terminologie. i omowic elementarne obwody zawieraja.ce diody,
tranzystory polowe, bipolarne a takze obwody ze sprz?zeniem zwrotnym. Oprocz
wiadomosci teoretycznych i przykladow symulacji komputerowych skrypt zawiera
cwiczenia rachunkowe, ktore pozwalajq. na ilosciowe zilustrowanie wlasciwosci ukiadow,
a w niektorych przypadkach lepsze zrozumieiiie zalozeri projektowych. Pytania
powtorkowe zwracaj% uwage na zasadnicze elementy tresci merytorycznej
poszczegolnych rozdzialow i moga. bye pomocne w porza_dkowaniu wiedzy przed
egzaminem.
Program studiow inzynierskich w IFE zawiera zmniejszona^ liczbe. godzin zajee z
ukiadow elektronicznych w porownaniu ze studiami magisterskimi na Wydziale
Elektrotechniki i Elektroniki PL. Dla lepszego wykorzystania przydzielonych limitow
czasowych i zwi?kszenia skutecznosci przekazu wiedzy, wyklady swoje ilustrowaiem
dose obficie przyktadami obhezen za pomoca. programu SPICE. W tym celu
prowadzilem wyklady w laboratorium komputerowym, w ktorym kazdy ze studentow
mogl samodzielnie realizowac symulacje omawianych ukiadow. Ta forma zajec
dydaktycznych dowiodla swojej uzytecznosci i jest godna polecenia innym,
zainteresowanym wykladowcom. Przyklady wybranych programow w j?zyku SPICE
zawarlem w skrypcie. Zachecam Czytelnikow do korzystania z tych przykladow, a takze
do pisania wlasnych programow i wykonywania symulacji ukiadow. Symulacje takie nie
moga. zastapic, bez wa_tpienia, doswiadczeri praktycznych i pomiarow rzeczywistego
obwodu. Sa_ one jednak mniej kosztowne, trwaj% krocej, a w przypadku projektowania
ukiadow scalonych sa. jedyna. forma, sprawdzania poprawnosci projektu przed faza.
realizacji. Zainteresowanym mog? udost?pnic darmowa. edukacyjna_ wersje. programu
PSpice (dwie dyskietM 3.25"). Przy tej okazji chcialbym podziekowac Wydawnictwu
Prentice Hall, ktore przyslalo mi ten program wraz z ksia^zka. [12] do oceny, kiedy
pracowalem jako wykladowca w Monash University w Australii.
Z faktu, iz studenci IFE — sluchacze moich wykladow — z niecierpliwoscia. pytali o
kolejne porcje notatek wnosz?, ze napisanie skryptu bylo potrzebne. Wypelnia on
okreslona. iuk? w krajowym pismiennictwie akademickim, dotyczaca. zwlaszcza
materialow optacowanych w jezykach obcych, co jest szczegolnie wazne w okresie
otwierania si? Polski na swiat Z drugiej strony, polskoj?zyczni studenci IFE powinm
miec mozliwosc zapoznania si? z rodzima. terminologie w dziedzinie ich studiow. Na ich
potrzeby opracowalem slownik angielsko-polski terminow uzywanych w skrypcie,
zawarty w Dodatku. Trzeba podkreslic, ze lektura niniejszego skryptu nie moze zastaj>ic
prawdziwie pogl?bionych studiow przedmiotu. W tym celu odsylam Czytelnikow do
-5-
dziel obszetniejszych, z ktotych kotzystalem przygotowujajc notatki do moich wykladow.
List? tych prac zawieta spis literatury.
Na koniec chc? zaznaczyc, ze optacowanie niniejsze nie mogloby powstac gdyby moja
Zona i moj Syn nie zgodzili si? na to, abym 30 weekendow toku 1997 sp?dzil z edytotem
tekstu. Setdecznie Im za to dziekuj?. Jestem towniez wdzi?czny Dziekanowi Wydzialu
Elekttotechniki i Elekttoniki Politechniki Lodzkiej, Panu Ptofesotowi Doktotowi Janowi
Leszczyriskiemu za ufundowanie honotatium, ktote pozwoli mi na modetnizacj? mojego
komputeta. Dzi?ki temu zmniejsz? troch? dystans do czolowki tego fascynuja.cego i
ptzeiazaja_cego wyscigu powszechniej komputetyzacji, u ktoiej podstaw znajduje si?
post?p w dziedzinie elekttoniki. No doubts it is...
Zycz? Panstwu ptzyjemnej i owocnej lektuty Andrzej Materka.
Lodz, wlutym 1998.
Preface / Przedmowa
This textbook is a systematized collection of the notes of lectures that I had a pleasure to
deliver in 1995-1997 to a number of student groups at the International Faculty of
Engineering (IFE), Technical University of Lodz. It describes the principle of operation
and properties of basic analog electronic circuits.
The material included in this textbook covers the huge variety of known electronic
circuits to a very limited degree only. I have made efforts to introduce fundamental
concepts, define terminology, and discuss elementary circuits that comprise diodes, fieldeffect and bipolar transistors, as well as circuits with feedback. Besides the theory and
computer simulation examples, the lecture notes include problems and exercises for the
student, which help quantitatively illustrate various circuit properties and provide better
understanding of design specifications in some cases. Review questions'provided turn the
reader's attention to essential elements of the theory presented in individual chapters;
they can be helpful to sort things out before exams.
The 4-year BEng study program at IFE allows less class time for Analog Electronics than
the 5-year MSc studies at the TUL's Faculty of Electrical and Electronic Engineering do.
To make better use of the student contact-hours allocated, I kept amply illustrating my
lectures with computer simulation examples, using the SPICE program. To achieve that,
a computer laboratory was chosen to be the place to give the lectures, where almost
every listener had an access to a PC terminal to individually exercise the virtual
experiments with the circuits under study. This form of carrying on the teaching has
proven its usefulness and deserves recommendation to other interested lecturers. I
integrated examples of the SPICE code with the text of the lecture notes. I encourage
the readers to use these examples and to write their own programs for circuit simulation.
Simulations can not, of course, replace practical experiments and measurements of
physical circuits. They are, however, less expensive, last shorter, and in the case of
integrated circuits are the only means of verifying the circuit correctness before its
fabrication. The interested readers can copy a free educational version of the PSpice
program (two 3.5 inches floppies) from my resources. On this occasion I wish to thank
the Prentice Hall publishing house representative who sent me this program along with
the book [12] for evaluation when I was with Monash University, Melbourne, Australia,
working as a lecturer.
From the fact that IFE students - the listeners to my lectures — kept impatiently asking
me about consecutive printouts of the lecture notes I gather writing the textbook was
needed. It fills up a gap in the country's academic literature that concerns especially
foreign-language material that is particularly important in the period of Poland's opening
to the outside world. On the other hand, those of students who are Polish native
speakers should be given an opportunity to familiarize with their mother-tongue
terminology in the field of their studies. I then elaborated a short English-Polish
dictionary of selected terms utilized in the textbook, which is included in the Appendix.
It should be emphasized that reading this textbook can not replace truly thorough studies
of ^he subject. To do such studies, I recommend the readers to refer to more extensive
-7-
books that I used to prepare the lecture notes. The list of these works is provided at the
end of the text
Finally, I wish to point out that this textbook would not appear if my wife and my son
would not agree that I might spend 30 week-ends in 1997 with the text editor. I do
sincerely thank them for this opportunity. I am also grateful to the Dean of the Faculty
of Electrical and Electronic Engineering, Professor Jan Leszczynski, for founding a
scholarship that will help upgrade my computer. Then I will be able to cut a litde bit die
distance to the leaders of the fascinating and frightening race of widespread
computerization that is actually based on the progress in electronics. Bez wa.tpienia tak
jest...
I wish you enjoyable and fruitful reading,
Andrzej Materka.
Lodz, in February 1998.
Analog Electronics /Introduction
1. Introduction
In these lecture notes we shall study electronic devices and their interconnections that form discrete
or integrated circuits (ICs). These circuits are devised to perform a variety of functions within
electronic systems' of interest The well-known examples of contemporary electronic systems are
Walkman radio, TV set, PC computer, TV satellite receiver, computerized monitors for patients in
intensive-care units, cellular phones, and the like. There exist also a variety of complex systems,
which do not operate based mainly on electrons' movement but nevertheless would not function
properly without an embedded electronic circuitry. These include aircraft, photocopying machines,
weather satellites, etc. Electronic systems control fuel mixture and ignition timing to maximize
performance and minimize undesirable emissions from automobile engines. It is in fact impossible
to overestimate the role of electronics in the modem society. Perhaps there is no human activity
area, ranging from daily life to space exploration, in which we don't utilize the electronic devices.
The foundations of electronics were established by the observations of M. Faraday (1791-1867) and
the discovery of electron by J. J. Thomson in 1897, but its development proceeded relatively slowly
until the importance of radar became apparent at the beginning of World War II. During that
period and extending until about 1955, the expanding field of electronics depended heavily on the
principle of electron emission from the hot cathode of a vacuum tube. By modern standards,
vacuum tube electronics was expensive, bulky, hot, unreliable and even dangerous because of the
high voltages present in the tubes. With the discovery of the transistor by J. Bardeen, W. H. Brittain
and W. B. Schockley in 1948, the stage was set for the solid-state electronics explosion of today.
This technological explosion is often compared to the industrial revolution.
Since the invention of transistor the technology has progressed from an individually prepared
laboratory device (a tube) to millions of components (diodes, transistors) on a single silicon wafer
only a millimeter a side. Coupled with this astounding miniaturization has been a corresponding
decrease in the price per component For example, a current popular microcomputer chip,
containing more than one million components, costs about the same as one of the electron tubes of
earlier electronics. At the same time, with the increased internal complexity and overall functionality
of solid-1 tate chips, the number of external components is highly reduced as compared to vacuum
tube electronic circuits. Also, the temperature of device operation is decreased. These two main
factors lead to a much higher reliability of solid state electronic systems.
1.1 Signals and Their Spectra
The main function of most electronic circuits is either to process or to generate signals. Signals
contain information about a variety of things in our physical world. For example, information about
the weather is contained in signals that represent the air temperature, pressure, the wind speed, etc.
The voice of a radio announcer reading the news into the microphone provides an acoustic signal
that carries information about world affairs. To monitor and diagnose the status of the human
body, biopotentials are measured on the surface of the skin to represent the activities of the heart
(ECG - the electrocardiogram), the brain (EEG - electroencephalogram), the muscles (EMG electromyogram) and others.
Processing of signals is needed to extract information from them, for an observer (be it a human or
a machine). Electronic circuits and systems most conveniently perform the signal processing. For
this to be possible, the signals, which are not originally in the form of an electrical signal, must first
-9-
be converted into a voltage or current Devices known as transducers accomplish this process. A
variety of transducers exist, each suitable for one of the various forms of physical signals. For
instance, a microphone is in effect a pressure transducer. It is not our purpose here to study
transducers; rather, we shall assume that the signals of interest already exist in the electrical domain.
For the purpose of our study the signals will be represented by one of the equivalent forms shown
in Fig. 1.1. In Fig. 1.1(a) the signal is represented by a voltage source v//) having a source resistance
R r In the alternate representation of Fig. 1.1(b), the signal is represented by a current source /,(/)
having a source resistance Rr Although the two representations are equivalent, the one shown in
Fig. 1.1(a) (known as the Thevenin form) is preferred when R, is low. The representation of Fig.
1.1(b) (known as the Norton form) is preferred when Rs is high.
From the discussion above it should be apparent that a signal is a time-varying quantity that can be
represented by a graph such as those shown in Fig. 1.2. In fact, the information content is
represented in changes in signal magnitude as time progresses. The overall level of the magnitude is
one of the signal characteristic features. Table 1.1 gives examples of the range of magnitudes of
different signals. In general, waveforms are difficult to characterize mathematically. In other words,
it is difficult to describe succincdy an arbitrary looking waveform, such as one of those of Fig. 1.2.
TABLE 1.1
MAGNITUDE AND INTERNAL RESISTANCE O F SELECTED SIGNAL SOURCES
SIGNAL SOURCE
ECG
Dynamic microphone
Radio antenna
Ionic chamber
RANGE OF
MAGNITUDES
<1mV
<10mV
0.1uV-1mV
10fA-10nA
INTERNAL RESISTANCE
1kft-1Mto
100fl
50-300Q
>1GQ
An extremely useful characterization of a signal is in terms of its frequency spectrum. Such a
description of signals is obtained through the mathematical tools of Fourier series and Fourier
transform. They provide a model for a signal, which is a sum of sine-wave signals of different
frequencies, amplitudes and phase angles. (This makes the sine wave a very important signal in the
analysis, design and testing of electronic circuits.) The Fourier series is utilized to represent signals,
which are periodic functions of time. On the other hand, die Fourier transform is more general and
can be used to obtain die frequency spectrum of a signal whose waveform is an arbitrary function
of time. The frequency spectrum of a signal is the amplitude of its sine-wave components expressed
as a function of their frequencies. Examples of signals and their spectra are shown in Fig. 1.3. Note
diat periodic signals possess discrete spectra as illustrated by Fig. 1.3(a), whereas the spectrum of an
arbitrary waveform is a continuous function of frequency, as shown in Fig. 1.3(b). In other words,
for periodic signals of angular frequency CO^, the spectrum consists of discrete frequencies (at the
fundamental frequency (O0 and its harmonics) and die spectrum of a non-periodic signal contains, in
general, consists of components of all possible frequencies.
-10-
The essential parts of the spectra of practical signals are usually confined to relatively shon
segments of the frequency axis. For instance, die spectrum of audible sounds such as speech and
music extends from about 20Hz to about 20kHz - a frequency range known as die audio band.
This observation is very useful in the processing of such signals and important for electronic circuit
designers. This is because the properties of signal processing circuits have to match the signal
properties. For example, there is no point in amplifying the signal in the frequency range in which
there is no signal component of significant energy. This would mean amplification of noise and
interference, which corrupt the information. Sometimes the range of frequencies occupied by a
signal can be changed. In radio communication, this is done so that different transmitter occupies
-11-
different frequency ranges. Then a receiver can separate the desired signal from the others by the
use of frequency-selective electrical circuits called filters.
When we approach the design of an electronic circuit to process a signal, one of our first questions
should be "What is the frequency range of the signal?" For example, we will see that ICs known as
operational amplifiers can be very useful but they are limited to fairly low frequencies, usually below
1MHz. If we need an amplifier for Channel 10 television we can rule out the use of operational
amplifiers. Table 1.2 provides examples of the frequency ranges of some signals of practical interest.
We conclude this section by noting that a signal can be represented either by the manner its
waveform varies with time or in terms of its frequency spectrum. The two alternative
representations are known as the time-domain and the frequency-domain representations,
respectively. The frequency representation of a signal v(t) will be denoted by the symbol V(o>).
TABLE 1.2 FREQUENCY RANGES OF SELECTED SIGNALS
SIGNAL
Electrocardiogram
Audible sounds
Video signals
AM radio broadcasting
TV broadcasting
FM radio broadcasting
GSM cellular phone
Satellite TV broadcasting
FREQUENCY RANGE
0.05-100Hz
20 Hz-20kHz
0-6 MHz
150- 285kHz (LW)
525 - 1600kHz (MW)
48.5 - 56.5MHz (channel 1 VHF)
222 - 230MHz (channel 12 VHF)
470 - 478 MHz (channel 21 UHF)
790 - 798MHz (channel 61 UHF)
66 - 73MHz, 87.5 - 108MHz
890 - 915MHz, 935 - 960MHz
11.7 - 12.5GHz
1.2 Analog and Digital Signals
The voltage waveforms depicted in Fig. 1.2 are analog signals. The name derives from the fact
that such a signal is analogous to the physical signal that it represents. The analog signal is a
continuous function of time. Its magnitude can take on any value. Electronic circuits that process
such signals are known as analog circuits. A variety of analog circuits will be studied in this
subject.
An alternative form of signal representation is that of sequence of numbers, each number
representing the signal magnitude at an instant of time. The resulting signal is called a discrete-time
signal or, shortly, discrete signal The signal is converted from analog to discrete form in the
process of sampling. This process is illustrated in Fig. 1.4. The time instants tv /„ /?,... are marked
along the time axis. The magnitude of the signal is measured (sampled) at each of these time
instants.
Yet another useful form of signal representation is that of digital signal. It is obtained from a
discrete signal by encoding the magnitude of signal samples with the use of finite-word-length
binary code. In the example shown in Fig. 1.4, each sample value is represented by a 3-bit code
word, corresponding to the amplitude zone into which the sample falls. Each sample value is
converted into a code word, which in turn can be represented by a digital two-level waveform as
shown in the lower panel of the figure. The digital signal is thus a sequence of encoded numbers.
-12-
The circuit for conversion of signals in this manner is called an analog-to-digital converter
(ADC). Conversely, a digital-to-analog converter (DAQ converts digital signals back to analog
form. Now if we represent the magnitude of each of the signal samples in Fig. 1.4 by a number
having a finite number of digits, then the signal magnitude will no longer be continuous; rather, it is
said to be quantized or digitized. The resulting digital signal then is simply a sequence of gn^nîypH
numbers that represents the magnitudes of the successive signal samples.
An example of analog, discrete and digital signal for a real world ECG waveform is shown in Fig.
1.5. Please note that the discrete signal of Fig. 1.5(b) is defined only at the sampling instants. It no
longer is a continuous function of time. However, since the magnitude of each sample can take any
value in a continuous range, a discrete signal is still an analog signal. Perfect (error-free)
reconstruction of analog signal is possible given the discrete-time signal, provided it has been
sampled at sufficiendy high rate. Obviously, there is no way to recover the original analog signal
from the digital signal. We will discuss these issues in more detail below.
1.5
1
vffltmV]
0.5
0
*5,
0
0.1
0.2
0.3
0.4
tim*[s)
0.5
0.6
0.7
0.8
30
40
50
60
sampling instance indax, n
70
80
'l.5
v(n) [mV]
10
20
Figure 1.5 Sampling of continuous-time analog signal (upper panel) results in the discrete-time signal (lower
panel). Quantization of the discrete signal produces the following sequence of numbers:
{0,0,5,11,14,14,11,7,4,1,0,1,1,1,2,3,3,3,3^6^0,51,90,105,77,27,-12,-28,-24,-10,
-2,1,1,1,0,0,0,0,0,0,1,2,3,4,5,7,8,10,11,12,12,14,18^2^6,31,36,41,46,51,55,56,57,57,
56,54,49,43,36,29,23,17,12,8,5,2,1,0,0} which represents the digital signal.
The rate at which a signal must be sampled depends on the frequency content of the signal If a
signal contains no components with frequencies higher than fy, the signal can be exacuy
-13-
reconstructed from its samples, provided that the sampling rate is selected to be more than twice fb.
(This is known as Shannon-Kotelnikov theorem.) For example, audio signals have a highest
frequency of less than 20kHz. Therefore, the minimum sampling rate that should be used for audio
signals is 40 kHz. Practical considerations require selection of a sampling frequency somewhat
higher than the theoretical minimum. For instance, audio compact-disc technology converts audio
signals to digital form with a sampling rate of 44.1kHz. Naturally, it is desirable to use the lowest
practical sampling rate to minimize the amount of data (in the form of code words) to be stored,
transmitted or manipulated.
The second consideration important in converting analog signals to digital form is the number of
magnitude zones to be used. Exact signal amplitudes cannot be represented, because all amplitudes
falling into a given zone have the same code word. Thus, when a DAC converts the code words to
form the original analog waveform, it is only possible to reconstruct an approximation to the
original signal - the reconstructed voltage is constant within each zone. This is illustrated in Fig. 1.6.
Thus some quantization error exists between the original signals and the reconstruction. Using a
large number of zones can reduce this error, which requires a longer code word for each sample
The number N of amplitude zones is related to the number of bits k by
N = 2*
(1.1)
Thus if we are using an 8-bit {k=S) ADC, there are N=f—256 amplitude zones. In compact-disc
technology, 16-bit words are used to represent sample values. With this number of bits, it is rather
difficult for a listener to detect the effects of quantization error on the reconstructed audio signal.
Reconstruction
CMfclMtftiS
I
"
jj
*Î/
.—Analog
I \
/
signal
1^
*"
,p - f- - t- - i
I
~
Quantisation
error
Y
Figure 1.6 Quantization error occurs when analog signal is reconstructed from its digital form
Electronic circuits that process digital signals are called digital circuits. The digital computer is a
system constructed mostly of digital circuits. Digital processing of signals has become quite popular
primarily because of the tremendous advances made in the design and fabrication of digital circuits.
Digital processing of signals is economic and reliable. Furthermore it allows a wide variety of
processing functions to be performed - functions that are either impossible or impractical to
implement by analog means. Nevertheless, most of the signals in the physical wodd are analog
Also, there remain many signal-processing tasks that are best performed by analog circuits. They
refer especially to high-frequency signals, but apply even to digital circuits themselves, which in fact
are analog dynamical systems that we interpret as digital ones. It follows that a good electronics
engineer must be proficient in both forms of signal processing. Such is the philosophy adopted in
the contemporary electronic engineering study programs.
The advantages of digital electronics are either coming through the use of digital electronics or are
related to limitations of analog electronics. They are listed below.
(1) Advantages of digital signal processing that originate in the features of digital electronic systems
• flexibility (digital circuits are programmable),
• lower sensitivity to external (e.g. temperature) and internal (e.g. aging and drift) effects,
-14-
• accuracy can be controlled by selecting the word length to represent signal samples,
• circuits are reproducible (no trimming or tuning during manufacture),
• circuits are easier to manufacture in IC technology (no large Ls or Cs).
(2) Advantages of digital signal processing related to limitations of analog electronics
• "ideal memory" to store signals for an infinite time is possible with the digital techniques;
thus very low frequency signals can be processed with no need for .large Ls and Cs,
• linear phase filters - not available in analog electronics,
• circuits for exact compensation of two effects,
• adaptive systems,
• precise signal transforms,
• possibility of processing 2D signals (images).
The main
•
•
•
•
disadvantages of digital electronics are as follows:
more supply power required (passive digital circuits do not yet exist),
restricted to low-frequency applications,
when used in analog environment, often complex AD and DA converters are required
difficulties with AD and DA conversion of very weak and very strong signals;
complicated analog pre- and post-amplifiers are required in such cases,
• the same information (e.g. music signal) requires larger bandwidth as a digital signal than
it does as the analog one.
The analog circuits are
• less accurate,
• sensitive to noise,
• advantageous for high-frequency, small- and large-signal applications.
Modem and future systems contain both analog and digital circuits. We call them mixed-sigi* «J
systems.
1.3 Electronic System Block Diagrams
Electronic systems are composed of subsystems or functional blocks. These functional blocks can
be categorized as amplifiers, filters, signal sources, wave-shaping circuits, digital logic
circuits, power supplies and converters. Briefly, we can say that amplifiers increase the power
level of signals, filters separate desired signals from undesired signals and noise, signal sources
generate waveforms such as sinusoids or square waves, wave-shaping circuits change one waveform
into anomer (sinusoid to square wave, for example), power supplies provide necessary dc (direct
current) power to the other functional blocks, and converters change signals from analog form to
digital form, or vice versa. In section 1.6 we will consider the external characteristics of amplifiers in
some detail.
The block diagram of a typical AM radio is shown in Fig. 1.7, as an example of a simple electronic
system. Notice that mere are three amplifiers and two filters. The local oscillator is an example of a
signal source, and a peak detector is a special case of wave-shaping circuit The complete system
description would include detailed specification of each block. For example, the gain, input
impedance, output impedance, the bandwidth (range of frequencies where the gain does not drop
below a specified value) of each amplifier would be given. (We define these terms later in uiis
chapter.) Each functional block in turn consists of a circuit composed of resistors, capacitors,
transistors, integrated circuits, inductive coils, and other devices.
-15-
The main goal of this text is to introduce basic electronic circuits comprising diodes, bipolar
transistors, field-effect transistors and opamps, in terms of their function and principle of operation.
At present, due to the ever-increasing complexity of electronic circuits and systems, no circuit can
be designed properly without the use of computers and adequate CAD software. We will then use
one of the most popular programs for circuit simulation, called PSpice, to investigate the properties
of the circuits under study. This approach, integrating the theory and simplified mathematical
analysis with computer laboratory exercises will, we hope, prepare the readers for the circuit design
methodologies they will "encounter as practicing engineers. A set of practical laboratory exercises is
provided also to further strengthen the knowledge and to familiarize with measurement techniques
typical for low-frequency analog electronics.
1.4 Information-Processing Electronics versus Power Electronics
Many electronic systems fall into one or more of the following categories: communication systems,
medical electronics, instrumentation, control systems, and computer systems. A unifying aspect of
these categories is that diey all involve collection and processing of information-bearing signals.
Thus the primary concern of many electronic systems is to extract, store, transport or process
information in a signal.
Often, systems are also required to deliver substantial power to an output device. Certainly, this is
true in an audio system for which power must be delivered to a speaker to produce the desired
sound level A cardiac pacemaker uses information extracted from the electrical signals produced by
the heart to determine when to apply a stimulus in the form of a minute pulse of electricity to
ensure proper pumping action. Although the output power of a pacemaker is very small, it is
necessary to consider the efficiency of its circuits to ensure long battery life.
Many systems are concerned mainly with the power content of signals rather than information. For
example, we might want to deliver ac (alternating current) electrical power, converted from dc
supplied by batteries, to a computer even when the ac line power fails.
1.5 Behavior of Circuit Components versus Frequency
The way that a component behaves and the theoretical model we must use to represent it depend
on the frequency of operation. For example, a simple 1000Q resistance can model the 1000Q0.25W carbon-film resistor at frequencies of a few kilohertz, but at several hundred megahertz the
more complex circuit model shown in Fig. 1.8 is needed for good accuracy. The small capacitance
in parallel with the resistance has such a high impedance at low frequencies that it can be neglected.
-16-
However, it cannot be neglected at high frequencies. Similarly, the inductance has very low
impedance and can be neglected at low frequencies. (The inductance is associated with the magnetic
field surrounding the leads when current flows in the resistor. Thus the exact inductance value
depends on die length of the wire leads used in making connection to the remainder of the circuit.)
We see later that the circuit models used for transistors at high frequencies include capacitances that
can be neglected at low frequencies.
•A/W
1000 ohm
w
nH
Figure 1.8 Circuit model for a 1000ft 0.25W carbon-film resistor
Even the construction methods for a circuit depend on its operating frequency range. Many circuits
intended to operate at low frequencies, say below 100 kHz, can be constructed by plugging
individual components into a prototype board and wiring them together without much regard to
the length of connecting wires or the distance between them. On the other hand, circuits intended
to operate at several hundred megahertz must be carefully constructed with regard to the layout and
lead length.
It is somewhat difficult to make a definite boundary between "low" frequencies for which stray
inductance and capacitance are not troublesome and "high" frequencies, for which they must be
considered. In a high-impedance circuit for which the components have impedance of several
megaohms or more, stray capacitance will be significant at lower frequencies than for a lowerimpedance circuits with impedance less than 100Q. In general, however, we do not need to
consider stray wiring effects below about 100kHz. Usually, we must take them into consideration
above about 10MHz.
1.6 Summary
Main function of electronic circuits is to process or generate signals. Either the Thevenin form or
the Norton form can represent an electrical signal source. The sine-wave signal is completely
characterized by its peak value (or rms value which is the peak/ V2 ), its frequency, and its phase
with respect to an arbitrary reference time. A signal can be represented either by its waveform
versus time, or as a sum of sinusoids. The latter representation is known as the frequency spectrum
of the signal. Analog signals have magnitudes that can assume any value. Electronic circuits that
process analog signals are called analog circuits. Sampling the magnitude of an analog signal at
discrete instants of time and representing each signal sample by a number, results in a digital signal.
Digital signals are processed by digital circuits. Modem and future systems contain both analog and
digital circuits. Systems are composed of subsystems or functional blocks. The functional blocks are
signal sources, amplifiers, niters, wave-shaping circuits, digital logic circuits, power supplies and
converters. Electronic systems fall into the category of either information-processing or powerprocessing electronics. Any electronic system is made up of components (resistors, capacitors,
inductors, semiconductor devices, etc.). Electrical properties of components depend on the signal
frequency.
BIBLIOTEKA GtOWNA PL
W36^ S
I
-17-
Analog Electronics /Amplifiers
2 Amplifiers: Behavioral Description
In this chapter we introduce most important functional blocks that are employed in almost every
electronic system, namely signal amplifiers. We will define specifications and characteristics of
amplifiers, which describe their behavior as seen by an external observer. In this system-like
approach we will not discuss internal structure of amplifiers, leaving this topic for the later chapters.
2.1 Basic Concepts
Signal amplification is conceptually the simplest signal-processing task. The need for amplification
arises because transducers provide signals that are said to be "weak", that is in the microvolt (uV)
and millivolt (mV) range. An example is a signal from a microphone, which is of about lmV peak
as one speaks to this transducer. Such a weak signal cannot produce any noticeable acoustic effect if
used to drive a loudspeaker. Much stronger signal is needed for a loudspeaker in order to obtain a
louder version of the sound entering the microphone. Thus the signal from the microphone is used
as the input to an amplifier with a voltage gain of 10000 to produce an output signal which is a peak
value of 10 V. This "strong" signal is applied to the loudspeaker which produces a loud replica of
the microphone's input.
The concept of signal amplification is illustrated in Fig. 2.1. The signal source produces a signal t\(t)
that is applied to the input terminals of the amplifier, which generates an output signal
vo(0 = 4,v,(0
(2-1)
across the load resistance RL connected to the output terminals. The constant A^ is called the
voltage gain of the amplifier. Often, the voltage gain is much larger than unity, but we will see later
that useful amplification can take place even if the magnitude of A, is less than unity.
Sometimes A^ is a negative number, so the output voltage is an inverted version of the input, and
he amplifier is then called an inverting amplifier. On the other hand, if A^ is a positive number,
we have a noninverting amplifier. These notions are illustrated in Fig. 2.2.
Equation (2.1) is a linear relationship; hence the amplifier it describes is a linear amplifier. A linear
amplifier does not introduce any distortions to sinusoidal amplified signals. The output signal is an
exact replica of the input, except of course for having larger amplitude (and possibly a nonzero
phase shift with respect to the input, which is discussed in Section 2.6). Linearity is the muchneeded feature of amplifiers so that the information contained in die signal is not changed and no
new information is introduced. Any change in waveform is considered distortion and is obviously
undesirable.
-18-
Analog Electronics/Amplifiers
Figure 2.2 Input signal to an amplifier (upper panel), output signal of a noninverting amplifier of gain A, =8,
output signal of an inverting amplifier of gain Ay =-8\
The signal amplifier is a two-port network. This is clearly seen in Fig. 2.1 where the two input
terminals are distinct to the two output terminals. A more common situation however is illustrated
in Fig. 2.3 where a common terminal exists between the input and the output The common
terminal is used as a reference point and is called the circuit ground.
Figure 2.3 An amplifier with a common terminal (ground) between the input and the output ports.
Voltage amplification can be modeled as a voltage controlled voltage source as illustrated in Fig. 2.4.
Because real amplifiers draw some current from the signal source, a realistic model of an amplifier
must include a resistance r\ across the input terminals. Furthermore, a resistance R^ must be
included in series with the output terminals to account for the fact that the output voltage of an
amplifier is reduced when load current flows.
The input resistance R; of the amplifier is the equivalent resistance seen when looking into the
input terminals. As we will find later, the input circuitry can sometimes include capadttve and
inductive effects, and we would then refer to the input impedance. For example, the input
amplifier of a typical oscilloscope has input impedance containing a 1-Mfi resistance in parallel with
a 47-pF capadtance.
Figure Z4 Model of an electronic amplifier
The voltage-controlled voltage source in Fig. 2.4 models the amplification properties of the
amplifier. Notice that the voltage produced by this source is the input voltage times a constant A°.
-19-
If the load is an open circuit, R^—°o, there is no drop across the output resistance R^ then vQ =
A%vt. For this reason the constant A% is called the open-circuit voltage gain. To summarize, the
voltage-amplifier model has three parameters: the input impedance, the output impedance and the
open-circuit voltage gain.
As shown in Fig. 2.4, the input current it is the current delivered to the input terminals of the
amplifier and the output current to is the current flowing through the load. The cuttent gain Ai of
the amplifier is the ratio of the output current to the input current
For linear amplifiers, the input current can be expressed as the input voltage divided by the input
resistance. For linear load, the output current is the output voltage divided by the load resistance.
(Linear amplifiers will be investigated in this chapter unless stated otherwise.) Thus we can find the
current gain in terms of the voltage gain and the resistance as
in which
A= -
(2-4)
is the voltage gain with the load resistance connected. Usually, A^ is smaller in magnitude than the
open-circuit voltage gain A°, because of the voltage drop across the output resistance.
The power delivered to the input terminals by the signal source is called the input power P^ and the
power delivered to the load is the output power PQ. The power gain G of an amplifier is the ratio
of the output power to the input power
G =£
(2.5)
Because we are assuming that the input impedance and the load are purely resistive, the average
power at either set of terminals is simply the product of die root-mean-square (rms) current and the
rms voltage. Thus we can write
Notice that we have used uppercase symbols, such as V0 and J0, for the rms values of the currents
and voltages. We use lowercase symbols, such as v0 and i^ for the instantaneous values. Of course,
since we have assumed that the instantaneous output is a constant times the instantaneous input,
the ratio of the rms voltages is the same as the ratio of the instantaneous voltages, and both are
equal to the voltage gain of the amplifier.
Exetcise 2.1 An amplifier has an input resistance l\ = 2000 Q, an output resistance of 25 Q, and
an open-circuit voltage gain of 500. The source has an internal voltage of Vs =20mV and a
resistance Rt =500Q. T h e load resistance is R L =750. Find the voltage gains Ay= VJV\ and A^=VJ Vs, the current gain and the power gain. Ans. ^ = 3 7 5 , ^ = 3 0 0 , ^ = 1 0 4 , G=3.75-106.
Exetcise 2.2 Assume that we can change the load resistance in Exercise 2.1. What value of load
resistance maximizes the power gain? What is the power gain for this resistance? Ans. R L =250,
G=5-106.
-20-
There exist other models of amplifiers, alternative to the voltage model. The input and output
resistances are the same for all models. They differ by the type of controlled source at the output.
Current amplifier model employs a current controlled current source, transconductance
amplifier uses a voltage-controlled current source and the transresistance amplifier is the one
with current-controlled voltage source at the output. The choice of the particular model is the
matter of analytical convenience or feasibility of measurement of the model parameters. An
amplifier can be modeled by any of the four models, provided that neither of the resistances (input
or output) is zero or infinity. For example, if JRjÔ, then t»,=0 and the voltage gain A^— v0/i\ is not
defined.
One can easily show that if the input impedance of an amplifier is much higher than the internal
impedance of the source, the voltage produced across the input terminals is-nearly the same as the
internal source voltage. On the other hand, if the input impedance is very low, the input current is
neady equal to the short-circuit current of the source. From this point of view, voltage amplifiers
should be designed to have large input impedance and current amplifiers should have very low
input impedance. Taking the output impedance level into consideration, we can force a desired
voltage waveform to appear across a variable load by designing the amplifier to have very low
output impedance compared to the load impedance. An example is an audio amplifier with a
variable number of loudspeakers connected to i t On the other hand, we can force a given current
waveform through a variable load by designing an amplifier to have very high output impedance
compared to the load impedance. Here an example could be the amplifier in optical communication
system where a light-emitting diode (LED). It is used to produce a light wave whose intensity is
proportional to a message signal such as a voice waveform. Because LED has nonlinear relationship
between voltage and current, light intensity "is not" proportional to the voltage across this device,
whereas it is proportional to the current flowing through the diode. Thus LED should be driven
from a current amplifier in this example.
We see that certain applications call for amplifiers with very high or very low input impedance
(compared to the source) and very high or very low output impedance (compared to the load). Such
amplifiers are classified as follows.
An ideal voltage amplifier (voltage-controlled voltage source) senses the open-circuit voltage of
die source and produces and amplified voltage across the load - independent of the load impedance.
Thus the ideal voltage amplifier has infinite input impedance and zero output impedance.
Figure 26 An ideal current amplifier (CCCS)
-21-
Analog Electronics /AmpMers
An ideal current amplifier (current-controlled current source) senses the short-circuit current of
the source and forces an amplified version of this current to flow through the load. Thus an ideal
current amplifier has zero input impedance and infinite output impedance.
O
4
O
Figure 2.7 An idealttansconductanceamplifier (VCCS)
An ideal transconductance amplifier (voltage-controlled current source) senses the open-circuit
voltage and forces a current proportional to this voltage to flow through the load. Thus it has
infinite input impedance and infinite output impedance.
Figure 2.8 An ideal transresistance amplifier (CCVS)
An ideal transresistance amplifier (current-controlled voltage source) senses the short-circuit
current of the source and produces a voltage proportional to this current to appear across the load.
Thus the ideal transresistance amplifier has a zero input impedance and zero output impedance.
The amplifier models considered here are unilateral; that is signal flow is unidirectional, from input
to output Most real amplifiers show some reverse transmission, which is usually undesirable but
must nonetheless be modeled. We will return to this point later.
Recall that we originally defined the gain of an amplifier to be the ratio of the output signal to the
input signal (2.4). This is true for amplifiers that do not contain any energy-storing element.
However, if any time delay or linear distortion occur, the ratio of output to input is a function of
time rather than a constant Thus we should not try to find the gain of an amplifier by taking the
ratio of the instantaneous output to input. Instead we recognize that gain is a function of frequency
and take the ratio of phasors for a sinusoidal input signal to find the (complex) gain at each
frequency
2.2 Cascaded Amplifiers
Amplifiers are built using active devices - most often transistors. We will see later that the gain
obtainable from a single-transistor amplifier is limited. Thus there is often a need to combine a
number of amplifiers in order to amplify a weak signal to the desired level. This can be achieved by
connecting the output of one amplifier to the input of another, as shown in Fig. 2.9. This is called a
cascade connection of-the amplifiers. The overall voltage gain of the cascade connection is given
by
Av = ^
(2.7)
Multiplied and divided by fol this becomes
-22-
"
(2.8^
v n v0l
However, referring to Fig. 2.5, we see that v^voV
Therefore we can write
(2.9)
'
v
,i
v
,2
We note also ^3OSAA^X=VOX/ »A is the gain of the first stage andy4 v2 =f o2 / tfo is the gain of the second
stage, so we have
Av=AvlAv2
(2.10)
Thus the overall gain of cascaded voltage amplifiers is the product of the voltage gains of the
individual stages. (Of course, it is necessary to include loading effects in computing the gain of each
stage. Notice that the input resistance of the second stage loads the first stage.) Similar rules can be
derived for the number of cascaded amplifiers bigger than two can.
Exercise 2 J Show that the overall current gain of a cascade connection of amplifiers is the product
of the current gains of the individual stages and that the overall power gain is the product of
individual power gains.
2.3 Power Supplies and Efficiency
For appropriate operation of the internal circuitry of an amplifier, power must be supplied to it
from an external power supply. The power supply typically delivers current from several dc
voltages, see an example in Fig. 2.10. The average power supplied to the amplifier by each voltage
source is the product of the average current and voltage. The total power supplied is the sum of the
powers delivered by each source. For example, the total average power supplied to the amplifier of
Fig. 2.10 is
(2.11)
*s - 'AAÂ +*BB*B
We have seen in Exercise 2.1 that the power gain of an amplifier can be very large. Thus the output
power delivered to the load is much greater than the power taken from the signal source. This
additional power is taken from the power supply. For example, a stereo audio system converts part
of the power taken from the power supply into signal power that is finally converted to sound by
the loudspeakers. Part of the supplied power is also dissipated as heat in the internal circuits of the
amplifier. This dissipation is an undesirable effect since the dissipated power is lost in most cases
and makes the battery discharged too fast in case of mobile battery-powered devices. We usually try
to minimize this effect when designing the internal circuitry of amplifiers.
-23-
_ Analog Electronics /Amplifiers
Figure 2.10 TTie power supply delivers power to the amplifier from
several external constant voltage supplies.
From energy conservatiqn law, the sum of the power entering the amplifier from the signal
P{ and the power from the power supply P s must be equal to the sum of the output power
the power dissipated P d .
pi+ps
= p0+pd
(2.12)
:>wer is negligible compared to the other terms in this
percentage of the power supplied that is converted in
n=
P.
(2.13)
P,+Ps
The power and efficiency values given in Exercise 2.4 below are typical of one channel of a stereo
amplifier under high output test conditions. Maximi2ing the efficiency is one of the important
design goals for battery-operated devices such mobile phones, hearing aids, implantable heart
pacemakers, etc., as well as for high-power equipment, such as motor drivers, power regulators,
DC-DC converters, etc.
Exercise 2.4 Find the input, output, supply and dissipated power in the amplifier shown in Fig. 2.6,
4
assuming the following values of its parameters: vs = 1 m V ^ , J^ = 0, ^ = 100 kQ, A% = 10 , R^ =
2 Q , R L = 8 Q , Vj^ = 15 V, JA = 1 A, Vm = 15 V, JB = 0.5 A. Calculate the efficiency of the
amplifier. A m . Pk = 10 pW, P0 = 8 W, Ps = 22.5 W, Pd = 14.5 W, rj = 35.6%.
Most amplifiers discussed in this lecture notes are small-signal circuits. The power involved is very
small and the efficiency was not a concern. We do not consider power amplifiers, or output
stages that are expected to provide large signal power requited by their loads. Examples of loads
for power amplifiers are stereo and public address loudspeakers, deflection coils in video monitors,
and servomotors in X-Y plotters. Powers supplies are also good examples of power circuits; their
function is to provide the dc power required by digital and analog components of all lands.
Several special concerns preoccupy the designer of an amplifier that must deliver a large amount of
power to the load. One is the amplifier efficiency, as discussed above in this Section. Equation
(2.13) shows that, for given output power, low efficiency directly means greater demand for
supplied power. Furthermore, efficiency suggests a second issue - the ability of the circuit
components to dissipate heat Any power that does not leave the circuit as the load power
contributes to the heating of transistors and resistors. In power circuits special efforts should be
-24-
made to ensure that components are not desttoyed by excessive heat. Finally, the large signal
amplitudes necessary for large output signal power make nonlinear distortion (see Section 2.9) a
matter of concern. Small signal models, based upon linearization of the device characteristic at some
jj-point (Sections 3.4, 4.6, 5.7 and 5.11), no longer apply here. Large-signal dynamic models have to
be used for circuit analysis and design, such as SPICE models discussed in Sections 3.8, 4.10 and
5.14.
Power amplifiers are classified according to the fraction of the time an output power transistor
conducts the current during one signal cycle. Class A amplifiers have output transistors in which
signal current is nonzero all the time. The FET amplifiers discussed in Sections 4.7 and 5.13 are all
class A amplifiers. The dc current of a class A amplifier is the same, no matter the signal is zero or
nonzero. For greater efficiency, class B amplifiers employ transistors that are active only half time otherwise they are cut-off. At zero signal value, there is no dc current through- a class B output stage
- and no power consumption. An example of PSpice simulation of a class B bipolar transistor
amplifier is presented in Section 6.1.3. So-called crossover distortion appears in class B amplifiers
(see Section 6.1.3), so for their reduction class AB amplifiers are designed. In class AB circuits,
transistors conduct slightly more than half the signal cycle. Class AB circuits have efficiencies
approximately the same as for class B circuits, but produce less distortion.
Class C amplifiers deliver large amount of output power at high efficiency by employing an output
transistor that conducts output current for only small fraction of a cycle. The resulting short,
periodic pulses of output current excite a resonant circuit, which suppresses the distortion
components that arise from the nonlinear operation of the transistor.
Class D amplifiers produce binary output waveforms of very high power with efficiencies
approaching 100% by using transistors as switches. In analog applications, the class D amplifier
includes a modulator that first transfers information from waveform amplitude into a more suitable
form as pulse width. A lossless filter than removes undesired high-frequency terms from the output.
Applications of power amplifiers, especially class C and D circuits, are somewhat specialized.
Therefore, they are not discussed further in this text
2.4 Decibel Notation
As we have noticed from the Exercises, the gain of an amplifier can take on values spanning a very
large range of magnitudes. To facilitate visualization of these extreme values, amplifier gain is often
expressed with a logarithmic measure. Specifically, the voltage gain A^ can be expressed in decibels
(dB)as
^=201og|^v|dB
(2.14)
and the current gain A, can be expressed as
^=201ogM,|dB
(2.15)
Since power is related to voltage (or current) squared, the power gain G can be expressed in
decibels as follows
G ^ g ^ O l o g G dB
(216)
A power gain of G=100 converts to 20dB, unky gain converts to OdB, and so on. The absolute
values of the voltage and current gains are used because in the case of inverting amplifiers AôxA^
can be negative values. A negative gain Av means that there is a 180°-phase difference between die
input and output signals; it does not imply mat the amplifier attemiates the signal On the other
-25-
AmJofElectronica/AinptiSen
hand, an amplifier whose voltage gain is -20 dB is, in fact attenuating the input signal by a factor of
10 (that is ^ = 0 . 1 ) .
Recall that the overall gain for cascaded amplifiers is the product of the power gains of the
individual amplifiers. Wbetifagm an e>pnssedmdecfals, the gons ofcascaded stagum
the properties of the logarithm function. To illustrate this point we have
G = GXG2
(2.17)
When expressed in decibels, this becomes
G* = lOlogG - lOlogCG.Gj) = lOlogCG,)* 101og(G2) dB
(2.18)
Note, referring back to (2.6), that power gain in decibels is equal to voltage gain in decibels only
when RL=R}.
Electronics engineers often use decibel notation for voltages, currents, powers, or other quantities.
To do so, a reference level must be stated or implied The quantity to be expressed in decibels is
divided by the reference value and the ratio is converted to decibels by taking 20 times me
logarithm of the ratio for voltages or currents. Ten times the logarithm of the ratio is taken for
powers. Some commonly used reference levels are 1 volt (dBV), 1 mW (dBm), and 1 watt (dBW).
For example, 40 dBV is a designation of 100 V, -10 dBm is for 0.1 mW, -40 dBW is also 0.1 mW,
and so on.
2.5 Frequency Response
From Chapter 1.1 we know that the input signal to an amplifier can always be expressed as the sum
of sinusoidal signals. So far we have considered the gain parameter to be a constant However, since
physical amplifiers contain capatiuve and inductive elements, the gain is a function of frequency.
Furthermore, the amplifier affects the amplitude as well as the phase of the input signal
It follows then that an important characterization of an amplifier is in terms of its response to input
sinusoids of different frequencies. Such characterization of amplifier performance is known as the
amplifier frequency response.
26
Analog Ehcttoaka/AmpWka
Figure 212 Measuring thefrequencyresponse of a Knear amplifier (a), phase characteristic of a two-transistor
small-signal amplifier (b) whose amplitude characteristic is shown in Fig. 2.7
Figure 2.12a shows a linear voltage amplifier fed at its input with a sine-wave signal of amplitude V{
and frequency CD. As the figure indicates, the signal measured at the amplifier output is also
sinusoidal with exactly me same frequency 0). This is always true for linear circuits. However, the
output sinusoid will have in general different amplitude and a different phase relative to the input.
The ratio of the amplitude of me output sinusoid (VJ to the amplitude of the input sinusoid (V$ is
the amplifier gain or transmission at the test frequency 0). Hie angle <p is the phase of the output
sinusoie relative the input If we denote the amplifier transmission or transfer function by ^(0))
then
(2.19)
(2.20)
arg{4, (o))} m /Ay ( o ) = <p(a>)
The response of a linear amplifier to a sinusoid of frequency Q) is described by \Ay{(fy | and <p{o)).
Now, to obtain the complete frequency response we simply change the frequency of the input
sinusoid, wait for any transients to setde, and measure the new value of AJcfy. The end result will
be a table and/or graph of gain magnitude \AJd) | versus frequency and a table and/or graph of
phase an$e <p(o}) versus frequency. These two plots together constitute the frequency response of
the amplifier. The first is known as the magnitude or amplitude response (amplitude
characteristic), and the second is the phase response (phase characteristic). Examples of
amplitude and phase characteristics for a two-transistor small-signal amplifier are shown in Fig. 2.7
and 2.8(b), respectively.
If one plots the magnitude of the gain of a typical amplifier, a plot such as one of those shown in
Fig. 2.11 results. Note that the gain magnitude is constant over a wide range of frequencies known
-27-
as the midband region. This is typical for wideband amplifiers. Wideband amplifiers are used for
signals that occupy a wide range of frequencies, such as audio signals or video signals (see Table
1.1). In some cases, the frequency response of an amplifier is deliberately limited to a small
bandwidth compared to the center midband frequency. Such an amplifier is called a nattowband
or banpass amplifier. Bandpass amplifiers are frequency selective circuits used in radio receivers to
amplify the signal from one transmitter and reject the signals from other transmitters in adjacent
frequency ranges.
Figure 2.13 Gain versus frequency for wideband amplifiers: ac-coupled amplifier (a) and
dc-coupled amplifier (b)
Usually we specify the approximate useful frequency range of an amplifier by giving the frequencies
at which the voltage (or current where appropriate) gain magnitude is l / \ 2 times the midband gain
magnitude. These are known as half-power frequencies (cut-off frequencies) because the output
power level is half the value for the midband region if the constant-amplitude, variable-frequency
sinewave test input is used. Expressing the factor 1 /"V2 in decibels results in -3.01 dB. Thus at the
half-power frequencies, the voltage or current gain is approximately 3dB lower than the midband
gain. The bandwidth B of an amplifier is the distance between the half-power frequencies, {fH -f{).
This is illustrated in Fig. 2.13.
In the cases illustrated by Fig. 2.13(a), the gain drops to zero at dc (zero frequency). Such amplifiers
are called ac-coupled, because only ac signals are amplified. These amplifiers are often constructed
by cascading a number of amplifiers that are connected together by coupling capacitors, so that
dc voltages of signal source, and individual amplifier stages are separated from each other. This is
illustrated in Fig. 2.14 where; caparitive coupling prevents a dc input component from affecting the
first stage, dc voltages in the first stage from reaching the signal source and the second stage, and dc
voltages in the second stage from reaching the first stage and the load. Sometimes magnetic
transformers are used for coupling, which also leads to an ac-coupled amplifier with a zero gain at
dc.
-28-
Other amplifiers have constant gain all the way down to the dc, as shown in Fig. 2.13(b). They are
said to be dc-coupled or direct coupled. Amplifiers that are realized as integrated circuits are
often dc coupled because the capacitors or transformers needed for ac coupling cannot be
fabricated in integrated form.
Electrocardiograph amplifiers are deliberately ac-coupled because a dc voltage of nearly a volt often
occurs in the input due to contact potentials developed on the electrodes placed on the skin. The ac
signal generated by the heart is on the order of lmV, and therefore the gain of the amplifier is high typically 1000 or more. A lV-dc input would cause the amplifier to produce an output of 1000V. It
would be difficult and undesirable to design an amplifier capable of such large outputs. Therefore, it
is necessary to ac-couple the input circuit of an electrocardiograph to prevent the dc component
from overloading the amplifier.
Amplifiers for video signals need to be dc coupled because video signals have frequency
components from dc to about 6MHz, see Table 1.1. Dark pictures result in a different dc
component than bright on the average pictures. It is necessary to dc-couple a video amplifier in
order to preserve dc component and to obtain proper brightness of the image.
As indicated in Fig. 2.13, the gain of an amplifier always drops off at high frequencies. This is
caused either by small capacitance in parallel with the input or output terminals or by small
inductances in series with the signal path in the amplifier circuitry. This is illustrated in Fig. 2.10.
Recall that impedance of a capacitor is inversely proportional to frequency, resulting in effective
short circuit at sufficiendy high frequencies. The impedance of an inductor is proportional to
frequency, so it becomes an open circuit at very high frequencies.
Figure 2.15 Capacitance in parallel with the signal path and inductance in series with the signal path reduce
gain in the high-frequency region
Some of these small capacitances occur because of stray wiring capacitance between signal-carrying
conductors and ground. Other capacitances are integral parts of active devices (transistors)
necessary for amplification. Small inductances result from the magnetic fields surrounding the
conductors in the circuit For example, a critically placed piece of wire 1-cm long can have enough
inductance to limit severely the frequency response of an amplifier intended to operate at several
GHz.
2y
Analog Electronic* /AupttiBen
Fot frequencies up to several MH2, the circuit and device capacitances ate the main source of
frequency response roll-off, either at low or high frequencies. To illustrate their impact on
frequency response we will consider an amplifier that can be represented by a real transconductance
amplifier model (Figure 2.16). Such a model is an ideal transconductance amplifier as shown in
Figure 2.7, augmented with input nonzero resistance r, and output finite resistance r* It is an
adequate low-frequency representative of amplifying devices, such as field-effect and bipolar
transistors. For the field-effect transistor, the input resistance rt is infinite whereas for the bipolar
transistor a base resistance is added in series with the input terminal as will be discussed in Chapters
4 and 5, respectively.
Figure 2.16 Lowfrequencynon-ideal transconductance amplifier model
The input to the amplifier is voltage source it with internal resistance R». Resistance RL represents
the load They are shown in Fig. 2.17. The symbol ^n used in Figs 2.16 and 2.17 to denote the shortcircuit transconductance coefficient is equivalent to A& in Fig. 2.7. To illustrate the effect of the
source resistance on the frequency response, we will use a voltage gain definition slightly different
to that given by Equation (2.1), namely
(2-21)
4 . = 7T
Figure 217 Transconductance amplifier
There is no capacitive or inductive element in the schematic diagram of Fig. 2.17. Therefore, the
bandwidth of this circuit is infinite. Indeed, the output voltage
V0=-ZmVi
r0RL
r0+R]L
does not depend on frequency. The voltage gain of the amplifier in Figure 2.17 can be expressed as
Ays ~ ~8lH
r0Ri
r0+RLr{+Rs
(2.23)
"" Ay,>so
The coefficient Ano in Equation (2.23) is a constant as plotted in Figure 2.18(a) wherein = 50 mS,
R, = 600 Q, r, = 5 kO, r0 = 100 kQ, and RL = 5.6kft were assumed. The phase characteristic in
Figure 2.18(b) is also constant and equal to 180 degrees that means the amplifier under
consideration is an inverting one. Of course, the circuit discussed is an idealized model of real-world
amplifiers. Nevertheless, for medium frequency range it is an adequate model of transistor
amplifiers.
Low frequency effects. To illustrate the effect of coupling capacitances on the frequency response
of amplifiers we consider the circuit of Figure 2.19. It contains a coupling capacitor C, in series with
the input resistance rf The output voltage is described by Equation (2.22) and the input voltage can
be expressed as
-30-
Analog Ekctroak* /AnpUtten
Vi=Vt
Rs+rt +
1
JartCs
l+JG>(Rs+rty:s
JG>(Rs+ri)Cs
n
= K* rt,+r,
1+ ./<»(*,+r,)C,
(2.24)
= V,
Ja>*L
Rs+ril+ja>TL
Figure 219 Ac-coupledttansconductanceamplifier
where
U=(*S+ri)Cs
(2.25)
is the low-frequency time constant of the amplifier. Combining Equations (2.22) and (2.26), one
obtains the effective voltage gain of the amplifier in Fig. 2.19
n
JQ>rL
r0RL
Ays ~ 8n
Rs+ril + ja>TLr0+Rl
(2.26)
~~Ayso
1+JOTL
Equation (226) indicates that in this case the voltage gain is a single-pole function of frequency.
The voltage gain magnitude
(2.27)
Vl+(a>rJ 2
-31-
tends to zero for frequencies approaching zero. Thus an ac-coupled amplifier does not provide a dc
gain. This is shown in Figure 2.20a where Cs= \ \x¥ was assumed with the other circuit parameters
taking the same values as for Figure 2.18.
The gain increases with frequency. For high frequencies, the gain magnitude is equal to AyM. Then
one can say the ac-coupled amplifier acts as a highpass filter. At the lower cut-off frequency, jL, the
gain magnitude is equal to A\,so I-J2 . The same results (except for different expression to describe
the low-frequency time constant) would be obtained if a coupling capacitor is connected in series
with the load resistance RLIt is easy to show that for a single-pole transfer function of the type given by Equation (2.26), the
lower cut-off frequency is inversely proportional to the low-frequency time constant
h=^—
(2-28)
Using Equation (2.26) one can find expression that describes the phase characteristic of the
amplifier in Figure 2.19
ZA\,S =270°
-tg-\coTL)
(2.29)
. -Jl i
= - 9 0 -tg- (a>r L )
The phase characteristic (2.29) of the ac-coupled amplifier, computed using PSpice program is
plotted in Figure 2.20b.
High frequency effects. Capacitances in parallel with the signal path (as well as capacitances
connected between the output and input terminals of inverting amplifiers that will be discussed later
on) make the amplifier gain decreasing with frequency. To illustrate this effect, capacitance C0 is
connected in parallel to the load resistance of the amplifier, as shown in Figure 2.21. The output
voltage is now given as
-32-
Analog Electronics /AmpliBets
vQ
jcoC0
= -gmVi
R0+—t—
JoCo
= -gmVi
=
Ro
\+jaC0R0
(2.30)
n
R
~Sm's ri+R \+jaC R0
s
0 0
-_«T
V
1-
R
o
where
(2.31)
T
H ~ RoCo
is the high frequency time constant of the amplifier and
_ rQRL
Rn =
r0+RL
(2.32)
Figure 2.21 A transconductance amplifier with parallel capacitance at the output
Substituting Equation (2.23) into (2.31), dividing both sides by Vs and making use of the definition
(2.21), one obtains the voltage gain
4,=-,
K
°
(2-33)
that is a single-pole function of frequency. Its magnitude
(2.34)
^\ + {0)tH)2
is equal to Aô at low frequencies and decreases to zero as the frequency approaches infinity. This
is a lowpass filter characteristic as shown in Figure 2.22a. At the higher cut-off frequency,^, the
I4«(<»)l=-
gain magnitude is equal to A\so I -j2. The same results (except for different expression that
describes the high-frequency time constant) would be obtained if a capacitor is connected in parallel
with the input resistance r,.
It is easy to show that for a single-pole transfer function of the type given by Equation (2.26), the
higher cut-off frequency is inversely proportional to the high-frequency time constant
fH=^—
(2-35)
Using Equation (2.33) one can find expression that describes the phase characteristic of die
amplifier in Figure 2.19
^ = 1 8 0
-li
°-\g-\<QT
H)
(2.36)
-33-
Antdog Electronics /Amplifiers
The phase characteristic (2.36) of the amplifier in Figure 2.21 with parallel capacitance C = 10 pF,
computed using the PSpice program, is plotted in Figure 2.22b.
2.6 The Miller Theorem
It is often the case in transistor circuits that a capacitor is connected between the output and input
ports of the amplifier. Such a capacitor may represent a base-collector junction capacitance of a
bipolar device or drain-gate capacitance of FET or MOSFET field-effect devices. As an example, a
simplified equivalent circuit of such an amplifier is shown in Figure 2.23, where g„ = 50 mS, R, =
600 CI, r,, = 5 kQ, r0 = 100 \£i, R^ = 5.6kQ and CF = 10 pF are assumed.
Figure 2.23 A transconductance amplifier with a feedback capacitor
The capacitive element connected between the input and output terminals causes difficulties in
analysis since its presence leads to higher-order nodal equations. The difficulties can be overcome
by applying Miller theorem to the circuit This theorem is a basis to one of approximate methods of
transistor amplifier analysis. To derive it, we will consider a general amplifier with feedback
impedance as shown in Figure 2.24a.
y,*
v, o
1
O"o
-OVo
\ >
(a)
(b)
Figure 2.24 Original amplifier with feedback impedance (a), equivalent amplifier with feedback element split
into two parts (b)
-34-
For the original circuit we have
j _VJ-VQ JTi-AVi
=Vj<X-A)
1
z
z
(2.37)
z
I
_Vo-Vir_Vo-Vo/A^V0(l-A)
Z
Z
-AZ
Similarly, for the circuit of Figure 2.24b we can write
(2.38)
2
(2.39)
Z
ml
(2.40)
m2
Now we require the circuit of Figure 2.24b to be equivalent to the original circuit of Figure 2.24a.
Z
i
i
The two circuits are equivalent if I\ = I\ and 1^ — 12 • After some manipulations we obtain the
impedances diat satisfy die equivalence requirement
Z
Z
(2.41)
m\ =
\-A
-A
2
(2.42)
m2=Z
\-A
The term A in the above equations is me original amplifier gain with die impedance Z in place.
Evaluating die value of A is usually not easy. Therefore, gain is often calculated for the amplifier
with feedback impedance Z open-circuited. This introduces some approximation in the analysis
based on the Miller theorem. It should be stressed, however, that Miller theorem itself does not
involve any approximation.
The gain of a typical transistor amplifier is negative number, A < 0, of a reasonably large module,
\A\ > > 1 . If Z is resistive, the impedances Zml and Zm2 are also resistive, with Zmi being much
smaller than Z and Zm2 being approximately equal to Z. If Z is a capacitive element, Z=l/')(oC, the
impedances Zmi and Zm2 are also capacitive
Cml=C(l-A)
(2.43)
Cm2=Cl-^j
(2.44)
Figure 2.25 Circuit equivalent to the amplifier of Fig. 2.23
Figure 2.25 shows the equivalent circuit for the amplifier of Figure 2.23. The approximate values of
die Miller capacitances are
Cml=CF(l-Avo)
(2.45)
Cm2=CF1-^(2-46)
-A
vo
where Am = Vo/V{. Now one can derive die formula for the voltage gain of die amplifier. The
output voltage is given by
-35-
Analog Electronics /AmpliGcra
(2.47)
"o ~ Sm"i, .
l + JG>T2
:das
where Ro is defined by Equation (2.32) and the output circuit time constant is described
as
(2.48)
T2 = R0C„2
In turn, for the input voltage we have
1
(2.49)
where
T
l ~
and
*1 =
R C
\ m\
(2.50)
V/
(2.51)
*s+n
Combining Equations (2.47), (2.49) and (2.23), one obtains the voltage gain
L_
A = _ , __J
(2.52)
l + JCDTi l + JG)T2
which is a two-pole transfer function. The time constants are related to the angular firequendes
COX = —
(2.53)
fi>2=—
(2.54)
*2
called break firequendes, since for CO exceeding a value of any of these firequendes the slope of the
amplifier amplitude characteristic increases by 20 dB/decade. Using (2.53) and (2.54) one can
rewrite Equation (2.52) as follows
AyS — ~AVS0
(2.55)
-TCO
1+ 7— ll + y
CO
•I'
One can perform the multiplication in the denominator of (2.55) to obtain
A>5 = ~A\s
VSO
/
(2.56)
1
0>1
+
1
CO
Q}2)
G>\<U2
For suffidently lowfirequendesthe right-hand term in the denominator is negligible
CO
« 1
coxco2
Then we have
A
(2.57)
1
s.—A
l + jco
s-A
1
1
1 \
—+ —
(2.58)
K*>1 *>2)
VSO
l+y—
<on
where
1
1
CO,
>o
l
(2.59)
°>\ <°2
-36-
Analog Electronics /AmpliGers
relates the approximate higher cut-off frequency fflb = (oH to the break frequencies CO, and 0)2. By
using Equations (2.53) and (2.54) one obtains
a>0 s ——
=r, + r 2
T0
(2.60)
It should be noted that Equation (2.60) underestimates the actual value of the higher cut-off
frequency, which is the effect of dropping the right-side term in the denominator of Equation
(2.56). The actual value of 0)H is higher than Oh and lower than minimum of CO, and cor It is then
reasonable to write
o)0 <O>H <min{fi>i,0)2}
(261)
The latter formula can be easily extended to three- and more-pole amplifiers.
Using the numerical values assumed above for the elements of our circuit, we have A.™ = 265.2, C,
= 2662 pF, C2 = 10.04 pF, R, s 536 Q and R, = 5.3 kQ. They give z, = 1.426 us, r2 = 52.83 ns, T0 =
1.479 us, which are respectively equivalent to co, = 701.36 krad/s, Q)2 S 18.93 Mrad/s, and flfc =
676.3 krad/s. Knowing t h a t ^ = <oH/(2n) one can write using Equation (2.61)
107.6 kHz <fH < 111.3kH2
(2.62)
This compares very well with the actual value of the lower cut-off frequency being equal to 107.6
kHz as obtained from PSpice simulation.
The accuracy of the higher cut-off frequency prediction is very good in this case, because one of the
time constants is much lower than the other, namely r, » T? As an effect of this property, the
Inequality (2.57) is satisfied with a large margin. Lower accuracy is achieved typically when r, = r?
Still, the Miller-theorem-based technique is a valuable tool for fast evaluation of the bandwidth of
amplifiers with caparitive feedback. This technique will be used in Chapters 4 and 5 to investigate
the properties of basic transistor amplifiers.
Please note that in the discussed case, a relatively small feedback capacitance (10 pF) has been
transformed into a large Miller capacitance (2662 pF) connected to the input of the amplifying
device. For large value of Cni, the input time constant becomes large, which according to Equation
(2.54) decreases the bandwidth of the amplifier. The capacitance Cm\ increases with the magnitude
of the voltage gain - see Equation (2.45). Thus with the feedback capacitance, the larger the gain,
the lower the bandwidth. The gain, in turn, is proportional to the load resistance. If large bandwidth
is of primary concern, one should keep the load resistance as small as possible. An inverting
amplifier circuit that allows achieving large gain and low load resistance simultaneously is a twotransistor amplifier known as "cascode". It will be discussed in Chapters 4 and 5.
2.7 Linear Distortion
If the gain of an amplifier has a different magnitude for the various frequency components of the
input signal, a form of distortions known as amplitude distortion occurs. As an example, suppose
the input signal to an amplifier is a waveform that contains two sine-wave components, one of
amplitude 3 mV and frequency 1kHz and the other of amplitude -2 mV and the frequency of 3kHz:
v 7 (0 = [ 3 s i n ( 2 0 0 0 ^ ) - 2 s i n ( 6 0 0 0 ^ ) ] m V
(2.63)
The gain of the amplifier at 1kHz is 10Z0° and the gain at 3kHz is 2.5Z0 0 . Apparently, the gain is
not the same for the two signal components. The plots of the input and the output signals are
shown in Fig. 2.26. It is seen that the shape of the output signal is different than the shape of the
input signal due to the amplitude distortion.
-37-
Zero phase characteristic of an amplifier results in an output waveform that is identical to the input
(up to the scale factor). On the other hand, if the phase shift between the output and input signals
of an amplifier is proportional to frequency, the output signal is a time-shifted version of the input,
but their shapes are the same, so we do not say any distortions take place.
Figure 2.26 Linear amplitude distortion: input signal (upper panel), output signal (lower panel).
If the phase shift of an amplifier is not proportional to frequency, phase distortion occurs. As an
example suppose the signal (2.63) is applied to uie inputs of three amplifiers A, B, C, which have a
constant gain magnitude of 10. The amplifiers have different phase characteristics, as specified in
Table 2.1
TABLE 2.1 TRANSFER FUNCTION O F EXAMPLE AMPLIFIERS
Amplifier
A
B
C
GainatlkHz
10Z0°
10Z-450
10Z-450
Gain at 3kHz
10Z0°
10Z-1350
10Z-450
Apparently, amplifier A has a zero-phase response. The phase response of amplifier B is
proportional to frequency with a proportionality constant of-45 degrees per kilohertz, and amplifier
C has the phase response which is not proportional to frequency (or more precisely, it is nonzero
constant, so the proportionality factor is zero). The plots of the output signals of the amplifiers are
shown in Fig. 2.27. It is clear that the amplifier A does not change the shape of the signal.
(However, it cannot be realized physically, as there is no delay between the input and the output compare with Fig. 2.26.) Amplifier B also does not introduce any signal distortion, as the shapes of
the input and output are the same. (Note the time delay between the signals.) Amplifier C that does
not have its phase proportional to frequency produces severe distortion of the signal.
We conclude that to avoid linear signal distortions, an amplifier should have constant gain
magnitude and a phase response that is linear versus frequency for the range of frequencies
contained in the input signal. For an audio amplifier it is required to have a constant gain for the
frequency range from 20Hz to about 20kHz. However, since it turns out that the ear is not sensitive
to phase distortion (at least for monophonic signals), we would not require the phase signals of an
audio amplifier to be strictly proportional to frequency. Since the shape of the waveform ultimately
determines the brightness of various points of an image, either amplitude or phase distortions of
television signals would severely affect the image quality. Therefore we require the gain magnitude
of a video amplifier to be constant and the phase response to be proportional to frequency in die
whole range from dc to 6MHz.
-38-
2.8 Pulse Response
Often one needs to amplify a pulse signal such as the waveform v(l) shown in Fig. 2.28a. Pulses
contain components that are spread over a wide range of frequencies; therefore amplification of
pulses calls for a wideband amplifier. A typical amplifier output pulse is shown in Fig. 2.28a,
denoted by v(7). The output waveform differs from the input in several important respects: the
pulse is delayed, it displays overshoot and ringing, the leading and trailing edges are gradual rather
than abrupt, and if the amplifier is ac-coupled, the top of the output pulse is tilted.
The gradual rise of the leading edge of the amplifier response is quantified by giving the rise time /„
which is the time interval between the point t10 at which the amplifier achieves 10% of the eventual
output amplitude K-and the point /j^ at which the output is 90% of the steady-state value. This is
illustrated in Fig. 2.29.
-39-
Analog Electronics/Amplifiers
ho
ho
'
Figure 2.29 Rising edge of a typical ac-coupied broadband amplifier output pulse
(Note: no tilt is shown. When it is present, some judgement is necessary
to estimate thefinalamplitude Vj).
The rounding of the leading edge can be attributed to the roll-off of gain in the high-frequency
region. A rule-of-thumb relationship between the half-power bandwidth B and the rise time tr of a
wideband amplifier is
tr s ~Y
(2-64)
This relationship is not exact,for all types of amplifiers; it is accurate for firs' order (single pole)
circuits. Analogously, the fall time U can be defined for the trailing edge of amplifier's pulse
response. Equation (2.64) approximates the value of tr as well, at least for linear wideband
amplifiers.
Another aspect of the output pulse shown in Fig. 2.28 is overshoot and ringing, which are also
related to the way the gain behaves in the high-frequency region. An amplifier that displays
pronounced overshoot and ringing usually has a peak in its amplitude characteristic, as shown in
Fig. 2.28b. The frequency of maximum gain approximately matches the ringing frequency. Because
both rise time and overshoot are related to the high-frequency response, there is usually some tradeoff between these specifications. In a particular design, component values that reduce rise time
often lead to more overshoot and tinging. Pulse amplifiers designed for fast rise time typically
display about 10% overshoot because a higher amount of overshoot and associated ringing are
usually undesirable.
Figure 2.30 Pulse response that does not display ringing (a) and corresponding amplitude characteristic (b) of
an amplifier
Example 2.1 A television picture (25 pictures per second, 625 lines per picture) is produced by a
TV camera that is able to reproduce about 765 distinguishable image elements within a 52^s active
-40-
Analog Electtoaica/AxapHSets
interval during each picture line. Estimate the bandwidth required for the video amplifier embedded
within this camera.
Solution. We estimate the bandwidth by the use of equation (2.22). Each picture element occupies
die interval of 52)ls/765=68ns. Assume 80% of this interval is allowed to be the rise time: tr
=54.4ns. Thus we obtain
0.35
,A
n
5 = - — — = 6.4 MHz
(2.65)
54.4/w
This is very close to die actual video bandwidth used in Europe.
Figure 2.31 Pulse responses (a) and amplitude characteristics (b) of ac-coupled amplifiers.
(Tis the input pulse duration and r represents the shortest time constant of the coupling circuit)
v(2): T« T, v(3): T=T, v(4): T» T.
The tilt at the top of the output pulse, shown in Fig. 2.31a occurs if the amplifier is ac-coupled and
originates from charging of coupling capacitors during the pulse. (After all, if the pulse lasted
indefinitely, it would be the same as a new dc level at the input Since ac-coupled amplifiers do not
pass me dc signal, output voltage would eventually return to zero.) Tilt (sag) is specified as a
percentage of the initial pulse amplitude
(2.66)
AF % = — - 1 0 0 %
where AV and V are defined in Fig. 2.32. As the duration of the input pulse increases (or die
amplifier lower cut-off frequency is raised by changing the coupling circuits to have shorter time
constants), output waveforms of consistently increased tilt result as shown in Fig. 2.31b.
For small amount of tilt, the following approximate formula relates the percentage tilt to the lower
cut-off (half-power) frequency^
AV% = 2 0 0 ^ 7
(2.67)
where Tis the duration of the pulse.
-41-
Analog Electronics /AmpliSera
Figure 232 Pulse response that displays tilt
Example 2.2 A cathode ray oscilloscope (CRO) is used to measure the pulse response of me
amplifier shown in Figure 2.21. The equivalent impedance of the oscilloscope, including the cable
connected between amplifies output and oscilloscope's input, is represented by a 1-MQ resistance
in parallel with a 100-pF capacitance. This is illustrated in Figure 2.33. Find the output pulse rise
time, as obtained from the measurement Compare with the value obtained for the circuit of Figure
2.21.
Figure 233 A transconductance amplifier with parallel capacitance and
Oscilloscope + cable equivalent connected at the output
Solution. We will use Equation (2.56). For the amplifier under consideration, the bandwidth equals
to the higher cut-off frequency. To find the higher cut-off frequency for the circuits in Figure 2.21
and 2.33, one can use Equation (2.35) with appropriate values for the equivalent resistance and
capacitance to calculate the time constant In the case of Figure 2.21, we have
— = —+ —
(2.68)
R
r
o
o &L
which for RL = 5.6 kQ and r0 = 100 kfi gives Ro = 5.303 kfl. The equivalent capacitance is equal to
Co = 10 pF. The time constant is the product RoG>, which is tti = 53.03 ns. This corresponds to the
upper cut-off frequency of/k = 3 MHz and the rise time of £ = 0.12 ^ls. In the case of Figure 2.33,
we have
1
1
1
1
(2.69)
•+ • = — +
R
Ro ro RL
CRO
which for the assumed values gives Ro = 5.275 kQ. The equivalent capacitance is equal to (G> +
CCRO)=
HO pF. The time constant is the product RO(G)+CCRO), which is m = 580.2 ns. This
corresponds to the upper cut-off frequency offii = 274 kHz and the rise time of U = 1.28 |J.s.
One can see from the results of Example 2.2 that a substantial reduction of the apparent cut-off
frequency can be experienced when measuring it by the oscilloscope, direcdy connecting die
oscilloscope input to the amplifier output by a coaxial cable. Lower value of the cut-off frequency
means that such an oscilloscope measurement setup introduces errors to the rise time and cannot
be used to evaluate amplifier pulse response. The rise and fall times will apparency be much longer
-42-
than their true values. To get rid of this undesired effect a device known as probe is used. The
oscilloscope probe is an RC voltage attenuator as shown in Figure 2.34, where Vî denotes the
voltage at die amplifier output and V2 is the voltage at the oscilloscope input Typically, a probe
introduces 10 times attenuation of the input signal. If the probe capacitance Cp is properly adjusted,
the attenuation is constant over a very wide frequency range. Such a probe is said to be
compensated. For a compensated ptobe with 10-times attenuation, the input capacitance is 10
times lower than the oscilloscope + cable capacitance. Thus the apparent reduction in the higher
cut-off frequency is not as severe as it is with the CRO directly connected to the amplifier.
Figure 234 Schematic diagram of the oscilloscope probe
Example 2.3 Find the values of Rp and Cp of the oscilloscope probe in Figure 2.34 such that the
attenuation V2/V\ is equal to 10 for all frequencies. Assume RCRO = 1 Mft and CCRO = 1 0 0 pF.
Solution. One can easily verify that the transfer function of the probe is described by
Av=^ =
P ^
V
( 2 - 70 )
n
Rr,
+R l + J*>RCROCCRO
P
1 + jaRpCp
^R°
It will be a constant equal to
*CRO
Aw =
l
RCRO +
^
(2J1)
R
p
for all frequencies if the following condition is satisfied
RpCp = RcROCCRO
( 2 - 72 )
Putting Avo = 0.1 in Equation (2.71) one obtains.Kp = 9 MQ, which substituted into condition
(2.72) gives Cp = 11.1 pF. It is suggested to the reader to find out that the input capacitance of the
probe + oscilloscope arrangement is, under these circumstances, equal to Cm = 10 pF, whereas the
parallel resistance Rin that loads the amplifier output is Rin = 10 MQ.
Example 2.4 Find the upper cut-off frequency of the amplifier in Figure 2.33, as measured using a
10-times attenuating compensated probe.
Solution. We replace the oscilloscope equivalent RCROCCRO in Figure 2.33 by a parallel connection
of Rin and Cm that represent the probe of Figure 2.34. Thus the equivalent output resistance of the
amplifier is now equal to
1 1 1 1
— =—+— +—
(2.73)
R0 ro RL
Rin
which for the assumed values gives Ro = 5.300 kfl. The equivalent capacitance is equal to (Co +
Cm)= 20 pF. The time constant is the product Ro(G+Cm), which is m = 106 ns. This corresponds
to the upper cut-off frequency ofjk = 1.5 MHz and the rise time of £ = 0.23 ^s. The latter value is
much closer to the true value of 0.12 |is than the rise time of 1.28 |4.s, as measured straight by the
oscilloscope with no probe. This illustrates the advantage of using die probe for wideband amplifier
characterization.
-43-
2.9 Nonlinear Distortion
Consider again an ideal resistive amplifier. Define its transfer characteristic as a ratio of the
instantaneous output to instantaneous input For an ideal amplifier, the transfer characteristic can be
plotted as a straight line on the output-input plane. For real amplifiers, diis characteristic remains
linear over only a limited range of input and output voltages. For an amplifier operated from two
power supplies the output voltage cannot exceed a specified positive limit and cannot decrease
below a specified negative limit. Thus the output becomes saturated for large positive and large
negative inputs. The resulting transfer characteristic is shown in Fig. 2.35, with the positive and the
negative saturation levels denoted as L + and L_, respectively. Each of the two saturation limits is
usually within 1 or 2 volts of the corresponding power supply. The output signal becomes distorted
when me input exceeds the respective values 2,+ / Ay and L_ I Ay. Obviously, in order to avoid
distorting the output signal waveform, the input signal swing should be kept within the linear range
of operation
L_/A,Zv,ZLJA,
(2.74)
Fig. 2.35 shows the input waveforms (1) and (2) and the corresponding output waveforms. We note
that the peaks of the larger waveform have been dipped off because of the saturation effect In
some applications, like audio signal amplification, clipping is perceived as a rather severe distortion.
Even small departures from linearity can be considered to be very serious in such cases. Assume the
output-input relationship of a nonlinear amplifier can be written as
vo(0 = M-(') + ^2[v,(')]2 +4J[V,(01 3 +•••
(2.75)
where Au A& A3 and so on are constants selected so diat the equation (2.75) matches the curvature
of the nonlinear transfer characteristic. Obviously, for a linear amplifier, A2-A3
=...= 0. Consider
the case for which the input signal is a sinusoid given by
v / ( 0 = ôcos(fl>or)
(2.76)
Now, we find an expression for the corresponding output signal. Substituting equation (2.76) into
(2.75), applying trigonometric identities for [cos(flJ^]*, collecting terms and defining V0 to be the
sum of all of the constant terms, V, to be the sum of the terms with frequency 0)^, and so on, we
find that
v 0 ( 0 = V0Q + Vol cos(<V) + Vo2 cos(2<V) + Vo3 cos(30)ot) + •••
(2.77)
Figure 2.35 An amplifier transfer characteristic that is linear except for output saturation
-44-
The desired output is the VQ\ cos(co0t) term, which we call the fundamental component. The
KJQ term represents a shift in the dc level (which does not appear at the load if it is ac-coupled). In
addition, terms at multiples of the input frequency have resulted from the second and higher order
nonlinear terms of the transfer characteristic (2.75). These terms are called harmonic distortion.
The 2co0 term is called second harmonic, the 3co0 term is die third harmonic, and so on.
Harmonic distortion is objectionable in a wideband amplifier because the harmonic can fall in die
frequency range of the desired signal In an audio amplifier, harmonic distortion degrades die
aesthetic qualities of the sound produced by the loudspeakers.
The second harmonic distortion factor D2 is defined as die ratio of the amplitude of the second
harmonic to the amplitude of the fundamental.
£)2=^2
(2.78)
V l
°
Similarly the third-order harmonic distortion factor, and so on, are defined as
B-J-03
3
v0{
D434-
(2.79)
vol
The total harmonic distortion (THD) denoted by D is defined as
D = VA 2 +^2+A 2 +-
(Z8°)
We can often find THD expressed as a percentage. A well-designed audio amplifier might have a
THD specification of 0.01% (i.e. D- 0.0001) at rated output power.
Notice diat THD specification of an amplifier depends of die amplitude of the output signal
because the degree of nonlinearity of the transfer characteristic is amplitude-dependent. Certainly,
any amplifier clips me output signal if the input becomes large enough. When clipping occurs, THD
becomes large.
Harmonic distortion is usually not a problem for bandpass amplifiers if the bandwidth is narrow
enough so mat harmonics fall outside the frequency range of the desired signal. However, amplifier
nonlinearity causes another type of distortion that can be very troublesome even for narrowband
amplifiers. This is intermodulation and crossmodulation distortion that appear when die
amplifier is excited by a sum of two or more sinusoidal components of close frequencies. Some of
die intermodulation components generated due to amplifier nonlinearity fall into the original
frequency band. The crossmodulation, which is the transfer of the amplitude of one input signal to
a term wim a frequency of anodier input signal, is a very serious problem for radio receivers. These
topics, however, fall outside the scope of this texdbook.
Exercise 2.5 For the bipolar transistor amplifier shown below, write the PSpice code, run .AC
analysis and plot amplitude and phase characteristics in the frequency range from 1Hz to 10MHz,
100 points per decade. Identify the midband frequency range of this amplifier and find its halfpower bandwidth. Calculate its voltage, current and power gains at 1kHz. Explain me role of die
capacitors C: and C0 in the circuit
-45-
RS
l>
CS |
I
cc
s
jg~""
"••IT
Exercise 2 . 5 - bipolar transistor common-amitter amplifier
Vs 1 0 ac InV
Rs 1 2 Ik
Cs 2 3 luF
RBI 7 3 330k
RB2 3 0 100k
Ql 5 3 4 Q2N2222A
.LIB EVAL.LIB
RE 4 0 10k
470uF
CE 4
12k
RC 7
luF
Co 5
10k
RL 6
VCC 7 0 10V
.ac dec 100 1Hz 10MEGHZ
.end
Exercise 2.6 For the bipolar transistor amplifier of Exercise 2.5, modify the PSpice code to run the
transient and Fourier analyses for the time interval of 5ms, assuming that the input signal is a
sinusoidal waveform of frequency 1 kHz and amplitude 1 mV. Repeat the simulation 10 times, each
time for a different value of the input signal amplitude, covering the range from 2 mV to 20 mV.
For each value of Vs:
(a) plot the waveforms of v(6) and -A*ik on a common diagram and compare their shapes,
(b) find the corresponding value of THD, as recorded in the output file.
(c) make the plot of the total harmonic distortion of the output signal as a function of input signal
amplitude.
2.10 Summary
Linear amplifiers obey the- superposition principle. Their transfer characteristic, output voltage
versus the input voltage, is a straight line with a slope equal to the voltage gain. Linear amplification
can be obtained from a device having a nonlinear transfer characteristic by employing dc biasing
and keeping the input signal amplitude small Depending on the signal to be amplified (voltage or
current) and on the desired form of output signal (voltage or current), there are 4 basic amplifier
types: voltage, current, transresistance and transconductance amplifiers. Amplifiers can be cascaded
to increase the gain available, provided loading effects are taken into account Amplifiers increase
-46-
the signal power and thus require dc power supplies for their operation, to convert the dc current
energy from the power supply into the ac current energy of the signal amplified. Sinusoidal signals
are invariant to linear operators; they are used to measure the frequency response of linear
amplifiers. Amplifiers are classified according to the shape of their amplitude characteristic as
lowpass, bandpass or highpass. The banpass amplifiers can be narrowband or wideband. Accoupled amplifiers are used to filter out dc component of the input signal and to break the dc
current path between the signal source and the amplifier input, as well as between the amplifier
output and the load. Ac-coupled amplifiers can never be classified as lowpass; they are either
highpass or, more realistically, bandpass. In every physical amplifier there are always stray wiring or
component capacitances in parallel with the signal path present At high frequencies they act as lowimpedance shunting elements that reduce the amplifier gain. Thus every physical amplifier has a
limited bandwidth. The amplifier gain remains almost constant over the midfrequency band. It falls
off at high frequencies where stray capacitances of components and devices no longer have high
reactance. For ac amplifiers the gain falls off at low frequencies as well, because the coupling
capacitors no longer have very low reactance. The amplifier bandwidth is the frequency range over
which the gain remains within 3dB of the value at midband. The limits of the bandwidth are the
frequencies^, and^. (For dc amplifier only^fr is meaningful). Variations of the amplitude and phase
characteristics of an amplifier with frequency may cause distortion to the signal shape. Amplifiers
do not distort signals if their gain is constant and phase is proportional to frequency for the range of
frequencies contained in the input signal. If the frequency response of an amplifier is inadequate for
a particular signal, there will be linear distortion — either amplitude distortion or phase distortion, or
both. The shape of the unit step and pulse responses of the amplifier is related to its transfer
function. Large bandwidth amplifiers produce pulses with a short rise time. Small values of the
lower cutoff frequency give low percentage tilt. If the input signal amplitude is not sufficiently small,
nonlinear distortion occurs due to inherent nonlinearities of amplifying devices. Nonlinear
amplifiers do not obey the superposition principle. For sinusoidal input, sine waves of frequencies
different to the input signal frequency appear in the output signal spectrum. Their amplitudes
relative to the amplitude of the input sine-wave signal are the measure of nonlinear distortion
introduced by an amplifier.
-47-
Analog Electronics /Diodes
3. Diode Circuits
The simplest and most fundamental element of semiconductor circuits is the diode. Just like a
resistor, it has two terminals; but unlike the resistor, it is a nonlinear element This chapter is
concerned with the study of basic circuits that contain diodes. In order to understand the essence
of diode function, we begin with a fictitious element, the ideal diode. We then introduce the real
semiconductor junction diode, explain its terminal i-v characteristic and provide techniques for
the analysis of diode circuits. The latter task involves the important subject of diode modeling,
both for large-signal and small-signal applications.
Of the many applications of diodes, their use in the design of rectifiers (which convert ac to dc) is
the most common. Therefore, we shall study rectifier circuit in some detail and briefly look at a
number of other diode applications.
The semiconductor diode is a two-terminal device that incorporates a pn junction. The pn
junction is the basis of many other solid-state devices, including the bipolar junction transistor.
Thus an understanding of the/>» junction is essential to the study the material of this chapter.
3.1 The Ideal Diode
The ideal diode is a two-terminal device having the circuit symbol of Fig. 3.1(a) and the i-v
characteristic shown in Fig. 3.1(b). One of its terminals is called the anode and the other is the
cathode. The voltage v across the diode is referenced positive at the anode and negative at the
cathode. The current is referenced positive when it flows from the anode to the cathode. The
terminal characteristic of the ideal diode can be interpreted as follows: If a negative voltage is
applied to the diode, no current flows and the diode behaves as an open circuit [Fig. 3.1(c)].
Diodes operated in this mode are said to be reverse biased, or operated in the reverse direction.
An ideal diode has a zero current when operated in the reverse direction and is said to be cut-off.
(c)
(d)
Figure 3.1 The ideal diode: circuit symbol (a), i-v characteristic (b), equivalent circuit in the reverse
direction (c), equivalent circuit in the forward direction (d).
On the other hand, if a positive current is applied to the ideal diode, zero voltage-drop appears
across the diode. In other words, the ideal diode behaves as a short circuit when biased in the
forward direction [Fig. 3.1(d)]. It passes any current with zero voltage-drop. A forward-biased
diode is called to be turned on or simply on.
-48-
It is evident from Fig. 3.1(b) that the i-v characteristic is far away from a linear relationship. It
contains two straight-line segments at 90 degrees one to another; thus it is highly nonlinear,
although one may say it is piecewise linear. Other variations of piecewise-linear diode
descriptions will be considered later in this chapter.
'D
V
D
'D
.. _n
v D =0
lD=0
VD
(d)
Figure 3.2 Rectifier circuit (a): equivalent circuit for vs>0 (b), equivalent circuit for v^0 (c),
input waveform (d, upper panel),output waveform (d, lower panel).
The fundamental application of the diode, one that makes use of its severely nonlinear i-v curve,
is the rectifier circuit shown in Fig. 3.2a. The circuit consists of a series connection of a diode D
and a resistor R. Let the input voltage be the sinusoid shown in Fig. 3.2d, and assume the diode
to be ideal. During the positive half-cycles of the input sinusoid, the positive vs will cause current
to flow through the diode in its forward direction. It follows that the voltage drop across the
ideal diode will be zero. Thus the circuit will have the equivalent shown in Fig. 3.2(b), and the
output )ltage vQ will be equal to the input voltage rs. On the other hand, during the negative
half-cyciits of pf, the diode will not conduct. Thus the circuit will have the equivalent shown in
Fig. 3.2c, and the output voltage will be zero. Note that while vs alternates in polarity and has a
zero average value, v0 is unidirectional and has a finite average value, or a non-zero dc
component. Thus the circuit from Fig. 3.2a rectifiers the signal and hence is called a rectifier. It
can be used to generate dc from ac. We will study more complex rectifier circuits later in this
chapter.
Exercise 3.1.For the circuit in Fig. 3.2a, sketch the transfer characteristic vQ versus vs.
Exercise 3.2 For the circuit in Fig. 3.2a, sketch the waveform of the voltage across the diode.
12V
-49-
Example 3.1 Fig. 3.3 shows a circuit for charging a 12-V battery. If vs is a sinusoid with 24-V
peak amplitude, find the fraction of each cycle during which the (ideal) diode is on. Also find the
peak value of the diode current and the maximum voltage that appears across die diode and,
finally, find the average current that charges the battery.
Solution The diode conducts when ?s exceeds 12V, as shown in Fig. 3.3(b). The conduction
angle is 20, where 6 is given by 24cos(6[)=12. Thus #=60° and the conduction angle is 120°, or
one-third of a cycle. The peak value of the diode current occurs when the input voltage is
maximum, and is given by 7 d =(24-12)/10 = 1.2A. The maximum reverse voltage across the diode
occurs when vs is at negative peak and is equal to 24V+12V = 36V. The average current which
1 *
charges the battery is given as 1^ = —— J Id cos(g)d£ = 0.33 A, where £ is the integration
1 IT
In
*
variable.
+5V
00
(b)
Figure 3.4 Diode logic gates: (a) OR gate; (b) AND gate
Diodes together with resistors can be used to implement digital logic functions. Fig. 3.4 shows
two diode logic gates. To see how these circuits function, consider a positive logic system in
which voltage values close to 0 correspond to logic 0 (or low state) and voltage voltages close to
+5V correspond to logic 1 (or high state). The circuit in Fig. 3.4a has three inputs, vA, vB, and vc.
It is easy to see that diodes connected to +5V will conduct, thus clamping the output voltage vY
to a value equal to +5V. This positive output voltage will keep the diodes whose anodes are low
(at zero voltage) cut off. Thus the output will be high if one or more inputs are high. The circuit
in Fig. 3.4a therefore implements the logic OR function, which in Boolean notation is expressed
as
Y=A+B+C
-50-
Analog Electronics
/Diodes
Similarly, the reader is encouraged to show that the circuit in Fig. 3.4b implements the logic
AND function, that is
Y = ABC.
In analysis of circuits containing ideal diodes we may not know in advance which diodes are on
and which are off. Thus we are forced to make a considered guess. Then we analyze the circuit to
find the currents in the diodes assumed to be on and the voltage across the diodes assumed to be
off. If the current iD through the diodes assumed to be on is positive and the voltage vD across the
diodes assumed to be off is negative, our assumptions are correct and we have solved the circuit.
Otherwise, me must make another assumption about the diodes and try again. After a little
practice, our first guess is usually correct, at least for simple circuits.
+I0V
+I0V
r -10V
(b)
00
Figure 3.5 Circuits for Example 3.2
Example 3.2 Assuming the diodes to be ideal, find values of J and K i n the circuits of Fig. 3.5.
Solution In these circuits it might not be obvious at first sight whether none, one or both diodes
are conducting. In such a case we make a plausible assumption, proceed with the analysis and
then check whether we end up with a consistent solution.
For the circuit in Fig. 3.5a, we shall assume that both diodes are conducting. It follows that VB=0
and V=0. The current through D2 can now be determined from
ID2 = (10V-0V)/10k = 1mA.
Writing the first Kirchhoff s law equation for node 8 we obtain
1+lmA^ (0-(-l 0V))/5k = 2mA.
Thus I=2mA-lmA = 1mA is larger then zero and thus diode D l is conducting as originally
assumed, and the final result is J=lmA, K=0.
For the circuit in Fig. 3.5b, if we assume that both diodes are conducting, then VB=0 and V=0.
The current through D2 is obtained from
ID2 = (10V - 0V)/5k = 2mA
The node equation at B is
I+2mA = (OV - (-10V))/10k
which yields I—1mA. The current is negative, so this result is not consistent with the assumption
that diode D l is on. We start again, assuming that D l is off and D2 is on. The current through
diode D2 is given by
ID2 = (10V-(-10V))/(10k+5k)=1.33mA
and the voltage at node B is
VB = -10V +10k*1.33mA = 3.3V.
Thus the voltage across D l is negative (the anode is at 2ero and the cathode is at +3.3V), which
means diode D l is off as assumed, and the final result is 7=0 and K=3.3V.
Exercise 3.3 Find the current I and voltage V for the circuits shown in Fig.3.6.
Ans. (a) 2mA, 0V, (b) 0mA, 5V, (c) 0mA, -5V, (d) 2mA, 0V, (e) 3mA, 3V, (f) 4mA, IV.
-51-
(e)
(0
Figure 3.6 Circuits for Exercise 3.3
Exercise 3.4 Find the state of the ideal diodes in the circuits of Fig. 3.7
lkQ
4kO
|6kfi
5mA
1
©
10V
T3V
Figure 3.7 Circuits for Exercise 3.4
Exercise 3.5 Fig. 3.8 shows a circuit for an ac voltmeter. It utilizes a moving-coil meter that
gives a full-scale reading when the average current flowing through it is 1mA. Hie moving-coil
meter has a 50 Q resistance. Find the value R that results in the meter indicating full-scale reading
when the input sine-wave voltage rs is 20V peak-to-peak. Ans. R=3.133 kfi.
o—W-
Moving-coil
'oJ meter
Figure 3.8 Circuit for Exercise 3.5
3.2 Terminal Characteristics of Semiconductor Diodes
In this section we study the characteristics of real diodes - specifically semiconductor junction
diodes made of silicon. Fig. 3.9 shows an example of the i-v characteristic of a silicon junction
diode. As indicated, the characteristic curve consists of three distinct regions:
1. The forward-bias region determined by v>0.
-52-
Analog Electronics / Diodes
2. The reverse-bias region, determined by v<0.
3. The breakdown region, determined by v< Vz.
The three regions of operation are described in the following.
Figure 3.9 The i-v characteristic of a silicon-junction diode (a), the diode i-v relationship with negative
current scale expanded and negative voltage scale compressed.
The forward-bias region. The forward-bias region of operation is entered when the terminal
voltage v is positive. In the forward region, the i-v relationship is closely approximated by the
Shockley equation
i = /J«p(-£-)-l]
(3-1)
In diis equation, Is is the saturation current, which is constant for a given diode and given
temperature. A better name for Is is the scale current - it is direcdy proportional to the crosv
sectional area of the diode semiconductor structure. Thus doubling the junction area results in a
diode with double the value of Is and, as (3.1) indicates, double the current / for a given forward
voltage v. For low-power or "small-signal" diodes, which are small-size, diodes intended for lowpower applications, Ty is in the order of 10"15A. The value of Is is a very strong function of
temperature. As a rule of thumb, Is of a silicon diode doubles in value for every 10° rise in
temperature.
The voltage VT in (3.1) is a constant called the thermal voltage, given by
VT = kT/q
(3.2)
where k is Boltzmann's constant equal to 1.38-10-23 joules/kelvin, T i s the absolute temperature
in kelvins and q is the magnitude of electronic charge equal to 1.602-10"19 coulomb. At room
temperature, (20°C=293K) VT is 25.2 mV. In rapid, approximate circuit analysis we can use
P/'r=25mV at room temperature.
The constant n in the diode equation has a value between 1 and 2, depending on the material and
the physical structure of the diode. Diodes made using the standard integrated-circuit fabrication
process exhibit n=\ when operated under normal conditions (low and moderate current). Diodes
available as discrete two-terminal components generally exhibit n-2.
For appreciable current /' in the forward direction, specifically for i»Is, equation (3.1) can be
approximated by the exponential relationship
v
« = /sexP(—)
(3.3)
The relationship between the voltage and the current in the forward direction takes the
logarithmic form, as can be obtained from (3.3)
-53-
v = nVT\n{—)
(3.4)
The exponential relationship of the current / to the voltage v holds over many decades of current
(a span of as many as seven decades - that is a factor of 1CK7- can be found). This is quite a
remarkable property of junction diodes (and bipolar transistors as well), one that has been
exploited in many interesting applications. The diode i-v characteristic is then most conveniently
plotted on a semilog paper. One can show that using the vertical, linear axis for v and the
horizontal, log axis for i, one obtains from (3.3) a straight line with a slope of 2.3«K T per decade
of current.
Since both Is and VT in (3.3) are functions of temperature, the forward i-v characteristic varies
with temperature as illustrated in Fig. 3.10. At a constant diode current, the voltage drop across
the silicon diode decreases by approximately 2mV for every kelvin increase in temperature. This
property has been exploited in the design of electronic thermometers.
A glance at the i-v characteristic in the forward region (Fig. 3.9) reveals that the current / is
negligibly small for v smaller than about 0.5V. This value is usually referred to as the knee
voltage. It should be emphasized, however, that this apparent threshold in the diode
characteristic is simply a consequence of the exponential relationship. Another consequence of
this relationship is the rapid increase of /'. Thus for a "fully conducting" diode; the voltage drop
across it lies in a narrow range, approximately from 0.6 to 0.8V for silicon diodes. This gives rise
to a simple model of a diode where it is assumed that a conducting diode is replaced by a 0.7-V
battery. This constant-voltage-drop model is discussed later on in this chapter. Diodes with
different ratings (i.e. different areas and different 1$) will exhibit the 0.7-V drop at different
currents. For instance, a low-power switching diode may be considered to have 0.7V drop at
/'=lmA, while a higher power diode may have a 0.7V drop at /-1A.
-54-
The reverse-bias region. The reverse-bias region is entered when the diode voltage v is made
negative. Equation (3.1) predicts that if v is negative and its absolute value is a few times larger
than VT (25mV), the exponential term becomes negligibly small compared to unity and the diode
current becomes
i=-Is
(3.5)
that is, the current in the reverse direction is constant and equal to Is-.'m magnitude. This is the
reason behind the term saturation current: the current saturates when-1 v\>Vj, v<0. In fact, the
saturation current increases with | v\ for real silicon diodes, due to the charge carriers generationrecombination phenomena that occur in the depletion layer. Moreover, real diodes exhibit
reverse currents that, though quite small, are much larger than Is. For instance, a low current or
1-mA diode whose Is is of the order of 10"14 A to 10"15 A could show a reverse current of the
order of 1 nA. A good part of the reverse current is due to the leakage effects. These leakage
currents are proportional to the junction area, just as Is is. Note that because of the very small
magnitude of the current these details are not clearly evident on the diode i-v characteristic such
as on that of Fig. 3.9.
The breakdown region. The third distinct region of diode operation is the breakdown region,
which can be easily identified on the diode i-v characteristic in Fig. 3.9. The breakdown region is
entered when the magnitude of the reverse voltage exceeds a threshold value specific to the
particular diode and called the breakdown voltage. This is the voltage at the left-hand-side
"knee" of the i-v curve in Fig. 3.9 and is denoted Kz, where the subscript Z stands for Zener
(regardless of the Zener or avalanche breakdown phenomenon takes actually place within the
device as is discussed below in this Section).
Zener diodes are intended to operate in the reverse breakdown region of the diode characteristic.
Most desirable characteristics of Zener diodes in many of their applications are:
1. Precise voltage in the breakdown region.
2. Breakdown voltage that is independent of temperature.
3. Extremely low dynamic impedance in the breakdown region (i.e. nearly vertical
current versus voltage characteristic).
Two mechanisms are responsible for breakdown oipn junctions under reverse bias. One is called
avalanche. As the reverse-bias voltage is increased, the width of the depletion region increases.
Furthermore, in the center of the depletion region, the electric field becomes more intense. This
field accelerates minority carriers from both sides of the junction that diffuse into the depletion
region. In moving through the material, the charge carriers repeatedly collide with the atoms in
die crystal lattice.
Under certain conditions of doping and applied voltage, the carriers gain sufficient energy
between collisions to break a covalent bond. This liberates two additional free charge carriers,
one hole and one electron, which are accelerated, eventually releasing additional charge carriers.
The term avalanche is rather descriptive of the process. The result is a large current flowing
through the reverse bias diode.
Avalanche does not occur until the field is strong enough so that the charge carriers can obtain
enough energy between collisions to break a covalent bond. A junction very lightly doped on
both sides has a very thick depletion region. Thus the field (volts/meter) is not sufficiently
intense until the applied voltage is very high.
The average distance that frees charge carriers travel between collisions decreases with
temperature. Therefore, at higher temperatures a slightly higher voltage must be applied before
avalanche breakdown occurs.
-55-
The second mechanism causing breakdown is called the Zener effect. It occurs in abrupt
junctions having high doping levels. For such junctions, the depletion region is very narrow
because die charge density of ionized dopant atoms is high and a large amount of charge can be
stored in a very thin layer. As reverse bias is applied, the field in the depletion region increases in
intensity. When the field strength is of the order of 1 V divided by the crystal lattice spacing, it is
possible for covalent bonds to be broken by the field. In other words, the forces are so strong
that electrons are pulled loose from their bonds.
The energy gap between the top of the valence band and the bottom of the conduction band
becomes slightly smaller with increased temperature. Hence the force required to break the
covalent bonds is slighdy smaller at higher temperatures. As a result, if the Zener effect is
responsible for breakdown, the breakdown voltage tends to become smaller with increased
temperature. This is opposite to the case for avalanche breakdown.
As a rule, the Zener effect is responsible if the breakdown voltage magnitude is less man 6 V, and
the avalanche effect is responsible if the breakdown voltage is greater than 6 V. For diodes
having breakdown voltages of approximately 6 V, a mixture of both mechanisms can occur. It is
possible to obtain diodes having breakdown voltages of about 6 V with very small temperature
coefficients, because the temperature effects of the two breakdown types offset one another.
Dynamic impedance also tends to be a function of breakdown voltage, reaching a minimum for
breakdown voltages of about 6 V. Thus, as circuit designers, we tend to select 6-V diodes.
As can be seen from Fig. 3.9, in the breakdown region the reverse current increases rapidly, and
the associated increase in the voltage drop is very small. Diode breakdown is normally not
destructive, provided that the power dissipated in the diode is limited by external circuitry to a
"safe" level. This safe value is normally specified on the device data sheets. It therefore is
necessary to limit the reverse current in the breakdown region such that the product of the dc
current and the dc voltage is lower than the permissible power dissipation.
The fact that the diode i-v characteristic in breakdown is almost a vertical line enables it to be
used in voltage regulation (stabilization). This subject will be studied in Section 3.6.
3.3 Analysis of Diode Circuits
In this section, we shall study methods for the analysis of diode circuits. We shall concentrate on
circuits in which the diodes are operating in the forward-bias region. Operation in the breakdown
region is considered in Section 3.5.
Consider the circuit shown in Fig. 3.11 consisting of a dc source VDDJ a resistor R and a diode.
We wish to analyze this circuit to determine the diode current ID and the diode voltage VD.
R
ID
Figure 3.11 A simple diode circuit
-56-
The diode is obviously biased in the forward direction. Assuming that VDD is greater than 0.1V or
so, the diode current will be much greater than Is and we can represent the diode i-v characteristic
by the exponential relationship
/D = / S e x p ( - ^ - )
(3.6)
The other equation that governs the circuit operation is obtained by writing a Kirchhoff loop
equation, resulting in
ID=(VDD-VD)/R
(3.7)
Assuming that the diode parameters Is and n are known, Eqs. (3.6) and (3.7) are two equations in
the two unknown quantities ID and Vp. Two alternative ways for obtaining the solution are
graphical analysis and iterative analysis.
Graphical analysis. Graphical analysis is performed by plotting the relationship (3.6) and (3.7)
on the i-v plane. The solution can then be obtained as the coordinates of the point of intersection
of the two graphs. A sketch of the graphical construction is shown in Fig. 3.11; the curve
represents the exponential diode equation (3.6) and the straight line represents (3.7). Such a
straight line is known as the load line, a name that will become more meaningful in later
chapters. The load line intersects the diode curve at point Q, which represents the operating
point of the circuit. Its coordinates give the values of ID and VD.
Graphical analysis aids in the visualization of circuit operation. However, the effort involved in
performing such an analysis, particularly for complex circuits, is too great to be justified in
practice.
VDD/R
0-point
o
vD
vDD
Figure 3.12 Graphical analysis of the circuit in Fig. 3.11
Iterative analysis. Equations (3.6) and (3.7) can be solved using a simple iterative procedure, as
the one illustrated in the following example.
Example 3.3 Determine the current ID and the diode voltage VD for the circuit in Fig. 3.11 with
KDD=5V and R=lk. Assume t h a t ^ l O 1 0 , «=1.7.
Solution Taking a look at Figure 3.11 one may arrive at conclusion that the diode is biased into
the forward direction. Thus Equation (3.6) describes the diode current Taking a natural
logarithm of both sides of (3.6) one obtains
VD=»VTki(ID/Q.
Now, substituting (3.7) for ID in the above formula gives
VD=»VT]n((VDD-VD)/(RIs))
The last equation can be rewritten for iterative calculations, such that the diode voltage in
iteration (k+1) is related to its value in the previous iteration (k) as follows
VDW = »VT]n((VDD-VD<»)/(RIs))
-57-
Analog Electronics
Assume the initial guess tor the diode voltage is VD
values of the current in successive iterations are
k
0
1
2
3
4
vj>
0.5V
0.748942V
0.746524V
0.746524V
/Diodes
—0.5V. Its values and the corresponding
J «
0.013mA
4.500mA
4.251mA
4.251mA
...
Thus the third iteration yields JD=4.251mA and KD=0.74624V. Since these values are not much
different from the values obtained after the second iteration, no further iterations are necessary.
Figure 3.13 The effect of diode series resistance
The iterative analysis procedure utilized in the example above is simple and yields accurate results
after two or three iterations. Nevertheless, there are situations in which the effort and time
required to perform the iterative calculations are still greater than can be justified. Specifically,
when one is doing a pencil-and-paper design of a relatively complex circuit, rapid circuit analysis
is a necessity. Through quick analysis, the designer is able to evaluate various possibilities before
deciding on a suitable circuit. To speed up the analysis process, one must be content with less
precise results. This, however, is seldom a problem, as the more accurate analysis can be
postponed until a final or almost final design is obtained. Accurate analysis of the final or almost
final design can be performed with the aid of a computer-analysis program such as SPICE. The
results of such an analysis can then be used to further refine or "fine-tune" the design.
Diode models. At high current levels, the ohmic resistance of the semiconductor forming the
junction becomes significant. Addition of a series resistance Rt to the diode modeled by the
Shockley equation (3.1) can account for this. The modified version of (3.4) becomes
-58-
(3.8)
v-«FI.In(7-) + « J
Typical low-current diodes have Revalues ranging from 10 to 100Q. The resistance R, decreases
with the diode cross-sectional area. Thus for high-current diodes it is of the order of 0.1Q. The
effect of die diode series resistance on the i-v characteristic is illustrated in Fig. 3.13. For a fixed
value of current, the voltage drop across the diode external terminals increases with the series
resistance value.
Slope=l/r£)
(c)
Figure 3.14 Approximating the diode forward characteristic with two straight lines (a), (b);
equivalent circuit representation of the piecewise-linear model (c).
Although the exponential i-v characteristic plus a series resistance is an accurate model in the
forward region, its nonlinear nature complicates the analysis of diode circuits. The analysis can be
greatly simplified if we can find piecewise-linear relationship to describe the diode terminal
characteristics. An attempt in this direction is illustrated in Fig. 3.14, where the exponential curve
is approximated by two straight-line segments, line A with zero slope and line B with a slope of
//r D . It can be seen that for this particular diode, over the current range from 0 to 10mA the
voltages predicted by the straight-line model differ from those predicted by the exponential
model by less than 50 mV. Of course, the choice of two lines is not unique; one can obtain a
closer approximation by restricting the current range over which the approximation is required.
The straight-lines (or piecewise-linear) model of Fig. 3.14 can be described by
»D = °>
h=
V
D ^ VDo
Do
(3.9)
vD>VDo
where VD, is the intercept of line B on the voltage axis and rD is the inverse of the slope of line B.
For the particular example shown, VD, = 0.65 V and rD = 20 Q.
-59-
The equivalent circuit shown in Fig. 3.14c can represent the piecewise-linear model described by
(3.8). Note that an ideal diode is included in this model to constrain iD to flow in the forward
direction only. This model is also known as the "battery-plus-resistance" model.
Example 3.4 Repeat the problem in Example 3.3 utilizing the piecewise-linear model whose
parameters are VDt — 0.65 V, rD = 20 Q.
Solution Replacing the diode in Fig. 3.11 with the equivalent circuit model of Fig. 3.14c results in
the circuit in Fig. 3.15, from which we can write the current ID
ID = {VDD - VJ/(R + rD) =4.26mA
The diode voltage can now be computed
VD = VD. + roID = 0.7353V
Note that the values obtained using the simplified model are not very much different from the
accurate values obtained in Example 3.3.
Figure 3.15 The circuit of Fig. 3.11 with the diode replaced by its
piecewise-linear model of Fig. 3.14c.
An even simpler model of the diode forward characteristic can be obtained if we use a vertical
straight line to approximate the fast-rising part of the exponential curve, as shown in Fig. 3.16a.
The resulting model simply says that a forward-conducting diode exhibits a constant voltage drop
VD. The value of VD for silicon diodes is usually taken to be 0.7V. The equivalent circuit shown
in Fig. 3.16c can represent the constant-voltage-drop model. This model is the one most
frequently employed in initial phases of analysis and design. Finally, note that if we employ the
constant-voltage-drop model to solve the problem in Examples 3.3 and 3.4 we obtain
IQ = (VDD -VD)IRID
= 4.3 mA which is not too differ^ it from the values obtained with the
more elaborate models.
V,D
-60-
(b)
Ideal
VD
(c)
Figure 3.16 Development of the constant-voltage-drop model of the diode forward characteristic with
two straight lines (a), (b); equivalent circuit representation of the model (c).
In applications that involve voltages much greater than the diode voltage drop (0.6 - 0.8V), we
may neglect the diode voltage drop altogether while calculating the diode current The result is
die ideal diode model, which we studied in Section 3.1.
The question of which diode model to use in a particular application is one a circuit designer
faces repeatedly, not just with diodes but with every circuit element. One's ability to select
appropriate device models improves widi practice and experience.
3.4 The Diode Small-Signal Model at Low Frequencies
We will encounter many examples of electronic circuits in which dc supply voltages are used to
bias a nonlinear device at an operating point and a small ac signal is injected into the circuit. We
often split analysis of such circuits into two parts. First, we analyze the dc circuit to find the
operating point. In this analysis of bias conditions, we must deal with die nonlinear aspects of the
device. In the second part of the analysis, we consider the small ac signal. Since virtually any
nonlinear characteristic is approximately linear (straight) if we consider a sufficiendy small
portion, we can find a linear small-signal equivalent circuit for the nonlinear device to use in
die ac analysis.
The small-signal linear equivalent circuit is an important analysis approach that applies to many
types of electronic circuits. In diis section we demonstrate the principles with a simple diode
circuit. In the next chapter we use similar techniques for transistor amplifier circuits.
Now we will show that in the case of a diode, the small-signal equivalent circuit consists of a
resistance. Consider a conceptual circuit in Fig. 3.17a and the corresponding graphical
representation in Fig. 3.17b. A dc voltage Kp, represented by a battery, is applied to the diode;
and a time-varying signal iy(/), assumed (arbitrarily) to have a triangular waveform, is added to
(superimposed on) the dc voltage VD. In the absence of the signal ty(/) die diode voltage is equal
to Vp, and correspondingly the diode will conduct the current ID given by the Shockley equation
(3.1), which for VD»VT, (i.e. for sufficiently large forward current) simplifies to
/D = /Sexp(-^T)
(3-10)
When die signal ta(/) is applied, the total instantaneous diode voltage vD(t) will be given by
vD(t) = VD+vd(t)
(3.11)
Correspondingly, the total instantaneous diode current iD(t) will be
K +V
' < ( 'fl
(3,2,
which can be rewritten
-61-
Figure 3.17 Development of a diode small-signal model
Using (3.10) we obtain
/D(0 = / D exp(^J
(3.14)
Now if the magnitude of the signal vt(t) is kept sufficiently small such that
v,(0 «
(3.15)
1
nVT
then we may expand the exponential of (3.14) into the Taylor series and truncate the series after
the first two terms to obtain approximate expression
V
(
(3.16)
iD(0 = ID ! + • ' ' ^
nVT J
This is the small-signal approximation. It is valid for signals whose amplitudes are smaller than
about lOmV.
From (3.16) we have
iD(0 =
(3.17)
ID+^-vd(t)
Thus superimposed on the dc current ID we have a signal component directly proportional to the
signal voltage m(/). That is
where
(3.19)
<*(')=-F-Ô
nVT
The quantity relating the signal current it(t) to the signal voltage ty(/) has the dimensions of
conductance, and is called the diode small-signal conductance. The inverse of this parameter
is the diode small-signal resistance, incremental resistance, or dynamic resistance r&,
-62-
r<=-f-
(3-20)
Note that the value of rd is inversely proportional to the diode dc current JD.
Let us return to the graphical representation in Fig. 3.17b. It is easy to see that using the smallsignal approximation is equivalent to assuming that the signal amplitude is sufficiently small such
that the excursion along the i-v curve is limited to a short, almost linear segment. The slope of
this segment, which is the slope of the i-v curve at the operating point J2, is equal to the smallsignal conductance. The reader is encouraged to prove that the slope of the i-v curve at i=ID is
equal to 7D/(«KT), which is 1 /rd, that is,
rd=\/
di„
dvD
(3.21)
Now, if we denote by VDo the point at which the tangent intersects the vD axis, we can describe
die tangent by the equation
;D=l[vD-FDo]
(3.22)
This equation is the model for the diode operation for small variations around the bias or
quiescent point Q, in the forward region. The model can be represented by the equivalent circuit
shown in Fig. 3.18, from which we can write
vD = VDO + iDr„ =VDo+ IDrd + idrd =VD+ idrd
(3.23)
=
Thus, as expected, the signal voltage across the diode is given by Vd id rd. We conclude that the
small-signal approximation allows one to separate the dc analysis from the signal analysis.
Eliminating all dc sources and replacing the diode with its small-signal resistance rd performs the
signal analysis. The obtained small-signal equivalent circuit is then solved using linear circuit
analysis techniques.
Ideal Vfo rd
Figure 3.18 Equivalent circuit model for the diode for small changes around a bias point Q.
The incremental resistance r& is the inverse of the slope of the tangent at Q and the VDO is the intercept of
the tangent with the VD axis (see Fig. 3.17).
Note: Perhaps we should review at this moment the notation we have used for the diode
currents and voltages, because we use similar notation throughout the book.
•
vD and iD represent the total instantaneous diode voltage and current. At times, we
may wish to emphasize the time-varying nature of these quantities and then we
use vD{t) and iD(t).
•
VD and ID represent the dc diode current and voltage at the quiescent point
•
Vd and id represent the (small) ac signals. If we wish to emphasize their timevarying nature, we use Vd{t) and id(t)Now we consider an example of linear equivalent circuit analysis for a simple but useful smallsignal diode circuit. The circuit is a voltage-controlled attenuator. It is shown in Fig. 3.19. The
input to the circuit is a small ac voltage vs(t) and the output v0(t) is an attenuated version of the
input. The amount of attenuation depends on the value of the dc control voltage Vc.
-63-
Notice that the ac signal to be attenuated is connected to the circuit by a coupling capacitor.
The output voltage is connected to the load R^ by a second coupling capacitor. However, the
coupling capacitors are open circuits for dc. Thus die j2-point of die diode is unaffected by the
signal source or the load. Furthermore, the coupling capacitors prevent (sometimes-undesirable)
dc currents from flowing in the source or the load.
Figure 3.19 Variable attenuator using a diode as a controlled resistance
Because of the coupling capacitors, we only need to consider Vc, R^ and the diode to perform
the dc analysis to find the =<2-point. The dc circuit is shown in Fig. 3.20. We can use any of the
techniques discussed earliest in this chapter to find the jg-point. Once it is known the j2-point
value of the diode current ID can be substituted into Equation (3.20) to determine the dynamic
resistance of the diode.
Fig. 3.20 Dc circuit equivalent to Fig. 3.19 for jg-point analysis
Now we turn our attention to the ac signal. The dc control source should be considered as a
short circuit for ac signals. In general, dc voltage sources have ac components of current, but no ac voltage the dc voltage source is a short circuitfor ac signals. This is an important concept that we will use many
times in drawing ac equivalent circuits.
The equivalent circuit for ac signals is shown in Fig. 3.21. The capacitors have been replaced by
short circuits, which means the signal frequency is assumed sufficiendy high for the capacitors'
reacMnce to be much smaller than resistance R and R^ connected in series to capacitors C, and
C2, respectively. The diode has been replaced by its dynamic resistance. This circuit is a voltage
divider and can be analyzed by ordinary linear circuit analysis. The parallel combination of R^, R L
and rd is denoted as R^ given by
1
(3.24)
Rp
\IRc+\IRL+\lrd.
Then the attenuation of the circuit is
^S
(3.25)
vs
R + Rp
Of course, Avs is less than unity.
-64-
Figure 3.21 Small signal equivalent circuit for Figure 3.19
Exercise 3.6 Suppose that the circuit of Fig. 3.19 has R = 100 Q , R^ - 2 k Q , and R L = 2 kQ.
The diode has n - l a n d is at a temperature of 300K. For purposes of j2-point analysis assume a
constant diode voltage of 0.6 V. Find the j2-point value of the diode current and Avs for (a) Vc 1.6 V and (b) Vc = 10.6 V. Ans. (a) 1D = 0.5 mA,Av
= 0.331, (b) 7 D = 5 * 1 ^ , ^ = 0.0492.
An application for voltage-controlled attenuator occurs in tape recorders, as an example. A
problem frequently encountered in recording a conversation is that some persons speak quietly,
whereas others speak loudly. Furthermore, some may be far from the microphone, whereas
others are close. If an amplifier with fixed gain is used between the microphone and the tape
head, either the weak signals are small compared the noise level or the strong signals drive the
recording nonlinear so that severe distortion occurs.
Figure 3.22 The voltage-controlled attenuator is useful in maintaining a
suitable signal amplitude at the recording head
A solution is to use a voltage-controlled attenuator placed between the microphone and a highgain amplifier in a system like that shown in Fig. 3.22. When the signal being recorded is weak,
the control voltage is small and very little attenuation occurs. O n the other hand, w h e n the signal
is stron^, the control voltage is large, so the signal is attenuated, preventing distortion. Rectifying
the output of the amplifier generates the control voltage. The rectified signal is filtered by a longtime-constant
R C filter so that the attenuation responds to the long-term average signal
amplitude rather than adjusting too rapidly. With proper design, this system can provide an
acceptable signal at the recording heads for a wide range of input signal amplitudes. Eventually,
we will discuss all of the circuits required in this system.
-65-
Analog Electronics /Diode Circuits
3.5 Rectifier Circuits
Now that we have introduced the diode and some methods for analysis of diode circuits, we
consider some practical circuits. First, we consider several types of rectifiers that convert ac
power into dc power. These rectifiers form the basis for electronic power supplies and batterycharging circuits. Other applications for rectifiers are in signal processing, such as demodulation
of a radio signal, or measuring the average amplitude of a signal from a microphone, as
considered in the previous section. Another application is precision conversion of an ac signal to
the dc in an electronic voltmeter.
Half-wave rectifier circuits. A half-wave rectifier with a sinusoidal source and resistive load is
shown in Fig. 3.23a. (The same circuit has already been analyzed in Section 3.1, for the ideal
diode case.). When the source voltage is positive, the diode is in the forward-bias region. If an
ideal diode is assumed, the source voltage appears across the load. For a typical real diode, the
output voltage is less than the source voltage by an amount equal to the drop across the diode,
which is about 0.7V for silicon diodes at "room temperature". This can be seen in Fig. 3.23b.
(a)
(b)
Figure 3.23 Half-wave rectifier with resistive load (a); input and load voltages versus time (b)
If the source voltage is negative, the diode is reverse b;.. :d and no current flows through the
load. Even for typical real diodes only a very small reverse current flows. Thus only the positive
half-cycles of the source voltage appear across the load. One can use a half-wave rectifier to
charge a battery, as shown in Fig. 3.3a. Current flows in that circuit whenever the instantaneous
ac source voltage is higher than the battery voltage. As shown in the figure, it is often necessary
to add resistance in series with the diode to limit the magnitude of the current. When the ac
source voltage is less than'the battery voltage, the current is zero. Thus the current flows only in
die direction that charges the battery.
Half-wave rectifier with smoothing capacitor. Often, we want to convert an ac voltage into a
nearly constant dc voltage to be used as a power supply for our designs. One approach to
smoothing the rectifier output is to place a large capacitor across the output terminals of the
rectifier. The circuit and waveforms of current and voltage are shown in Fig. 3.24. When die ac
source reaches a positive peak, the capacitor is charged to the peak voltage (assuming an ideal
diode). Then when the voltage drops below the voltage stored on the capacitor, the diode is
reverse biased and no current flows through the diode. The capacitor continues to supply current
to the load, slowly discharging until the next positive peak of the ac input. As shown in the
figure, current flows through the diode in pulses that recharge the capacitor.
-66-
«
(b)
Figure 3.24 Half-wave rectifier with smoothing capacitor (a); voltage and current waveforms (b)
Because of the charge and discharge cycle, the load voltage contains a small ac component called
ripple. Usually, it is desirable to minimize ripple, so we choose the largest capacitance value that
is practical. In this case, the capacitor discharges for nearly the entire cycle, and the charge
removed from the capacitor during one discharge cycle is
QsJJ
(3.26)
where IL is the average load current and T is the period of the ac voltage. Since the charge
removed from the capacitor is the product of the change in voltage and the capacitance, we can
also write
Q=VrC
(3.27)
where Vr is the peak-to-peak ripple voltage and C is the capacitance. Equating the right-hand
sides of (3.27) and (3.26) allows one to solve for C
/ T
c=-£-
(3.28)
In practice, Equation (3.28) is approximate because the load current varies in time and because
the cap; itor does not discharge for a complete cycle. However, it gives a good-starting value for
the capacitance required in the design of power-supply circuits. We will return to the subject of
power-supply design after we have introduced transistor circuits and operational amplifier
feedback circuits.
The average voltage supplied to the load if a smoothing capacitor is used is approximately
midway between the minimum and maximum voltages. Thus, referring to Fig. 3.24, the average
load voltage is
V.
(3.29)
vL=vm-vD-
An important aspect of rectifier circuits is the peak inverse voltage (PIV) across the diodes. Of
course, the breakdown specification of the diodes should be greater in magnitude than PIV. For
example, in the half-wave circuit with a resistive load, shown in Fig. 3.23, the PIV is Vm.
Addition of a smoothing capacitor in parallel with the load increases the PIV to (approximately)
2Vm. Referring to Fig. 3.24, for the negative peak of the ac input, we see that the reverse bias of
the diode is the sum of the source voltage and the voltage stored on the capacitor.
Full-wave rectifier circuits. Several full-wave rectifiers are in common use. One approach
uses a center-tapped transformer and two diodes as shown in Fig. 3.25a. This circuit consists of
-67-
Analog Electronics/Diode Circuits
two half-wave rectifiers widi out-of-phase source voltages and a common load. The diodes
conduct on alternate half-cycles.
Figure 3.25 Center-tapped-transformer full-wave rectifier
Besides providing the out-of-phase ac voltages, the transformer also allows adjustment of Vm by
selection of turns ratio. This is an important function because the ac voltage available is often not
suitable for direct rectification - usually either higher or lower dc voltage is required.
A second type of full-wave rectifier uses the diode bridge shown in Fig. 3.26a. When the ac
voltage is positive at the top of the secondary winding, current flows through diode A, then
through the load and returns through diode B as shown in the figure. For the opposite polarity,
i.e. during the alternate half-cycle, current flows through diodes C and D. Notice that in either
case the current flows in the same direction through the load. For real diodes, the peak voltage
across the load is lower in the case of the diode bridge rectifier circuit compared to the centertapped transformer rectifier circuit. This is because two diodes are connected in series with the
load in the diode bridge circuit and, consequently, twice the voltage drop across a forward-biased
diode is subtracted from the ac source voltage to obtain the load voltage. A single voltage drop
across the diode is subtracted in the case of the circuit in Fig. 3.25a. This property is illustrated in
Fig. 3.26b and Fig. 3.25b.
If one side of die load is connected to die ground as shown in Fig. 3.26a, neither of the ac source
terminals of the diode bridge can be connected to the ground. The transformer with floating
secondary winding helps to realize this condition. (If both the ac source and the load had a
common ground connection and no transformer was used, part of die circuit would be shorted.)
If one wishes to smooth the voltage across the load, a capacitor can be placed in parallel with the
load similar to the half-wave circuit discussed earlier. In the full-wave circuits, the capacitor
-68-
discharges for only a half-cycle before being recharged. Thus the capacitance required
maintaining a given level of ripple is only half as much in the full-wave circuit as for the halfwave circuit. Therefore, Equation (3.28) can be modified to obtain
C = -LIV.
(3.30)
for the full-wave rectifier with a capacitor filter.
-5.9U +
*•
n UUMIU)
•. UC2)
(b)
Fig. 3.26 Diode-bridge full-wave rectifier
Power supply circuit. A diode rectifier forms an essential building block of the dc power
supplies required to feed electronic equipment. A block diagram of such a power supply is shown
in Fig. 3.27. As indicated, the power supply is fed from the 220-V (tms) 50-Hz ac line, and it
delivers a dc voltage VL (usually in the range of 5 to 20V) to an electronic circuit represented by
the load block. The dc voltage VL is required to be as constant as possible in spite of variations
in the ac line voltage and in the current drawn by the load. Normally, the ripple at the output of a
rectifier circuit, with a smoothing capacitor of practical value, are to big and do not fulfill the
requirements of most electronic equipment Further means are necessary to stabilize (regulate)
the voltage across the load.
-69-
The first block in the power supply is the power supply. It consists of two separate coils wound
around an iron core that magnetically couples the two windings. The primary winding, having
N 7 turns, is connected to the 220-V ac supply. The secondary winding, having N2 turns, is
connected to the rectifier circuit We denote the turns ratio by n=N1/N2. Thus an ac voltage vs of
220/ ft V(rms) develops between the two terminals of the secondary winding. By selecting the
appropriate turns ratio n for the transformer, the designer can step the line voltage down to the
value required to yield the particular dc voltage output of the supply. For instance, a secondary
voltage of 8-V rms is usually required for a dc output of 5V. This can be achieved with the
«=28:1 turns ratio.
It should be noted that to simplify the discussion, zero-resistance transformer windings ate
assumed for the rectifier circuits analyzed in this Section. Thus the results obtained are
approximate only; however, they can still be sufficiently accurate for low-power applications.
In addition to providing the appropriate sinusoidal amplitude for the dc power supply, the power
transformer provides electrical isolation between the electronic equipment and die power line
circuit. The isolation minimizes die risk of electric shock to the equipment user.
The diode rectifier converts the input sinusoid vs to a unipolar output that can have a pulsating
waveform indicated in Fig. 3.27. Although this waveform has a nonzero average, or a dc
component, its pulsating nature makes it unsuitable as a dc source of electronic equipment, hence
die need for a filter that suppresses high-frequency components of the rectifier's signal In its
simplest form, the filter is a smoothing capacitor, as discussed above. The filter block in Fig. 3.27
significantly reduces the variations of the rectifier output.
The filtered rectifier output, though much more constant than without the filter, still contains a
time-dependent component, known as ripple. To reduce the ripple and to stabilize the magnitude
of the dc output voltage of the supply against variations caused by changes of in load current, a
voltage regulator is employed. Such a regulator can be implemented using a Zener diode, which
will be discussed, in the following section. Alternatively, an integrated circuit (IC) regulator can
be used.
Exercise 3.7 A power-supply circuit is needed to deliver 0.1 A and 15V (average) to the load. The
ac source available is 220V rms with a frequency of 50Hz. Assume that the full-wave circuit of.
Fig. 3.26 is to be used with a smoothing capacitor in parallel with the load. The peak-to-peak
ripple voltage is to be 0.4V. Allow 0.7V for forward diode drop. Find the turns ratio n needed
and the approximate value of die smoothing capacitor. Verify your results using SPICE. Ans.
»=18.74, C=2500nF.
Exercise 3.8 Repeat Exercise 3.7 using the circuit of Fig. 3.25 with a smoothing capacitor in
parallel with the load resistance. (Define the turns ratio as the ratio of primary turns to the
secondary, turns between the center tap and one end.) Verify your results using SPICE. Ans.
»=19.58, C=2500nF.
3.6 Zenor-Diode Voltage Regulator Circuits
The circuit shown in Fig. 3.28 is used to provide a nearly constant output voltage from a variable
source. (For its proper operation, it is necessary for the minimum value of the variable source
voltage to be somewhat larger than the desired output voltage.) A Zener diode having a
breakdown voltage equal to the desired output voltage is used. The resistor R limits die diode
-70-
current to a safe value so that the Zener diode does not overheat Moreover, as will be shown
later, the larger the value of R, the lower the output voltage variation for a given diode.
'o
Figure 3.28 A simple regulator circuit that provides a nearly constant
output voltage from a variable input voltage
Assuming that the characteristic of the diode is available, one can use a load-line construction
(see Section 3.2) to analyze the operation of the circuit. As before, we use Kirchhoff s voltage law
to write an equation relating vD to ijy. In this circuit, the diode operates in the breakdown region
with negative values of vD and /p. For the circuit of Fig. 3.28, one obtains
Vss + RiD+vD=0
(3.31)
This is the equation'of a straight line, so location of any two points is sufficient to construct the
load line. Inspection of (3.31) shows that the slope of the load line is -1/R. Thus a change of the
input voltage changes the position but not the slope of the load line. The intersection of the load
line with the diode characteristic yields the operating point.
Example 3.5 The voltage regulator circuit of Fig. 3.28 has R = 1 kfl and uses the Zener diode
having the characteristic shown in Fig. 3.29. Find the output voltage for Vss — 15 V. Repeat for
^=20V.
Solution The load lines for both values of Vss are shown in Fig. 3.29: The output voltages are
determined from the operating points where the load lines intersect the diode characteristic. The
output voltages are found to be «y=10.0V for VSS=\5V and t^lO.SV for VSS=20V. Thus a 5 V
change in the input voltage results in only a 0.5-V change in the regulated output voltage.
irfmA)
vjbOO
Figure 3.29 Graphical analysis of the circuit of Fig. 3.28
Actual Zener diodes are capable of much better performance than the one illustrated by Example
3.5. The slope of the Zener diode characteristic has been accentuated in Fig. 3.29 for clarity.
Actual Zener diodes have a more vertical slope in breakdown.
The circuit of Fig. 3.28 is known as a shunt regulator because the Zener diode is connected in
parallel (shunt with the load. The circuit is fed with a voltage that, as indicated in Fig. 3.27, is not
very constant; it includes a large tipple component from a rectifier circuit. The load can be a
simple resistor or a complex electronic circuit.
Analog Electronics / Diode Circuits
The function of the regulator is to provide an output voltage vL that is as constant as possible in
spite of the ripples in Vss and the variation in the load current IL. Two parameters can be used
to measure how well the regulator is performing its function: the line regulation and the load
regulation. The line regulation is defined as the change in VL corresponding to 1-V change in
Vss
AV,
Line
regulation =
(3.32)
AKss
and is usually expressed in mV/V. The load regulation is defined as the change in
corresponding to a small change in IL
Load
regulation = ——
VL
(3.33)
Variable
supply
VL=-vD
(b)
0
Figure 3.30 Piecewise linear model (a) for the Zener diode, valid for VD<VZThe shunt regulator equivalent circuit (b) with die Zener. diode replaced with its model.
Replacing the Zener diode can derive expressions of these performance measures by its
equivalent piecewise-linear circuit model, shown in Fig. 3.30a. Using this model, the shuntregulator circuit can be represented by its equivalent circuit shown in Fig. 3.30b. Straightforward
analysis of this circuit yields
Rr,
R
VL =V — — + V,
(3.34)
L
~h
^R+r
R + rz ~ R + rz
In the equation, only the second term on the right-hand side is a desirable one. The first and diird
terms represent the dependence on the input voltage and the load current, respectively, and thus
should be minimized. It follows that r2 should be kept low compared to R in order to obtain the
best circuit performance.
Fig. 3.31a shows an example of a half-wave rectifier using a 1N4148 low-power diode with a
smoothing capacitor of C — 47 uF and a lN750-type 5-V Zener diode regulator. It is seen in Fig.
3.31b that in spite of the capacitor voltage ripple, the output voltage is almost constant; it is equal
to the breakdown voltage of the 1N750 diode.
Fig. 3.31c contains the SPICE code used to simulate the circuit operation that resulted in the
waveforms plotted in Fig. 3.31b. Models for the diodes are included in the library file
EVAL.LIB; their names are, respectively, D1N4148 and D1N750.
-72-
3.7 Wave-Shaping Circuits
A wide variety of wave-shaping circuits find application in electronic systems. One of them is
in function generators used to generate electrical test signals for laboratory work. In a function
generator, a switching oscillator is used typically to generate a square wave. This square wave is
then passed through a circuit that integrates it, resulting in a triangular waveform. Then the
trianguk: waveform is passed through a carefully designed wave-shaping circuit to produce
sinusoidal waveform. All three waveforms are available to the user. We will consider the design
of such a function generator later. Numerous examples of wave-shaping circuits can be found in
transmitters and receivers for television, as another example. In this section we discuss a few
examples of wave-shaping circuits that can be constructed using diodes.
Clipper Circuits. Diodes can be used to form clipper circuits in which a portion of input
signals waveform is "clipped" off. For example, the circuit of Fig. 3.32a clips off any part of die
input waveform above 6V and below -9V. (We are assuming ideal diodes.) If the input voltage is
between -9V and 6V, both diodes are off and no current flows through them. Then there is no
voltage drop across R and the output voltage v0 is equal to the input voltage vs. On the other
hand, if vs is larger than 6V, diode A. is on and the output voltage is 6V because the diode
connects the 6-V battery to the output terminal. Similarly, if vs is lower than -9V, diode B is on
and the output voltage is -9V. The output voltage resulting from 15V-peak triangular input is
shown in Fig. 3.32b. The transfer characteristic of the circuit is shown in Fig. 3.32c.
-73-
Figure 3.32 Clipper circuit (a), its input and output waveforms (b) and
transfer characteristic (c)
The resistor R is selected large enough so that the forward diode current is within reasonable
bounds (usually, a few milliamperes) but small enough so that the reverse diode current results in
a negligible voltage drop. Often, we find that a wide range of resistance values provide
satisfactory performance in a given circuit.
In Fig. 3.32 we have assumed ideal diodes. If low-current silicon diodes are used, we expect a
forward drop of about 0.6V, so we should reduce the battery voltage to compensate.
Furthermore, batteries are not desirable for use in circuits if they can be avoided, because may
need periodic replacement. Thus a better design uses Zener diodes instead of batteries. Practical
circuits equivalent to Fig. 3.32a are shown in Fig. 3.33. 'Tie Zener diodes are labeled with their
breakdown voltage. The circuits shown in Fig. 3.33 have learly the same performance as the
circuits of Fig. 3.32.
/f=2kii
/f=2kn
vo(0
(b)
. 00
Figure 3.33 Circuit of Fig. 3.32a with batteries replaced by Zener diodes and
allowance made for a 0.6 forward diode drop (a); simpler circuit (b)
Exercise 3.9 Sketch the transfer characteristic for the circuits of Fig. 3.34a and 3.34b. Allow for
a 0.6V forward drop for die diodes. Sketch the output voltage if *>s(/)=15sin(ey/).
-74-
(a)
(b)
Figure 3.34 Clipper circuits for Exercise 3.9
Clamp Circuits. Another diode wave-shaping circuit is the clamp circuit that is used to add a
dc component to an ac-input waveform, so that the positive (or negative) peaks are forced to
take a specified value. In other words, the peaks of the waveform are. "clamped" to a specified
voltage value. A simple clamp circuit example is shown in Fig. 3.35. In this circuit, the positive
peaks of die input are clamped to -5V. As an application example, clamp circuits are used to
restore die dc component (black picture level) of video signals transmitted through ac-coupled
channels.
In the circuit of Fig. 3.35 the capacitor is a large value, so it discharges only very slowly and we
can consider the voltage across the capacitor to be constant. Because the capacitance is large, die
capacitor has very small impedance for the ac-input signal. Thus the output voltage of the circuit
is given by
v o (0 = v 5 ( 0 - ^ c
(3-35)
If die positive swing of the input signal attempts to force the output voltage to become more
positive than -5V, the diode conducts, charging die capacitor and thus increasing the value of Vc.
Thus the capacitor is charged to a value that adjusts the maximum positive value of the output
voltage to -5V. A large resistor R is provided so that the capacitor can slowly discharge. This is
necessary so that the circuit can adjust if the input waveform changes to smaller peak amplitude.
Of course, one can change the voltage to which the circuit clamps by changing the battery
voltage in Fig. 3.35. Reversing the direction of the diode causes the negative peak to be clamped
instead of the positive peak. Switched, or pulse operating clamp circuits are actually used in the
video e upment.
v„(V)
Figure 3.35 Example clamp circuit (a); output waveform for
triangular, 0 dc, 10-V peak-to-peak input (b)
Selection of R and C values for clamp circuits is a compromise. On one hand, we want die
capacitor to have very small impedance compared to the resistor for the ac signal. This is
necessary because we want the ac part of the output waveform to be identical to the input. On
the other hand, if we make the RC time constant too long, the circuit takes a long time to adjust
to changes in input amplitude. For now we can choose R to be a fairly large resistor, say about
100 kQ, so diat the peak diode currents are not required to be large (more than a few
milliamperes). Then we pick C so that the RC time constant is large compared to the period of
die ac input signal, say by an order of magnitude. (In the case of video signal, the discharge
-75-
Antdog Electronics / Diode Circuits
period is equal to 64 |is, which is the duration of a single picture line.) This gives a clamp circuit
with approximately the desired clamping action. Then we can simulate the circuit and adjust
values until the performance is satisfactory. Finally, we can construct the circuit to measure its
performance.
3.8 Switching and High-Frequency Behavior of the pn Junction
We have seen that the pn junction conducts little current when reverse biased and easily conducts
a lot of current when forward biased. In many applications, such as high-speed logic circuits and
high-frequency rectifiers, diodes that can switch rapidly between the conducting and
nonconducting states are extremely desirable. Unfortunately, thepn junction displays two charge
storage mechanisms that introduce delays and slow down the switching. Both of these
mechanisms can be modeled as nonlinear capacitances.
Due to the presence of the charge-storage effects, one has to distinguish between static and
dynamic properties of any electronic device, using the diode as the simple example. The diode's
familiar i-v characteristic is a static characteristic. For a one-port the word "static" implies that
operation is described by an algebraic equation, iD—fiv^j, that relates corresponding dc voltage and
current values, this is Equation (3.1) in the case of ideal pn junction. A device's static
characteristic also relates time-varying voltages and currents, but only if the time variations are
not "too fast". Electronic devices always contain internal capacitances that modify device
behavior for fast signals. Therefore, these devices are described by differential equations, which
reduce to algebraic equations for sufficiently slow signals. Thus "static" means that the device
variables are changing at such low rates that their time derivatives in the differential equation are
small enough to ignore. At high frequencies, derivatives of the circuit variables cannot be
neglected and the device capacitances significantly affect the circuit performance. Before we
consider charge storage in pn junctions, we briefly review conventional linear capacitors.
A capacitor is constructed by separating two conducting plates by an insulator as shown in Figure
3.36a. If voltage is applied to the capacitor terminals, charge flows in and collects on one plate.
Meanwhile, current flows out of the other terminal and a charge of opposite polarity collects on
the other plate. Positive charge accumulates on the plate to which the positive voltage is applied.
This is illustrated in Figure 3.36b.
Q
Conducting
plates
Insulating
dielectric
++++++++++++
Figure 3.36 Parallel-plate capacitor (a), applying voltage to a capacitor causes a charge +J2 to accumulate
on one plate and -Q to accumulate on the other plate (b)
The magnitude of the net charge Q on one plate is proportional to the applied voltage V. Thus
we have
Q=CV
(3.36)
For a parallel-plate capacitor such as that shown in Figure 3.36, the capacitance is given by
-76-
-f
(3.37)
where A is the area of one plate, d is the distance between the plates, and e is the dielectric
constant of the material between the plates. Often, the dielectric constant is expressed as
e = ere0
(3.38)
12
where e r is the relative dielectric constant and Eo = 8.85xl0" F/m'is the dielectric constant for
vacuum. [Actually, (3.37) is an approximation valid for d much smaller than both the length and
the width of the plates.] Notice that the capacitance of the parallel-plate capacitor is proportional
to the area of the plates and inversely proportional to the distance between the plates.
Depletion Capacitance. Now consider ike. pn junction under reverse bias. As the magnitude of
the voltage applied to the junction is increased, the field in the depletion region becomes
stronger, and the majority carriers are pulled back farther from the junction. This is illustrated in
Figure 3.37.
The charge in the depletion region is similar to the charge stored on parallel-plate capacitor.
Unlike the parallel-plate capacitor, a larger distance separates each additional increment of charge
stored in the depletion region. Thus the reverse-biased junction behaves as a capacitor, but its
equivalent plates move apart with the voltage, so the capacitance is not constant. The stored
charge is not proportional to the applied voltage. This capacitance is called the depletion
capacitance. Because the relationship between the stored charge and the applied voltage is not
linear, we say that the depletion capacitance is nonlinear.
C
j =
(3.39)
dvD
in which dQ is the differential of the charge stored in one side of the depletion region, and dvD is
the increment in the voltage which caused the charge increment. C\ is the capacitance of the
diode for a small ac signal superimposed on a dc operating point.
It can be shown that the incremental depletion capacitance is given by
BIBLIOTEKA GLOWNA PL
QWb^ 5
-77
(3.40)
C
J=-
~\m
1V
J0]
in which Cjo is the incremental depletion capacitance for zero bias, vD is the voltage across the
diode (which is negative for reverse bias), V® is the built-in barrier potential (typically about 1 V),
and m is called the grading coefficient. For a linearly graded junction m — 1 / 3 , and for an abrupt
junction m— 1/2.
The zero-bias depletion capacitance Cjo is approximately proportional to the area of the junction.
Thus it is larger for high-power rectifiers, which must be physically large to accommodate high
power dissipation. The value of Cjo also depends on doping levels. In highly doped junctions, a
large amount of charge can be stored close to the junction - similar to the parallel-plate capacitor
with small plate separation. Thus we find high values of CJO for highly doped junctions and low
values for lighdy doped junctions.
A reverse-biased diode can be used in circuits as a variable, voltage-controlled capacitor. A
control signal reverse biases the diode, and as the control signal changes, the capacitance of the
diode varies accordingly. -Diodes designed to have smoothly varying capacitance over a wide
range of control voltage are called variable-capacitance diodes or varicaps. They can be used
in. an JLC resonant circuit to vary the resonant frequency. Variable capacitance diodes can be used
to design bandpass filters for which a control filter vary the center frequency, e.g. to tune in to an
AM radio station. Varicaps are also useful in the design of voltage-controlled oscillators in which
the frequency of oscillation depends on the diode capacitance. An application of this is the
automatic frequency control (AFC) circuit of an FM radio. Manufacturers offer diodes intended
for these applications having zero-bias capacitance Cp ranging from 10 to 1000 pF.
For the MV2201 variable-capacitance diode (Motorola), approximate values of the parameters of
formula (3.40) are CJO = 15 pF, m = 0.43, and KJO = 0.75 V. A plot of the depletion capacitance
versus bias voltage using these parameters is shown in Figure 3.38.
Figure 3.38 Depletion capacitance versus bias voltage for the MV2201 varicap diode
Diffusion capacitance. Another basic charge-storage mechanism occurs when thepn junction is
forward biased. For simplicity, we consider an abrupt junction with much heavier doping on the
p-side than on the »-side (Le. NA»N^).
Sometimes this is called a/>+» junction, where "plus"
refers to heavy doping of the />-material. For such a diode under forward bias, die current
-78-
crossing the junction is due mainly to holes crossing from the/>-side to die /x-side. The current of
electrons that are injected from the /r-side to the/>-side can be neglected.
P
P
n-type
n-type
Stored
charge
Stored
charge
(a)/D=J,
(b)/D=^>I,
Figure 3.39 Hole concentration versus distance for two values of forward current
Consider the hole concentration of the forward-biased p*n junction shown in Figure 3.39. The
charge associated with the' holes that have crossed the junction is stored charge and is
represented by the areas indicated in the figure. The charge is stored because a finite time passes
before an average hole disappears due to electron-hole recombination. This time is called
minority carriers lifetime. As the forward current is increased, more holes cross the junction and
the stored charge increases. Because this charge is associated with b a i n that ice diffusing across
the junction, we call the effect diffusion capacitance.
It can be shown that the incremental diffusion capacitance is given approximately by
in which TT is a parameter known as the transit time of the minority carriers. For the p+n
junction, xT— Z"p is the lifetime of the holes on the »-side of the junction. On the other hand, f*. r
the n+p junction, we have TT = rn, which is the lifetime of the free electrons on the/>-side. For \
junction with comparable doping levels, rT is the weighted average of both lifetimes. Finally, I0 s
the j2-point diode current, and VT is the thermal voltage.
9.2V It
JT) c 40*B-9»IMlote>
Figure 3.40 Diffusion capacitance versus voltage for the 1N4148 diode
Notice that the diffusion capacitance is proportional to the diode current. Thus the diffusion
capacitance, like the current, increases rapidly when the voltage VD exceeds approximately 0.6 V
for silicon devices at room temperature. A plot of the diffusion capacitance- of the 1N4148
switching diode is shown in Figure 3.40. Under forward bias conditions, the diffusion capacitance
is much larger than the depletion capacitance. (For the 1N4148 device, the depletion-capacitance
-79-
parameter Cjo is approximately equal to 2 pF.) However, diffusion capacitance is negligible for
reverse bias.
Small-signal diode model at high frequencies. A small-signal equivalent circuit for the pnjunction diode is shown in Figure 3.41. The resistance R* represents the ohmic resistance of die
material on both sides of the junction; r& is the dynamic resistance of die pn junction that is
discussed in Section 3.4. Its value is given by Equation (3.20), which is repeated here for
convenience
nVT
r
*=-T
(3-42)
1
D
Cj is the depletion capacitance and Cdif is the diffusion capacitance.
All die equivalent circuit parameters except R» depend on the bias point Under reverse-bias
conditions, Cdif is zero, and ra is an open circuit. Hence die equivalent circuit simplifies as shown
in Figure 3.41b.
This equivalent circuit is vahd for the ptt junction over a wide range of frequencies, provided that
small-signal conditions apply. At very high frequencies, lead inductances and stray capacitances of
the diode physical structure are added to provide adequate accuracy.
Diodes are most often used with large signals, and their nonlinear behavior must be taken into
account. Computer modeling does this most easily. We illustrate this widi a few examples.
(a) Forward bias
Figure 3.41 Small signal linear circuits for the/>«-junction diode at high frequencies
Large-Signal Diode Switching. Consider the circuit shown in Figure 3.42. The waveform of
die source voltage t>s is shown in Figure 3.43a. Until / = 10 ns, v, is +50 V and the diode is
forward biased. At / = 10 ns, the source voltage jumps rapidly to -50 V, reverse biasing the diode.
rj]
tf=5kO
ryj
Figure 3.42 Circuit illustrating switching behavior of a/>«-junction diode
-80-
Analog Electronics /Diode
-81-
Circuits
(d) Voltage across the diode (expanded scale)
Figure 3.43 Waveforms for the circuit of Figure 3.42
The resulting diode current is shown in Fig. 3.43b. As one might expect, the diode current is
approximately (50 V)/(5 \£l) = +10 mA until / = 10 ns. Then the source voltage jumps to -50 V.
Instead of dropping immediately to zero the diode current reverses to IR = -10 mA. At
approximately / = 18.3 ns, -the current begins to fall in magnitude and approaches zero at / = 25
ns. In the interval immediately after the source reverses polarity, the diode continues to act as it
were forward biased. This is called the storage interval ft, as labeled in Figure 3.43.
We can explain the behavior of the diode as follows. (To simplify the discussion, we assume a
diode that is heavily doped on the/>-side compared to the «-side.) When forward bias is applied,
holes flow across the junction into the »-side. The holes are minority carries there that diffuse
into the «-side and eventually recombine with free electrons. When tfe reverses polarity, the holes
stored in the »-side can again cross the junction back to the />-side. Until the supply of excess
holes on the «-side is exhausted, current can easily flow in the reverse direction. This explains the
storage interval of the diode current waveform.
It can be shown that the storage interval for a/>»-junction diode is given by
(3.43)
*,Z
B(l + B)
in which TT is the transit time for the minority carriers and B= \IR/IF\ • IF is the forward current
before switching and JR is the reverse current during the storage interval. The storage time
becomes shorter at larger ratios B.
After the excess holes have all recrossed the junction (or recombined with free electrons), the
depletion capacitance of the diode is charged through the resistor. Thus, after the storage
interval, we see an approximate exponential transient for the current in Figure 3.43b. (Since the
depletion capacitance is nonlinear, the transient is not precisely exponential, as would be the
transient in a linear RC circuit.) The interval for this transient, called transition time, is denoted
by ft. By definition, the end of the transition interval occurs when the reverse diode current has
reached a specific value, typically I R /10.
The total time interval for the diode to become an approximate open circuit, called the reverse
recovery time, is denoted by ftr. It is the sum of the storage time and the transition time.
'«•='*+'/
(3-44)
The transition time ft depends on the circuit resistance. Even though the depletion capacitance is
nonlinear, the transition time is proportional to the circuit resistance.
-82-
The diode voltage is shown in Figure 3.43c. To be able to see the details of the diode voltage
during forward bias, an expanded-scale waveform is also plotted in Figure 3.43d. Notice that the
diode voltage remains positive during the storage time, showing that the diode continues to act as
if it were forward biased, even though the current is in the reverse direction. The terminal voltage
of the diode does fall somewhat when the current reverses, because the voltage drop across the
ohmic resistance R, of the diode reverses polarity. Prior to / = 10 ns, the terminal voltage is the
sum of the junction voltage and the ohmic drop. On the other hand, between / = 10 ns and t 18.3 ns the terminal voltage is the junction voltage minus the ohmic drop.
Exercise 3.10 Consider the parallel-plate capacitor shown in Figure 3.36. The plates have
dimensions of 20 |wm x 30 \ltn. (These dimensions are typical of the area of an integrated-circuit
/>»-junction diode.) The relative dielectric constant of the material is ft = 11.9. (This is the value
for silicon.) The capacitance is 1 pF, which is a typical zero-bias depletion capacitance for a lowpower diode. Find the distance between the plates. (The answer is the approximate zero-bias
thickness of the depletion region.) Ans. d= 6.32x10-8 m.
Exercise 3.11 A certain abrupt-junction diode has a zero-bias depletion capacitance of 5 pF and
a built-in barrier potential of 0.8 V. Compute the depletion capacitance for a reverse-bias voltage
of (a) 5 V, (b) 50 V. Ans. 1.86 pF, 0.627 pF.
Exercise 3.12 A certain diode has a transit time of 10 ns. Find values for the small-signal
resistance and diffusion capacitance at ID = 5 mA. Assume the emission coefficient n - \ and a
temperature of 300 K. Ans. r& - 5.2 Q., Cm = 1920 pF.
Exercise 3.13 Consider the circuit of 3.42 with R changed to 50 kfi and with the source
waveform of Figure 3.43a. (a) Think about the circuit and sketch the current versus time. Make
use of results shown in Figure 3.43, use (3.43) to estimate / r Like the time constant of an RC
circuit, /, is approximately proportional to R. (b) Write a PSpice program to obtain the plot of the
diode current. Compare the results. Ans. The current waveforms are similar to Figure 3.43b with
IF = 1 mA and IR = -1 mA, 4 = 8 n s and A = 50 ns.
Exercit : 3.14 Consider the circuit of 3.42 with the input voltage "being equal to -50 V in die
time interval from 0 to 10 ns, then jumping up to reach 50 V at 10.01 ns, staying at 50V until 30
ns and returning to -50 V at 30.01 ns. Write a PSpice program to obtain the plot of the diode
current. Explain.
TABLE
3.1 SPICE PARAMETERS FOR I N J U N C T I O N DIODES
R,
SPICE
notation
IS
N
BV
IBV
RS
Default
value
1.0E-14A
1
oo V
1.0E-3A
0Q
c*
CJO
OFyfV
0.5
1.0 V
0s
Text
notation
I
n
V,
h
m
yt
brr
M
V]
TT
-83-
SPICE ParatL. jis for Diodes. Figure 3.41a is the SPICE equivalent circuit for th&pn junction
diode. Table 3.1 lists nine parameters required to model the static and dynamic behavior of the
diode. The first four, describe the static properties of the diode, as expressed by the Schockley
equation (IS, N) and the following equation that approximates the breakdown component of the
diode current
iD = -Iz exp -
VZ+vD
(3.45)
in terms of parameters (VB and IBV), where VB = Vz is the breakdown voltage and IBV = Iz is
die current that flows through the diode at vD = Vz. The remaining five parameters (RS, CJO, M,
YJ, TT) describe the dynamic properties of the diode, following Equation (3.40) for depletion
capacitance and (3.41) for diffusion capacitance. The default values of the two key dynamic
parameters, CJO and TT, are both zero. This means that tbcdefault diode model in SPICE is a static
model. To simulate dynamic effects, these default values must be overridden by nonzero values in
.MODEL statement. An example diode model for the 1N4148 low-power general-purpose diode
can be specified as follows:
.model D1N4148 D(I»-0.1p Rs=16 CJO=*2p Tt=12n Bv«100 Ibv=0.1p)
3.9 Special Diodes
3.9.1
Schottky diodes
The junction between certain metals (e.g. aluminum) and lightly doped n-type material forms a
Schottky diode. A typical structure is shown in Figure 3.44. (On the other hand, a metallic
contact with heavily doped »-type material results in an ohmic contact.)
The detailed theory of the Schottky diode is somewhat different from that of the pn junction.
However, the form of the results is the same - the current in the Schottky diode is given by the
Shockley equation. The saturation current 7, is much higher for Schottky diodes than for pn
junctions of the same size. A typical value for a Schottky diode is 7S = 10"10 A, whereas for a
typical IC pn junction it is 7S = 10"16 A. Because of the larper value of 7S, the forward voltage of
the Schottky diode is significandy smaller (about 0.4 V compared to 0.7 V for a silicon pn
junction diode).
Schottky
contact
j—
/
-v
Ohmic
contact
Figure 4.44 Schottky diode
Another important difference is that the Schottky diode does not display charge storage when
being switched from forward conduction to reverse-bias conditions. This is because for forward
bias electrons rather than holes are injected from metal anode into the n material. Thus majority
carriers (i.e. electrons) carry the current, and storage of minority carriers does not occur. Hence
switching tends to be faster for the Schottky diode. This fact can be used to advantage in die
design of fast logic gates. An example of a bipolar transistor inverter with a Schottky diode is
discussed in Chapter 6.
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Analog Electronics/Diode Circuits
3.9.2 Light-Emitting Diodes and Photodiodes
Light emitting diodes (LEDs) are junctions constructed of special semiconductor materials
(compounds) such as gallium arsenide-phosphide. In these devices,' the injected minority carriers
that result from forward biasing give up energy in the form of radiated light when they
recombine. Wavelength ranging from ultraviolet through the visible and into die infrared bands is
obtainable by use of different semiconductors and doping impurities. LEDs are widely used as
display devices, for transferring information into optical fibers, and for optical isolation as
discussed shortly
'D
[••
+
V
D
0.7V
Figure 3.45 Volt-ampere curves of photodiode/solar cell
A photodiode is a junction in which photons of energy in incident light break covalent bonds,
adding the drift of these new carriers to the existing reverse saturation current. Figure 3.45 shows
die volt-ampere curve of such a diode as a function of increasing incident light intensity. In tiie
third and fourth quadrants the device functions as a (dependent) current source controlled by
light. Is is the amount of current that can flows through the diode in dark, at a given temperature
Operation in the third quadrant represents passive conversion of light intensity information in
electrical information. This is the photodiode functioning as an optical to electrical transduceApplications include light meters and communication systems that receive information coded in
the form of light. The obvious circuit model is a light-intensity-controlled current source.
Diodes designed for fourth quadrant operation are called photovoltaic cells or solar cells.
Points on the curves in the fourth quadrant correspond to voltage and currents of opposite sign,
implying an active device that delivers power to an external circuit. The curves emphasize that the
solar cell is an active device for voltages not exceeding 0.7 V. Series connection of many solar
cells produce large dc voltages. Parallel connection of solar cells of large area generates high
output currents. Thus series-parallel interconnections can convert relatively large amounts of
solar power into electrical power at convenient voltage levels. Solar cell efficiency (electrical
output power divided by solar input power) is typically of the order of 10 - 1 5 %.
Figure 3.46a shows how the light-emitting diode and the photodiode are combined in a useful
device called an optical isolator. The signal th(t) is transferred to the load by means of light, with
no physical connection whatsoever between the input and output circuits. The optical isolator is
useful in many applications, including computer interfaces for biomedical instrumentation. In
these applications, we want to transmit information while protecting a human subject and
delicate equipment from dangers imposed by high voltages in the input circuit Figure 3.46b
illustrates the principle. Notice that the hazard voltage that exists in the circuit on the left is not
transferred to the load circuit on the right. Transformers also give isolation; however, die optical
isolator operates at frequencies down to dc and is smaller, lighter and less expensive.
-85-
3.9 Summary
The semiconductor diode is basically a pn junction. Such a junction is formed in a single
semiconductor material crystal, most often silicon crystal. In the forward direction, the ideal
diode conducts any current forced by the external circuit while displaying a zero voltage drop.
The ideal diode does not conduct in the reverse direction; any applied voltage appears as reverse
bias across the diode. The unidirectional current flow property makes the diode useful in the
design of rectifier circuits. The diode i-v characteristic is described by Schockley equation; the
forward conduction of practical silicon diodes is accurately described by the relationship i =
7sexp[f/(«l/T)]. A silicon diode conducts a negligible current until the forward voltage is at least
0.5 V. Then the current increases rapidly, with the voltage drop increasing by 60 or 120 mV
(depending on the value of ») for every decade of current change. In the reverse direction, a
silicon diode conducts a current of the order of 1 nA. This current is much greater than 7s and
increases with the magnitude of reverse voltage. The diode i-v characteristic and parameters
depend on temperature. Diodes designed to operate in the breakdown region are called Zener
diodes. They are employed in the design of voltage regulators whose function is to provide a
constant dc voltage tfiat varies litde with variation in power supply voltage and/or load current.
A hierarchy of diode models exists, with the selection of an appropriate model dictated by the
application. Analysis of diode circuits depends on viewing the diode as a 3-state device that, at a
particular time operates in its ON, OFF or breakdown state. For each state, we replace the diode
by a simple equivalent that approximates the diode's i-v characteristic and we then solve the
resulting linear circuit to check our state assumptions. In many applications, a conducting diode
is modeled as having a constant voltage drop, usually about 0.7 V. A diode biased to operate at a
dc current ID has a small-signal resistance ra = nVT/ID. With the diode modeling and analysis
techniques, we can describe many useful circuits such as half- and full-wave rectifiers, limiters,
clamping circuits, voltage doublers and voltage regulators. These perform a variety of useful
functions such as turning ac into dc, shifting waveform levels, modifying the shapes of time-
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varying waveforms, and producing dc that is relatively independent of changes in loading and
source voltage. There are two sources of parasitic capacitance within the pn junction: depletion
capacitance, associated with the layers of bound ions in the depletion region, and diffusion
capacitance, associated with storing excess minority charge carriers just outside the depletion
region. When signals change slowly in diode circuits, these capacitances are insignificant, and the
diode satisfies its static equation (i.e. Schockley equation). For rapid.transitions or fast signals,
however, a more appropriate diode description is needed, which is a differential equation that is
nonlinear when posed in terms of diode current or voltage. Because of diode capacitances, delays
occur in diode switching, and waveform processing predicted upon static diode theory
deteriorates at high frequencies. SPICE models for diodes were introduced. By understanding
how SPICE parameters relate to the diode parameters we use in circuit analysis, we can use
simulation to study with increased accuracy and precision both static and dynamic operation of
diodes. With SPICE we can formulate more realistic circuit descriptions, relying on computation
to verify and reinforce our understanding of diode circuits. Accuracy of computer simulation,
such as obtained from SPICE, is limited by the approximations we made when building models
for real-world, physical devices. A Schottky diode is based on metal-semiconductor junction. It
features fast switching and lower voltage drop when forward biased as compared to pn junction
diode. Light-emitting diode, the photodiode and optical isolator are examples of optoelectronic
devices useful in telecommunications and instrumentation technology.
Notes
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Analog Electronics / FET Circuits
4 Field-Effect Transistor Circuits
Now we turn our attention to devices that can amplify an input signal. In this chapter we
consider the field-effect transistor (FET). In the first few sections of this chapter we describe
the external characteristics of several types of FETs. Then we consider some simple but
important circuits that use FETs, useful in amplifiers and logic circuits.
4.1 The n-Channel Junction FET
The simplified physical structure of an ^-channel junction field-effect transistor (FET) is
shown in Fig. 4.1a, and the circuit symbol is shown in Fig. 4.1b. The device consists of a channel
of »-type semiconductor with ohmic (nonrectifying) contacts at each end. These contacts are
called the drain and the source. Alongside the channel, there are regions of ^-type
semiconductor electrically connected to each other and to the gate terminal.
(a)
(b)
Figure 4.1 Simplified physical structure (a) of an ^-channel FET; its circuit symbol (b)
The /w»-junction between the gate and the channel is a rectifying contact similar to />»-junction
diodes discussed in Chapter 3. In almost all applications, this junction is reverse biased, so
virtually no current flows in the gate terminal. Hence the gate is negative with respect to the
channel in normal operation of an »-channel FET. (Recall that the/>-side is negative with respect
to »-side for reverse bias of a/>»-junction.)
Applying reverse bias voltage between gate and channel causes the depletion layer of the gatechannel junction to become wider. The greater the reverse bias, the wider this nonconductive
layer becomes. Eventually, for VQS<V?, the nonconductive layer extends all the way across the
channel, and we say that pinch-off has occurred, where Vp is the pinch-off voltage which is a
parameter of the transistor. Typically, it is a few volts in magnitude and is negative for ^-channel
devices. As the voltage vGS is varied from 0 to Vp, the cross-sectional area of die conductive path
from drain to source decreases toward zero. Therefore, the resistance between the drain and the
source depends on the gate-to-channel bias.
In normal operation of art «-channel device, we apply a positive voltage to the drain with respect
to the source, as shown in Fig. 4.2. Cvirrent flows into the drain, through the channel, and out of
die source. Because the resistance of the channel depends on the gate-to-source voltage, die
amount of drain current that flows is controlled by the gate-to-source voltage.
-88-
'D
Figure 4.2 Circuit for discussion of drain characteristics of the ^-channel FET
Suppose that vGS is zero. Then, as vDS increases, iD increases as shown in Fig. 4.3. The channel is
a bar of conductive material with ohmic contacts at the ends - exactiy the type of construction
used for ordinary resistors. Therefore, it is not surprising the iD is proportional to vDSi for small
values of t>DS.
VG5=0
Figure 4.3 Drain current versus drain-to-source voltage for
zero gate-to-source voltage
However, for larger values of vDS, drain current increases more and more slowly. This is because
the end of the channel closest to the drain is reverse biased by the (vDS + \ vGS\)=vDS voltage. As
vDS increases, the depletion layer becomes wider, causing the channel to have higher resistance.
After the pinch-off voltage is reached, the drain current becomes nearly constant for additional
increases in p,DS-
Cutoff vcs<VP
Figure 4.4 Drain characteristics of an ^-channel JFET
A complete family of drain characteristics of a small-signal FET is shown in Fig. 4.4. For negative
values of vGS, the gate-to-channel junction is reverse biased even with vDS=Q. Thus the initial
channel resistance is higher. This is evident in Fig. 4.4 where the initial slope of the curves (i.e.
for vDS =0) is smaller for values of vGS closer to the pinch-off voltage. Thus for small values of
vDS, the FET behaves as a resistance between the drain and source. Furthermore, the resistance
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value is under the control of vGS. If vGS is less than the pinch-off voltage, vGS<V?) the resistance
becomes an open circuit, and we say that the device is in cutoff
As in the case of vG^=-Q, the drain current for other values of PGS eventually becomes constant as
vDS is raised, due to pinch-off at the drain end of the channel The region where the drain current
is constant is called the saturation region. The region for which iD depends on vDS is called the
linear region of the FET. These regions are labeled in Fig. 4.4.
Now we give the first-order equations for the drain current in terms of the voltages for the three
regions of operation. (There are several second-order effects not taken into account by these
equations. We discuss these effects as the need arises.)
Cutoff region. An ^-channel FET is in cutoff if
vGS<Vp
In the cutoff the drain current is zero
/D=0
Linear region. An ^-channel FET is in the linear (triode) region if
vGS>Vp
and if
V
GD=(VGS-VDS)>VP
(4.1)
(4.2)
(4.3)
(4-4)
In the linear region, the drain current is given by
iD=K[2(vas-VJ.)vDS-vls]
(4.5)
2
The constant K has units of current per volt . Study of this equation for a fixed value of vGS
shows that it describes a parabola passing through the origin of the ip-Vps plane. Furthermore,
the maximum of the parabola is on the boundary between the linear and saturation regions.
Saturation region. An ^-channel FET is in the saturation region if
vG5 > V,
and if
V
GD=(VGS-VDS)<VP
(4.6)
(4.7)
In the saturation region, the drain current is given by
iD=K(vGS-Vpf
(4.8)
A plot of Equation (4.8) is shown in Fig. 4.5. This plot represents the so-called transfer
characteristic of the FET in saturation.
The drain current in the saturation region for vGS =0 is denoted as IDSS and is usually specified on
manufacturers' data sheets. Substituting vGS =0 into (4.8) we find that
Ioss=KV^
(4.9)
Solving for K, we have
* =7 T
(4-10)
If values are given for IDSS and Vp, the static characteristics of the JFET can be plotted. Typical
values for a small-signal »-channel JFET are: IDSS =5-rl0mA and KP=-3V.
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Analog Electronics/FET Circuits
IDSS
+ *GS
Figure 4.5 Plot of fa versus VDS in the saturation region of the FET
Breakdown. As we mentioned earlier, there are several effects not modeled by the device
equations we have given. An example of one of these effects occurs if the reverse bias between
gate and channel becomes too large - then the junction experiences breakdown and the drain
current increases very rapidly. Usually, the greatest reverse bias is at the "drain end of the channel,
so breakdown occurs when V^Q exceeds the breakdown voltage V B in magnitude. Because vDG =
V
DS ' VGS> breakdown takes place at smaller values of vDS as PGS takes values closer to pinch-off.
This is illustrated in Fig. 4.6. We seldom operate FETs in breakdown; however, breakdown
voltage value VB should be taken into design consideration to avoid the device entering into this
operation region.
4.2 Me il-Oxide-Semiconductor FETs
Another important class of devices is the metal-oxide-semiconductor field-effect transistor
(MOSFET). There are two types, known as depletion MOSFETs and enhancement
MOSFETs. Each of them can be realized as an w-channel or a/(-channel device. Basically, all of
these FETs have very similar characteristics. Once you master one type, such as the »-channel
JFET, it is much easier to assimilate the relatively minor differences between them. Circuit
symbols for the »-channel and for the />-channel MOSFET devices are shown in Fig. 4.7. The
circuit symbols for/>-channel are the same as the circuit symbols for n-channel devices except for
the directions of die arrowheads. The MOSFET terminal labeled as B denotes the substrate or
the body contact.
Drain current versus vGS in the saturation region for »-channel devices is shown in Fig. 4.8. The
depletion MOSFET has output characteristics nearly identical to those of die JFET. The main
difference between die »-channel JFET and the »-channel MOSFET is in the fact diat the
MOSFET can be operated with positive values of vGs. (This is usually not done with the JFET
because it would result in forward bias of the gate-to-channel junction and the corresponding
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increase of gate current.) The equations we have given in Section 4.1 for the ^-channel JFET also
apply for the //-channel depletion MOSFET.
Source
(h)
<g>
Figure 4.7 //-channel FETs circuit symbols (a, b, c),/(-channel FETs circuit symbols (d, e, f),
simplified physical structure of the //-channel enhancement MOSFET (g)
and the //-channel depletion MOSFET (h)
No current flows in the //-channel enhancement MOSFET for vGS less than a certain positive
value known as the threshold voltage which is denoted by VA. The equations we have given in
Section 4.1 for the //-channel JFET also apply for the //-channel enhancement MOSFET if the
(positive) threshold voltage Vth replaces the (negative) pinch-off voltage Vp. In particular, the
enhancement MOSFET is cutoff for vGS < V&.
The characteristics of the /(-channel FETs are the same as for the respective »-channel devices
except that voltage polarities and current directions are inverted. If we continue to reference the
drain current into the drain, the algebraic signs of the currents and voltages must be inverted for
/(-channel devices. As a consequence, for the /(-channel devices the drain current is negative and
the pinch-off voltage Vp is positive for the JFET and for the depletion MOSFET. The threshold
voltage Vfo for the/(-channel enhancement MOSFET takes on negative values.
Figure 4.8 Transfer characteristics for FETs:
(a) //-channel JFET and depletion MOSFET, (b) //-channel enhancement MOSFET,
(c) /(-channel JFET and depletion MOSFET, (d)/(-channel enhancement MOSFET.
-92-
Gate protection. Because of their construction, MOSFETs have extremely high input
impedance between gate and channel - in excess of 1000MQ. This high impedance is due to a
thin insulating (silicon dioxide) layer placed between gate and channel. In handling these devices,
it is easy to develop sialic electric voltages greater than the breakdown voltage of the gate
insulation. Breakdown of the insulating layer is destructive, usually resulting in a short circuit
between gate and chat?
To alleviate this problem, the gate terminals are usually protected by back-to-back Zener diodes
as shown in Fig. 4.9. If the device is exposed to a static electric charge, one of the Zener diodes
breaks down (depending on die voltage polarity, the other diode is on), providing a
nondestructive discharge path. Usually, die diodes are fabricated on die same chip of
semiconductor as the FET.
Variation in FET parameters with temperature. It is well known mat the semiconductor
material mobility decreases with temperature. Because of more frequent collisions of charge
carriers with the rapidly moving lattice ions. It turns out that the constant K in (4.5) and (4.8),
which are valid for both JFETs and MOSFETs (with Vp = Kth where appropriate), is
proportional to the carrier mobility. Consequendy, their transfer characteristics change as in Fig.
4.9b and 4.9c. It follows that the output characteristics for both devices crowd more closely as in
Fig. 4.9d, juat opposite of bipolar transistor characteristics that are considered in Chapter 5.
The threshold and pinch-off voltages decrease by about 2 mV/°C with increases in temperature,
causing the curves in Fig. 4.9b and 4.9c to shift to the left with temperature increases as they
droop downward.
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4.3 Load-Line Analysis of a Simple JFET Amplifier
In this section we analyze the JFET amplifier circuit shown in Fig 4.10 by use of the graphical
load-line approach. The batteries bias the JFET at a suitable operating point so that amplification
of the input signal vs{f) can take place. We will see that the input voltage vs(t) causes vGS to vary
with time, which in turn causes tD to vary. The changing voltage drop across RD causes an
amplified version of the signal to appear at the drain terminal.
Figure 4.10 Simple JFET amplifier circuit
VGS(0
Applying Kirchhoff s voltage law to the input loop, we obtain the following expression:
= V,(0-KGC
(4.11)
As an example, we assume that the input signal is a 1-V peak 1-kHz sinusoid and that VGG is IV.
Thus we have
vC5 (/) = sin(2000^) - 1
(4.12)
Writing a voltage equation around the drain circuit, we obtain
VDD = RDiD{t) + vDS{t)
(4.13)
For our example, we assume that RD=l]sQ and VDD=20V, so Equation (4.13) becomes
20 = * D (/) + v D S ( 0
(4.14)
where we have assumed that iD(t) is in milliamperes. A plot of this equation on the drain
characteristic of the transistor is a straight line called the load line. To establish the load line, we
first locate two points on it. Assuming zero drain current, iD — 0, in (4.14) we find that vDS —2QW.
These values plot as the lower right-hand end of the load line shown in Fig. 4.11. For a second
point, we assume that vDS = 0, which yields / D =20mA when substituted into (4.14). This pair of
values (yDS — 0 and iD = 20 mA) plots as the upper left-hand end of the load line.
If vs{t) = 0, Equation (4.11) yields vGS = -VQQ — -1 V. Therefore, the intersection of the curve for
PGS = -1- V with the load line is the quiescent operating point. The quiescent values are 7p = 9 mA
andKD=llV.
The maximum and minimum values of the gate-to-source voltage are J^cj-max = 0 V and ViGSnan
= -2V [see Equation (4.12) and Figure 4.12]. The intersections of the corresponding curves with
the load line are labeled as points A and 23, respectively, in Fig. 4.11. At point A, we find that
^nrmin= 4 V and 1 ^ = 16 mA. At point B, we find that VDSm2x = 16 V and 1 ^ = 4 mA.
94-
The plots of vs(t) and vDS(t) versus time are shown in Fig. 4.12. Notice that the peak-to-peak
swing of the drain-to-source voltage is 12V, whereas the peak-to-peak swing of the input signal is
2V. Furthermore, the ac voltage at the drain is inverted compared to the signal at the gate.
Therefore, this is an inverting amplifier. Apparently, one can calculate the circuit gain as equal to
A^ = -12/2 = -6, where the minus sign is due to the inversion.
Figure 4.11 Drain characteristic and load line for the FET amplifier
Notice, however, that the output waveform shown in Fig. 4.12 is not a symmetrical sinusoid Wte
the input. For illustration, we see that starting from the j2-point at VD = 11 V, the output voltage
swings down to Vi>mâ — 4 V for a change of 7 V. On the other hand, the output swings up to
16V for a change of only 5V from the j2-point on the positive going half-cycle of the output We
cannot properly define gain for the circuit because the ac output signal is not proportional to the
ac input Apparendy the FET is a nonlinear device. Nevertheless, die output signal is larger th n
die input signal even if it is distorted.
The distortion is due to the fact that die characteristic curves for the FET are not uniformly
spaced. Of course, if much smaller input signal were applied, we would obtain amplification
without appreciable distortion.
The rather modest gain (A^ = -6) that we see in this circuit is typical of RC-coupled FET
amplifiers. In general, BJT amplifiers have much larger voltage gains. However, if we consider
current gain, FET circuits have larger gains than most BJT circuits. For example, we have
considered the input current for the circuit of Figure 4.10 to be zero. An infinite current gain
would result if one attempts to calculate current gain as the ratio of the ac current in RD to tb-i
input current. On die other hand, the current gain of a BJT ranges from about 10 to several
hundred.
The amplifier circuit we have analyzed in this section is fairly simple. Practical amplifier circuits
are usually much more difficult to analyze by graphical mediods. Later in this chapter we develop
a small-signal equivalent circuit for the FET, and then we can use mathematical circuit-analysis
techniques instead of graphical analysis. Usually it is more useful for investigation of practical
amplifier circuits. However, graphical analysis of simple circuits provides an excellent way to
understand die basic concepts of amplifiers.
-95-
4.4 The Self-Bias Circuit
Analysis of amplifiers is undertaken in two steps. First, we analyze the dc circuit to determine the
j2-point In this analysis, the nonlinear device equations and/or curves are used. Then, after the
bias is found, we use a linear, small-signal equivalent circuits to find the input resistance, voltage
gain, and so on. In this section and die next, we consider analysis and design of dc bias circuits
for FETs.
The two-battery bias circuit used in the amplifier of Fig. 4.10 is not practical. Usually one battery
is available only. Even more significant problem is that FET parameters vary considerably from
device to device. For a given type of JFET, IDSS may vary by a ratio of 5:1. Furthermore, the
pinch-off voltage is different from device to device.
Plots of iD versus vGS are shown in Fig. 4.13 for extreme FETs, all of which have die same
manufacturer's type number. The range of variation shown is typical. Notice that if VGS were the
same for all devices, a considerable variation in ID would occur. Some devices would be biased at
one end of the load line and others at the opposite end. On the other hand, to obtain the
maximum symmetrical swing of the output voltage without severe distortion, we require the
operating point to be near the middle of the load line for all devices. Thus a fixed-bias circuit
that maintains the same value of VGS independent of the device parameters is not suitable for
mass production.
Figure 13 Transfer characteristics of extreme devices for JFETs having the
same type number. Bias with fixed VGS results in a large variation of h.
-96-
A more practical circuit, known as the self-bias circuit, is shown in Fig. 4.14. The resistor Rs is
essential to the operation of this bias circuit. The resistor R^ is usually a large value (several
megaohms) that maintains the dc voltage at the gate close to the ground and still provides high
input impedance at the gate. Since only a small current (InA or less) flows in the gate, the dc
voltage drop across RQ is negligible. In applications where input signal is not applied to gate, RQ
can be replaced by a short circuit. The drain resistance RD is required to be present if we want an
amplified signal to appear at the drain. However, in some circuits, the output is taken from the
source terminal, and then we would replace RD by a short circuit.
DD
Figure 4.14 Self-bias circuit used for JFETs and depletion MOSFETs
The drain current flows out through the source and through the source resistor Rs, creating a
voltage drop. Writing a voltage equation around the gate-source loop of Fig. 4.14 and neglecting
the drop across RQ we have
vGs=-Rsh
(4-15)
A plot of this equation is called the bias line and is shown in Fig. 4.15, which also shows iD
versus vGS for extreme devices of a given type. The operating point is at the intersection of the
bias line and the device transfer curve. Notice that VGS is smaller in magnitude for the lowcurrent device than for the high-current device. Thus the self-bias circuit adjusts VGS to
compensate for changes in the device, thereby reducing variations in ID compared to a fixed-bias
circuit.
Bias line
VQS=-RS'D
VGS
Figure 4.15 Graphical analysis of the self-bias. The device-to-device variation of drain current is
much less than for the fixed-bias circuit.
The device curves shown in Fig. 4.15 are valid only if device operates in the saturation region.
Usually, in FET amplifier circuits, operation in the saturation region is desired. However, we
should check to make sure that VDS is large enough for operation in the saturation region before
accepting results based on this analysis.
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Exercise 4.1 Design a self-bias circuit for an »-channel FET having IDSS=4toA and VP=-2V.
The circuit is to have R D =2.2kQ, VDD=20V, and Ij^ltnA. Use standard 10%-tolerance resistor
values. Ans. Ry=270Q.
Exercise 4.2 Analyze the self-bias circuit designed in Exercise 4.1. Repeat the analysis for a highcurrent device having 7 Dxr =8mA and VP=-4V. Ans. VQ^-OSGV,
J D =2.07mA, VDS=U.9V;
VGS=-l.l2V,ID=4.UmA,
VDS=9.11V.
4.5 The Fixed- plus Self-Bias Circuit
The self-bias circuit gives fair performance in maintaining a fixed ID from device to device, but
sometimes better performance is needed. The fixed- plus self-bias circuit shown in Fig. 4.16
provides a solution.
For purposes of analysis, we replace the gate bias circuit with its Thevenin equivalent, as shown
in Fig. 4.16b. The Thevenin voltage is
X
Vr.=Vn
Q
(4.16)
DD
R,+R,
and the Thevenin resistance RQ is the parallel combination of Rf and R2. Writing a voltage
equation around the gate loop of Fig. 4.16b, we obtain
VG = v GS + RsiD
(4.17)
Notice &at we have assumed that the voltage drop across R^ is zero. Now, if we assume that the
transistor is in the saturation, we have
iD=K(vGS-VPf
(4.18)
Simultaneous solution of (4.17) and (4.18) yields the operating point (provided that it falls in the
saturation region). Then we can find vDS by writing a voltage equation around the drain loop
v D S =r D 0 -(tf 0 + * s ) / ,
(4.19)
(a)
(b)
Figure 4.16 Fixed- plus self-bias circuit. Original circuit (a),
Gate t>ias circuit replaced by its Thevenin equivalent (b).
Figure 4.17 shows the graphical solution of Equations (4.17) and (4.18). Notice that higher values
of VG result in smaller variation in ID for extreme devices because die bias line becomes closer to
horizontal. (Also notice that VG=0 corresponds to the self-bias circuit.) However we must not
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choose VG too high because this raises the voltage drop across Rs and sufficient voltage must be
allocated for vDS and RD.
Exercise 4.3 Design for V^SV a fixed- plus self-bias circuit for the JFET of Exercise 4.1
having IDSS=4mA and VP=-ZV. The circuit is to have R p =2.2kQ, VDD=20V, and I^lmA.
Use
standard 10%-tolerance resistor values. Ans. Rj=2.7kfi.
Exercise 4.4 Analyze the self-bias circuit designed in Exercise 4.3. Repeat the analysis for a highcurrent device having IDSS=SmA and VP=-4V. Ans. F GJ -=-0.564V, 7 D =2.06mA, VDS=9.9V;
VGS=-\.76V,ID=2.50mA,
VDS=7.73V.
Figure 4.17 Graphical solution for the fixed- plus self-bias circuit.
Note that ID is nearly independent of the device if VG yz large.
Another advantage of the fixed- plus self-bias circuit is that it also works for enhancement
MOSFETs. On the other hand, the self-bias circuit is not suitable for enhancement MOSFETs
because the gate must be more positive than the source (assuming »-channel devices), which is
not possible in the self-bias circuit.
Example 4.1 Design a bias circuit for an «-channel enhancement MOSFET having K /4 =4V and
K=10-3A/V2. The power supply voltage is VDD=\5V. A jg-point of ,KDJ-=5V and / p =5mA is
desired. The circuit is to be used in an amplifier with the ac output taken from the drain terminal.
Solution. Since an enhancement device is specified, we must use the fixed- plus self-bias circuit
because the self-bias circuit would give a zero drain current regardless of the resistor value
selected. The circuit is shown below. Values must be selected for R,, R^, RD and Rs.
Figure 4.18 Circuit of Example 4.1
Since the supply voltage is 15V and a bias value VDS=5V is specified, a total of 10V remains to
be divided between RD and Rs. The drain current flows through both resistors, so we have
-99-
RS+RD= {yoD ~^DSr ID = 2 ^ • Equation (4.17) shows that larger voltages across Rs result
in higher values for VG. As we have seen, large values of VG lead to good bias stability. Thus we
are led to allocate a large portion of the available voltage to the drop across Rs. In the extreme
case, this would produce R D =0 and Ry=2k£2. However, since an ac output signal is to be
developed at the drain, R D =0 is not acceptable. Thus we select R D =Rj-=lkfi. (This choice is
somewhat arbitrary - perhaps in a complete amplifier design, the ac performance required would
dictate particular choice.)
Assuming the MOSFET is operating in the saturation region, we have
Io =
K{Vcs-Vlh)2
Substituting values and solving, we find that VGS= 6.24V. Thus the gate voltage is given by
VG=VGS +
RsID=n.2V
Now we must find values for R, and R^, so that the 15-V supply voltage divides with 11.2V
across R^ and 15-11.2=3.8V across R ; . If we make somewhat arbitrary choice that
R,+R 2 =1.5MQ, then R,=380kQ and R2=\A2MQ
provide the desired voltage division.
Therefore we choose the closest standard values, namely R,=390kfi and R 2 =1.1MQ
Exercise 4.5 (a) Analyze j h e circuit designed in Example 4.1 to find of VDS and ID. The answer
should verify that the operating point achieved is close to the design objectives, (b) Repeat the
analysis for an «-channel enhancement MOSFET having Vtb=5V and K=2taA/Vz. Ans.
7D=4.87mA, K DJ =5.27V; J D =4.56mA, VDS=5.STV.
Exercise 4.6 A certain «-channel enhancement MOSFET has Vlt=2V and K=2mA/V2 The bias
circuit (as shown in Fig. 4.18) has VDD=20V, R 7 =R 2 =1MQ, R D = l k O and Rj-=2.2kQ. Find VDS
and ID. Ans. J D =3.07mA, VDS=\0.2V.
Exercise 4.7 Find the largest value of R D that can be used in the circuit of Exercise 4.6 if the
MOSFET is to remain in saturation. Assume that R?, R2, and Rs remain fixed. Ans.
RDmax=3.91kQ.
4.6 The Small-Signal Equivalent Circuit
In the preceding two sections, we considered dc bias circuits for FET amplifiers. Now we
consider the relationships, between the signal currents and voltages for small changes from Qpoint. As usual, we denote total quantities by lowercase letters with uppercase subscripts such as
/D(/) and vG5(t). The dc j2~point values are denoted by uppercase letters with uppercase subscripts,
such as JD and VGS. The signals are denoted by lowercase letters with lowercase subscripts such as
*d(/) and %i(^. The total current or voltage is the sum of the j2-point value and the signal. Thus we
can write
iD(0 = ID+iAO
(4-20)
and
v G S (0 = f G S + v f f ( 0
(4-21)
In the following we assume that the FETs are biased in the saturation region, which is usually die
case for amplifier circuits. Equation (4.8), repeated here for convenience, gives the total drain
current in terms of the total gate-to-source voltage.
-100-
iD=K(vGS-VPf
Substituting (4.21) and (4.20) into (4.8), we obtain
(4.8)
+ v^ -VP]2
(4.22)
ID+id(0
= tfycs
The right-hand side of (4.22) can be expanded to obtain
ID + /,(/) = K(VGS -VPf
+2K(VGS -VP)Vgs
+ Kv2m
(4.23)
However, the j2-point values are also related by (4.8), so we have
Io=K{VGS-VP)2
(4.24)
Therefore the first term on either side of Equation (4.23) can be cancelled. Furthermore, we are
interested in small-signal conditions for which the last term on the right-hand side is negligible
and can be dropped [i.e. we assume that |f G j(^| is much smaller than | Vcr^pW- With these
changes, Equation (4.23) becomes
(t) = 2K(vGS-VP)vgs
(4.25)
We define the transconductance of the FET as
g„=2K(VGS-VP)
(4.26)
Then Equation (4.25) can be written as
(4-27>
'„(') = S.v,(0
The gate current of the FET is negligible, so we have
/f(0 = 0
(4.28)
The small-signal equivalent circuit shown in Fig. 4.19 can represent equations (4.27) and (4.28).
Thus die FET is modeled by a voltage-controlled current source connected between the drain
and source terminals. The model has an open circuit between gate and source.
Figure 4.19 Small-signal equivalent circuit for FETs
Solving Equation (4.24) for the quantity (VGS - VP) and substituting into Equation (4.26), we
obtain
gm = 24H~D
(4.29)
Then if we use Equation (4.10) to substitute for K, we have
which is often a convenient formula for computing the transconductance of a JFET or depletion
MOSFET at a given j2-point. Of course, IDSS does not apply for enhancement MOSFETs, and
Equation (4.29) applies for them.
-101-
Figure 4.20 Small signal equivalent circuit that accounts for the dependence of ID on ws
Furthermore, the first-order equations we have given for the FET do not include a term to
account for the small effect of (&. on the drain current Previously, we assumed that the drain
characteristics are horizontal in the saturation region, but this is not exact — the drain
characteristics slope slightly upward with increasing Pd>. If we wish to account for the effect of ?&
in the small-signal equivalent circuit, a resistance r& called the drain resistance is added between
the drain and source as shown in Fig. 4.20. Equation (4.27) becomes
r
U = Smv** +
ds
(4.31)
Figure 4.21 Determination of ra from the FET drain characteristics using (4.33)
The definition of the reciprocal of n (which actually is the drain conductance gd=l/rd) is the
partial derivative of drain current with respect to %., evaluated at j^-point:
1
Si n
SvDS
(4.32)
Q-foial
which can be approximated as
1
A/
"
-t*a
(4-33)
- uo
where AID is an increment of drain current centered at the j2-point. Similarly, Avds, is an
increment of drain current centered at the jg-point. Equation (4.33) is useful to evaluate the drain
resistance from the output characteristic of the FET plotted for a constant vGS=VGs. This is
illustrated in Fig. 4.21.
Similarly, an alternative definition of the transconductance ^ is the partial derivative of ID with
respect to vGS, evaluated at j2-point:
on
Sv,GS
(4.34)
Q-poinl
which can be approximated as
-102-
ML
bm —
(4.35)
AvGS
where A t ^ is an increment of drain current centered at die jg-point. Equation (4.35) is useful to
evaluate die transconductance from die FET output characteristics plotted for two different
values of gate-to-source voltage. This idea is illustrated in Fig. 4.22.
+ VDS
Figure 4.22 Determination of g^ from die FET drain characteristics using (4.35)
The device equations and the equivalent circuit that we have derived from them describe only the
static behavior of die device. For rapidly changing currents and voltages, the additional
capacitances are required for an accurate model. A small-signal FET model suitable to represent
die JFET behavior for high-frequency operation is shown in Fig. 4.23.
Thrê capacitances are added to the circuit of Fig. 4.20 to obtain the circuit of Fig. 4.23. The
gate-source and gate-drain capacitances, Cgs and Cgd, respectively, represent die junction
capacitance of die gate-to-channel junction. These capacitances are nonlinear — they depend
respectively on gate-drain and drain-source dc voltages, Cgs = J{VGs) a n d Qd = J^YGD)- Th e
function J{.) has die form of Equation (3.40) with vD = VGS for Cgs and vD = VDS for Cgd. The
capacitance Cds. is mainly the stray capacitance between the drain and source terminals, so it is a
linear (constant) capacitance to a first approximation.
It should be noted diat small-signal high-frequency model for a MOSFET device is a little more
complicated than the circuit of Fig. 4.23. Namely it comprises diree other capacitances,
connected between the gate, source and drain terminals and die bulk terminal, respectively. These
capacitances are nonlinear (junction) capacitances, unlike die Cgs and Cgd capacitances diat are
stray capacitances for the MOSFET. Such a model will be discussed in Section 4.11 for die
purpose of SPICE simulation of MOSFET circuit behavior. For approximate, simplified analysis
die circuit shown in Fig. 4.23 can be used for both JFET and MOSFET devices.
Figure 4.23 Small-signal high-frequency JFET equivalent circuit
The small-signal JFET equivalent circuits shown in Fig. 4.20 and 4.23 will be used in die next
section to analyze properties of basic types of FET amplifiers at low and high frequencies.
-103-
4.7 Basic Small-Signal FET Amplifier Circuits
The circuit diagram of a common-source amplifier is shown in Fig. 4.24. The ac signal to be
amplified is supplied by %(/). The coupling capacitors C1 and C2 as well as the bypass capacitor Cs
axe. intended to have very small impedance for the ac signal. In this section we then carry out a
midband analysis of the amplifier, in which we assume that these capacitors are short circuits for
the signal. Later when we consider the frequency response of amplifiers, we include the
capacitors. The resistors R^, Rs, and RD form a self-bias network, and their values are selected to
obtain a suitable j2-point. The amplified output signal appears as the voltage drop across the load
resistor RL.
The small-signal equivalent circuit for the amplifier is shown in Fig. 4.25. The input coupling
capacitor has been replaced by a short circuit. The FET has been replaced by its small-signal
equivalent, which is shown in Fig. 4.20. Because the bypass capacitor Cs is assumed to be a short
circuit, the source terminal of the FET is connected direcdy to ground - which is why the circuit
is called the common-source amplifier (the source is common to die input and the output). The dc
supply voltage source is considered to be a short circuit for the ac signal. Consequendy, the
resistor RD appears connected from drain to ground.
Figure 4.24 Common-source amplifier
Next we consider the voltage gain of the amplifier. Refer to the small-signal equivalent circuit
and notice that the resistances rd, RD, and RL are in parallel. We denote the equivalent resistance
by
R
-
=
(4.36)
Figure 4.25 Small signal equivalent circuit for the common-source amplifier
The output voltage is the product of the current from the controlled source and the equivalent
resistance.
(4'3?)
v0 = -ig.vJR'L
-104-
Analog Electronics /FET Circuits
The minus sign is necessary because of the reference directions selected (i.e. the current g[avs
flows out of the positive end of the positive end of the voltage reference for p j .
The input voltage and the gate-to-source voltage are equal to each other
Now if we divide Equation (4.37) by (4.38), we find the voltage gain, which is given by
Av=^
(4.39)
= -gmR1
The minus sign in the expression for the voltage gain shows that the common-source amplifier is
inverting.
The input resistance of the common-source amplifier is given by
(4.40)
R, = — = Rr
h
This impedance forms part of the bias network, but its value is not critical. Practical values
change from 0 to perhaps 10 MQ in discrete component circuits. Thus we have a great deal of
freedom in design of the input resistance of a common-source amplifier. This is not true for BJT
amplifier circuits. One should note, however, that the effective voltage gain
defined as the ratio of the output voltage to the signal source voltage depends on the resistance
jR^. Namely, one can easily demonstrate that
Rr
Rr.
(4.42)
K = -g« R + R, RL=AV R + Rr
It follows from Equation (4.42) that with the decreasing gate resistance, the effective voltage gain
decreases, approaching zero when r^—>0. Therefore, high-intemal-resistance sources require
larger values of RQ to maintain large values of the effective voltage gain.
Figure 4.26 Circuit used to find output resistance Ro
To find the output resistance of an amplifier, we disconnect the load, replace the signal source by
its internal resistance, and then find the resistance looking into the output terminals. The
equivalent circuit with these changes is shown in Fig. 4.26.
Because there is no source connected to the input side of the circuit, we conclude that fgs=0.
Therefore, the controlled current source produces 2ero current and appears as an open circuit.
Consequendy, the output resistance is the parallel combination of RD and ra.
R
°
=
VRD
+
( 4 - 43 )
l/rd
Exercise 4.8 Find the voltage gain, input resistance and output resistance for the circuit of Fig.
4.24. Also find «(/) and vQ(t). Assume %(/) = 0.1sin(20007t/) V, R = 100 kQ, J ^ = 1 MQ, RD = 2.7
-105-
kQ, R L = 10 \sQ, rd = oo, IDSS= 8 mA, VP= -2 V and ID= 2 mA. Ans. A„= -8.5, ft = 1 MQ, Ro =
2.7 kQ.
Exercise 4.9 Find the voltage gain of the amplifier of Exercise 4.8 if an open circuit replaces R^.
Ans. A,,= -10.8.
Exercise 4.10 Find the value of Rs of Exercise 4.8. Ans. Rs = 500 Q.
Exercise 4.11 Consider the circuit of Fig. 4.24 with the bypass capacitor replaced by an open
circuit. Draw the small-signal equivalent circuit. Then assuming that ra is an open circuit for
simplicity, derive an expression for the voltage gain in terms of gm and the resistors. Ans.
Exercise 4.12 Evaluate the gain expression found in Exercise 4.11 using the values given in
Exercise 4.8 and the value of Rs found in Exercise 4.10. Compare the results with the voltage
gain found in Exercise 4.8. Ans. Av = -2.84.
Another amplifier circuit known as source follower is shown in Fig. 4.27. The ac signal to be
amplified is supplied by the %(/) signal source and R is the internal resistance of the signal source.
The coupling capacitors C1 cause the ac input to appear at the gate of the FET. The capacitor C2
connects the load to the source terminal of the FET. (In the midband analysis of the amplifier, in
which we assume that the coupling capacitors behave as short circuits.) Later when we consider
the frequency response of amplifiers, we include the capacitors. The resistor Rs provides a path
for the dc current flowing out of the source terminal of the FET.
Figure 4.27 Source follower
The resistor R^ provides a path for the gate leakage current. One reason for using a source follower is to
obtain high input impedance, and we would pick a large value for R^. The largest resistors available
are on the order of 10 Mfi. Even with such large values, the dc voltage drop caused by leakage
current is usually negligible, so we can consider the bias value of the gate-to-source voltage to be
zero. As a result, the bias value of the drain current is ID = IDSS. Since IDSS demonstrates
considerable device-to-device variation, the bias current of this circuit is not well controlled.
(This situation could be corrected by returning R^ to ground, forming the self-bias circuit
discussed in Section 4.4, but this causes a significant reduction of the input resistance.) Even
though the circuit of Fig. 4.27 has poor bias stability, it achieves extremely high input resistance,
so it is sometimes useful.
The small-signal equivalent circuit is shown in Fig. 4.28. The coupling capacitors have been
replaced by short circuits, and the FET has been replaced by its small-signal equivalent. Notice
-106-
that die drain terminal is connected to direcdy to ground because the dc supply becomes a short
in the small-signal equivalent. Therefore, the source follower is also called the common-drain
amplifier. The FET equivalent circuit is drawn in different configuration than shown in Fig. 4.25,
but it is the same electrically.
Drawing the small-signal equivalent for an amplifier circuit is an important skill for electronics
engineers. Test yourself to see if you can obtain the small-signal circuit starting from Fig. 4.27.
Figure 4.28 Small-signal equivalent for the source follower
Now we derive an expression for the voltage gain of the source follower. Notice mat rd, Rs and
RL are in parallel. We denote the parallel combination by
1
(4.44)
l/rd+l/RD+URL
*
*
=
•
The input current must flow through Rg. Therefore, the current flowing through RL is
''y+SmV-Tkus
v„ = K{L + g^v)
We can write the following voltage expression:
(4-45)
Vy
v
l = V
gS +' Vv0
(4.46)
Finally,» otice that the voltage across Rg is v^, so we have
v„, = RQif
(4.47)
Equations (4.45), (4.46) and (4.47) form the set needed to solve for the voltage gain. First we use
Equation (4.47) to substitute into Equation (4.45), resulting in
(4.48)
Now if we substitute Equation (4.47) and (4.48) into (4.46), we obtain
vi=RGii
(4.49)
RL{ii+gmRGit)
+
Finally, if we divide Equation (4.48) by (4.49), we find the voltage gain
A
=VQ-
R
(4.50)
L(l + gm*G)
A few simple checks can be performed on this expression. First, voltage gain is dimensionless.
Checking we see that the expression given on the right-hand side is indeed dimensionless. (Recall
diat the units of g^ are Siemens, i.e. mA/V.) Another simple check is to notice that if ^ = 0 , the
controlled source in Fig. 4.28 becomes an open circuit. Then the equivalent circuit becomes a
-107-
resistive voltage divider. Substituting g^ = 0 into the voltage-gain expression results in the
voltage-division ratio of the resistive circuit. Checks such as this are useful for detecting errors in
writing equations or in the algebra.
Notice that the voltage gain given in Equation (4.50) is positive and is less than unity. However,
in most circuits, it is only slightly less than unity. It follows then, that the output voltage follows
the input voltage that justifies the name sourcefollower for the circuit in Fig. 4.27. To summarize, the
sourcefollower is a noninverting amplifier with a voltage gain slightly lower than unity.
The input resistance can be found from Equation (4.49) by dividing both sides by r,
Ri = Vj- = RG^RL{\
+
(4.51)
gmRG)
We see that the input resistance can be very large compared to R^.
To find the output resistance, we remove the load resistance, replace the signal source with its
internal resistance, and look into the output terminals. It is helpful to attach a test source vx to the
output terminals as shown in Fig. 4.28. Then the output resistance is found as
R
R
"~i
(4.52)
- ^
where Ix is the current supplied by the test source as shown in the figure. It can be shown that
the output resistance is given by
R„ =
;
—
—
\
—ir~
(4-53)
1 . + .&A
Rs+rd+
RG+R
R + RG
This can be quite low, and another reason for using a sourcefollower is to obtain low output impedance.
Figure 4.29 Equivalent circuit used to find die output resistance of
the source follower.
Exercise 4.13 Find the voltage gain, input resistance, output resistance, current gain and power
gain of the source follower shown in Fig. 4.27 if R = 100 kQ, B^ = 10 MQ, Rs = 1 kQ, RL = 2.2
kQ. The FET has ra = oo, IDSS = 1 6 mA, VP = -2 V; assume it operates in saturation. Ans. Av =
0.917, ft = 120 MQ, R„= 59.4 Q,A,= 5-104, G = 4.49-101
From the results of Exercise 4.13 we see that even though the voltage gain of the source follower
is less than unity, the output power is much greater than the input power because of the very
high input resistance.
Exercise 4.14 Derive expressions for the voltage gain, input resistance and output resistance of
the source follower shown in Fig. 4.30. Calculate their respective numerical values if R = 100 kQ,
-108-
RG= 1 0 M Q , R j = 1 k Q . R ^ 2.2 kO. The FET has rd=co,IDss=
operates in saturation.
8 mA, Vp= -2 V; assume it
Exetcise 4.15 Derive expressions for the voltage gain, input resistance and output resistance of
the common-gate amplifier shown in Fig. 4.31. Calculate their respective numerical values if R =
100 kQ, R c = 10 MQ, Ks= 1 kfi, Ro= 2.7 kQ, R L = 10 kO. The FET has rd = oo, IDSS= 8 mA, VP
= -2 V; assume it operates in saturation.
TABLE 4.1 PROPERTIES OF BASIC FET AMPLIFIERS
Circuit
configuration
Common
source
Common
drain
Common
gate
Amplifier
type
Voltage gain
at midband
Input
resistance
Output
resistance
Bandwidth
Inverting
-x-R',.
Medium
Medium
Noninverting
si
Large
Large/,
very large
Small
Very wide
Noninverting
A*«.
Small
Medium
Wide
Looking at the results obtained from Exercises 4.8, 4.14 and 4.15, one can compare the
properties of the basic FET amplifier configurations. They are summarized in Table 4.1. Please
note that although the common-source and common-gate amplifiers have the same magnitude of
voltage gain, the common-source circuit is an inverting amplifier, unlike the common-gate
configuration. The common-gate one offers larger bandwidth, as will be discussed next in dais
section.
Example 4.1 Find the higher cutoff frequency of the common-source amplifier of Fig. 4.24.
Assume R = 1 kQ, Rc = 10 MQ, Rs= 1 kft, RD= 5.6 kfi, R^= 10 kft, rd= oo, IDSS= 16 mA, and
VP = -2 V, Qs = 10 pF, Cgd = 3 pF, Gh = 2 pF.
Solution. To find the circuit small-signal transconductance, one has to determine the j2-point of
the FET by using the techniques of Section 4.4. In the case of the circuit shown in Fig. 4.24, the
gate-to source voltage equals to HGS = -1.407 V. This corresponds to ^ = 2.372 mS. The
equivalent circuit of the amplifier, valid for high frequencies is shown in Fig. 4.32. To build this
-109-
circuit, its model of Fig. 4.23 replaced the FET in Fig. 4.24, and the coupling and bypass
capacitances were replaced by respective short circuits.
Figure 4.32 Small signal high-frequency equivalent circuit for the common-source amplifier
The capacitance Cgd in Fig. 4.32 is connected between the input and the output of the amplifying
device and thus makes the derivations difficult. To simplify the analysis we will use the Miller
theorem of Section 2.6. First, the midband voltage gain should be evaluated between the
terminals to which the feedback capacitance is connected. We will use the circuit of Fig. 4.33 to
find this gain, as
V
A = -?- = -gmRL
gs
Since R'L = RD\ \ RL = 3.59 kQ, we find A = -8.52. According to Miller theorem, the capacitance
Cgd may be split into two parts
Cm]=Cgd(\-A)=
28.55 pF,
\-A
'ml = cgd -A = 3.35 pF.
These capacitances are connected, respectively, from the gate to source, and drain to source
terminals, as shown in Fig. 4.34. The resistance RQ has been neglected as being much larger than
the signal-source resistance R and the reactance of capacitances (Cgs + Cmi) at high frequencies.
Figure 4.34 Simplified equivalent circuit for die common-source amplifier
To find the higher cut-off frequency, one has to evaluate the effective gain of the amplifier
-no-
The output voltage can be described as
ô ~
Sm'gs
l + 7«*/.(CA+Cm2)
=
~ Sm" gs
(4.54)
R'L
1 + JQ)/0)2
where Q)2 =1/R^C^
+Cm2)
source voltage is equal to
1
V =V
Y l +
jcoR(Cgs+Cml)
* gs
= 1/(3.59 kQ x 5.35 pF) s 1/18.94 ns = 52.8 Mrd/s. The gate-
(4.55)
1
=v1 + jco/coi
where <QX =1/R(C g s +Cml)
= 1/(1 kfl x 38.55 pF) £ 1/38.55 ns = 25.9 Mrd/s. Using the
approximate analysis techniques of Section 2.6, knowing t h a t ^ = 0)H/(27t) one can write using
Equation (2.62)
2.77 MHz <fH < 4.13 MHz
(4.56)
This result compares very well with the PSpice-computed value of 3.0 MHz.
Example 4.2 Use PSpice to find the higher cut-off frequency of the common-drain amplifier
shown in Fig. 4.30. Assume R = 1 kQ, 1^ = 1 Mi), Ry = 1 kQ, RL = 10 kQ. The FET has rd = oo,
IDSS = 1 6 mA, Vp = -2 V, Cgs = 10 pF, Cgd = 3 pF, Cds = 2 pF; it operates in saturation.
Solution. One can find that the FET transconductance in this Example is ^n = 2.372 mS and R'L
= 909.1 Q. The high-frequency equivalent circuit is shown in Fig. 4.35a. The corresponding
PSpice code is presented in Fig. 4.35b.
The PSpice simulation run for the circuit of Fig. 4.35a p r o d u c e d ^ 2 27.7 MHz. The higher cutoff frequency of the common-drain amplifier is then much larger than the value of 3.0 MHz
obtained for the common-source amplifier.
*
(a)
common-gate amplifier
V 1 0 ac lmV
R 1 2 Ik
Rs 2 0 Ik
Cgs 2 0 lOpF
Cds 2 3 2pF
Cdg 3 0 3pF
gm 3 2 0 2 2.372m
Rd 3 0 5.6k
RL 3 0 10k
.ac dec 100 10Hz lOOMEGHz
.end
Figure 4.36 Small-signal high-frequency equivalent for the common-drain amplifier
Example 4.3 Use PSpice to find the higher cut-off frequency of the common-gate amplifier
shown in Fig. 4.31. Assume R = 1 kQ, Rs = 1 kQ, RD ~ 5.6 kQ, RL = 10 kQ. The FET has rd =
oo, IDSS = 1 6 mA, Kp = -2 V, Cgs = 10 pF, Cgd = 3 pF, C<k - 2 pF; it operates in saturation.
Solution. One can find that the FET transconductance in this Example is ^ = 2.372 mS and R'L
= 3.59 kQ. The high-frequency equivalent circuit is shown in Fig. 4.36.
The PSpice simulation run for the circuit of Fig. 4.36a p r o d u c e d ^ =11.2 MHz. The higher cutoff frequency of the common-drain amplifier is then larger than the value of 3.0 MHz obtained
for the common-source amplifier.
*
From the three basic configurations of single-FET voltage amplifiers, the bandwidth of the
common-drain circuit is the largest one. The smallest bandwidth is offered by the commonsource circuit, which is explained by the pronounced Miller effect. There is a possibility of
building an amplifier that has a large input impedance and gain of the common-source circuit and
maintains the wide bandwidth of the common-gate amplifier. This is a cascode 2-FET amplifier
shown in Fig. 4.37. In this amplifier, the input stage is a common-source circuit whose load
resistance is very small being the input impedance of the output stage that is a common-gate
circuit. Thus lower device has a small voltage gain, slight Miller effect and wide bandwidth. The
upper device offers large bandwidth and converts the current from the lower device into large
output voltage. Investigation of the properties of this amplifier is left as an exercise to the reader.
-112-
Exercise 4.16. Consider the amplifier shown in Figure 4.37. Both FETs have V& = 3 V, K = 0.6
mA/V 2 , n = oo, Cgs = 10 pF, Cgd = 3 pF, Qs = 2 pF, and operate in saturation. Assume R = 1
k£2, R D = 5.6 k£2, R L = 10 kfi. Find device Q-points and transconductance. Draw the small-signal
equivalent circuit valid for medium and high frequencies (replace the capacitances by short
circuits). Run .ac PSpice analysis to find the amplitude characteristic of the amplifier. Determine
the effective gain and bandwidth of the whole cascode as well as gain and bandwidth of its
individual stages.
24V
Figure 4.37 2-MOSFET cascode amplifier
In the next few sections, we turn our attention to other applications for the FET, such as voltagecontrolled resistance and CMOS logic circuits.
4.8 The FET as a Voltage-Controlled Resistance
Besides its use as an amplifier or as a switch (see Section 4.9), the F E T is useful as a voltagecontrolled resistance. In this application, the bias point is chosen at the origin of the output
(drain) characteristics as illustrated in Fig. 4.38.
10
VDSOO
Figure 4.38 When used as a voltage-controlled resistance, the FET is biased at the origin
If the gate-to source voltage is greater than the pinch-off voltage, the device operates in the linear
(triode) region, and the drain current is given by Equation (4.5), which w e repeated here for
convenience
iD=K[2(vGS-VP)vDS-v2DS]
The transconductance of the device can be found by application of Equation (4.34), which is
-113-
dir
dvGS
Q-pmnt
Applying this equation to (4.5) we have
°«
(4.57)
"»\Q-poml
But the jg-point is vDS= VDS=Q, so we have^n=0. Another way to obtain this result is to recall that
gm is a measure of vertical spacing of the drain characteristics. However, as shown in Fig. 4.38, all
the characteristic curves pass through the origin, and the spacing is zero for that point.
Figure 4.39 Small-signal equivalent circuit for a FET operated at
VDS=0
Because gm=0, the controlled current source of the small-signal equivalent circuit for the FET
biased as shown in Fig. 4.38 becomes an open circuit. Thus as shown in Fig. 4.39, die smallsignal equivalent circuit simply becomes a resistor ra connected between the drain and source
terminals. This resistance can be found by application of Equation (4.32), which is repeated here,
for convenience
1
dir
8vDS
Q—poinl
Applying this equation to (4.5), we obtain
(4.58)
a
Evaluating for vDS=0 and rearranging, we find that
1
2K(VGS-VP)
(4.59)
which is valid provided that Vcs is above pinch-off. Of co. rse, if is less than Vp, die FET is in
cutoff and ra=oo.
Figure 4.40 Voltage-controlled FET attenuator
Thus we see tiiat if the FET is biased at the origin of the drain characteristics, it behaves as a resistor connect
from drain to source, the value of which is controlled by the gate-to-source voltage. This conclusion can also b
made by inspection of Fig. 4.38, where die curves are approximately straight lines intersecting die
origin, and their slopes depend on vGS.
One application of the FET as a variable resistance is in die voltage-controlled attenuator circuit
drawn in Fig. 4.40. The resistor R and die resistance r<i of the FET form a voltage divider, so die
output voltage is given by
-114-
Analog Electronics / FET Circuits.
v
o=vs
?d
(4.60)
R + rd
The control voltage is applied to the gate of the FET. If the control voltage is less than the
pinch-off voltage, ra = oo and no attenuation occurs. However, as the control voltage is raised
above pinch-off, rd becomes smaller and the attenuation becomes greater.
The circuit of Fig. 4.40 is an alternative solution to the semiconductor diode voltage-controlled
attenuator circuit discussed in Section 3.4. The FET-based attenuator is often used to stabilize
die amplitude of RC sine-wave oscillators.
Exercise 4.15 Suppose that R = 10 kQ in the voltage-controlled attenuator in Fig. 4.40 and a
FET having IDSS = 16 mA, VP= -4 V is used. Compute values of ra and Av = *b/i% for a number
of values of vGS= -Vc— 0, -1, -2, -3 and -4V. Compare the results with those obtained in Exercise
3.6. ADS. rd = 125 fi, 167 Q, 250 Q, 500 Q and oo;Av =0.0123, 0.0164, 0.0243, 0.0476 and 1.0,
respectively.
4.9 The CMOS Analog Switch
The CMOS analog switch also known as a transmission gate is shown in Fig. 4.41. We will
see that it acts as a switch that either connects points A and B through a low resistance or
disconnects mem, depending on the digital control signal Vc.
We assume that the logic levels for Ve are +VDD (high) or -VDD (low). Note mat the control
signal is connected directly to the gate of the NMOS and to the input of the logic inverter. The
inverter output is connected to the gate of the PMOS. The input signal vs to be connected to the
load R L can be either analog or digital and is assumed to range between -VDD and +VDD. The
NMOS substrate is connected to the negative supply. We assume the FETs are identical except
for polarity. The threshold voltage for the NMOS is V^ = V^ and for the PMOS is V^ = VA. Furthermore, we assume that K A is less than V D p.
Figure 4.41 CMOS analog switch
If Vc = -VDD, both FETs are cut off (provided that vs is within - K D D and + VDIj). Thus for Vc
low, an open circuit appears between A and B.
Now consider Vc = +K D D . To start, we assume that the input voltage vs is positive, so current
flows from left to right through the FETs. With PS = 0 both the NMOS and PMOS are
conducting. Then as v% increases, current flows through the FETs, raising the output voltage.
-115-
Usually, the resistance of the FETs is low enough so that vQ is approximately equal to vr (The
FETs operate in the triode region.) Notice that the gate-to-source voltage of the NMOS is
VGSN=VC-V0
(4.61)
However, since we are assuming that Vc = VDD, we have
V
GSN=VDD-V0
(4.62)
When the output voltage exceeds VDD-VA, the NMOS becomes cut off, but the PMOS is
heavily conducting. Thus point A is connected to point B by a low-resistance path for all values
of v% between zero and VjrjD.
Similar reasoning, with source and drain terminals interchanged, applies for negative values of the
input voltage. Thus for Vc high, point A is connected to point B regardless of the polarity of vs.
For v0 between -VDD+VA and VDD-VA, both FETs are on. Outside this range, only one of the
FETs is on.
The resistance of a FET is nonlinear. However, for an analog signal, it is undesirable for die
resistance between points A and B to be nonlinear because this leads to distortion of the output
voltage. Fortunately, it can be shown that the nonlinearities of the PMOS and NMOS cancel in
the range of voltages for which both transistors are on. This is an advantage of the CMOS switch
as compared to circuits having only a single transistor. For an analog input signal taking both
positive and negative values, we should choose positive and negative logic level so that both
transistors are on for the range of input voltages expected.
Exercise 4.16 Suppose that the analog switch of Fig. 4.41 uses enhancement MOSFETs having
\VA\ =1V and I K ^ l m A / V 2 . Also, VDD=SV and RL is large enough so that vQ is
approximately equal to vs. The control signal is Vc = VDDt so die gate is in the on state. Find the
small-signal resistance rd of the NMOS for vs=0, 1, 2, 3, 4, and 5V. Repeat for the PMOS. Find
the effective incremental resistance of each device between points A and B for each input
voltage. (Hint. vDS=0, so Equation (4.56) can be used with appropriate changes in notation.)
4.10 CMOS Logic Circuits
In this section we briefly discuss an important logic family that uses complementary metaloxide-semiconductor (CMOS) FETs. The term complementary implies that both /(-channel and pchannel devices are employed.
NMOS inverter with resistive pull-up. Before considering CMOS circuits, we discuss die
simpler inverter circuit shown in Fig. 4.42. The transistor is an »-channel enhancement device
(NMOS) having a threshold voltage V^. The load capacitance represents the input capacitance
of driven circuits (such as gates, inverters).
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The drain characteristics of the NMOS transistor are shown in Fig. 4.43. The input voltage Vm is
applied to the gate, so we have vGS= V^. For the moment, we assume that the load capacitance is
an open circuit. Using the values R D =10kQ and VDD=10V we construct the load line shown in
Fig. 4.43.
Notice that if the input voltage is less than the threshold voltage of the transistor (assumed to be
K th =3V in this example), the transistor is cut off. Then the circuit operates at point A, and the
output voltage is K 0 = VDD.
As the input voltage is raised above threshold, the point of operation moves up the load line.
When f^n= VDD, the circuit operates at point B, and the output voltage is low. Thus the circuit
operates as a logic inverter (low input corresponds to high output and vice vend).
Fig. 4.44a shows the transfer characteristic of the MOS inverter with resistive load. Points A and
B are marked, as discussed above. With logic one at the input (vGS = 10 V), the output voltage is
equal to 0.61 V and is considered output logic zero for this inverter circuit At the same time, the
drain current is equal to 0.94 mA. Thus the inverter circuit considered draws a power from a
power supply when its input is logic one. Part of this power is dissipated in die drain resistor [PR
= (10V-0.61V)x0.94mA = 8.8mW] and the rest is dissipated inside the MOS transistor structure
(PT = 0.61Vx0.94mA = 0.6mW). This causes the power loss that shortens the battery life in
portable equipment and leads to heating of the circuit.
-117-
In selecting the value of the pull-up resistor Rp, we encounter conflicting objectives. On die one
hand, we want to make die resistor large because this leads to a small current the transistor is on.
This, in turn, means a smaller demand on the power supply and less heating of die circuit On die
other hand, we want to make RD small, so that when the FET switches off, the load capacitance
is quickly charged. (Usually it is important for logic transitions to take place quickly.)
The CMOS inverter. A solution to diis conflict is to use an enhancement ^-channel MOS
(PMOS) transistor in place of die pull-up resistor as shown in Fig. 4.45. (An additional benefit is
diat die PMOS takes much less chip area dian a resistor and dierefore is advantageous for IC
implementation.)
vDD=\0\
11
In the following discussion we assume diat except for the differences in voltage polarity and
current direction, die NMOS and PMOS have identical characteristics. The direshold of die
NMOS is Vfan = Vfa which is a positive value, and the threshold voltage for the PMOS is V^ =
-Vfa. Also we assume, as is often the case, that the supply voltage VDD is greater that twice die
direshold voltage magnitude. (In the illustrations, we assume that K th =2V and VDD=\OV.)
Notice in Fig. 4.45 that the source terminal of the PMOS is connected to VDD and that the drain
is connected to the inverter output The gate-to-source voltage of the PMOS is given by
vGSP=^-rDD
(4-63)
When Vm = VDLh die gate-to-source voltage of the PMOS is zero, so it is cut off. Then it acts as
a very high value of R^, and virtually no current flows from the supply. On the other hand, when
Vm = 0, we have PGSP = -VDD, and the PMOS can deliver a large drain current to charge the load
capacitance. Since the NMOS is cut off for UQ^ = V^ = 0, no current flows after die
capacitance is charged.
An important advantage of CMOS logic is that, except during logic transitions, either the NMOS
or the PMOS is cut off, and no current flows. Thus the static power consumption (i.e. the
power consumption of a logic circuit when the logic states are not changing) is virtually zero.
For this reason, CMOS is an attractive choice for battery-operated circuits such as portable
computers.
Now we consider load-line constructions for the CMOS inverter. Recall that in the case of a
resistive load, the load line is straight - the volt-ampere characteristic of a resistor. However, for a
PMOS pull-up transistor, the load line is not straight; instead, it is a characteristic curve of the
PMOS. Furthermore, the resistor results in a fixed load line, but the line (actually, it is a curve)
for the PMOS pull-up changes as Vm changes. The load lines are shown for several values of Vin
on the characteristics of the NMOS in Fig. 4.46.
-118-
Figure 4.46 Load-line analysis of CMOS inverter.
For the NMOS V^ - 2V and for the PMOS K,hp=-2V.
For Vm - 0, die PMOS is highly conductive, but the NMOS if cut off, and the point of
operation is at A. Point B illustrates the operating point for a value of Vm greater than the
threshold voltage of the NMOS but less than VDD/2. At point B the NMOS is in saturation and
the PMOS in the triode region.
When Vm = VDD/2 = 5V, the load line and the NMOS characteristic intersect not at a single
point but along a line from C to D in Fig. 4.46. Thus as Vm increases through VDD/2, the
operating point switches abrupdy from C to D.
Point E illustrates an operating point for a value of Vm between VDD/2 and VDD-VA. At point E
die PMOS is in saturation, whereas the NMOS is in the linear region. For Vm = KOD, the PMOS
is cut off, and operating point is F.
-119-
K A . The transfer characteristic falls abruptly for Vm = VDD/2. The CMOS inverter closely
approximates the ideal transfer characteristic for a logic inverter.
The current flowing from the supply through the transistors of the CMOS inverter, assuming an
ideal open-circuit load is shown in Fig. 4.47b. Notice that if Vm = 0 or Vm = VDD> then the
current is zero. Maximum current flow occurs for Vm = VDD/2. The maximum current value
depends on the supply voltage and on the value of transistors' K.
The CMOS N O R and N A N D gates. The circuit diagram of a two-input CMOS N O R gate is
shown in Fig. 4.48a. The source and drain terminals of the FETs are not labeled in the figure.
The devices are physically symmetrical, so either end can be considered to be the source, with the
other becoming the drain. Usually, we consider the source of the PMOS devices to be the
terminal that current enters (i.e., the top en of this circuit). Similarly, we consider the sovirces of
the NMOS devices to be the end that current leaves (Le., the bottom terminals of this circuit).
Designation of source and drain is convenient in analysis of the circuit; however, the physical
construction of a device is the same regardless of which end is the source or drain.
Now we consider the operation of the circuit shown in Fig. 4.48a. If both inputs A and B are
low, the PMOS transistors' M1 and M2 are conductive, and both the NMOS transistors M3 and
M4 are off. Consequently, the output is high. If either A or B or both are high, one or both of the
PMOS devices are cut off. Furthermore, at least one of the NMOS devices is conductive.
Consequendy, the output is low. Consequendy, this circuit performs the N O R logic function. A
two-input CMOS NAND gate is shown in Fig. 4.48b.
A/? "H MM
Mj
(»)
(b)
Figure 4.48 CMOS two-input logic gates: (a) NOR gate, (b) NAND gate
Exercise 4.16 For the circuit of 4.48b, prepare a table showing all possible combinations of
inputs (each input can be" high or low), the corresponding state of each transistor, and the
corresponding output. Indicate the state of each transistor either as on for operation in the triode
or saturation region or as o^for operation in cutoff.
Exercise 4.17 Draw the circuit diagram of a three-input CMOS N O R gate.
Exercise 4.18 Prepare a table showing the regions of operation (saturation, triode or cutoff) of
the PMOS and the NMOS for each labeled point on the inverter transfer characteristic shown in
Fig. 4.47.
Exercise 4.19 A CMOS inverter is constructed with symmetrical devices having V^ = 3V and
Vfo. = -3V. Sketch the transfer characteristic to scale if VDD = 15V.
-120-
Exercise 4.20 If the devices have \K\ =lmA/V 2 , find the supply current through the inverter of
Exercise 4.20 if Vm = VDD/2. Assume an open-circuit load. Ans. 20.25mA.
4.11 FET Dynamic Circuit Models
Figure 4.49 shows a circuit model for an ^-channel JFET and the corresponding SPICE
parameters. Current source iD describes the static characteristic of the device. In pinch-off, the
drain current of an ^-channel JFET is given by
iD = K(vGS - Vpfil + Xvos)
(4.64)
in which K is a proportionality factor that depends on a particular device, Vp is the pinch-off
voltage, and the parameter X accounts for the slope of the output characteristic curves in
saturation. In die preceding sections of this chapter we have assumed X = 0 to simplify
discussion. The nonzero value of X leads to a nonzero value of the small-signal output
conductance r<i (see Fig. 4.21). It can be shown that the output conductance can be calculated as
where ID is the dc value of the drain current For the transconductance of a JFET with nonzero
X, Equation (4.29) should be modified as follows
gm=2^KID(l
+ XVDS)
(4.66)
where VDS is the dc drain-source voltage.
Figure 4.49 Dynamic model of a discrete »-channel JFET
The diodes add to the model the reverse saturation current Js that flows through the input circuit of
the physical device. If the gate is not reverse biased in simulation, these diodes conduct that is not
the expected operation mode of a JFET. Since the gate is usually reverse biased relative to the
channel, die diodes have nonlinear depletion capacitances Ca and CGD. The depletion capacitance
of the diode is given by Equation (3.40). Making appropriate changes in notation, we have
1 - 'GS
m
^0 .
C
DS0
C
DS =•
1-
DS
(4.68)
m
-121-
(Additional depletion capacitances C„ and CBD connect source and drain to the substrate in IC
structures.) Ohmic resistances RQ and Ry complete the model. Below there are two examples of
PSpice device statement for JFETs:
Jl Dnode Gnoda Snoda JModName
.MODEL JModNana NJF(VTO>-4 BETA=lE-3)
Jl Dnode Gnoda Snode J2N3819
.LIB EVAL.LIB
TABLE 4.2
Text
notation
vP
SPICE PARAMETERS FOR N-CHANNEL JFKD
SPICE
notation
VTO
BETA
Parameter name
Typical
value
-3V
600E-6A/V 2
Default
value
-2V
100E-6 A/V2
Pinch-off voltage
K
Transconductance
coefficient
LAMBDA Channel-length
2E-3 V 1
0
X
modulation coefficient
IS
Saturation current
2E-12 A
1.0E-14A
I
N
1
n
Emission coefficient
1
RD
Ohmic drain resistance
0
\Cl
RD
RS
Ohmic source resistance
0.1 Q
0
Rs
CGD
Gate-drain depletion
3pF
0
^GDO
capacitance (zero bias)
3.3 pF
0
CGS
Gate-source depletion
^GSO
capacitance (zero bias)
0.5
Junction grading factor
M
0.333
m
0.5
V
Built-in
barrier
potential
I
V
PB
fo
"Typical values of the parameters are shown for a discrete general-purpose device
An enhancement MOSFET structure is presented in Fig. 4.50. Notice that the length JL and
width W of the channel are labeled on die figure. In the saturation region, the drain current is
given by
iD = j
K(vGS
- Vth)2{\ + XvDS)
(4.69)
(Previously, we assumed for simplicity that W/L. = 1 and X = 0.) Notice that the current depends
on the width-to-length ratio of the channel. The device designer can vary this ratio to obtain
devices best suited for various functions in a circuit It turns out that the small-signal
transconductance depends on the W/L, ratio as follows
8m = 2^KID^(\
+ XVDS)
(4.70)
whereas the formula (4.65) is valid for MOS as weU. Keep in mind that the equation for ^n and gd
are valid only for operation in the saturation region.
Of course, it is desirable to construct devices with small dimensions so that a large number of
mem fit in a .given chip area. Another advantage of smaller devices is that the device capacitances
are smaller. Provided that the width-to-length ratio is maintained, the current available to charge
and discharge these capacitances is independent of device size. Thus digital circuits constructed
-122-
with smaller MOS transistors can switch faster. Both the speed and complexity of digital MOS
circuits can increase as the device dimensions become smaller.
Figure 4.50 ^-Channel enhancement MOSFET structure with channel length L and channel width W
SPICE uses Equation (4.69) to relate the current to the voltages, except that ohmic resistances
are added in series with the source and the drain. This is shown in Fig. 4. In the MOSFET of Fig.
4.51 there are depletion capacitances CBS and CBD between the substrate and reverse-biased ntype wells. To describe these capacitances, SPICE uses zero bias capacitances CBS and CBD,
grading coefficient MJ and bulk junction potential PB. The SPICE model also includes the
exponential volt-ampere behavior of the body-drain and body-source junctions. These junctions,
characterized by the reverse saturation current IS, must be kept reverse biased in the simulation (as in the
physical device) by connecting the substrate to the most negative point in the circuit.
Figure 4.51 »-channel MOSFET dynamic circuit model
Fig. 4.51 also suggests capacitance, CGB, between gate and substrate, with the oxide for its
dielectric. It turns out that in places where the gate slighdy overlaps the drain and source
materials there are additional capacitances C G5 and CGD. SPICE computes values for CGS, CGD, and
CGB when we specify the value of oxide thickness, TOX, on the .MODEL line and when W and
L are specified on the element lines of the transistors. TOX = 0.1U, W = 1U, and L = 1U suffice
as rough estimates when actual values are unknown. Resistors Rs and RD represent the voltage
drops by current flowing to the external contacts D' and S'. SPICE includes temperature variation in
threshold voltage and pn junction parameters; however, variations in mobility with temperature
are absent in basic versions of SPICE.
-123-
TABLE
Text
notation
SPICE
notation
Kh
VTO
KP
4.3 SPICE PARAMETERS FOR MOSFETs*)
Parameter name
Threshold voltage
Transconductance
2|K|
coefficient
LAMBDA Channel-length
X
modulation coefficient
RD
Ohmic drain resistance
Ro
RS
Ohmic source resistance
Rs
"Typical values are given for devices having W/L, = 1.
Typical
Value,
NMOS
IV
30E-6 A / V 2
Typical
Value, PMOS
1E-2 V-1
1E-2 V 1
10 Q
10 Q
ion
ion
-IV
12E-6 A/V 2
Table 4.3 lists SPICE model parameters for static operation. The capacitances Ccs, CCD, and CCB
have been described in the text above. Relationships of the type (4.67) and (4.68) describe the
capacitances CfBand CDB. Notice that the SPICE transconductance coefficient is
KP = 2|K|
Much work has been expended to obtain SPICE models based on device dimensions and process
parameters. Three of the resulting models are incorporated into PSpice. A particular model can
be selected in the .MODEL statement by the LEVEL=1, 2, or 3 parameter. The default is
LEVEL =1. Detailed discussion of these models is beyond the scope of these lecture notes. The
device statement for MOSFETs takes the form
Mdevice Dnode Gnode Snoda Bnoda MNama L»VALUE W-VALUE
.MODEL MNama NMOS(VTO=l KP=30U LAMBDft-0.01 RD=10 RS-10)
Notice that the first character of a MOS device must be M.
Example 4.4 Write a PSpice code to plot the output characteristics for an NMOS having the
model parameters given in Table 4.3. The device dimensions are JL = 20 um and W — 50 urn.
Assume that the substrate is connected to the source. Allow the drain-to-source voltage to range
from 0 to 20 V in 0.1-V steps and gate-to-source voltage to range from 0 to 10 V in 1-V steps.
Solution. The program listing is
NMOS characteristics example
Ml 2 1 0 0 NAME L=20u W*50u
.MODEL NAME NMOS (VTO=l KP=30u LAMBDA»0.01 RD=10 RS=10)
VGS 1 0 dc 5V '
VDS 2 0 dc 10V
.DC VDS 0 20 0.1 VGS 0 10 1
.END
The reader is suggested to reconstruct the schematic diagram of the simulated circuit from the
code.
After running the program and starting PROBE, we request a plot of ID(M1). The resulting plot
of the output characteristics is shown in Fig. 4.52.
-124-
n IBtmi
Figure 4.52 Output characteristics for Example 4.4
4.12 Summary
Field-effect transistors are 3-state devices that serve as dependent sources in analog applications
and as controlled switches in digital circuits. In one kind of FET, the control gate is insulated
from the conducting channel; another FET class has a gate that is in physical contact with the
channel by electrically separated by a reverse biased or Schottky junction. All ^-channel FETs
share the same state definitions, equations, and circuit models, except for minor notational
differences. Mathematically, the transfer characteristics of all «-channel FETs are the right-hand
branches of parabolas; they differ only in the algebraic sign of the threshold or pinch-off voltage.
Equations for /(-channel FETs are identical to those of //-channel devices; however, output
characteristics plot in the second quadrant instead of the first. The circuit models differ only in
the reference directions of the drain currents. The transfer characteristics are of all />-channel
devices are left branches of parabolas; individual differences are only in the algebraic sign of the
threshold or pinch-off voltage. In analysis of FET circuits we encounter two general classes of
problems: finding ^-points when transistor states are known and finding j2-points when the
states are unknown. The tools and procedures for solving these problems are load lines,
equivalent circuits and guessing and verifying states. The infinite input resistance of the FET
tends to simplify circuit analysis; however, FET circuits usually require us to solve a quadratic
equation and select the solution that has physical meaning. The FET can operate as a voltagecontrolled linear resistor. The most significant second-order effects in FET are breakdown
associated with reverse-biased junction that places an upper limit on the useful active region,
channel length modulation that causes a nonzero positive slope of output characteristics in the
saturation state, and the decrease of the threshold voltage and channel resistance with
temperature. Internal parasitic capacitances limit the FETs ability to operate at high frequencies.
Some are depletion capacitances associated with reverse-biased junctions; others are linear
capacitances associated with insulated gates. Because the minority charge carriers do not take
significant part in the current conduction in FETs, these devices lack large diffusion capacitances
associated with stored minority charge carriers. SPICE models can simulate the nonlinearities of
the static transistor models, and also include most second-order effects of FETs. Because of high
input impedance and output characteristic that resemble resistor volt-ampere curves in both first
and second quadrants, FETs make excellent switches. Bidirectional transmission gates
constructed from FETs are widely used in both digital and linear applications. For small signals, a
-125-
FET biased in die saturation region functions as a voltage-controlled current source with a
transconductance ^n = 2K(KcrVth) and the output resistance r0 = 1/(AJD). The common-source
configuration provides a high voltage gain and a very high input impedance, but a limited highfrequency response. A much wider bandwidth is achieved in the common-gate configuration but
its input impedance is low. The source follower (common-drain amplifier) provides a voltage gain
less than unity but features a low output resistance. Integrated-circuit MOS amplifiers utilize
MOS transistors as amplifying and as load devices. In CMOS technology, both n- and/(-channel
enhancement MOSFETs are used, thus providing the circuit designer with considerable
flexibility. The most important things to remember about FETs are their output and transfer
characteristics, the state definitions, the procedure to find their jg-point given bias circuit, and the
load line concepts, including load lines for CMOS inverter sketched in Fig. 4.46.
Notes:
-126-
Analog Electronics/BJT Circuits
5 Bipolar Transistor Circuits
Bipolar junction transistors (BJTs) are constructed as layers of semiconductor materials (most
often silicon) doped with suitable impurities, producing either »-type or/Mype layer. A simplified
physical structure of an integrated circuit npn transistor is shown in Fig; 5.1a. The actual transistor
action takes place along die A-A' line shown in Fig. 5a. The idealized structure of the bipolar
transistor, corresponding to its cross-section along the A-A1 line is presented in Fig. 5b, where a
layer of p-type material is placed between two layers of the »-type material. This structure can be
seen as two pn junctions made close together in a single crystal of semiconductor. The current in
one junction affects the current in the other junction.
The layers of the bipolar transistor are called the emitter, the base, and the collector, as shown
in Fig. 5.1b. The circuit symbol of an npn transistor is shown in Fig.5c, including reference
directions for the terminal currents and voltages.
Recall that a pn junction is forward biased with applying positive polarity to the p-side. On the
other hand, reverse bias occurs if the positive polarity is applied to the /|-side.
In normal operation of a BJT as an amplifier, the base-collector junction is reverse-biased and the
base-emitter junction is forward biased. In the following discussion we assume that the junctions
are biased in this fashion unless stated otherwise.
The Shockley equation gives die emitter current iE in terms of the base-to-emitter voltage vBE:
'£ = JES e x p ^ - l
(5.1)
This is exacdy the same equation as for the current in a junction diode given in Equation (3.1),
except for changes in notation. The emission coefficient n is made equal to unity since this is the
appropriate value for most junction transistors. Typical values for the saturation current 1^ range
from 10 12 A to 10"16A, depending on the size of the device. Recall that at a temperature of 300K,
the thermal voltage VT is approximately 26mV.
Of course, Kirchhoff s current law requires that the current flowing out of die BJT is equal to the
sum of the currents flowing into it. Thus, referring to Fig. 5.1c, we have
i£=iC+iB
(5.2)
-127-
Analog Electronics /BJT Circuits
This equation is true regardless of the bias conditions of the transistor junctions.
We define the parameter a for the transistor as the ratio of the collector current to the emitter
current.
'C
(5.3)
E
Values of a range from 0.9 to 0.999, with 0.99 being very typical. Equation (5.3) indicates that
the emitter current is partly supplied through the base terminal and partly through the collector
terminal. However, since a is nearly unity, the collector supplies most of the emitter current.
a =
l
Substituting Equation (5.1) into (5.3) and rearranging, we have
rVBE
ic = cdgs e x p ( — — ) - l
(5.4)
For vBE greater than a few tens of a volt, the exponential term in the bracket is much larger than
unity. Then the 1 inside the bracket can be dropped. Also, we define the scale current as
Is=a*ES
"
and Equation (5.4) becornes (for the normal bias conditions)
(5-5)
(5.6)
V
T
Solving Equation (5.3) for i0 substituting into Equation (5.2), and solving for the base current,
we obtain
iB=(\-a)iE
(5.7)
Since a is slighdy less than unity, then only a very small fraction of the emitter current is supplied
by the base. Using Equation (5.1) to substitute for iE, we obtain
(5.8)
iB=(l-a)/£S exp(^)-l
We define the parameter fi as the ratio of the collector current to the base current. Taking the
ratio of Equations (5.4) and (5.8) results in
o
'C
a
(5.9)
iB
l-«
Values for/? range from about 10 to 1000, and a very common value is /£=100. We can write
P
-128-
ic=PiB
(5.10)
Note that since /? is usually large compared to unity, the collector current is an amplified version of the
base current. Current flow in an npn BJT is illustrated in Fig. 5.2.
As in the case of the FET, the bipolar transistor is a three-terminal device. There are then three
basic circuit configurations that employ the BJT as an amplifier, i.e. a circuit that has one input
and one output. In each of these configurations, one of the transistor terminals is common to the
input and to the output. The common-emitter configuration for an npn BJT is shown in Fig. 5.3.
The battery connected between the base and the emitter supplies a positive voltage vBE that
forward biases the base-emitter junction. The V& battery produces a positive voltage at the
collector with respect to the emitter. Notice that the voltage across the base-collector junction is
given by
VBC=VBE~VCE
(511)
Thus if VCE is greater than vBE, the base-collector voltage vBC is negative, which is reverse bias.
<
*'c
Figure 5.3 Common-emitter circuit configuration for the npn BJT
The common-emitter characteristics of the transistor are plots of the currents iB and ic versus the
voltage vBE and v^. Representative characteristics for a low-power silicon device are shown in
Fig. 5.4.
The common-emitter input characteristic shown in Fig. 5.4a is a plot of iB versus vBE, which
are related by Equation (5.8). Notice that the input characteristic takes the same form as the
forward characteristic of a junction diode. Thus, for appreciable current to flow, the base-toemitter voltage must be approximately 0.6V. Just as for a junction diode, the base-to-emitter
voltage, for a given current, decreases with temperature by about 2mV/K.
The common-emitter output characteristic shown in Fig. 5.4b is a plot of ic versus v^ for
constant values of iB. The transistor illustrated has /£=100. As long as the collector-base junction
is reverse-biased (ffcÔ, or equivalently vCE>t,BE)>we have
; C =/B B =IOOZ B
As VCE becomes less than vBE, the base-collector junction becomes forward-biased, and eventually
the collector current falls as shown at the left-hand edge of the output characteristics.
Refer to Fig. 5.4a and notice that a very small change in the base-to-emitter voltage vBE can result
in an appreciable change in the base current iB, partially if the base-emitter junction is forward
biased, so some current (say, 40|iA) is flowing before the change in vBE is made. Provided that v^
is more than a few tens of a volt, this change in base current causes a much larger change in the
collector current i0 because i^fii^ In suitable circuits, the change in collector current is
converted into a much larger voltage change than the initial change in vBE. Thus the BJT can
amplify a signal applied to the base-emitter junction.
-129-
5.1 Load-Line Analysis of a Common-Emitter Amplifier
A simple amplifier circuit is shown in Fig. 5.5. The power-supply voltages VBB and Vcc bias the
device at an operating point for which amplification of die input signal ik(t) is possible. We will
show that the amplified version of the input signal appears between the collector and die ground.
~
Figure 5.5 Common-emitter amplifier
The load-line technique will be used to analyze die circuit Applying the Kirchhoff s voltage law
to die base loop, we obtain
VBB + V , ( / ) = RbhiO + VBEiO
(5-12)
-130-
Figure 5.6 Load-line analysis of the amplifier of Fig. 5.5:
(a) input (load line shifts downwards for a smaller value of«%), (b) output.
A plot of Equation (5.12) is shown in Fig. 5.6a as the load line on the input characteristic of the
transistr -. To establish the load line, we must locate two points. If we assume that iB=0, then
Equatio (5.12) yields vBE=VBB+t>s- This establishes the point where trie load line intersects the
voltage axis. Similarly, assuming that vBE=Q results in iB = (fflB+«fc)/.Rb, which establishes the
load-line intercept on the current axis. The load line is shown in Fig. 5.6a.
Equation (5.12) represents the constraints placed on the values of iB and vBE by the external
circuit. In addition, iB and vBE must fall on the device characteristic. The values that satisfy both
constraints are the values at the intersection of the load line and the device characteristic.
The slope of the load line is -1/Kb. Thus the load line shifts position but maintains a constant
direction as t% changes in value. For example, the lower load line in Fig. 5.6a is for a smaller value
of t>s than that for the upper load line.
The quiescent operating point or Q-point corresponds to %(/)=0. Thus as the ac input signal t%(/)
changes in value with time, the instantaneous operating point swings above and below the Qpoint value. Values of iB can be found from the intersection of the load line with the input
characteristic for each value of t%.
-131-
After the input circuit has been analyzed to find values of iB, a load-line analysis of the output
circuit is possible. Referring to Fig. 5.5, we can write a voltage equation for the collector loop,
through Vcc, R& and the transistor from collector to emitter. Thus we have
YcC = Rcic+vCE
(513)
This is plotted on the output characteristic of the transistor in Fig. 5.6b.
Now, with the values of iB that we have already found by analysis of the input circuit, we can
locate the intersection of the corresponding output curve with the load line to find values of ic
and «>££. Thus as t% swings through a range of values, iB changes, and the instantaneous operating
point swings up and down the load line on the output characteristic. Usually, the ac component
of V& is much larger than the input voltage; hence the amplification takes place.
Examination of Fig. 5.6a shows that as v&{t) swings positive, the value of iB increases (i.e. the
intersection of the load line with the input characteristic moves upward). This in turn causes the
instantaneous operating point to move upward on the output load line, and v^ decreases in
value. Thus a swing in the positive direction for ik results in a (much larger) swing in the negative
direction for v^. Therefore, as well as being amplified, the signal is inverted. In other words, the
common-emitter amplifier is an inverting amplifier.
>«T
:
r
:.—.,—-^
Figure 5.7 Output of the amplifier of Exercise 5.1 for ?s(/)=1.2sin(2000iu)
demonstrating gross distortion
Exercise 5.1 Assume that the circuit of Fig. 5.5 has Vcc = 10 V, VBB = 1.6 V, Rb= 40l£i and
R,. = 2 ls£l. The input signal is a 0.4-V peak 1-kHz sinusoid given by t>s(f) = 0.4sin(200(br/). Use
PSpice to plot the device-characteristics assuming fip= 100 and Js = le-14 A. Using the load-line
technique find the maximum, minimum and Q-point values for vCE. Use PSpice to plot the
waveforms of vjt) and v^t). Ans. pCanki s 3 V, V^ = 5 V, VCmaai s 7 V.
It is not apparent in the waveforms that you have plotted in Exercise 5.1, but the output signal is
not a precise sine wave like the input The amplifier is slightly nonlinear because of the curvature
of the characteristics of the transistor. Therefore, as well as being amplified and inverted, the
signal is distorted. Of course, distortion is not usually desirable. Fig. 5.7 shows the output of the
amplifier of Fig. 5.5 and Exercise 5.1 if the input signal is increased in amplitude to 1.2 V peak.
The distortion is obvious.
Notice that the positive peak of v^ has been clipped at KCC=10V. This occurs when iB and ic
have been reduced to zero by the negative peaks of the input signal, and the instantaneous
-132-
operating point moves down to the voltage-axis intercept of the output load line. When this
happens, we say that the transistor has been driven into cut-off.
The negative-going peak of the output waveform in Fig. 5.7 is dipped at v^ = 0.2 V. This occurs
because iB becomes large enough so that operation is driven into the region at the upper end of
the output load line, where the characteristic curves are crowded-together. We call this the
saturation region.
Reasonably linear (undistorted) amplification occurs only if the signal swing remains in the active
region between saturation and cutoff on the load line. An output load line is shown in Fig. 5.8,
including labels for the cutoff, saturation and active regions.
J IC(Q1)
W2
Figure 5.8 Amplification occurs in the active region. Clipping occurs when the instantaneous operating
point enters saturation or cutoff. In saturation, VCE = 0.2V.
Exercise 5.2 Repeat Exercise 5.1 if Vs{t) = 0.8sin(2(XXbt^. Ans. Pamm = 1 V, V^ = 5 V,
Vcsrv^S.SV.
Exercise 5.3 Repeat Exercise 5.1 if tk(t) = 0.8sin(2000it/) and VBB = 1.2 V. Ans. ^cûn S 3 V , V^
s 7 V, Pep* s 9.8 V.
5.2 The pnp Bipolar Junction Transistor
So far we have considered the npn BJT, but an equally useful device is the pnp bipolar junction
transistor, in which the base is a layer of »-type material between />-type emitter and collector
layers. For proper operation of an amplifier, the polarities of the dc voltages applied to the pnp
device must be opposite to those of the npn device. Furthermore, current flows in the opposite
direction. Aside from the differences in voltage polarity and current direction, the two types of
devices are nearly identical
A diagram of the structure of zpnp BJT and its circuit symbol are shown in Fig. 5.9. Notice that
the arrow on the emitter of the pnp transistor points into the device, which is the normal
direction of the emitter current For the pnp transistor we can write the following equations,
which are exactly the same as for the npn transistor.
'C = ctiE
(5.14)
iB=(\-a)iE
(5.15)
ic=fiB
(5.16)
-133-
and
i£=iC+iB
(517)
Equations (5.14) through (5.16) are valid only if the base-emitter junction is forward biased (vBE
negative for i-pnp) and the base-collector junction is reverse biased (y^ positive for a. pup). As for
the npn transistor, typical values are a £ 0.99 and fi= 100.
and
These equations are identical to Equations (5.1) and (5.8) for the npn transistor except that - v#£
has been substituted for vBE (because vBE takes negative values for ihepnp device). As for the npn
device, typical values for 1^ range from 10"12 to 10",6A, and at 300K we have V-f226tiN.
The common-emitter characteristics ofapnp transistor are exacuy the same as for the npn except
that the values on the voltage axes are negative. A typical set of characteristics is shown in Fig.
5.10.
134
z.«
i
1
-12M
! ItVB)
(b)
Figure 5.10 Common-emitter characteristics for zpttp BJT: (a) input, (b) output
Exercise 5.4 Find the values of alpha and beta for the transistor having the characteristics shown
in Fig. 5.10. Ans. a = 0.98, fi - 50.
Exercise 5.5 Use load-line analysis to find the minimum, maximum and Q-point values of iB and
VCE for the amplifier circuit shown in Fig. 5.11. Use the characteristics shown in Fig. 5.10. Does
this pnp BJT common-emitter amplifier invert the signal? Ans. igg^ = 5 |xA, IB = 25 |iA, tg^x =
48 uA, VOa^ £ -8.3 V, K C E S -5.3 V, Pcimax s -1.8 V.
ic
Rc=3kCl
5.3 Secondary Effects
Collector breakdown. The description we have given so far is only a first-order model of the
BJT. Real transistors exhibit many secondary effects that can be important in circuit design. For
example, the common-emitter output characteristics of a real transistor are shown in Fig. 5.12.
Notice that the collector current increases very rapidly as the collector-emitter voltage v^
approaches 30V in this case. This is due to the reverse-bias breakdown of the collector-base
junction. Usually, we try to avoid having BJTs enter the collector-breakdown region because high
currents and voltages can result in high power dissipation that leads to overheating and
destruction of the device. Collector breakdown voltages range up to several hundred volts,
depending on the device type.
Base width modulation. Another difference between the first-order BJT model and real
transistors is that even before collector breakdown is reached, the collector current increases with
collector-to-emitter voltage. For example, notice the positive slope of the curves in the active
region of Fig. 5.12a. The slope is more pronounced at higher currents. This effect is attributed to
die base width modulation effect known from the physics of the BJT operation.
-135-
The shading of Fig. 5.12c and 5.12d suggests how the two depletion regions extend into the/>type base region during forward active operation. The base-emitter depletion region is narrow,
the reverse-biased base-collector junction has a wider depletion region. For increased v^, the
depletion region of the base-collector junction extends still further into the base, which reduces
the width of the base region. Thus carriers injected from the emitter spend shorter time for die
transit through the base and therefore a lower number of them recombine in the base region.
This brings a closer to one. Since /? is a sensitive function of a, the separation of the commonemitter output characteristics increases noticeably with f^, as illustrated in Fig. 5.12a and 5.12b.
For a constant base current, the collector current increases with v^. This is known as the Early
effect. (The change in common-base output characteristics whose spacing is determined by a, is
hardly noticeable because the percent change in a is small.)
If straight lines extend the collector characteristics in the active region, they (approximately) meet
at a point on the negative v^ axis as shown in Fig. 5.12b. The magnitude of the voltage at the
intersection is called the Early voltage, denoted by VA.
4
«+— '
c mm
si>
m
(c)
(d)
Figure 5.12 Common-emitter output characteristic that shows collector breakdown (a);
extensions of die active-region collector characteristics intersect at -VA (b),
base width modulation (c) and (d)
Variation of fi with Q-point. In the first order model of the BJT, the collector characteristic
curves are uniformly distributed in the active region. Real transistors tend to have characteristics
that are crowded closer together at very low and very high currents.
-136-
At low values of Ic, recombination of charge carriers within the depletion region of base-emitter
junction becomes significant. These phenomena manifest themselves by the emission coefficient
n in the Shockley equation for base current having a value close to n = 2 which decreases to n = 1
for medium collector currents (e.g. for / ^ l m A in Fig.5.13a) whereas the emission coefficient for
the collector current stays constant, n — 1. Thus for low currents, the ratio of base current to
collector current is lower than for medium currents.
At high values of 7C, the unspoken assumption that the carriers injected into the base and emitter
are not disturbing the original concentration of carriers is no longer valid. These high-injectiondensity effects, sometimes referred to as "current crowding", cause reduction of p. For large
collector currents, e.g. /^lOmA in Fig. 5.13b, high current density in the base region causes a
significant increase of the minority carrier concentration in the base which, in turn, produces a
masking effect to the built-in nonuniform impurity concentration in -the base. In particular, the
built-in electric field in the base is reduced by the injected charge and carrier transport efficiency
is reduced. This reduces the value of the current gain.
Earlier we defined fi = IC/IB, but since the curves are not uniformly spaced, the value of f) is not
constant for all points in the active region of real transistors. The variation of fi with collector
current is illustrated in Fig. 5.13a for a typical BJT, showing the current gain decrease both at low
and high currents. However, since the curves are generally broad and flat at moderate values of
Ia we usually assume that ft is independent of operating point as a first approximation.
-137-
(c)
(d)
Figure 5.13 Typical variation of ^with collector current (a),
input characteristics of a real transistor (b);
output characteristic at temperature Tt (c), T? > T/ (d)
Parameter variation with temperature. Input and output characteristics of the BJT change
with temperature, as do the BJT parameters. The common-emitter input characteristic translates
to the left with increased temperature, in a similar way to the diode forward characteristic shown
in Fig. 3.10. The base current is described by Equation (5.8). As in the diode, Vr = kT/q in the
exponent dominates the temperature dependence and causes the voltage VBE to decrease by 2
mV for every degree Kelvin increase in temperature (at constant base current).
The output characteristics increase in separation and translate upward with increasing
temperature as shown in Fig. 5.13c and 5.13d. This reflects the increase of j3 with temperature.
Minority carrier lifetime increases with temperature, increasing the carrier transport efficiency
through the base region and bringing a closer to one. According to Equation (5.9), /? is a
sensitive function of a, the result is a large increase in /?, as temperature increases. An empirical
relationship that predicts variations in /?is
flCn =
fiCrJ^\
(5.20)
where T and TR are temperatures in degrees Kelvin and B is a constant called the temperature
exponent. For a class of silicon bipolar transistors one may select B - 1.7, which makes (5.20)
predict that J3 approximately doubles from 27°C to 175°C.
To explain the upward translation, we need to notice that equation (5.10) is an approximation.
Namely, for higher accuracy it should have two terms as follows
where I^g is a small dc saturation current that doubles for every 5K increase in temperature.
From the three temperature-sensitive parameters, VBE actually proves to be the most
troublesome in practical circuits. Even at elevated temperatures, J ^ is usually too small to play a
major role in silicon transistors. Also, it is relatively easy to design circuits that work well for any
high value of fi. Many IC designs use the VBE drops of matched transistors to cancel each other
over wide temperature ranges.
Variation of/? from unit to unit. If we test many units of a transistor of a given manufacturer's
type number, we generally "find that J3 displays considerable variation in the value from unit to
unit. Typically, the ratio of the highest and lowest values of beta is 3:1. Furthermore, the value of
beta-varies significantly with temperature for a given transistor. Therefore, m must design cinuits
that junction properlyfor transistors having a wide range ofbeta values.
-138-
Dependence of input characteristics on VCE. The input characteristics of a real transistor are
shown in Fig. 5.13b. Notice that the input charactetistics ate not a single curve, as in the firstorder model, but instead consist of a family of curves. The effect of base-width modulation with
the varying vK voltage is mainly responsible for this property. However, assuming that v^ is
larger than a few tenths of a volt (i.e. the BJT is biased in the active region), the input
characteristic curves are very close together.
Charge storage effects. The characteristic curves show only die static operation of the device.
For rapidly changing signals, charge storage effects occur, and the instantaneous operation of the
BJT is not adequately described by the characteristic curves. These effects are important in the
design of high-frequency amplifiers and high-speed logic circuits. We consider these aspects later
in this chapter.
Even though real BJTs display many secondary effects that can be important in the design of
critical circuits, the first-order model is sufficient for many designs. Even in critical circuits,
designs often begins with die simple model. In die next few sections, we show some useful
circuits man be analyzed and designed by use of first-order models.
5.4 Large-Signal dc BJT Models
In the analysis or design of BJT amplifier circuits, we often consider the dc operating point
separately from the analysis of the (small) signals. This was illustrated for diode circuits and FET
circuits in previous sections. Usually, we consider the dc operating point first. Then we turn our
attention to the signal to be amplified. In this section, we present models for large-signal dc
analysis of BJT circuits. Then, in die next section, we show how to use these models to design
and analyze bias circuits for BJT amplifiers. Later we consider small-signal models used to
analyze the circuit for the signal being amplified.
It is customary to use uppercase symbols with uppercase subscripts to represent large-signal dc
currents and voltages in transistor circuits. Thus Ic and V^ represent the dc collector current and
collector-to-emitter voltage, respectively. Similar notation is used for the other currents and
voltage.
As we have seen, BJTs can operate in the active region, in saturation, or in cutoff. In the active
region, the base-emitter junction is forward-biased and the base-collector junction is reversebiased. (Actually, the active region includes the forward bias of the collector junction by a few
tenths of a volt.)
C
C
n
_
-139-
Q
Figure 5.14 BJT large-signal models (Note: Values shown are appropriate for
typical small-signal silicon devices at a temperature of 300K.)
Active-region model. Circuit models for BJTs in the active region are shown in Fig. 5.14a. A
current-controlled current source models the dependence of the collector current on the base
current. The constraints given in the figure for IB and K ^ must be satisfied to ensure validity of
the active-region model.
VC£(V)
vae(V)
(b)
Figure 5.15 Regions of operation on the characteristics of an npn BJT:
(a) output, (b) input characteristic.
0.2V
0.5V
(*)
Let us relate the active-region model to the device characteristics. Fig. 5.15 shows the
characteristic curves of an npn transistor. The base current IB is positive and pBH=0.7V for
forward bias of the base-to-emitter junction as shown in Fig. 5.15b. Also notice in Fig. 5.15a that
VCE must be greater than about 0.2V to ensure that operation is in the active region (i.e. above
the knees of the characteristic curves).
-140-
Similarly, for the pnp BJT we must have JB>0 and VCE<-0.2V for validity of the active region
model. (We have assumed that IB is referenced positive out of the base for the pnp BJT.)
Saturation-region model. The BJT models for the saturation region are shown in Fig. 5.14b. In
this region, both junctions are forward biased. Examination of die collector characteristics in Fig.
5.15a shows that VQ^Q.ZV for an npn transistor in saturation. Thus the model for the saturation
region includes a 0.2-V source between collector and emitter. As in die active region, IB is
positive. Also, we see in Fig. 5.15a that for operation below the knee of the collector
characteristic, the constraint is pHB>I(>0.
Cut-off region. In cutoff, both regions are reverse biased, and no current flows in the device.
Thus the model consists of open circuits among all three terminals as shown in Fig. 5.14c.
(Actually, if small forward-bias voltages up to 0.5V are applied, the .currents are non-zero but
often negligible, and we still use the cutoff model.) The constraints on the voltages for the BJT to
be in the cutoff region are shown in the figure.
Inverted mode. When the collector-base junction is reverse-biased and the base-emitter junction
is forward biased, we say that the transistor is operating in the forward or normal mode.
Sometimes, we encounter situations for which the base-collector junction is forward biased and
the base-emitter junction is reverse-biased. This is the opposite of the normal situation, and we
say that the transistor is operating in the inverted mode. Operation in the inverted mode is the
same as in the normal mode, but with the collector and emitter interchanged. Most devices are
not symmetrical, so alpha and beta take different, much lower, values for the inverted mode that
for the normal mode. For now, we concentrate our attention on operation in saturation, cutoff or
the normal active mode regions.
Exercise 5.6 A given npn transistor has /£=100. Determine the region of operation if (a)
7B=50uA and J ^ m A , (b) I B =50uA and VCE=5Vi (c) VBE=-2V and VCE=-1 V.
Exercise 5.7 A certain/>/#> transistor has /?=100. Determine the region of operation if (a) VBE—0.2V and K CT =5V, (b) 7B=50uA and Ic=2mA, (c) F C E =5V and 7B=50uA.
5.5 Large-Signal dc Analysis of BJT Circuits
In this section, we will use the large-signal BJT models presented in Section 5.4 to analyze
circuits.
(a)
(b)
(c)
(d)
Figure 5.16 Bias circuits for Examples 5.1 and 5.2: actual circuit (a),
equivalent circuits assuming operation in cutoff (b),
in saturation (c), in the active region (d)
-141-
In the dc analysis of BJT circuits, we first assume that the operation of the transistor is in a
particular region (i.e. active, cutoff, or saturation). Then we use the appropriate model for the
device and solve the circuit Next, we check to see if the solution satisfies the constraints for the
region assumed. If so, the analysis is complete. If not, we assume operation in a different region
and repeat until a valid solution is found. (This is very similar to the analysis of diode circuits
using ideal-diode model or a piecewise-linear model.)
This approach is particularly useful in the analysis and design of bias circuits for BJT amplifiers.
The objective of the bias circuit is to place the operating point in the active region so that signal
can be amplified. Because transistors show considerable variation of parameters, such as (3, from
unit to unit and with temperature, it is important for the bias point to be independent of these
variations.
The next several examples illustrate the technique and provide some observations that are useful
in bias-circuit design.
Example 5.1 The dc bias circuit shown in Fig. 5.16a has R B =200kQ, Rc=lkQ, and K CC =15V.
The transistor has /£=100. Solve for I c a n d VCESolution We will eventually see that the transistor is in the active region, but we start by
assuming that the transistor is cut off (to illustrate how to test the initial guess of operating
region). Since we assume operation in cutoff, the model for the transistor is shown in Fig. 5.14c,
and the equivalent circuit is shown in Fig. 5.16b. We reason that 7fl=0 and there is no voltage
drop across RB. Hence we conclude that V B£ =15V. However, in cutoff, we must have Kfl£<0.5V
for an npn transistor. Therefore, we conclude that the cutoff assumption is invalid. Next, let us
assume that the transistor is in saturation. The transistor model is shown in Fig. 5.14b. Then the
equivalent circuit is shown in Fig. 5.16c. Solving, we find that
/c=(^cr0.2)/Rc=H.8mA
and
/ B =(âr0.7)/R, = 71.5uA
Checking the conditions for saturation, we find that / B >0 is met, but /3IB>IC is not met.
Therefore, we conclude that the transistor is not in saturation.
Finally, if we assume that the transistor operates in the active region, we use the BJT model of
Fig. 5.14a, and the equivalent circuit is shown in Fig. 5.16d. Solving, we find that
4=(* / ar0.7)/R B =71.5uA
where we have assumed a forward bias of 0.7V for the base-emitter junction. (Some authors
assume 0.6V for low-power silicon devices. Others assume 0.7V.) In reality, the value depends on
the particular transistor and the current level. Usually, the difference is not significant.) Now we
have
lc=Ph = 7.15nA
Finally, VCE=VCC-R<IC = 7.85V
The requirements for the active region are K CE >0.2V and 7B>0, which are met. Thus the
transistor operates in the active region.
Exercise 5.8 Repeat Example 5.1 with /£=300. Ans. 7 c =14.8mA, K CE =0.2V (saturation).
-142-
^15V
vCT(V)
(b) P = 300
Figure 5.17 Load lines for Example 5.1 and Exercise 5.8
It is instructive to consider the load-line constructions shown in Fig. 5.17 for the last two
examples. For /£=100, the operating point is approximately in the center of the load line. On the
other hand, for /?=300, the operating point has moved up into saturation.
To use this circuit as an amplifier, we would want a Q-point in the active region where changes <-i
the base current cause the instantaneous operating point to move up and down the load line. In
saturation, the operating point does not move significantly for small changes in the base current,
and amplification is not achieved. Thus a suitable Q-point is obtained for /?=100 but not for
/?=300. Since we often find unit-to-unit variations in fi of this magnitude, this circuit is not
suitable as an amplifier bias circuit for mass production. (We could consider adjusting RB to
compensate for unit-to-unit variations in /?, but tibis is usually not economical.)
Sometimes the discussed circuit of Fig. 5.16a is called a fixed-base bias circuit because the base
current is fixed by Vcc and RB and does not adjust for changes in p. (Notice that if we want a
circuit that achieves a particular operating point on the collector load line, the base current must
change when /3 changes.)
Exercise 5.9 Repeat Exercise 5.8 for (a) ,£=50, (b) ,0=250. Ans. (a) 7 c =3.575mA, V^-X 1.43V,
(b) I(f=U.ZiaA, Va,=0.2V
Exercise 5.10 Assume that R ^ k O , F BE =0.7V and VCC=20V in the circuit of Fig. 5.16a. Solve
for the value of RB needed to place the operating point exacdy in the middle of the output load
line-for (a) P=100, (b) P=300. Ans. (a) R B =965kQ, (b) RB=2.9MQ.
-143-
Analog Electronics/BJT
Circuits
Exercise 5.11 Solve the circuit shown in Fig. 5.18 to find Ic and V^ if (a) >0^5O, (b) /£=150.
Ans. (a) 7c=:0.965mA, ^ = - 1 0 . 3 5 ^ (b) 7c=1.98mA, Va=-0JZV (transistor in saturation).
lOkfl
Figure 5.18 Circuit for Exercise 5.11
In the next example we consider a circuit that achieves an emitter current that is relatively
independent of /3.
W
(b)
Figure 5.19 Circuit of Example 5.2 (a) and its equivalent circuit (b) assuming
operation in the active region.
Example 5.2 Solve for 7C and V^ in the circuit of Fig. 5.19a if VC(~=\5V, VBB=5V, Rc=2kQ,
and yS^lOO. Repeat for ^=300.
Solution We assume that the transistor is in the active region and use the equivalent circuit
shown in Fig. 5.19b. Wr»«ing a voltage equation through VBB, the base-emitter junction and RH)
we have
F f l B =0.7 + 7ERH
This can be solved for the emitter current
h= {VBB-0.1)/RE=
2.15mA
Notice that the emitter current does not depend on the value of fi.
Next we can compute the base and collector current using Equations (5.10) and (5.2)
IE = IB+/3IB = (JS+1)IB
Solving for the base current, we obtain
7 B =/ E /09+l)
Substituting values, we obtain the results given in Table 5.1. Notice that 7B is lower for the higher
P transistor, and Ic is nearly constant.
- 144-
TABLE 5.1 RESULTS FOR EXAMPLE 5.2
1
I
1
p
JJOÂ)
100
300
21.3
7.14
J^mA)
2.13
2.14
^CE(V)
6.44
6.42
Now we can write a voltage equation around the collector loop tofind*VCE
Vc^RcIc+Vcz+RzIz
Substituting values found previously, we find that 1^=6.44V for /?=100 and
K CE =6.42Vfor^=300.
The Q-point for the circuit of Fig. 5.19a is almost independent of p. However, die circuit is not
usually practical for use in amplifier circuits. First, it requires two voltage sources, VBB and Vco
but often one source is readily available. Second, we may want to inject the signal into the base
(through a coupling capacitor), but the base voltage is fixed with respect to ground by the VBB
source. Because the VBB source is constant, it acts as a short-circuit to ground for ac signal
currents (i.e. the VBB source does not allow an ac voltage to appear at the base.)
5.6 Four-Resistor Bias Circuit
A circuit that avoids the objections discussed at the end of the previous section is shown in Fig.
5.20a. We call this the four-resistor BJT bias circuit. The resistors R, and R2 for a voltage
divider that is intended to provide a nearly constant voltage at the base of the transistor
(independent of transistor p). As we saw in Example 5.2, constant base voltage results in nearly
constant values of Ic and K^. Because the base is not directly connected to the supply or ground
in the four-resistor bias circuit, it is possible to couple an ac signal to the base through a coupling
capacitor.
The circuit can be analyzed as follows. First, the circuit is redrawn as shown in Fig. 5.20b. Two
separate voltage supplies are shown as an aid in the analysis to follow, but otherwise the circuits
in part (a) and (b) of the figure are identical Next, we find the Thevenin equivalent for the circuit
to die le t of the dashed line in Fig. 5.20b. The Thevenin resistance Rfl is the parallel combination
of R, and R2 given by
RIM
R
< 5 - 2,)
^1^T2
The Thevenin voltage is
R2
Vn
'B = V,
CC
(5.22)
The circuit with the Thevenin replacement is shown in Fig. 5.20c. Finally, its active-region largesignal model as shown in Fig. 5.20d replaces die transistor.
Now we can write a voltage equation around the base loop of Fig. 5.20d, resulting in
VB = RBIB + VBE + REIE
(5.23)
Of course, for low-power silicon transistors at room temperature, we have K B£ s0.7V. Now we
can substitute
/£«=(l+0)/a
and solve to find that
-145-
VB-VBE
(524)
''-Ri+Q+fiR,
Once IB is known, IE and Ic can be easily found. Then we can write a voltage equation around the
collector loop of Fig. 5.20d and solve for V^. This yields
VCE = VCC ~ Wc
~ &Eh
(5-25)
(b)
«
r
cc
07V
'|/ £ =(P+i)4
(c)
(d)
Figure 5.20 Four-resistor bias circuit (a), equivalent circuit show lg separate voltage sources for base and
collector circuits (b), circuit using Thevenin's equivalent for Vcc, R/ and R2 (c),
equivalent to part (c) with active-region transistor model (d).
Exercise 5.12 Find the values of Ic and V^ in the circuit of Fig. 5.20a for (a) /fc=100 and (b)
y9=300. Assume that VgÔ.TV and that the circuit elements take on the following values of
R,=10kQ, R,=5kQ, R ^ l k Q , R E =lkfi, F CC =15V. Ans. (a) J c =4.12mA, VCE=6.12V, (b)
7c=4.24mA, VCE=6.5\V.
Exercise 5.13 Repeat Exercise 5.12 for R,=100kQ and R 2 =50kQ. Compute the ratio of Ic for
/?=300 to Ic for /fc=100 and compare to the ratio of the currents found in Exercise 5.12. Ans.
The ratio of the collector currents is 1.21. On the other hand, in the previous exercise, the ratio
of the collector currents is only 1.029. Larger values of Rl and R2 lead t o larger changes in Ic
with changes in f3.
We "often use the four-resistor circuit of Fig. 5.20a for biasing BJTs in discrete-component
amplifiers. Now we consider the design of this type of bias circuit The principal problem of bias
circuit design is to achieve nearly identical operating pointsfor the BJTs even though P may vary by afactor of 3
-146-
Analog Electronics / BJT Circuits
more units. Furthermore, some circuits are required to function over a wide range of temperature,
which can cause significant variations in fi and VBE.
Notice that Ic and V^ are nearly independent of fi in the circuit of Fig. 5.20a and Exercise 5.12.
This is achieved by selecting values for R, and R2 that provide a nearly constant voltage to the
base. As the values of R, and R2 become larger, the Q-point exhibits larger changes with /?.
Comparing the results of Exercise 5.13 with those of Exercise 5.12 can see .this.
For the voltage divider to provide a nearly constant base voltage for different values of base
current, the resistors R, and R2 should be small in value. However, this leads to large currents,
possible overheating, and the need for a larger, more expensive power supply. Moreover, the ac
impedance seen at the base decreases with the decrease of the values of the resistors R, and R2,
which is undesirable in some applications. Thus we also wish to make R, and R2 as large in value
as possible. As a general rule, a good compromise is to choose R2 so that the current through it is 10 to 20 times
the largest base current expected.
Equation (5.24) shows that the base current is proportional to the difference between VB and
VBE. Recall that VBE decreases in value by about 2mV/K as temperature increases. Furthermore,
resistor tolerances cause VB to vary. If we design so that the difference between KB and VBE is
very small, these variations could result in troublesome changes in the jg-point. Therefore, we should
design so that VB is much larger than the changes expected in VBE due to temperature variation and the changes
in VB due to resistor tolerances.
Often, we choose VB to be one-third of the supply voltage, which is usually large enough to
ensure a sufficiently stable jg-point. Usually, VB is much larger than VBE, so that the drop across
RE is approximately equal to VB. A rule in common use is to design so that one-third of the
supply voltage is dropped across R^ one-third across the transistor (P^g) and one-third across
RE.
Consideration of the frequency response, peak signal swing, available component values and
various other matters place constraints on the j2-point and the selection of resistor values to be
used in the bias circxiit. Thus the design of the bias circuit is intertwined with the ac performance
of an i iplifier. We consider more aspects of bias design later in conjunction with amplifier
performance.
Figure 5.21 Circuit for Example 5.3
Example 5.3 Suppose that considerations of the frequency response of an amplifier have
dictated that the j2-point of the transistor should be at Ic = 2mA and that R^ = 4.7kQ. The
supply voltage is Vcc = 25V. The transistor to be used has /? ranging from 100 to 300. Design a
four-resistor bias circuit for this amplifier. Use standard 5%-tolerance resistor values.
-147-
Solution The diagram of the circuit to be designed is shown in Fig. 5.21. The drop across R^ is
Vc- IcRc= 9.4V. This leaves 25 - 9.4 = 15.6V to be allocated between V^ and the drop across
RE-
Suppose that we choose V^ = 10V leaving 5.6V across Rg. Then, since IE = Ia we have RE =
VE/IB = 2.8kQ. However, the closest nominal value is RE = 2.7kQ, so that is our choice. (See
Appendix A for nominal 5%-tolerance resistor values.)
Next, we compute the base voltage VB = VBE + VE = 0.7 + 5.6 = 6.3V. The base current is IB =
Zc/p. Since we want the maximum base current to be much less than I2 (so that VB does not
change excessively when IB changes due to variations in j$), we use the minimum p. Thus IBmax. —
J c /P = 20|oA. Then using the rule of thumb that I2 should be 10 times the maximum base
current, we have I2 = 0.2mA. Now, R2 = VB/I2 = (6.3V)/(0.2mA) = 31.5kQ, so we choose R2 =
33kQ, which is a standard value.
Next, we see that I, = IB + I2 = 0.22mA and V, = Vcc - VB = 18.7V. Finally, R, = V,/I, = 85kQ,
so we choose a close nominal value of R, = 91kQ. (We could just as well have chosen R, =
82k£l)
Thus our design calls for R^ = 2.7kn, R, = 91kfi, R2 = 33kQ and R^ = 4.7kQ. Of course, many
other choices could have been made in the design, resulting in different but equally useful values.
Usually, there are many rijjfrt answers to the design problems.
Exercise 5.14 Analyze the circuit designed in Example 5.3 to find the jg-point values of Ic and
Vcg that actually result with the nominal resistor values for (a) p=100; (b) P=300. Ans. (a) Ic =
2.00mA, VCE = 10.1V; (b) Ic = 2.13mA, V^ = 9.22V.
Exercise 5.15 In the four-resistor bias network, does Ic decrease, increase or stay the same for a
(small) increase in the value of (a) R^, (bJRg, (c) R„ (d) R^, (e) P?
Exercise 5.16 In the four-resistor bias network, does K ^ decrease, increase or stay the same for
a (small) increase in the value of (a) R& (tyRg, (c) R„ (d) R^, (e) P?
Exercise 5.17 Find the maximum and minimum values of Ic and V^ for the circuit designed in
Example 5.3. {Hint: Consider the combination of P and values of resistors within ±5% of the
nominal values.) Ans. Icmn. = 2.43mA, i ^ , = 1.77mA, VcEmax ~
1 2 . 1 ^ 1 ^ , = 6.76V.
Exercise 5.18 Suppose that Vcc = 20V, R^ = lkQ and a j2-point of Ic = 5mA is desired. The
transistor has P ranging from 50 to 150. Design a suitable bias circuit. Use standard 5%-tolerance
resistor values.
-148-
5.7 Small-Signal Equivalent Circuits
Now we turn our attention to small-signal currents and voltages in circuits containing BJTs. First
we establish the notation used in amplifier circuits. We denote total currents and voltages by
lowercase symbols with uppercase subscripts. Thus iB(t) is the total base current as a function of
time.
The Q-point currents and voltages are denoted by uppercase symbols with uppercase subscripts.
Thus IB is the dc base current if the input signal is set to zero.
Finally we denote the' changes in currents and voltages from the Q-point (due to the input signal
being amplified) by lowercase symbols with lowercase subscripts. Thus it(/) denotes the signal
component of the base current. Since the total base current is the sum of the Q-point value and
die signal component we can write
iB® = h + M
(5-26)
Similarly
• t a W ^ r e + Ô)
,(5-27)
The Q-point is established by the bias circuit as discussed in the previous section. Now we
consider how the small signal components are related to each other in the BJT. The total base
current is given in terms of the total base-to-emitter voltage by (5.8), repeated here for
convenience
iB = (1 -
(5.28)
ayigs e x p ( ^ ) - l
We are concerned with operation in the active region for which the 1 inside the bracket is
negligible and can be dropped. Substituting (5.26) and (5.27) into the modified (5.28) one obtains
h + ^ O = ( 1 " "VES e*P[VBE tVbeit))
VT
This can be written as
IB + ib(l) = (1 - aVus
,
=h
(5-29)
e x t f ^ e x p O ^ )
T
^ < \
(5
T
-M)
exp(-r—)
We are interested in small signals for which PBE is much smaller than VT at all times, | pfc | «
On the other hand, for small x, exp(xr) = 1 + x. Thus (5.30) can be written as
/ . + 4 » - / . + ^
f f l
VT.
(5.31)
If we cancel IB from both sides and define rx — Vr / IB, we have
W O - ^
(5-32)
Thus for small-signal variation around the Q-point, the base-emitter junction of the bipolar
transistor appears to be a resistance r* Using the relationship IB = I c /($, we have an alternative
formula for rn
pvT
r
7c=-r-
(5-33)
-149-
At room temperature, K^=26mV. A typical value of f$ is 100 and a typical bias current for a
small-signal amplifier is lc - 1mA. These values yield rn - 2600Q.
It is easy to show that the signal component of the collector current is given as
»C(0 = A,(0
'
(5.34)
Equations (5.32) and (5.34) relate the small-signal currents and voltages in a BJT. They lead to a
small-signal equivalent circuit of the BJT shown in Fig. 5.22a. This circuit is very useful in the
analysis of the BJT amplifier circuits. It turns out that the pnp transistor has exactly the same
equivalent circuit as the npn - even the reference directions for the signal currents and voltages
are the same. The resistance r„ is given by (5.33) for both transistor types.
An alternative small-signal equivalent circuit is shown in Fig. 5.22b. Instead of the currentcontrolled current source it uses a voltage-controlled current source to describe the signal
component of the collector current
»c(0 =
ft»vte(0
(5.35)
The coefficient g, is called the transconductance of the BJT, defined as
8m
d(
&>BE
(5.36)
VT
Q- point
Which one of the circuits shown in Fig. 5.22 is used for circuit analysis is the matter of
convenience.
Figure 5.22 Small-signal equivalent circuit for the BJT
5.8 The Common-Emitter Amplifier
In a BJT amplifier circuit, the power supply biases the transistor at an operating point in the
active region for which amplification can take place. For example, we can use the four-resistor
bias circuit discussed in Section 5.6. Coupling capacitors are used to connect the load and the
signal source without affecting the bias point.
We can analyze amplifier circuits to find gain, input resistance and output resistance by use of the
small-signal equivalent circuit. In this section, and the next we illustrate this procedure for two
important amplifier circuits.
Fig. 5.23a shows the circuit diagram of a common-emitter amplifier. The resistors R,, R2, RE
and R^ form the four-resistor biasing network. The capacitor C, couples the signal source to the
base of the transistor, and JC2 couples the amplified signal at the collector to the load RL. The
capacitor CE is called the bypass capacitor. It provides a low-impedance connection between
the emitter and ground, for ac.
The coupling and bypass capacitors are chosen large enough so that they have very low ac
impedance at the signal frequencies. For simplicity, in our initial small-signal ac analysis, we treat
-150-
the capacitors as short circuits. However, at sufficiendy low frequencies, the capacitors reduce
the gain of the amplifier, because their impedance increases in magnitude with decreasing signal
frequency.
Vc
VC
(a) Actual circuit
*' > B
(b) Small-signal ac equivalent circuit
'b=0 B
Source
"turned off
^MJ
(c) Equivalent circuit to find Z0
Figure 5.23 Common-emitter amplifier
Because the bypass capacitor grounds the emitter for ac signals, the emitter terminal is common
to the input source and to the load. This is the origin of the name common-emitter-amplifier.
The analysis we give here is valid for the midband region of frequency. In the low-frequency
region, the effects of the coupling and bypass capacitors must be considered. In the highfrequency region, a more complex transistor model must be used that includes the frequene
limitations of the transistor. We treat this high-frequency response later in this chapter.
Before we analyze the amplifier, it is very helpful to draw its small-signal equivalent circuit. This
is shown in Fig. 5.23b. The coupling capacitors have been replaced by short circuits and the
transistor has been replaced by its small-signal equivalent.
The dc power supply is replaced by a short circuit. This is appropriate because it has zero internal
resistance, so no ac voltage can appear on it.
Carefully compare the actual circuit of Fig. 5.23a with the small-signal ac equivalent shown in Fig.
5.23b. Notice that the signal source is connected directly to the base terminal because C, has
been treated as a short circuit. Similarly, the emitter is connected direcdy to the ground and the
load is connected to the collector.
-151 -
Notice that the top terminal of R, connects to the supply in the original circuit, but R, is
connected from base to ground in the equivalent circuit, because the power-supply voltage
source is treated as a short circuit to ground for ac signals. Notice also that R, ends up in parallel
with R^. Similarly, R^ and RL are in parallel. We find it convenient to define RB as the parallel
combination of R, and R2
Similarly, R^' is a parallel combination of B^ and RL
R'L = - ^ \ -
(5-38)
To find the voltage gain v0/v/ of the amplifier, first we note that the input voltage is equal to the
voltage across r„, given by
v, = v ^ = rKib
(5.39)
The output voltage is produced by the collector current flowing through R j /
vo=-RL0ib
(5.40)
The minus sign is necessary because of the reference directions for the current and voltage - the
current flows out the positive voltage reference. Dividing (5.40) by (5.39) gives the voltage gain
Rr
V
A = - ^ = -/*—
V-
r
v
'it
i
(5-41)
Notice that the gain is negative showing that the common-emitter amplifier is inverting. The gain
magnitude can be quite large - several hundred is not unusual.
The expression for gain given in (5.41) is the gain with the load connected. We found the opencircuit voltage gain useful to characterize amplifiers in Chapter 2. With RL replaced by an open
circuit, the voltage gain becomes
Avo=-/3^
(5.42)
Another important amplifier specification is the input impedance, which in this case can be
obtained by inspection of the equivalent circuit. The input impedance is the impedance seen
looking into the amplifier-input terminals. For the equivalent circuit of Fig. 5.23b, it is a parallel
combination of RB and rn.
Z,=^ =- ^ (5.43)
',
Rli+rx
In this case the input impedance is a pure resistance. Therefore we can find the impedance by
dividing the instantaneous voltage f, by the instantaneous current if Of course, if there were
capacitances or inductances in the equivalent circuit, it would be necessary to obtain the
impedance as the ratio of the phasor voltage and the phasor current.
The current gain Aj can be found by use of Equation (2.3). With some changes in notation, the
equation is
4-'-?- = 4%• li
(5.44)
R
L
The power gain G of the amplifier is the product of the current gain and the voltage gain
(assuming that the input and load impedance are pure resistive).
-152-
(5.45)
G = AjAv
The output impedance is the impedance seen looking back from the load terminals with the
source voltage vs set to zero. This situation is shown in Fig. 5.23c. With vs set to zero, there is no
driving source for the base circuit, so ib is zero. Therefore, the controlled source P/4 produces
zero current and appears as an open circuit. Thus the impedance seen from the output terminals
is simply Rc.
Z0 = Rc
(5.46)
Exercise 5.19 Find Av, Avo, Zn, A\, G, and Z 0 for the amplifier shown in Fig. 5.23a with Rs =
500Q, Ri=10kQ, R 2 =5kQ, R E =lkO, Rc=lkQ, R L =2kft, (3=100, VBE=0.TV, VCC = 15V. If
f,(/)=0.001sin(<at), find and sketch p,(t) versus time. Ans. Av=-106, ^4 vo =-158, Z m =53lQ, A>=28.1, G=2980, Z 0 =lkQ, fo=-54.6sin((0t) mV.
Exercise 5.20 Repeat Exercise 5.19 if P = 300. (Hint: do not forget that the Q-point changes
slighdy when beta changes.) Ans. Av=-\09, v4vo=-164, Z,= 1185Q, Ai=-64.5, G=7030, Z c =lkQ,
vo=-16Jsin((0t) mV.
5.9 The Emitter Follower
The circuit diagram of another type of BJT amplifier called an emitter follower is shown in Fig.
5.24a. The resistors Rf, R2 and RE form the biasing circuit. The collector resistance is not present
in diis circuit. Thus we have a four-resistor biasing circuit with R(- = 0. The input signal is applied
to this circuit through the coupling capacitor C,. The output signal is coupled from the emitter to
the load by the coupling capacitor C2.
Vc
VC
(b) Small-signal equivalent circuit
-153-
(c) Equivalent circuit used to find output impedance Z„.
Figure 5.24 Emitter follower
The ac equivalent circuit is shown in Fig. 5.24b. As before, we replace the capacitors and power
supply with short circuits. The transistor is replaced by its small-signal equivalent.
Notice that as a result, the collector terminal is connected directly to ground in the equivalent
circuit. The transistor equivalent circuit is oriented with the collector at the bottom in Fig. 5.24b,
but it is electrically the same as the transistor equivalent circuit we have used before. Because the
collector is connected direcdy to ground, this circuit is sometimes called a common-collector
amplifier.
The ability to-draw the small-signal equivalents for BJT circuits is an important skill for the
electronic-circuit designer. Carefully compare the small-signal equivalent in Fig. 5.24b to the ori
circuit. Better still; try to draw the small-signal equivalent circuit on your own starting from the
original circuit.
Notice that R, and R2 are in parallel in the equivalent circuit. We denote the combination by RB.
Also, RE and RL are in parallel and we denote the combination by RL'. In equation form we have
A1A?
(5.47)
RB
~RI+R,
and
Rr =
RrR
ERL
RE+RL
(5.48)
Next we find the voltage gain of the emitter follower. The current flowing through RL' is 4+Pv
Thus the output voltage is given by
v0=RL(\
+ {3)ib
(5.49)
Writing the voltage equation from the input terminals through r„ and then through the load to
ground, we have
Vi=Vb+RL(l
+ /B)tb
Division of (5.49) by (5.50) results in
(5.50)
A- =r + R {\ + P)
K
L
(5.51)
The voltage gain of the emitter follower is less than unity because the denominator of the
expression is larger than the numerator. However, the voltage gain is usually only slighdy less
dian unity. An amplifier widi voltage gain less than unity can sometimes be useful because it can
have a large current gain.
Also, notice that the voltage gain is positive. In odier words, the emitter follower is noninverting.
Thus if the input voltage changes, the output at the emitter changes by almost the same amount
-154-
and in the same direction as the input The output voltage follows the input voltage. This is the
reason for the name emitter follower.
The input impedance Zi can be found as the parallel combination of Rg and the input impedance
seen looking into the base of the transistor, which is indicated as Z« in Fig. 5.24b. Thus we can
write
_
RaZu
The input impedance looking into the base can be found by dividing both sides of (5.50) by ib.
Zit=T- = rx+BLL(\ + P)
(5.53)
l
b
The input impedance of the emitter follower is relatively high compared to other BJT amplifier
configurations. However, if very high input impedance is needed, we often have to resort to
more complex amplifiers using feedback. We consider this approach later. We have already seen
that field-effect transistors are capable of providing much higher input impedance than BJTs.
Once we have found the voltage gain and input impedance of the emitter follower, the current
gain and power gain can be determined by use of (2.3) and (2.5).
The output impedance of an amplifier is the Thevenin impedance seen from the output
terminals. To find the output impedance of the emitter follower, we remove the load resistance,
put the signal source to zero, and look back into the output terminals of the equivalent circuit.
This is shown in Fig. 5.24c. We have attached a test source vx that delivers a current ix to the
impedance we want to find. The output impedance is given by
Zo=T
(5-54)
l
x
(here again, the impedance can be expressed as the ratio of instantaneous time-varying quantities
because the circuit is purely resistive. Otherwise, we should use phasors.)
To find this ratio, we write equations involving vx and ix. For example, summing current at the
top of J? cr, we have
ib+flb+ix-jfe
(5-55)
We must eliminate ib from this equation before we can find the desired expression for the output
impedance. We do not want any circuit variables such as ih in the result - only transistor
parameters and resistor values. Thus we need to write another circuit equation.
First, we denote the parallel combination of R„ R, and R2 as
i
R*RR
Rs = „ '
„
(5.56)
The additional equation needed can now be obtained by applying Kirchhoff s voltage law to the
loop consisting of v^ rx and R/.
vx+r„ib+Rsib=0
(5.57)
If we solve Equation (5.57) for th substitute into (5.55) and rearrange the result, we obtain the
output impedance.
-155-
vx
Z
o~
f
1
~ \+p
*s+*K
(5-58)
1
R
E
This can be recognized as the parallel combination of R^ and the impedance
z
vx
(559>
Rs+ r-
«=tr^t
It can be shown that Zot is the impedance seen looking into the emitter of the transistor, as
indicated in Fig. 5.24c. The output impedance of the emitter follower tends to be smaller than
that of other BJT amplifier configurations.
Exercise 5.21 Compute the voltage gain, input impedance, current gain, power gain and output
impedance of the emitter-follower amplifier shown in Fig. 5.24a. Assume R, = lOkft, R, =
lOOkO, R2=100kQ, R E =2kQ, Ri=lkQ, B=200, KBE=0.7V, V C c=20V. Verify the results using
PSpice. Ans. ^4V=0.991, Zj=36.5kn, Z 0 =46.6Q, ^4=36.2, G=35.8
In general, the output impedance of the emitter follower is much lower and the input impedance
is much higher than those, of other single-stage BJT amplifiers. Thus we can use an emitter
follower if high input impedance and low output impedance is needed.
Notice that even though the emitter follower gain is less than unity, the current gain is large.
Thus die output power is larger than the input power and the circuit is effective as an amplifier.
If the emitter follower is cascaded with common-emitter stages, amplifiers with many useful
combinations of parameters are possible. Furthermore, there are several other useful amplifier
configurations using the BJT. Later we study additional circuit configurations and consider the
design of multistage amplifiers.
Exercise 5.22. Repeat Exercise 5.21 widi P=300. Compare the results. Verify the results using
PSpice. Ans.y4 v =0.991, Zi=40.1kQ, Zo=33.2Q, ^4=39.7, G=39.4.
5.10 Review of Small-Signal Equivalent-Circuit Analysis
Before we leave the important topic of small-signal equivalent-circuit analysis, we review die
technique and provide a few useful observations.
The first step in analysis is to draw die small-signal equivalent circuit by making the following
changes to the original circuit:
1) Replace the dc power-supply voltage sources by short circuits.
2) Sometimes we may encounter dc current sources. Replace these by open circuits. This is
appropriate because dc current sources force a constant current with no ac component to
flow.
3) If a midband analysis is desired, replace the coupling and bypass capacitors with short circuits.
However, if we want to find expressions for gain or impedance as a function of frequency or
perform a transient analysis, the capacitors should be included in the equivalent circuit (and
we should use phasors to represent currents and voltages in the ac analysis).
4) Sometimes we use inductors to provide a dc connection, but the inductance is picked large
enough so that it has very high impedance for the ac signal. (Usually, this technique is practical
-156-
only in circuits intended to operate at high frequencies.) Replace such inductors by open
circuits in the small-signal equivalent
5) Replace the transistor with its equivalent circuit If the circuit has several transistors, use
subscripts to distinguish the currents and parameters of different transistors.
It pays to be careful in drawing the equivalent circuit; analysis of an incorrect circuit is time and
effort wasted. Double-check your circuit before writing equations.
Once the small-signal equivalent circuit is finished, we turn our attention to finding expressions
for the gains and impedance of interest First, identify the pertinent currents and voltages and
label them on the equivalent circuit For example, in finding the voltage gain, the pertinent
variables are the input voltage vt and the output voltage tt>. On the other hand, for the input
impedance, we are concerned with vt and the current if
The output resistance is the Thevenin resistance of the amplifier. To find the output resistance,
we remove the load, turn off independent signal sources and look back into the output terminals
to find the resistance. Turning off the independent signal sources means replacing voltage
sources with short circuits and current sources with open circuits. Dependent sources, such as
me controlled current source of the transistor equivalent, are not turned off - the controlled
source models the effect of the transistor.
Often, it is convenient to attach a test voltage vx to the output terminals as we did in Fig. 5.24c to
find the output resistance of the emitter follower. Then the output resistance is the ratio of vx and
After drawing the small-signal equivalent circuit and identifying the pertinent current and voltage
variables, we use circuit analysis to write equations. Then we use substitutions to eliminate the
unwanted currents and voltages until we have an equation relating the two variables of interest
Make sure that the equations are not dependent. Otherwise, substitution results in cancellation of
all terms, so that you have 0=0 that indicates you must return to writing additional circuit
equations.
After we have found expressions for the gain or impedance, it is a good_idea to check to see that
me units of the expression are correct Voltage or current gain should be. unitless. Input or output
impedance should have units of ohm. In case the units do not check as expected, we should look
for errors in writing the original equations or for algebraic errors.
Small-signal equivalent circuit analysis is not as troublesome as it might seem from this
discussion. We have tried to mention all of the common problems encountered with this
technique so that you will not waste too much time if they come up. Many useful results can be
obtained with ease by the use of small-signal equivalent circuit analysis.
Possibly the thought process and viewpoints gained from the small-signal analysis technique are
just as important as the expressions that we derive using them. After all, if we only wanted the
formulas, we could resort to using a handbook. It is the understanding of the circuits obtained
that makes the technique so important
Example 5.4 A variation of the common-emitter amplifier is shown in Fig. 5.25a. Draw die
small-signal equivalent circuit and derive an expression for the voltage gain.
-157-
Figure J3.25 Variation of the common-emitter amplifier (a), and
its small-signal equivalent circuit (b)
Solution The small-signal equivalent circuit is shown in Fig. 5.25b. For convenience, we denote
the parallel combination of R^ and RL by RL'. To find voltage gain, we must write equations
involving t>, and v0. However, we find it necessary to involve 4, iy and L in the equations. We can
write
v,=W
(5-60)
because t>; is the voltage across rn. Summing currents at the collector node we have
if=j3ib+iy
(5.61)
The output voltage is
v0 = RLiy
(5.62)
Summing voltages around the outside of the circuit yields the fourth equation
(5.63)
Vi = RBif+v0
Equations (5.60) through (5.63) are the set we use to find the voltage gain. Before doing algebra,
we check to be sure that enough equations have been written. The variables ih i, and i must be
eliminated. Since a final equation relating t>( to v0 must result, a total of four equations is needed,
and that is exactly the number we have written.
Next, we proceed to eliminate the unwanted variables. First, we can solve Equation (5.60) for ib
and substitute into the other equations to obtain the set
If =
+ h,
' K
V0 = Rtfy
Vi = RBif+vo
The first equation in the last set can be used to substitute for ifi resulting in the equation set
v 0 = RLiy
vt=Ri
^
\r*
+
i 1+v
i-ly +V 0
-158-
Then we solve the first equation in this set for ip substitute into the second equation and use
algebra to form the ratio of v0 and t>,
v,
r„(RL+RB)
Next, we check to make it sure that our expression is unitless, as it should be for the voltage gain.
The check is satisfied. (Recall that P is unidess and rK has the units of resistance.)
Exercise 5.23 Derive the expressions for the input resistance and output resistance for the
circuit of Fig. 5.25a.
RcRpt
~r, RQ" *c + Rot
n+*B +(l + P)RL
RBRS + RBrn + Rsra
where Rot =
Ans. Ri =
RB+RL
r
(\ + P)Rs+r„
Exercise 5.24 The circuit shown in Fig. 5.26 is known as a common-base amplifier. Derive
expressions for the voltage gain, input resistance and the output resistance in terms of P, rx and
die resistor values.
Ans. Ay=p-}<-
where RL=
^
R, = R^/(I
+ J3),
R0=RC.
Figure 5.26 Common-base amplifier circuit
Exercise 5.25 Evaluate the expressions found in Exercise 5.24 if R»= 100ft, RE=5kft, R2=50kft,
Ri=100kft, Rc=5kQ, R L =lkft, Vcc=15V, KflE=0.7V and P=100. Find the output voltage if
v,=sin(fi)t)mV. Also evaluate the current and power gain. Ans. J c =0.799mA, r,r= 3254ft, A =25.6,
R,=32ft, Ro=5kQ, %(/)=6.21sin(a>t)mV,yl^0.819, G=21.
Exercise 5.26. Draw the small-signal equivalent circuits for the circuits shown in Fig. 5.27.
vc
vc
(a) Common-emitter amplifier widi unbypassed emitter resistor
-159-
5.11 The Common-Emitter Hybrid-Parameter Small-Signal Model
Another small-signal model for the BJT is shown in Figure 5.28. It is based on a set of two-port
circuit parameters, known as hybridparameters.
Figure 5.28 Common-emitter A-parameter small-signal equivalent circuit
This equivalent circuit is completely general for small-signal conditions. Given the proper values
of the four parameters, the ^-parameter model accounts for all the second-order effects in the
device. (This is, however, a linear model that does not account for nonlinear effects.) If the
parameters are allowed to be complex-valued functions of frequency, the model is valid for all
frequencies! However, the model parameters are related to the internal device physics in complex
ways, so their variation with frequency is not easy to understand. Consequendy, other models
that are more easily related to the device physics are usually used for high-frequency analysis as
will be seen in the following section.
In tdrms of die ^-parameters, the small-signal currents and voltages are related by
ne=hieib+Kevce
( 5 - 64 )
ic=hfeib+hoevce
( 5 - 65 )
-160-
Notice that hk has the units of resistance, bn and b^ are unitless, and h„ is a conductance.
Starting from Equations (5.64) and (5.65), we can express each parameter as a partial derivative,
evaluated at the operating point. For example, if we set ik — 0 in Equation (5.64) and solve for h^
we have
v
be
Avj£
(5.66)
AV
ce ib=0
CE
Thus we can find the value of h„ by making a small change in v with /fl held constant, and taking
die ratio of the resulting change in PBE and p^. In other words, h„ is the partial derivative of vBE
vidi respect to v^.
hro =
v
Expressions for the other three ^-parameters similar to Equation (5.66) can be found. These
expressions can be used to determine low-frequency values for the A-parameters for the static
characteristics. (This is similar to the procedure we used to determine low-frequency values for
the FET in the previous chapter.) It is important to understand that parameter values found from
the static characteristics of a device are valid only for low-frequency operation. The
characteristics do not account for capacitances.
-OE
Figure 5.29 The A-parameter equivalent circuit with h„ =0 and hK = 0.
The parameter hn accounts for the effect of base-width modulation on the input characteristic of
die device. Typically, its value is very small. Similarly, hx is a small conductance that accounts for
die upward slope of the output characteristics, which is also caused by base-width modulation.
As an approximation, we can set bn and hK to zero. (Since hK is a conductance, setting it to zero
causes it to become an open circuit.) With these changes, the ^-parameter equivalent circuit
reduces to the circuit shown in Figure 5.29. Except for different labeling of the parameters this is
the same as the equivalent circuit of Figure 5.22. Thus we have
K=*K
(5-67)
and
hfoSfi
(5.68)
We do not intend to make much use of the complete A-parameter circuit in the design or analysis.
However, data sheets sometimes contain information about the ^-parameters. We have included
this discussion primarily so that you will be familiar with thes> parameters when you encounter
them in the literature or on data sheets.
5.12 The Hybrid-* Model
A small-signal equivalent circuit for the BJT known as the hybrid-Tt is shown in Figure 5.30. This
model is motivated by the internal physics of the device. It includes charge storage effects and is
useful over a wide range of frequencies. The resistance rh called die base-spreading resistance,
accounts for the ohmic resistance of the base region. Typically, it is small compared to rx, ranging
from 10 to 100 Q for small-signal devices. Its vale is nearly independent of the operating point.
-161-
The resistance rK represents the dynamic resistance of the base-emitter junction as seen from the
"internal" base terminal. It is the same as the rK shown in Figure 5.22 and its value is given by
Equation (5.33). The resistance rM accounts for the effects of base-width modulation on the input
characteristic. In other words, rM represents feedback from the collector to the base. In this sense,
it plays virtually the same role that b„ plays in the /^-parameter equivalent circuit The following
approximate formula relates these parameters:
Ke=——
( 5 - 69 )
=—
The value of rM is very large - several megaohms is typical. For simplicity, we often replace it by
an open circuit At high frequencies, this is justified even further because rM is shunted by the
much lower impedance of Cf,.
OE
Figure 5.30 Hybrid-7i equivalent circuit
The resistance r, accounts approximately for the upward slope of the output characteristics of die
transistor. Thus it plays approximately the same role as hm of the ^parameter equivalent circuit.
We can write
r0=T"oe
Sometimes, to simplify analysis, we replace ra by an open circuit
(5.70)
The capacitance CM is the depletion capacitance of the base-collector junction. Its value depends
on the dc base-collector voltage V^ and the device type. Values are often given on data sheets as
Cobo or CVt. For example, the data sheet for the 2N2222A device lists Cbo value of 8 pF for V^
= -10V.
Sometimes the time constant of the RC circuit between the collector and base terminals is given
on the data sheet For example, the data sheet for the 2N2222A gives the value labeled as rbCf
This time constant is approximately equal to r/7^. Assuming that CM is known, we can use the
value given for the time constant to find rk.
The capacitance C„ accounts for the diffusion base capacitance and junction capacitance of the
base-emitter junction. The value of CK depends on the jg-point and the transistor type. Values
typically range from 10 to 1000 pF for small-signal devices.
Usually, data sheets do:not give values for CK directly. However, the transition frequency^ is
often given. The transition frequency is related to hybrid-7t parameters by the approximate
formula
-162-
The controlled source gj>x shown in Figure 5.30 accounts for the amplification properties of the
transistor. Using Equation (5.36) we can compute £, from knowledge of the j2-point (and
temperature). It is easy to demonstrate diat for low frequencies and rk = 0, rM = oo and r0 = <»,
the hybrid-it small signal model reduces to the equivalent circuit of Figure 5.22, where
gnPn=&b
Solvingfor^ taking account of the fact that rK = »K/ih we have
(5-72)
8m = J-
(5-73)
Example 5.5 Knowing from the data sheets for a typical 2N2222A transistor that bj, - P = 200,
b„ = 4X10"4, bj, = 200, and bK= 15 - 200 yS, determine values for the hybrid-Tt equivalent circuit
for a typical 2N2222A transistor at appoint of Ic = 10 mA and V^ = 10 V.
Solution First, we use Equation (5.36) to compute the transconductance
Ir
10mA A „ , , „
g„
= 0385S
Sm = — =
VT 26mV
We know mat bfi = p. Using Equation (5.32), we have
PVT 200x0.026 ,
r, = —~ =
= 520fi
*
Ic
0.01
Equation (5.69) yields
M
Ke 4 x l 0 ~ 4
The typical 2N2222A has rM somewhat higher than this. In any case, the value is so high that it
has litde effect on the performance of most circuits. Almost certainly, the unit-to-unit variation in
P will have a much larger effect. This is why we drop rM from the equivalent circuit to simplify
analysis.
We can use Equation (5.70) to find a value for r0. The data sheets give a range from 15 to 200
mS. Thus, r0 ranges from 5 to 66.7 kQ. We take
r o =20*O
as a typical value.
For Va - 10 V and IE = 0, the data sheet gives a matimum value of Cobo= CM of 8 pF. This is
the depletion capacitance of the collector junction and is nearly independent of emitter current
The capacitance does depend on Va, but the j2-point given in the problem statement yields
VCB =VCE -VBE =10K-0.6F = 9.4K
which is very close to the data-sheet specification. As is often the case, no information is given
concerning typical values of C+,. Thus we take the maximum value for use in the equivalent
circuit
CM=SpF
Next, we use the specification for the transition frequency fT to determine a value for C*
According to the data sheet, the minimum value offT is 300 MHz for V^ = 20 V and Ic = 20
mA. The transition frequency is a function of both V^ and Ia and the data sheet does not give
data for the bias point given in the problem statement However, examination of the data
published for other similar transistors indicate that^ should not vary by more than about 20%
-163-
due to the difference inj2-points. Therefore, we u s e ^ = 300 MHz in computing a value for C^.
Solving Equation (5.71) for CK and substituting values we find that
P
200
%pF = \96pF
2^x520x300x10'
Finally, the data sheet gives a maximum value for the collector-base time constant of 150 pF.
Thus we have
C
* ~ 2»Sr/r
-C„=-
/fcCy = 150xlO" 1 2
Solving for rb and substituting the value found for CM, we have
/i=190
Thus we have used the value published in the data sheet to find values for the parameters of the
hybrid-7t equivalent circuit The circuit and values are shown in Figure 5.31.
As you can see, determination of parameter values for a BJT model from the data sheet is not an
exact science. Many parameter, such as E5, show large unit-to-unit variation. Since we must design
circuits that work with all the devices of a given type, an exact model for a particular unit is not
important. Often, we use the worst-case device specification in finding a device model. If our
circuit design meets its goals with a range of device model parameters, including the worst case,
we can be reasonably sure that it can be mass-produced with an acceptable rejection rate.
The hybrid-7i model is useful for die analysis of single- and multistage amplifiers in a wide
frequency range. Slighdy modified, it is incorporated into the PSpice program to represent the
BJT properties for ac analysis.
5.13 Bipolar Transistor Behavior at High Frequencies
The hybrid-7l model can be used to determine the transistor behavior at high frequencies. To
derive the current gain p as a function of complex frequency s, a current source Ih(s) is connected
between the base and emitter, and a short circuit is placed between the collector and emitter
terminals, as shown in Figure 5.32. To simplify the analysis, the resistances ru and r, have been
replaced by open circuits.
r
B
b
B'
„cu
C
Figure 5 32 7 quivalent circuit used to derive die short-circuit current gain of die bipolar transistor as a
function of frequency
-164-
From the K.CL applied to the collector node, we have
lc(s) =
gmVA')-sCMV„{s)
i
\
(5.74)
For sufficiently low frequencies (still being in the high frequency range) the following inequality
can be fulfilled: &n»sCu, and accordingly Equation (5.74) can be simplified as follows
Ic(s) = gmVx(s)
(5.75)
On die other hand, from the KVL applied to the input loop one can obtain the relationship
between the base voltage and base current
Vx(s) =
*„
„
(5.76)
s
where &t=1/r& Substituting Equation (5.76) into (5.75) we obtain
ic(s)=—gm!i{s)
( 5 - 77 )
„ x
which allows us to derive expression for die short-circuit current gain
Multiplying numerator and denominator of Equation (5.78) by rK and making use of Equation
(5.73) we have
"W-1WC.+C,)
«"*>
where symbol /?, is used to denote low-frequency value of p. Putting s = j<D = j2rt/ and
introducing the definition of the beta cutoff frequency
1
(5.80)
we obtain a single-pole approximation to P as a function of frequency
/?(/)=
A
(5.81)
H
The magnitude of P is equal to
Po
Wi
i+
V
(5.82)
2
fpj
-165-
It follows from Equation (5.82) that the magnitude of the short-circuit common-emitter current
gain P decreases with frequency. This is illustrated by Figure 5.33. At f—fp the gain magnitude is
lower by v 2 (or 3 dB) compared to its low-frequency value p o . This gives an interpretation to
the beta cutoff frequency defined by Equation (5.80).
The parameter^ as described by Equation (5.71) is the common-emitter transition frequency
or unity-gain crossover frequency. In other words, fT is the frequency at which the magnitude
of P is equal to unity. Putting
A
|A/TO| =
1+
v
2
(5.83)
=1
\fpj
we obtain
#
2
2
V
V
= i+
fp)
(5.84)
\fp
\jpj
where the value oifi was assume much bigger than^g. Finally, we have
(5.85)
The value oifT can be found in transistor data sheets. For example, minimum value oifT for the
2N2222A transistor is specified as being equal to 300 MHz. One has to remember that in fact for
a real transistor | P(/^) | is usually larger than unity, due to the current flowing from the base to
collector terminals through capacitance CM, that was neglected under the assumption gm»sCfl.
The parameter^- is thus used to describe the P frequency dependence for frequencies lower than
fT, say foif<fT/\0.
At larger frequencies, the single-pole approximation (5.79) is no longer valid.
In these cases, another characteristic frequency^ is specified in transistor data books as the unity
current gain frequency. This is shown in Figure 5.33. However, we will not take it into
consideration when analyzing transistor amplifiers.
Example 5.5 Suppose the common-emitter amplifier in Fig. 5.23 employs a BC548C bipolar
transistor. Compute the effective voltage gain and input impedance of the amplifier at high
frequencies. Assume R, = 100 Q, R, = 68 kQ, R2 = 12 kQ, RE = 1.1 kQ, Re = 5.6 kft, RL = oo,
Vcc = 12 V. Verify the results using PSpice.
Solution. At high frequencies all the capacitances C„ Q , and CE may be replaced by respective
short circuits. To simplify the matter further we substitute
Rl=Rs\\Rl\\R2=Rs
(5-86)
RL = Rc || RL || r0
(5.87)
and draw the high-frequency equivalent circuit of the amplifier, as shown in Fig. 5.34.
Figure 5.34 Small-signal equivalent circuit of the common-emitter amplifier
at medium and high frequencies
-166-
It is easy to find in Fig. 5.23 that the dc base voltage is approximately equal to 1.8 V. Assuming
I/B£=0.7 V, we calculate the dc emitter voltage VE - 1 V and the emitter current IE £ 1mA. For a
typical, which is reasonably high, current gain f30 we may write
and
VCE=VCC-RCIC-REIE=5AV
From the transistor data book one can find at Ic= 1 mA and V^ = 5 V:
fT = 130 MHz, p0 = b(e = 600, rQ = 1/A* = 25 kQ. Similarly,
vCB=Vcc-Rck-Vcc-^nr=4-6WA] +K2
For this value of V^ one can find from the transistor data book:
CM=CC= 2.6 pF.
The transconductance can be calculated from Equation (5.36) to obtain gn = 38.5 mS. This can
be substituted to Equation (5.85) that can be solved for C„:
Q = 8m •CM s 4 4 . 5 p F .
2nfT
Also from Equation (533) we have r„ S 15.6 kQ. The parameter rh is estimated from the
collector-base time constant as rb= 25 £2.
To simplify the analysis, it is reasonable to apply the Miller theorem to the circuit of Fig. 5.34.
For this purpose we note that A - Vol V„ S -#nR'L- From Equation (5.87), R'L = 4.6 kfi, then A
s -177. Consequendy, Cm\ = 2.6x(l+177) s 462.8 pF and Cm2 = 2.6x178/177 = 2.6 pF. Now we
can draw the simplified equivalent circuit for the common-emitter amplifier, where
Ci = C^ + Cm] = 507.3 pF, C 2 = Cm2 = 2.6 pF. This circuit is shown in Fig. 5.35.
Figure 5.35 Simplified equivalent circuit of the common-emitter amplifier
The schematic diagram of Fig. 5.35 can be further simplified by using Thevenin's theorem to
modify the input circuit, where
(5.88)
r„ + Rs+rb
and is equal to 124 Q in the present example, and
r
n
(5.89)
v's=vs Rs+rb+r*
The resulting final equivalent circuit is shown in Fig. 5.36.
-167-
For the circuit of Fig. 5.36 we have
V -
Vs 1
1 + JaX'sCi
-y
(590)
r
n
*
* * j + ' & + ' * 1 + 7'®/®!
where ©j = l/(^Cj) s 15.9 Mrd/s, and
yo=-gmV^L7-^~r-
(5.91)
wherefi>2= 1/(^1^2) = 83.6 Mrd/s. Combining Equations (5.90) and (5.91) we may write
down the effective voltage gain of the common-emitter amplifier as:
A =YjL=
-Po*l>
1
!__
(5.92)
s
Vs Rs +1% + rx 1 + joaltox l + ja)/a>2
^
This is a transfer function of the type discussed in Section 2.6. Using Equation (2.60) and (2.61)
we have
2.12 MHz <fH < 2.53 MHz
(5.93)
Since a>,« Q)^ one can write an approximate expression
///=®1/(2^) = 1 / ( ^ C 1 )
(5.94)
Since the gain A in the Miller equations is proportional to the load resistance, the capacitance C,
•
<
increases with R^ . Then the upper cut-offfrequencyincreases with the Ri decrease. The largest
(
bandwidth is for Ri tending to zero:
On the other hand, it follows from Equation (5.88) that resistance Rs depends on the input
signal source resistance. It decreases with R, - • 0. Finally we note that the resistance rK = Ic/gn is
t
typically much larger than the base resistance r¥ We conclude that in the limit of Ri -» 0 and
Rs -> 0, the upper 3-dB frequency of the common-emitter amplifier is equal to the so-called
transverse cutoff frequency
fb
Jb
=
(5.96)
2nrb(C„+CM)
This frequency determines the largest bandwiddi of a common-emitter amplifier without
feedback. In the present example,_/J =135 MHz.
-168-
Figure 5.37 Amplitude characteristics of the common-emitter amplifier for different load resistances
With the load resistance decreasing, as the bandwidth increases, the midband gain decreases. This
follows from Equation (5.92) and is illustrated by Fig. 5.37, which is a result of PSpice simulation
performed for the circuit of Fig. 5.35. A summary of PSpice results is presented in Table 5.1.
TABLE 5.1 MIDBAND G A I N AND UPPER 3 - D B FREQUENCY
OF THE COMMON-EMITTER AMPLIFIER
Re
Avso
teak -.
kQ
v/v
fH
MHz
i.e»
. i«»
5.6
175
2.16
iee»
1
36.7
8.14
0.1
3.8
22.4
' " v -jJ*f
i«o> -
iee»i
o ttCttVlCg*)! » wtZZJ/MHiZ) a M « 2 V M B S 3 }
.:* frequency
Figure 5.38 Magnitude of common-emitter amplifier input impedance as a function of frequency
To find the input impedance dependence on frequency one can use the equivalent circuit in
Fig. 5.35, namely
Zie=rb+:
'x
(5.97)
\ + j6)/a)j
where 0)j =\/(r7rC\).
For low frequencies, the input impedance of the common-emitter
amplifier is equal to (r^ +rn)
that is 15.7 kQ in our example. Zie decreases with frequency, and
-169
since the break frequency depends on the Miller capacitance Cu it depends also on amplifier load
resistance. For high frequencies, the input impedance goes to rb. This property is sometimes used
to measure the base resistance of bipolar transistors. The PSpice-simulated dependence of the
magnitude of input impedance on frequency, for different load resistances is shown in Fig. 5.38.
Example 5.6 The common-base amplifier of Fig. 5.26 employs a BC548C bipolar transistor. Use
PSpice to compute the effective voltage gain of the amplifier at high frequencies. Assume R, =
100 Q, R, = 68 kQ, R2 = 12 kQ, RE = 1.1 kQ, Re = 5.6 kQ, RL = oo, Vcc = 12 V.
Solution. Since the^-point of the transistor is the same as calculated in Example 5.5, we can use
the above-obtained values of equivalent circuit elements: rn = 15.6 kQ, ^ = 38.5 mS, CK — 44.5
pF, CM - 2.6 pF, rQ = 25 kQ, rh = 25 Q. The equivalent circuit of this amplifier is shown in Fig.
5.39.
Figure 5.39 Small-signal equivalent circuit of the common-base amplifier
A PSpice simulation was performed for the circuit of Fig. 5.39. A summary of the simulation
results is presented in Table 5.2.
TABLE 5.2 MIDBAND G A I N AND UPPER 3 - D B FREQUENCY
OF THE COMMON-BASE AMPLIFIER
Re
Avso
fH
5.6
v/v 41.5
MHz 9.13
kQ
1
7.7
42.4
0.1
0.78
113.4
Comparing the contents of Table 5.1 and 5.2, one can notice that the midband gain of the
common-base amplifier is (for the same load resistance) lower than the gain of the commonemitter circuit. This is due to the fact that we compare the values of the effective voltage gain
Avs =V0IVS that takes into account the input signal attenuation in the circuit made of Rs and R,,
The input resistance R, for the common-base circuit is much smaller than that of the commonemitter circuit. Indeed, in the discussed example R, = 31 Q and thus only V* of the signal
amplitude excites the amplifier. On the other hand, the gain Av = V0 I Vi is the same for both
amplifiers. For R^ = 5.6 kQ it is equal to 176 in Examples 5.5 and 5.6. Notice that the commonbase is a noninverting amplifier, unlike the common-emitter circuit.
The bandwidth of the common-base amplifier is much larger than the bandwidth of the
common-emitter amplifier. This is also seen in' Tables 5.1 and 5.2. The properties of the
-170-
common-emitter and common-base circuit resemble those of the common-source and commongate amplifiers, respectively, discussed in Chapter 4. Main differences between them come from
the generally lower transconductance in FETs.
Example 5.7 The common-collector amplifier of Fig. 5.24 employs a BC548C bipolar transistor.
Use PSpice to compute the effective voltage gain and input impedance of the amplifier at high
frequencies. Assume R, = 100 Q, R, = 68 kO, R2 = 12 kQ, RE = 1.1 kfi, RL = oo, Vcc = 12 V.
Solution. The base-collector voltage VBC is larger in magnitude in this example as compared to
Examples 5.5 and 5.6. To account for this difference, we have CM = 2.1 pF. The other elements
take the previously determined values: r„ = 15.6 kQ, ^ = 38.5 mS, Cn — 44.5 pF, r0 = 25 kQ, rb =
25 Q. The equivalent circuit of this amplifier is shown in Fig. 5.40, where it was assumed that
Rs<<R, | | R2, for simplicity.
Figure 5.40 Small-signal equivalent circuit of the common-collector amplifier
The PSpice ac simulation was performed for the circuit of Fig. 5.40. The midband gain of this
amplifier is equal to 0.977. It is much closer to one than the gain of the common-drain amplifier
considered in Exercise 4.13 (0.917), because of the much larger transconductance of the bipolar
device (38.5 mS against 2.4 mS in the two examples discussed). The upper 3-dB frequency of the
common-drain amplifier is fH = 548 MHz. This is a very large value; however, it should be
mentioned that this value extends beyond the validity of the BJT hybrid-7C model. This model is
valid approximately t o / - / 1 0 that is about 13 MHz in the present example. To investigate the
high-frequency behavior of the common-drain amplifier, it is more appropriate to study the
frequency dependence of the input and output impedances.
The input impedance of the common-collector amplifier is capacirive, mainly due to the
capacitance CM present at the input (see Fig. 5.40). Then the magnitude \Zic\ decreases with
frequency. This is shown in Fig. 5.41. Very high at low frequencies, this impedance is reduced
twice in magnitude at 100 kHz in this example. (Larger values of the impedance break frequency,
in order of several MHz, can be obtained by operating the BJT for larger transconductance.) One
can conclude that the input impedance deteriorates much faster with increasing frequency than
the voltage gain does for this amplifier. In the limit, for /-•oo, the input impedance goes to
approximately rb, which is merely 25 Q in the discussed case.
-171-
Figure 5.41 Magnitude of the input impedance of the common-collector amplifier
as a function of frequency
One can demonstrate by using analytical derivations applied to the circuit of Fig. 5.40 that its
output impedance is inductive. Then, being very small at low frequencies, it increases in
magnitude with the frequency increases. This behavior is illustrated by Fig. 5.41. Th output
impedance has a small value of about 26 Q at low frequencies; however, its magnitude doubles at
about 50 MHz.
Exercise 5.27. Consider the amplifier shown in Figure 5.43. Both transistors are of the same,
2N2222A, type. Assume Rs = 1 k«, J^ = 1.2 kQ, Re = 1.2 kQ, ^ = 10 kQ. Find device Qpoints and transconductance. Draw the small-signal equivalent circuit valid for medium and high
frequencies. Derive formula for the midband effective gain of this amplifier. Run .ac PSpice
analysis to find the amplitude characteristic of the amplifier using the device model from
EVAL.LIB library. Determine me effective gain and bandwidth of the whole cascode as well as
gain and bandwidth of its Individual stages.
-172-
5.14 Large-Signal Dynamic Model for the Bipolar Transistor
In Section 5.4 we presented simple large-signal dc models for the BJT in the various operating
regions. These models are shown in Figure 5.14 and are appropriate for analysis of dc circuits. A
nonlinear large-signal model for the BJT, known as the Ebera-Moll model is shown in Figure
5.44. This model is suitable for computer simulation of the transistor static behavior in all the
regions of operation. The diodes DE and Dc model the base-emitter and base-collector junction,
respectively. In parallel with each diode is a current-controlled current source, and the controlling
current is the current in the other diode. This model is valid for low-frequency signals in all four
operating regions.
apiDE
BO
ORiDC
Figure 5.44 Ebers-Moll static model
We have used subscripts to distinguish aF and aR. Previously, we have been concerned mainly
with the forward active region, and we used the symbol a instead of a P In equation form
(5.98)
c
forward active region
In a similar fashion, we have
l
E
'C
(5.99)
reverse
active region
-173-
We have used the = sign in Equations (5.98) and (5.99) because these expressions neglect the
reverse leakage currents of DB and D0
Equation (5.9) gives (3 in terms of a. We can write similar equations forfipin terms of aFand for
/?R in terms of aR. Thus we have
PF=-^—
\-aF
and
(5.100)
( 5101 )
A-r£"
\-aR
Typical values for this parameters are a = aF = 0.99, 0 = fip - 100, aR = 0.5, fiR = 1. Let us
consider die physical basis for dieses parameter values, assuming an npn device. Transistors are
usually designed to be operated in die forward active region. Thus the doping of die emitter is
much heavier dian that of die base. Furthermore, the collector junction has a relatively large area
to collect the minority carriers (electrons) diffusing dirough die base. Moreover, due to
nonuniform base doping^ there is an electric field "built-in" in die base which accelerates
electrons toward die collector, reduces their transit time dirough me base and dius decreases die
electron loss due recombination. These features ensure that almost all of the carriers crossing die
emitter-base junction are swept into the collector. Thus in the forward active region, ic = iE, and
aF is close to unity.
In die reverse active region, die base-collector junction is forward biased and die base-emitter
junction is reverse-biased. The collector is not as heavily doped as die emitter is. Thus a
significant fraction of the current crossing die collector-base junction is due to holes crossing
from me base into die collector. Furthermore, the electrons mat are injected into die base region
take longer to diffuse to die emitter junction. The emitter junction is smaller and it takes longer
for die electrons to "find" the junction and be swept into the emitter. Consequently, more of die
electrons recombine in die base. This is why iE is typically only half of ic in die reverse active
region.
The currents in diodes Dc and DE of die Ebers-Moll model are given by the Schockley equation.
Thus we have
4tY
(5.102)
'DC = / c s e x p ^ ^ | - l
(5.103)
»D£=/£Sexp|^-)-l
and
The collector current and emitter current are given by
ic = aFiDE -ipc
and
'£ = -<*RiDC + *DE
Of course, the base current is given by
iB=iE-ic
It can be shown diat die following relation holds
ctflEs = aRIcs = / ,
-174-
(5.104)
( 5 1 °5)
(5.106)
(5.107)
where 7, is known as the device scale current,firstintroduced in Equation (5.5). Since aF = 1, we can
write
Is = IES
( 5 - 108 )
Therefore, these symbols are sometimes used interchangeably.
Figure 5.45 Dynamic BJT model
Base and Collector Resistance. All components in an integrated circuit are fabricated on a
single piece of semiconductor material called the substrate. A consequence is that external
connections to the base, emitter and collector are located at the top of the IC as shown in Fig.
5.1a. The transistor in thisfigureis a vertical transistor because current flows vertically through the
active region beneath the emitter material. Three parasitic resistances are thus important in the
planar bipolar transistor. Thefirst,the base spreading resistance rh is the ohmic resistance of the base
current path from the junction to the base contact at the surface. This parameter has been
accounted for in the hybrid-n model of the transistor. It is important in high-current transistors
and significantly influences amplifier frequency response. Next in importance is the ohmic
resistance of the collector, rc. Because ND in the collector is relatively low to reduce base width
modulation, the collector region has rather high resistivity. The result is a collector resistance of
the order 10 - 100 Q that may cause internal transistor saturation due to the ic rc voltage drop on
it A special highly conduction buried layer helps reduce rc. Of lesser importance than the other
two is re, the ohmic resistance in the highly doped emitter material. A typical value is 1 CI. Figure
5.45 shows a large-signal transistor model, which comprises the Ebers-Moll model with parasitic
resistances added.
Parasitic Capacitances. We know that diodes possess parasitic internal capacitances, nonlinear
Q versus V relationships that are transparent to slow signals but important when rapid changes in
voltage or current are imposed upon the device. Since the BJT contains two pn junctions, we
expect it to have similar dynamic limitations. Associated with each junction are depletion and
diffusion capacitances, which limit high frequency performance. In forward active region,
depletion capacitance is dominant at the reverse-bias collector-base junction. At the forwardbiased base-emitter junction, diffusion capacitance and depletion capacitances are both
important
-175-
The transistor depletion capacitances are exactly those we described in diode theory. Thus the
parameter values in Equation (3.40) and the capacitance characteristic within the transistor
depend upon the individual junction grading and geometry.
The diffusion capacitance for a transistor differs slightly from that of an isolated diode because of
the narrow base. For the forward active transistor, the minority charge concentration profile in
the base is triangular as in Figure 5.46. This is because base width W is much smaller than the
electron diffusion length, and because the electric field at the collector junction sweeps away
electrons as fast as they arrive, pinning down the concentration at the collector end of the base to
zero.
Emitter
0
Base
W
Collector
Figure 5.46 Excess minority charge stored in the base of a forward active npn transistor
For a base-emitter junction of cross-sectional area A, we use Figure 5.46 to calculate that the
quantity of minority charge stored in the base is
QFA=qA~n[Q)W
(5.109)
Diode theory tells us that the electron concentration at the emitter end of the base is controlled
by the base-emitter junction voltage vBE
where Ncb is the concentrations of acceptor ions in the base material. Substituting Equation
(5.110) into (5.109) gives the charge-voltage relationship that characterizes the diffusion
capacitance of the forward-active transistor
-fe)
qAW nf
(5.111)
2 Nab
yrT
This particular collection of stored charge represents electrons in transit from emitter to
collector. On the average, these injected electrons take TF seconds to traverse the base, where TF
is called the forward transit time. For integrated npn transistors, the transit time is of the order
of 1 ns, for integrated/>»p transistors TF = 30 ns. It can be shown that the forward transit time is
related to the transition-frequency by
QFA =
(5 1,2
*'-WT
- >
Similarly to the forward transit time, rR is the average time that a minority carrier spends in the
base region for operation in the reverse active region. It can be shown that
PFrF=PRrR
(5.113)
Figure 5.45 introduces a dynamic BJT model. We recognize it as the Ebers-Moll model of Fig. 5.44
augmented by nonlinear diffusion (subscript d) and depletion (subscript J) capacitances. Each
diffusion capacitance is a nonlinear Q versus V relation, which for the base-emitter diffusion
capacitance takes the form of Equation (5.111). In amplifiers, these internal capacitances reduce
-176-
the high-frequency gain. In digital circuits, the capacitances introduce switching delays that will
be considered in the following section.
SPICE Parameters for the BJT. Table 5.3 lists basic parameters required to model the static
and dynamic behavior of the transistor using PSpice. It includes second-order effects, such as the
Early effect and ohmic resistances. As shown in the table, the parameters fall into three groups.
First are the parameters that model the static characteristics of the device (IS, BF, BR, RB, RC,
RE, VAF). The second group of parameters models the depletion capacitances of the two
junctions (CJC, MJC, VJC, CJE, MJE, VJE). These parameters are similar to those discussed in
Section 3 for/w-junction diodes. The third group of parameters (TF, TR) models charge storage
in the base region.
TalDie 5.3 SPICE Model Parameters for the BJT*)
Text
SPICE
Parameter name
Typical
Default
notation
notation
value
value
IS
Scale current
1.0E-16A
1E-14A
I,
BF
Forward
beta
100
100
A
BR
Reverse beta
1
1
A
RB
Ohmic base resistance
10 Q
0
h
r
RC
Ohmic collector resistance
\Cl
0
c
Ohmic emitter resistance
0
RE
0.1 Q.
r,
100
V
VAF
Forward
Early
voltage
QO
vA
B-C
depletion
capacitance
10
pF
0
CJC
<=jc
(zero bias)
B-C junction grading factor
0.333
MJC
0.333
Mr
0.75 V
0.75 V
VJC
B-C built-in barrier potential
CJE
25 pF
0
B-E depletion capacitance
^
(zero bias)
MJE
B-E junction grading factor
0.333
0.333
Mm
0.75 V
B-E built-in barrier potential
0.75 V
VJE
0
TF
Forward transit time
500 ps _
*>
50 ns J
0
Reverse transit time
TR
„.
\ - ,._
*Typical values of the parameters are shown for a discrete general-purpose device
v..
v*.
5.15 Switching Behavior of the Bipolar Junction Transistor
Many transistor applications such as digital logic gates, interfacing circuits, power supplies and
communication circuits use the BJT as a switch operated by a control signal. In such applications
the transistor operates as a two-state device, with saturation corresponding to a closed switch.
In this section we discuss briefly the switching behavior of the BJT. The primary objective of this
section is to relate switching behavior to the internal device physics. Consider the simple RTL
inverter shown in Figure 5.47. We use the PSpice code as listed in the Figure to analyze the
circuit and generate the waveforms shown in Figure 5.48.
-177-
^cc= + 3V
BJT switching bahaviour
Vin 1 0 PWL(0 0 O.lu O O.llu 3 0.3u 3 0.31u O)
KB 1 2 5K
Ql 3 2 0 Q2N2222A
.LIB EVAL.LIB
RC 4 3 2k
Vec 4 0 3 V
.TBAN lOOn lu 0 In
.and
Figure 5.47 RTL inverter
The input voltage viH is the 3-V pulse shown in Figure 5.48a. Initially, the input voltage is zero,
and the transistor is in thecutoff region. Therefore, the base current is 2ero, the collector current
is zero and the output voltage vt is 3 V.
During the time interval from / = 0.10 )j.s to / = 0.11 |4.s, v-m rises rapidly to 3 V. The immediate
effect is to cause iB to increase rapidly. This is shown in Figure 5.48b. The current that flows into
-178-
Analog Electronics /BIT Circuits
the base charges die B-E junction depletion capacitance and dius raises the base-to-emitter
voltage.
Part of the base current flows through the collector junction capadtance and out die collector
lead (opposite to the usual current direction for an npn transistor in die active region, see Figure
5.38b). This current causes die output voltage to increase. Notice diat die output voltage actually
goes slighdy higher dian die supply voltage (which is 3 V).
Shortly after the beginning of die input pulse, tine base voltage rises high enough to forward bias
die emitter junction. Then electrons cross from the emitter into the base. These electrons diffuse
into me collector junction. Thus conventional current begins to flow into the collector, and die
collector voltage v, starts to drop. At about / = 0.19 ps, die transistor enters the saturation region
(P's >'o cf- Figure 5.48b). Then die collector voltage becomes approximately constant at a few
tendi of a volt.
The input switches back to zero during the time interval from / = 0.30 u.s to / = 0.31 us.
However, die output voltage remains low until approximately / = 0.52 u.s. The reason for diat is
die excess minority carriers (electrons) stored in die base region. When the transistor is driven
into saturation, bodi junctions are forward biased. Thus a large concentration of electrons builds
up in die base. Until diese electrons have been removed from the base, forward, current
continues to flow across die junctions. Notice that the base current actually reverses directions at
die end of die input pulse. This is due to stored charge flowing out of the base terminal.
At about / = 0.52 p.s, most of die excess charge in the base has been removed and die collector
current begins to fall as shown in Figure 5.48b, causing die output voltage to rise (Figure 5.48a).
However, die output voltage rises gradually because of die junction capacitances. The transistor
returns to die cutoff state at / > lu.s.
The circuit of Figure 5.47 acts as a logic inverter. When die input is low, die output eventually
becomes high. Similarly, when the input is high, the output eventually becomes low. Because of
the charge storage effects, changes in the output do not occur immediately when die input
changes. Of course, in most applications it is desirable for the switching delays to be as short as
possible Several aspects of die device construction influence the switching speed. For example,
reducing junction areas can reduce die device capadtances. Doping levels and junction grading
also affect junction capadtances. A tiunner base region leads to quicker diffusion of minority
charge carriers out of die base region. Selected impurities can be used to reduce die minority
carrier lifetime.
Often, BJT data sheets give specifications for switching time intervals for test circuits similar to
die RTL inverter. We define die start of a logic transition as die point at which 10% of die
voltage change has occurred. For example, the start of die leading edge of the 3-V input pulse is
die point at which die input pulse reaches 0.3 V. Similady, die start of the leading edge of die
output pulse is die point at which % has fallen to 2.7 V. These start points are labeled in Fig. 5.49.
Similady, we define die end of a logic transition as die point for which 90% of the voltage
(current) change has occurred.
-179-
4.6U-T
,
Start of leading edge of Vo
•»-N
if
•ts2.8U.
/
2.7V
A
/
/
End of
trailing
edge of Vo
End of leading edge of Vo /
z=*
[Start of leading edge of Vin
r
h
200ns
400ns
au(l) ou(3)
Tine
—
—
i
-
-
0.3V
-1
"
"I
660ns
Figure~5.49 Waveforms illustrating turn-on and turn-off rimes
Data sheets for BJTs intended for switching applications often specify the following switching
intervals:
• td is the delay time measured from start of the input leading edge to the start of the output
leading edge.
• tr is the rise time, measured from the start point to the end point of the leading edge of the
output pulse. Notice that the rise time is defined for the leading edge of the output pulse even
though this is the negative-going edge.
• /, is the storage time, measured from the start point on the trailing edge of the input pulse to
the start point on the trailing edge of the output pulse.
• îs the fall time measured between the start and end points of the trailing edge of the output
pulse.
Turn-on and turn-off times, respectively ton =td+tr
and t0ff = f, + / y , are given on transistors
data sheets. Typically, the switching times are much longer for power transistors than for smallsignal devices. This is due to larger junction capacitances and charge stored in a larger volume of
the base region.
Vaf*W
S c h o t t k y - c l a a p a d RTL i n v a r t a r
V i n 1 0 FML(0 0 O . l u 0 O . l l u 3 0 . 3 u 3 0 . 3 1 u
0)
RB 1 2 5K
Ca 1 2 2 0 p F
Cs=20pF
Ql 3 2 0 Q2N2222A
D l 2 3 DSH
.MODIL DSH D ( I « - l « - 8 CJO-2pF V J - 0 . 6 M - 0 . 5
. L I B EVAL.LIB
A (v
^
L
2N2222A
RC 4 3 2k
Vcc 4 0 3 V
.TRAM O . l u l u 0 I n
. «nd
Figure 5.50 RTL inverter with speed-up capacitor and Schottky clamp diode
-180-
TT-0)
Analog Electronics / BIT Circuits
Techniques for speeding-up the transitions of the RTL inverter are of much interest in many
applications. Two such techniques are illustrated in Figure 5.50. First, a speed-up capacitor has
been added in parallel with die base resistor Rj. Second, a Schottky clamp diode has been
added between the base and the collector terminal. Figure 5.51 shows waveforms generated by
the PSpice program shown in Figure 5.50.
*.4n
«.tes
fine
Figure 5.51 Voltage waveforms for the Schottky-clamped RTL inverter
Notice that die switching times for diis circuit are much smaller than for the simple RTL inverter
of Figure 5.37. The speed-up capacitor couples the leading edge of the input pulse to the base.
The voltage across the capacitor cannot change instantaneously. Thus the rapid increase in the
input pulse causes a rapid increase in vBE. Thus the transistor is forced to turn on very quickly.
Moreover, without the speed-up capacitor, the junction capacitances must be charged through
Rg, and vBE rises more gradually.
The Schottky diode prevents the transistor from entering the saturation region. When the
collector voltage reaches approximately 0.4 V, the diode conducts and reduces the portion of the
base current available to the transistor. Thus the output voltage is not allowed to fall below 0.4V
and the transistor remains in the active region. This greatly reduces the concentration of electrons
in the base region.
The important point for the circuit designer is that if fast switching is important, the circuit should be
designed so that the transistors do not enter saturation.
5.16 Summary
The bipolar junction transistor is a solid-state device consisting of two pn junctions fabricated in
close proximity on single-crystal semiconductor. A large-signal, static, nonlinear Ebers-Moll
model, embodied in an equivalent circuit and a pair of corresponding simultaneous equations,
help us understand the transistor in terms of the theory of pn junctions. For small-signal linear
applications, we use external dc sources to bias the transistor in its forward active state. For this
state, the Ebers-Moll predicts the characteristic curves and linear equations of a currentcontrolled current source. This leads to a simple linear equivalent circuit that we use to predict
how die transistor interacts with other circuit elements. To do the ac circuit analysis we replace
me transistor with its equivalent circuit. Parameters of this circuit depend on the device jg-point
that results from the particular bias. The BJT has two other states of major importance:
saturation, in which the transistor resembles a closed switch, and cut-off, where the transistor
-181-
represents an open switch. The third state, inverse active, is of little practical importance. Each of
die four states corresponds to operation in a particular region of the transistor output
characteristics, and each has a circuit model, related in an obvious way to the characteristic
curves. For amplifier applications the BJT is operated in the active state. Switching applications
make use of the cut-off and saturation states. A BJT operating in the active state provides a
collector current ic = isexp(J vBE\ / F T ) . The base current is iB = ; c / p , and the emitter current iE =
ic + iB. Also, ic = aiE and thus P = a / ( l - a ) and <X=P/(1+P). To ensure operation the active state,
the collector voltage of an npn transistor must be kept greater than the base voltage. For a pnp
transistor, the collector voltage must be lower than the base voltage. In some important problems
the state of the transistor is unknown. In these cases we must first assume a state for each
transistor, and dien use its equivalent circuit to check our assumptions. An important aid in
developing understanding and intuition about transistor circuits is a graphical tool, the load line.
A number of static second-order effects describe differences between real transistors and
idealized devices. These include temperature variations of parameters, nonzero output resistance,
junction breakdown, internal feedback, and parasitic resistances. At a constant collector current,
die magnitude of the base-emitter voltage decreases by about 2 mV for every °C rise in
temperature. With the emitter open-circuited (iE — 0), the Base-collector junction breaks at a
reverse voltage BV^Q that is typically > 50 V. For iE = 0, the breakdown voltage is less than
BVCM- In the common-emitter configuration the breakdown voltage specified is BV^g, which is
about half BVCBO. The emitter-base junction breaks down at the reverse bias of 6-8 V. There are
also dynamic transistor limitations embodied in the depletion and diffusion capacitances of the
junctions and between the collector and the substrate. Except for minor differences they are the
same nonlinear capacitances we encounter in diodes. SPICE transistor models enable us to
simulate both static and dynamic behavior of transistors, including all of the major nonlinearities.
We generally use our simplified and more intuitive transistor concept to design, and then follow
up with accurate computer simulations that include the second-order effects we consciously
ignored to make our initial design work tractable. We examine and evaluate the simulation results
in terms of our simple conceptual ideas and then redesign, if necessary. The dc analysis of
transistor amplifiers is gready simplified by assuming diat | VBE \ = 0.7 V. To operate as a linear
amplifier, the BJT is biased in the active region and the signal vh is kept small (vk « KT). Bias
design seeks to establish a dc collector current that is as independent of the value of p as
possible. For small signals, the BJT functions as a linear voltage-controlled current source with a
transconductance &n - IJ Vv T h e i n p u t resistance between base and emitter, looking into the
base is r„ = P/#n. For a common-emitter configuration, a high voltage gain and reasonable input
impedance are obtained, but the high-frequency response is limited. Including an unbypassed
resistance in the emitter lead can increase the input resistance of a common-emitter amplifier. In
the common-base configuration, a high voltage gain (from emitter to collector) and an excellent
high-frequency response can be obtained, but the input resistance is low. The CB amplifier is
useful as a current buffer. In the emitter follower (common-collector amplifier) the voltage gain
is less than unity, the input resistance is very high, and the output resistance is very low. The
circuit is useful as a voltage buffer.
-182-
Analog Electronics / Feedback Circuits
6 Feedback Circuits
Feedback consists of returning part of the output of a circuit or system to its input In an
amplifier with negative feedback, a portion of the output signal is returned in opposition to the
original input signal. In positive feedback, the feedback signal aids the original input Usually, in
amplifiers, negative feedback is more useful than the positive feedback. However, the positive
feedback is useful in the design of oscillators, which are considered later. The advantages of
adding a negative feedback to an amplifier are
• reduced sensitivity to parameter variations
• increased bandwidth
• reduced nonlinear distortion
• improved input and output resistances
One disadvantage is the reduction of gain; however, gain is easy to obtain with modern circuits,
so this is a small price to pay. The other disadvantage, a possibility of unwanted oscillations or
instability, is more serious, for oscillations make the circuit worthless as an amplifier.
Fig. 6.1 shows the general structure of & feedback amplifier. A nonfeedback amplifier with gain A
(the A circuit) delivers an output signal x, = Ax, to an external load. Instead of using signal x, as
the amplifier input, a feedback circuit (p circuit) produces a feedback signal xf = fix, to form the
actual input x> The feedback factor, P, used extensively in this chapter is unrelated to the bipolar transistor beta.
Figure 6.1 Feedback amplifier structure
Feedback theory applies to all four amplifier types named in Chapter 2.1,-but each type implies a
different interpretation of Fig. 6.1. For a voltage amplifier, A denotes voltage gain and xe and x,
are voltages. The external source must be a voltage source xs (ideally having zero internal
resistance) and there is a voltage subtraction at the input. The feedback factor P = x/x. is a
voltage gain (or attenuation) in this case.
For a current amplifier, A denotes current gain. The input comes from a source of current xs
(ideally having infinite internal resistance) and the current subtraction is required at the input.
The p denotes the current gain of the feedback circuits.
In a transconductance amplifier, A is transconductance gain with dimensions of amperes/volt.
Thus output x, is a current and the source xt is the voltage source; the subtraction at the input is
the voltage subtraction. For this case, P has a dimension of volts/ampere, or transresistance.
Finally, the transresistance amplifier has transresistance gain with dimensions of volts/ampere.
Thus output xc is a voltage and the external input source is the current source. The subtraction is
a current subtraction. For this case, p has a dimension of transconductance.
From Fig. 6.1 one can write
-183-
x0 = Axi = A(xs -x/)
= A(xs - p * 0 )
Solving for the gain of the feedback amplifier, Aj- = x0 I xs, gives
which applies to all four amplifier classes.
Reduction of Gain. For midrange frequencies, both A and p are real quantities having the same
algebraic sign. Equation (6.1) shows one of the prices we pay for negative feedback, a gain
reduction. For example, suppose A~\W and the fraction of x, that is fed back to the input is
P=0.01. Then the resulting feedback amplifier has gain of only
10000
nn
Usually, as in this illustration, ^ 4 P » 1 , and (6.1) gives an important approximation
Af=-
(6-2)
P
For a voltage amplifier, common practice is to obtain P from a resistor network of lowtemperature coefficient, such as from a voltage divider, giving
_ 1 Rx+R2
.
%
In an IC realization, such a P is easily controlled to within 1% or so by masking tolerances, and
the temperature variations cancel over a wide temperature range. Thus Afis quite constant, even
though A might vary greatly with temperature or other factors.
Improvement Factor. The denominator of (6.1), 1 + ^ P , is of considerable importance in
feedback theory. This factor by which the gain is reduced also turns out to be the degree of
improvement effected by introducing feedback. Ideally, parameter sensitivity, distortion,
bandwidth, input resistance and output resistance all improve by \+A$ when one adds feedback.
Thus the quantity is referred to as the improvementfactor.
6.1 Effects on Sensitivity, Bandwidth and Distortion
We will first develop an intuitive notion of how the feedback works. Suppose in Fig. 6.1, while
the input xt is held constant, output x, decreases in amplitude for any reason. Being proportional
to *•„ feedback signal xyalso decreases, thereby increasing x ; to compensate, at least partially, for
the original change. A complementary scenario opposes increases in x,. Thus negative feedback
opposes any change in x, caused by an event unrelated to x> If the cause is variation in an
internal parameter/), we say that feedback has reduced the sensitivity. If it is loss of gain at either
low or high frequencies because of amplifier capacitances, we say feedback has improved the
frequency response. If x, changes in a manner not proportional to xs because of nonlinearities
within the amplifier, we say feedback has reduced distortion. We now examine the details to
obtain a more quantitative understanding to use in analysis and design.
6.1.1 Effect of Feedback on Sensitivity
If the sensitivity of gain A to parameter/) is Sff, adding negative feedback results in an amplifier
of gain Aj with lower sensitivity; specifically,
-184-
Analog Electronics /Feedback Circuits
To prove this, we need only use the definition
s*f
p
_ P
Af
M
f _ P d[AI{\ + AP)}
dp
Af
dp
Since the circuits we design employ feedback p that is independent of p, the derivative gives
(1 + AP){dA I dp) - AfiidA I dp)
(1 + A0)2
Af
A
f
dA
A/3
P
(1 + AP)
2
(i + AJJ) J dp
Substituting for A, from (6.1) gives an equation that is equivalent to (6.3).
Exercise 6.1 A nonfeedback amplifier with A = 347 contains a critical resistor R. By laboratory
measurements we determine that SR =1.12. After adding negative feedback to reduce the
sensitivity, we find the amplifier gain is 24. Find the gain sensitivity of the feedback amplifier to
changes in K Ans. 0.077.
Example 6.1 We have an amplifier with a gain of 800. Sensitivity of gain to temperature change
is 0.1; however, our specification requires temperature sensitivity to be no more than 0.001.
Investigate the possibility of using negative feedback to bring the temperature sensitivity within
specification.
Solution. From (6.3), we need an improvement factor of
Since A = 800, P must satisfy
1 + 800^=100
or P = 0.124. The resulting feedback amplifier meets the temperature-sensitivity specification;
however, its gain is only 800/100 = 8.
+
Sensitivity of Cascaded Feedback Amplifiers. Example 6.1 shows that we can improve
sensitivity by sacrificing gain. Because specifications often involve both gain and sensitivity, it is
interestirg to see if we can come out ahead by using a cascade'of low-gain, low-sensitivity
feedback amplifiers.
Consider a cascade of n feedback amplifiers, each of gain Af. We denote the overall gain by G =
(A)", and now need to find the sensitivity of G to parameter/). By definition,
SG
p
EdG
p
,dAf_
v
G dp
G
J'
Substituting G = (A)* gives
cG
P
Sp niA/)
„,,
^
sn-ldA*
dp
m
=n
P
dAf
-d7 ^Hp-
We conclude that
Sf=nSp
which shows that sensitivity increases only in proportion to n as we cascade stages, whereas G
varies as die »-th power of gain. This suggests that we should be able to be better off using
feedback. Table 6.1 illustrates the point using numbers from Example 6.1. Cascading feedback
amplifiers is obviously an effective strategy.
-185-
TABLE 6.1 G A I N AND SENSITIVITY O F CASCADED FEEDBACK AMPLIFIERS
n
G
1
2
3
4
' 8
64
512
4096
^P
0.001
0.002
0.003
0.004
6.1.2 Effect of Feedback on Bandwidth
Upper Half-Power Frequency. Consider using feedback to increase the bandwidth of an
amplifier described by
A = A(co) = A0 , *"
(6.4)
where Ae is the midband gain. Substituting this expression into (6.1) gives
a>H/(ja) + a>H)
Af
f
— At
°l + A0[a>H/(j6>
= 4°
+ o>H)y3
ja> + <0H + Aoa>Hp
<oH
= A
°ja> + a,H(\ + AoP)\
\+AQP
+ Ao0
or
a>Hf
f = Aof J6) + a>
Hf
(6.5)
A
where
Anoft ==
l+ A B
and
a
W
= aH
®
+ A
°®'
Comparing (6.5) and (6.4) one can see that the feedback amplifier's gain function has the same
general form as the original; however, its upper half-power frequency is higher by (l+A$) and its
midrange gain is lower by the same factor. Figure 6.2a compares the two frequency response
curves. Notice the original and final gain-bandwidth products, Afi)H and A^Q)Hj. We see that the
gain-bandwidth product is preserved when one adds feedback. We conclude that for a one-pole
amplifier described by (6.4), negative feedback facilitates a direct trade-off of gain for bandwidth
as in Fig. 6.2a. Amplifiers with more complex gain functions do not give an exact trade-off;
however one can expect a dominant high-frequency pole to approximate such a trade-off.
Gain (dB)
Gain (dB)
<BH
(Off
(O
(OLf
(a)
(b)
Figure 6.2 Using negative feedback to improve bandwidth:
(a) high-frequency (b) low-frequency responses
-186-
Lower Half-Power Frequency. Consider a nonfeedback amplifier described by
id)
A = A(a>) = A0 /
(6.6)
Substituting into (6.1) gives
A
ja>l(J(Q + a>L)
=A
f
°\ + Ao[ja>/U<0 + Q>L)]fi
Algebraic manipulations similar to those for the high frequency case lead to
id)
AfJ = AofJ - H
°
ja) + <aLf
(6.7)
where
From Eqs. (6.6) and (6.7) we conclude that negative feedback lowers the lower half-power
frequency and midrange gain by (1+A$). Fig. 6.2b shows the Bode plot changes. A practical
consequence can be that coupling or bypass capacitors can be smaller for the same lower -3dB
frequency in a feedback amplifier. More complicated low-frequency gain functions than (6.6) are
not changed in exactly this fashion; however, if there is a dominant low-frequency pole,
reduction in Ct)L by (l+Afi) is usually a reasonable estimate. Wideband amplifiers approximated
by Eq. (6.6) at low frequencies and Eq. (6.4) at high frequencies have their passband extended at
each side by (Hv4„P).
Exercise 6.2 We need an amplifier with a rise time of 300 ns. The amplifier we have has rise
time of 3 us and gain of 40. Find the resulting gain and P, if we correct the rise time problem
with a negative feedback. Ans. 4, 0.225.
Example 6.2 We need an amplifier that meets the following specifications: gain > 80, fL < 50 Hz,
fH > 15 kHz, and sensitivity to power supply changes < 0.2. We have an amplifier with gain =
1000,^ = 400 Hz,fH — 9 kHz and its sensitivity to changes in supply voltage is 1.5. Determine
whether we can meet the specifications by adding feedback to the existing amplifier. If so, find an
acceptal e value of P and give final specifications for the feedback amplifier.
Solution. Assume that a pair of dominant poles characterizes the frequency response.
Considering each specification individually gives the following requirements:
gain:
6
1000
-—,^,,^80
1 +1000^
400
lower -3 dB frequency:
M
} -—, nnn •- < 50/fe
1 + 1000/?
upper -3 dB frequency: 9000(1 + 1000/ff) £ 15000/fe
sensitivity:
- — . „ „ „ „ ^ 0.2
1 + 1000^
These lead, respectively, to the requirements l+10 3 p < 12.5, 1+HPP > 8, l+10 3 p > 1.67, and
1+103P > 7.5. Sorting out these inequalities shows that any P that satisfies 8 < 1 + 1 0 ^ < 12.5 is
acceptable. To leave room for error, choose 1+103P = 10, which requires P = 0.009. Then the
final amplifier will have gain = 1 0 0 , ^ = 40 H z , ^ = 90 kHz and supply sensitivity of 0.15, all
within specifications.
y
-187-
6.1.3 Effect of Feedback on Nonlinear Distortion
Nonlinear distortion is normally not a problem in the small-signal amplifiers. However, in power
amplifiers, output signal must be large enough to develop a specified amount of power in the
load. This often requires signal swings large enough to traverse a significant portion of the
amplifier's nonlinear transfer characteristic. When this occurs, the output signal is the amplified
input signal, plus additive harmonic distortion components that arise within the amplifier itself.
Amazingly, negative feedback can be used to feed these distortion components back into the
input in such a way that they subtractfromthemselves. The feedback also causes some signal to
subtract from itself; however, this is the gain reduction we expect with negative feedback, and we
can compensate for it by adding a preamplifier. The bottom line is that negative feedback reduces
internally generated distortions by (\+A$). The following quasi-numerical development shows
how it works.
jco(0=10cos(a)0+d(/)
XJ(0=0.001COS(<IDO
(a)
XoC/HOcosCoO+O-OOWdW
Consider a linear model of a nonlinear amplifier shown in Fig. 6.3a. To be specific, suppose that
when the available input signal xt= 0.001cos(a)/) is applied, the output is the Fourier series
xo(t) = \0cos(ojt) + d(t)t
where distortion d(t) consists of Fourier harmonic terms and accounts for the distorted
appearance of the output. (The distortion term depends on the amplitude of the amplifier's
output signal.) Suppose design specifications require an output signal of 10cos(©/) for the given
x/t), but also require that the distortion be much less than d(i). In this type of problem we can
add negative feedback as in Fig. 6.3b to cancel the distortion. For Fig. 6.3b
-188-
xo(t) = d(t) +
or
d(t)
A[xs(t)-0Xo(t)]
sy
\ + A0
\ + A0 MO
Using P = 0.01, as an example, gives
x0 ( 0 = 0.0099 COS(GJT) + 99x s (r)
(6.8)
showing that distortion is reduced to 1 % of its former value. Notice, however, that the new
input xj(t} is multiplied by only 99 in (6.8) instead of the original 104. Because specifications
require 10cos(©i) at the output, from Eq. (6.8) the input to the feedback amplifier must satisfy
9 9 x s ( 0 = 10cos(fi#)
or x,= 0.101cos(©/), showing that the feedback amplifier requires a larger input voltage that the
nonfeedback amplifier for the same output (The corresponding transfer characteristic of the
feedback amplifier, x, versus x„ is much more linear than the original curve, but has lower slope.)
Since only 0.001cos(©/) is available for us for our ioût signal, we must add a preamplifier of gain
Ap at the input that satisfies
0.001^=0.101
or
4 , = 101.
Figure 6.3c summarizes the design, both in general notation and in the numerical values used as
examples. For this strategy to succeed, we must construct a preamplifier that does not produce
distortion. The key observation is that the signal amplitude required at the preamplifier output is
much smaller than the specified amplifier output voltage. That is, it is much easier to produce a
distortion-free output voltage of 0.101 V than 10 V.
Vcc = +15V
V
K £ £ = -15V
Figure 6.4 Nonlinear class-B power amplifier
Example 6.3: Crossover distortion. Distortion is mainly a problem in power amplifiers. A
simplified example of an audio-amplifier output stage is shown in Fig. 6.4. Notice that if vs = 0,
both transistors are in cutoff since there is no forward bias of the base-emitter junctions. In fact,
neither transistor conducts until v, swings outside the range from -0.6 V to 0.6 V (for typical
silicon power transistors). With both transistors cut off, the output voltage is zero. As v, swings
higher than 0.6 V, the npn transistor J2, turns on and supplies current to the load. In this case the
output voltage is given by, approximately,
v 0 = v 5 - 0.6
for
vs > 0.6
If vs is less than -0.6 V, the npn is off and the pnp transistor Q^ is in the active region. Then we
have
for
v,<-0.6
v0=vs+
0.6
-189 -
The transfer characteristic for the amplifier is shown in Fig. 6.5. Notice the nonlinearity in the
region around vt = 0. This nonlinearity causes ctossover distortion when conduction is changing
from one transistor to the other. Also, notice that the voltage gain, which is the slope of the
transfer characteristic, is approximately unity (except in the region around zero input).
5.W
-S.W +
-6.W
-4.MI
Figure 6.5 Transfer characteristic for the amplifier of Fig. 6.4
This circuit is an example of a class B amplifier, in which each device conducts for
approximately half of the signal cycle. Figure 6.6 shows the class B output stage driven by a
differential amplifier that has a differential gain of 1000. The feedback network consisting of R,
and R2 returns part of the output voltage to the inverting input of the differential amplifier.
Normally the switch would be in position JB, so the output voltage across the load is fed back.
However, we will also analyze the circuit with the switch in position A to illustrate the crossover
distortion of the output stage.
The feedback ratio P is given by
*2
= 0.1
A
Ri
+R-,
"o
- 4 •••
2
Since the gain of the differential amplifier is 1000 and the gain of the class B stage is
approximately unity, the overall open-loop gain isA= 1000. Thus we have A$ = 100, which is
much larger than unity. Consequendy, we expect to 6ndA/= 1/P = 10.
P=
V
RL=&n
F££=-15V
00
-190-
Figure 6.6 Class B power amplifier with feedback:
(a) circuit diagram, (b) model for die differential amplifier
The differential amplifier provides the means for subtracting the feedback signal v} from the
source voltage. We use the circuit model for the differential amplifier shown in Fig. 6.6b. The
SPICE program to analyze the circuit is listed below.
Class B Feedback Example
Vs 1 0 sin(0 0.2 1000)
82 4 0 Ik
Rl 2 4 9k; change first node to 3 for switch at B
EA 2 0 1 4 1000; voltage-controlled voltage source
Rin 1 4 1MEG
Ql 5 2 3 npnpower
Q2 6 2 3 pnppower
RL 3 0 8.0
Vcc 5 0 15V
Vee 6 0 -15V
.model npnpower npn(bf=150 Is=le-12)
.model pnppower pnp(bf=150 Is=le-12)
.tran lOOus 2ms 0 5us
.end
After executing this program, we request plots of the drive voltage V(2) at the bases of J2, and Q2
as well as the output voltage vt — V(3). The result is shown in Fig. 6.7a. In this case, the switch is
in position A, so the nonlinearity of the output stage is not included in the feedback loop. The
output signal waveform thus deviates from an ideal, distortion-free shape and, consequendy, a
significant distortion signal d(t) is a part of the output, as is also shown in Fig. 6.7a. The base
drive is sinusoidal, but the output demonstrates considerable crossover distortion.
-191-
Tlaa
(b) Waveforms for the switch in position B
Figure 6.7 Waveforms for the circuit of Fig. 6.6
Then we change the program to place the switch in position B, obtaining the results shown in
Fig. 6.7b. In this case the output voltage is almost free of distortion, d{t) = 0. Notice that the base
drive voltage V(2) has been predistorted to compensate for the nonlinearity of the output stage.
Also notice that, as expected, the output voltage is Aj = 10 times larger that the source voltage
signal.
Exercise 6.3 Suppose that we need to change the amplifier of Fig. 6.6, so that the gain Aj is
approximately 20. What changes do you suggest? Include component values.
Exercise 6.4 Change the resistor values in Fig. 6.6 to R2= lOfi and R, = 9990Q. What is the
approximate value of A$? Use a Spice program to find the output waveform for the switch in
position B and vs — 0.004sin(2000rc/). Is the feedback effective in reducing distortion in this case?
Explain.
6.1.4 Effect of Feedback on Noise
An undesirable aspect of amplifiers is that they add unwanted noise to the desired signal. Sources
of this noise include power-supply hum, coupling of signals from other circuits, and thermally
generated noise in resistors. Another noise is shot noise caused because current flow is not
continuous; instead, charge is carried in discrete quantities by individual electrons. Still another
source is microphone noise, which is an electrical signal arising from vibration of circuit
mechanical components.
Some source of noise can, in principle, be eliminated. For example, power-supply hum can be
reduced by additional filter circuitry in the power supply. However, some of the noise sources,
such as thermal and shot noise, stem from the basic natural processes and cannot be totally
eliminated. Thus all amplifiers add noise, but some amplifiers are much worse than others are. In
this section we wish to show that feedback can, under certain circumstances, reduce noise.
If d(t) introduced in Section 6.2.3 represents noise arising within the nonfeedback amplifier,
negative feedback reduces output noise just as it reduces distortion; however, it can be shown
that adding negative feedback results in no improvement in the input signal-to-noise-ratio of an
amplifier. Since this ratio is usually the critical factor, adding negative feedback is not a generally
effective noise reduction strategy.
-192-
Figure 6.8 Model that accounts for the addition of noise in amplifiers
The additive noise can be modeled as shown in Fig. 6.8. The amplifier gain is denoted as A,. To
quantify noise performance, engineers use the signal-to-noise tatio, which is the desired signal
power delivered to the load divided by the noise power. We denote the rms values of the signal
and noise by X, and X,. The rms signal delivered to the load in Fig. 6.8 is A,XS and the rms noise
is A,XM. If the signals are voltages, the powers delivered to the load are
Ps =
(
^
R,
(6.9)
(Axny
(6.10)
and
Ps =
The signal-to-noise ratio is given by
(6.11)
Equation (6.11) also applies if Xs and X, are currents.]
Figure 6.8 Model that accounts for the addition of noise-in amplifiers
Now consider the feedback amplifier shown in Figure 6.9. X, is the rms noise of amplifier A,.
Amplifier A2 has been added to increase the open-loop gain of the amplifier. Amplifier A2 is
assumed to be noise-free. This is a reasonable assumption if we have a situation in which
amplifier A, is very noisy and amplifier A2 is well designed, so its noise is very small. For
example, A, can be a very high power amplifier with a great deal of power-supply hum, but A2 is
supplied with a well-filtered power. (Perhaps the designer is trying to be economical by using less
filtering for the power to amplifier^!,.)
We analyze the system shown in Figure 6.9 to find an expression for the signal-to-noise ratio. We
can write
x2{t) = xs{t)-px0{t)
(6.12)
xl(t) = A2x2(t) + xn(t)
(6.13)
x0(t) = AlXl(t)
(6.14)
Substitution of Equation (6.12) into (6.13) and the result into (6.14) results in
x0{t) = Ax {A2[xs(t) - Px0{t)\ + xn{t)}
(6.15)
Solving for x,[t), we find that
-193-
*»<')=**(')!7^+X»(')!7M^
(616>
The first term on the right-hand side of (6.16) is the desired signal, and the second term is the
noise. As in Equation (6.11), the signal-to-noise ration is given by the ratio of the rms signal
squared divided by the rms noise squared, which can be shown to be equal to
SNR = [j£j
(A2f
(6.17)
Comparing this with the results given in Equation (6.11) for amplifier^, without feedback shows
the SNR has been increased by a factor of A\ •
Thus feedback is a powerful technique for reducing noise in some circumstances. Keep in mind,
however, that the result we have derived applies only if the preamplifier A2 can be assumed
noise-free. No improvement in SNR value is obtained when A2 = 1 in (6.17). In conclusion,
adding negative feedback is not a generally effective noise reduction strategy.
6.2 Feedback Types
If the feedback network samples the output voltage, we say the amplifier has voltage feedback.
On the other hand, if the feedback network samples the output current, the amplifier is said to
have current feedback.
The feedback signal can be connected either in series or in parallel with the signal source and
amplifier input terminals. Thus we have series feedback or parallel feedback, respectively.
Series feedback can be combined either with voltage sampling or current sampling. Similarly,
parallel feedback can be used either with current sampling or with voltage sampling. Thus we
have four types of feedback: series voltage, series current, parallel voltage, parallel current.
Keep in mind that the terms series and parallel refer to the input connections, whereas die terms
voltage and current refer to the output signal that is sampled. (Other books on the subject use the
terms differendy. Unfortunately, a uniform usage does not exist.) Figure 6.9 illustrates the four
types of feedback.
In complex circuit configurations, sometimes it is not clear whether we deal with current or
voltage feedback. A simple test is to open-circuit or short-circuit the load. If the feedback signal
vanishes for an open-circuit load, we have currentfeedback. Similarly, if the feedback signal vanishes for a shortcircuited load, we have voltagefeedback.
In series voltage feedback, it is natural to consider the input signal to be vt and the output signal
to be vt. Thus it is appropriate to model the amplifier as a voltage amplifier for which die gain
parameter is A,~vjvf This is indicated in Figure 6.9a.
In series current feedback, it is natural to consider the input signal to be vt and the output signal
to be if Thus we model the amplifier as a transconductance amplifier for which the gain
parameter is the transconductance gain Gm=ijvj. This is indicated in Figure 6.9b.
Similarly, for parallel voltage feedback, we model the amplifier as a transresistance amplifier with
gain E^,. Finally, for parallel current feedback, we model the amplifier as a current amplifier with
gain^.
-194-
The units of p are the inverse of the units of the gain for each type of feedback. Refer to Figure
6.9b, as an example. Notice that the units of the transconductance gain Gm are Siemens. Also, we
see that Vf=$ic. Therefore, p is a transresistance parameter with units of ohms.
Similarly, for parallel voltage feedback, the gain parameter is a transresistance, and the feedback
ratio p is a transconductance. For series voltage feedback A = Ar which is unidess, and P is also
unidess. Finally, for parallel current feedback, A = At and P are both unitless.
The effect of each type of feedback on input and output impedance of an amplifier is different.
In design, we select the type of feedback in accordance with the design objectives.
(b) Series current feedback
(c) Parallel voltage feedback
(d) Parallel current feedback
Figure 6.9 Types of feedback
-195-
6.3 Effect of Feedback Types on Input and Output Impedance
Now we examine the effect of series feedback on input impedance. The model for our discussion
is shown in Figure 6.10. The output signal x, is sampled by the feedback network, which
produces a feedback voltage signal vf = fix, connected in series with the source and the input
terminals of the amplifier. The original (before the feedback is added) input impedance of the
amplifier is R,. The input impedance of the amplifier with feedback is
%=7"
(6-18)
Writing a voltage equation around the loop in Figure 6.10, we obtain
v, = *,-/, + vf
(6.19)
But «y=P^ so we have
vs = R,is+0Ko
(6.20)
Also, the input voltage is given by
Vi = Riis
(6.21)
and the output is given by
x0 = Avt
(6.22)
where A=Ar is a voltage gain if x,=v„ or A—Gm is a transconductance gain if die output is a
current x=i,. Substituting Equation (6.21) into (6.22) and the result into (6.20), we obtain
v^Riis + A/SR^
(6.23)
which can be solved for the input impedance with feedback
Rif=f = Ri(l + Afi)
(6.24)
Recall that for negative feedback the factor (\+A$)
feedback increases input impedance.
is larger than unity. Thus negative series
-O-
Load
Figure 6.10 Model for analysis of the effect of series feedback on input impedance
Load
Figure 6.11 Model for analysis of the effect of parallel feedback on input impedance
Next, we consider the effect of parallel feedback on the input impedance. The model is shown in
Figure 6.11. It can be shown that the input impedance in this case is
-196-
Ri
Thus negative parallel feedback reduces input impedance.
Exercise 6.5 Derive Equation (6.25).
To find the output impedance of an amplifier, we turn off die input source, remove the load, and
look back into the output terminals. A model for the voltage feedback amplifier with these
changes is shown in Figure 6.12. A test voltage source vM has replaced the load at the output
terminals of the feedback amplifier. The output impedance with feedback is
v
test
(6.26)
hest
To simplify our analysis, we assume that the input impedance of the feedback network is infinite.
Thus the feedback network does not load the amplifier output.
R
of =
'Jest
Xs=0
+
v
test
Figure 6.12 Model for the analysis of output impedance widi voltage feedback
The output circuit of the amplifier is modeled by a controlled voltage source with gain parameter
A^ The subscripts of the gain parameter indicate that it is the open-circuit amplifier gain. If x,=f,
we have series voltage feedback and AK = A.,^. On the other hand, if x~ij, we have parallel
voltage feedback and AK = R ^ In any case, the resistance R, shown in Figure 6.12 is the output
resistance of the amplifier before feedback.
For die output loop of Figure 6.12 we can write
v
(6.27)
test ~ ôhest + ôcxi
However, we have
*/ = -Potest
(6-28)
Substituting Equation (6.28) into (6.27) and solving for the output resistance with feedback, we
have
v
R„
test
R
(6.29)
of =
1 + fiA,
hest
oc
Thus negative voltage feedback reduces the output impedance of an amplifier.
Next, we consider the effect of current feedback on output resistance. The model for this analysis
is shown in Figure 6.13. As before, the source signal x/is set to 2ero, the load is removed, and a
test source is connected to the output terminals. The feedback network is assumed to have zero
input impedance, so it produces no loading effects at the amplifier output.
197
+
^-40-^
x=0
&==&,
a
^"Irjf
Figure 6.13 Model for the analysis of output impedance with current feedback
The output of the amplifier is modeled by a controlled current source in parallel with the output
resistance. The gain parameter Ax has subscripts indicating that it is the gain of the amplifier with
a short-circuited load. For parallel current feedback, the gain is the short-circuit current gain Ax
= Aisf For series current feedback, the gain is the short-circuit transconductance gain Ax = Gau,
For the system of Figure 6.13 we can show that
v
test = R0(1 + J3ASC)
hest
Thus negative current feedback increases the output impedance of an amplifier.
R
of =
(6.30)
Exercise 6.6 Derive Equation (6.30).
6.4 Summary of the Effects of Various Feedback Types
We have seen that four types of feedback are possible. One effect of feedback is to stabilize and
linearize gain. (i.e. Aj tends to be independent of A). However, the particular type of gain
stabilized depends on the type of feedback. Table 6.1 shows the type of gain stabilized and
linearized for each type of feedback.
We have seen that (negative) series feedback increases input impedance, whereas parallel
feedback reduces input impedance. If yip is very large, the input impedance tends toward either
an open circuit or a short circuit. The formulas for input impedance are shown in Table 6.1.
TABLE 6.1 EFFECTS O F FEEDBACK
FEEDBACK
TYPE
Series voltage
x
t
GAIN
STABILIZED
x
f
'
t>,
*>.
A
A
V-\
+
Av0
INPUT
IMPEDANCE
OUTPUT
IMPEDANCE
Ro
RiQ + Av/?>
1 + AvocP
Gm
Series current
»,
i.
Parallel
voltage
',
».
Parallel
current
*,
'.
mf
~\ + Gmp
j?
Ri(l + Gm)T>
Ri
Ro
i + RmP
1 + RmocP
Rm
Ai
A
» - l + AiP
R0Q + GmscP)
Ri
1 + 4/?
-198-
Ro(\ + Aisc0)
IDEAL
AMPLIFIER
Voltage
Transconductance
Transresistance
Current
To reduce output impedance we could employ voltage feedback. On the other hand, to increase
output impedance, we could choose current feedback. Of course, in making these statements, we
assume negative feedback - the effect of positive feedback is the opposite. Table 6.1 also
contains formulas for the output impedance for each of the four feedback types.
We can summarize the effect of each type of feedback by stating that it tends to produce an ideal
amplifier of a certain type. For example, series voltage feedback increases input impedance,
reduces output impedance and stabilizes voltage gain. Thus series voltage feedback tends to
produce an ideal voltage amplifier. As summarized in Table 6.1, similar statements can be made
for the other feedback types.
6.5 Practical Feedback Networks
So far, we have modeled feedback networks as controlled sources. This approach simplified the
analysis and allowed us to focus on the main effects of the various types of feedback. However,
in practice we use simple networks of resistors (or in some cases resistors and capacitors). This
components are available with precise and stable values (over time and with temperature
changes) compared to the parameter values of active components (transistors). We employ
negative feedback, so the amplifier characteristics depend mainly on the feedback network,
thereby achieving amplifiers having precision and stability. Figure 6.14 shows examples of
feedback amplifiers using practical resistive feedback networks.
R
n Vf
2
(a) Series voltage, p =
=—
— (assuming U £ 0)
v0
R\ + R2
f = Rf (assuming U = 0)
(b) Series voltage, /? = -;—
-199-
l
n lf
(c) Parallel voltage, p = —
= —R—— (assuming v,- 2 0)
v
o
>L
n
f
_
if
R\
(d) Parallel current, p = T~ = ~ T
7T (assuming p,- = 0)
l0
R\ + K2
Figure 6.14 Examples of resistive feedback networks
Notice that we have modeled the source as a voltage source for series feedback and as a current
source for parallel feedback. This is consistent with Figure 6.9. The Thevenin model for the
source is more natural for series feedback because the feedback voltage jîs subtracted from the
source voltage v, in a series connection. The Norton model is more natural for parallel feedback
because the feedback current ij\& subtracted from the source current is in a parallel connection.
Each of the feedback amplifiers shown in Figure 6.14 has negative feedback. For example, in the
series voltage case shown in part (a) of the Figure, suppose that vs has a positive value. This
results in a positive voltage at the noninverting input. The amplifier, in turn, produces a positive
output voltage. The feedback network, composed of R, and R*, returns a fraction of the output
voltage to the inverting input. This reduces the input voltage vf Thus the feedback signal acts in
opposition to the original source signal, and we have negative feedback. If the inverting and
noninverting input terminals were interchanged, positive feedback would result. A similar
discussion applies to the remaining feedback amplifiers shown in Figure 6.14 b, c, and d.
We can identify series feedback and parallel feedback by examination of the circuit configuration
at the amplifier input. Study Figure 6.14a and b to verify that the signal source, the amplifierinput terminals and the output of the feedback network are in series. Also, verify the parallel
connection for Figure 6.14c and d.
To test for cutrent feedback, open-circuit the load so that the output current becomes zero. If
the signal returned to the amplifier input by the feedback network becomes zero, the amplifier
has current feedback.
-200-
To test for voltage feedback, short-circuit the load so that the output voltage becomes zero. If
the signal returned to the amplifier input by the feedback network becomes zero, the amplifier
has voltage feedback. Verify that the types of feedback are correcdy labeled in Figure 6.14 by use
of these tests.
Exercise 6.7 For each of the circuits shown in Figure 6.15, identify the type of feedback present
(negative-positive, series-parallel, and voltage-current?). Determine the value of the feedback
ratio, assuming zero input current and zero input voltage of the non-feedback amplifier, where
appropriate. What type of ideal amplifier results if A$ is very large? What is the gain of this ideal
amplifier? What value (0 or oo) do the input and output resistances approach? Ans. (a) Negative
series voltage, 3 = 1, ideal voltage amplifier, A4 = l , K , - = ° o , R , = 0
(b) negative parallel current, P = 1, ideal current amplifier, A^= 1, R, = 0, R, = oo,
(c) negative parallel voltage, P=-1/(3R), ideal transresistance amplifier, R^=-3R, R,=0, R„=0,
(d) negative series current, P=R/2, ideal transconductance amplifier, G^ = 2/R, R,= oo, R = oo,
(e) negative series voltage, p = l / 1 2 , ideal voltage amplifier, >4^= 12, R, = oo, R, - 0.
-201-
6.6 Stability of Feedback Amplifiers
In the above discussion, the nonfeedback amplifier (A circuit) was considered memoryless, for
simplicity. Therefore, a zero phase shift between its output and input was assumed. Because of
nonzero phase shift within any real amplifier, it is possible that a component of the feedback
signal at some particular frequency/ is actually added rather than subtracted from die signal
source. If this component is sufficiently large, the result is sustained oscillation at frequency/ the circuit has become an oscillator or signal generator instead of an amplifier, and we say the
circuit is unstable.
AiS)
Figure 6.16 Loop gain in a feedback amplifier
For concrete illustration of how oscillations can occur, consider a series voltage circuit of Figure
6.16, where sinusoidal voltages are indicated as phasors. With the switch in position 1, we have
the feedback amplifier; position 2 gives the nonfeedback amplifier. Usually, P is a constant and
does not contribute to the phase shift At midband frequencies, A((0) is real. Therefore, with die
switch in position 1, the feedback phasor, Vf=\SA(<Q)Vi, has the same angle as Vf For sinusoidal
excitation, this means that all voltages in die KVL equation, vs(t) = v,-(f) + Vy (/), have identical
phase angles; hence the subtraction
v,(0 = v 5 ( 0 - v / ( 0
(6.31)
required for negative feedback.
Now, with the switch in position 2, suppose signal
v,(/) = Vsin(2rf0t) = v,-(0
(6.32)
is applied to the nonfeedback amplifier. Further, suppose that/ is such that A(fy is complex, that
is
A(f0) = \A(f0)\zMf0)
(6.33)
The output of the dependent source is then
vf ( 0 = $A(f0)\v
sin[2nf0t +
ftf0)]
(6.34)
If die condition
*K/ O ) = 180°
(6.35)
happens to be satisfied at the excitation frequency/, men according to Equation (6.34), vit) is
inverted relative to ?,(/), giving
V/
( 0 = -/3[A(f0)\Vsm[2nf0t}
= -jÂ(f0)\vs(t)
(6.36)
Under these conditions, if the switch suddenly goes to position 1, the feedback signal reinforces,
or adds to vs(t). The KVL "subtraction" at the input now gives
v,(0 = v,(/) - vf{t)
= v,(/) + ^ ( / o ) K ( 0
(6.37)
-202-
When this occurs, we say we have positive feedback at frequency/. It is possible for feedback
to be negative for midrange components and positive for frequencies outside the -3dB passband
of A(f) where additional phase shift occurs.
With the switch in position 2 and sinusoidal excitation at frequency/, suppose in Equation (6.36)
that in addition to ^ ( / 0 ) = 180°, we also have
( 6 - 38 >
Mfoi * i
This amplitude condition leads to two interesting possibilities. Equality in Equation (6.38)
implies
vf (/) = V sm{l7rf0t +180°) = -V sm{2nf0t +180°)
(6.39)
When this condition is satisfied, we can use Vj to replace Vf That is, once the circuit is running,
we can flip the switch to position 1, turn off the external signal and use V{ as the input signal.
Thereafter, there would be a sustained sinusoidal output at frequency/. Circuits that operate like
this are called non-self-starting oscillators.
When the inequality occurs in Equation (6.38), Vf is identical in frequency and phase to Vs but
greater in amplitude. In this case, the sinusoid of frequency/ increases in strength until circuit
nonlinearities limit its amplitude. The result is a sustained oscillation with a periodic output
waveform of fundamental frequency/. A circuit deliberately designed to operate in this way is
called a self-starting oscillator. If such oscillations occur in an amplifier, we say the amplifier is
unstable.
Taking into account the complex-valued gain of the nonfeedback amplifier, the closed-loop gain
of a feedback amplifier, as given by Equation (6.1), can be described by
Af(f)
A{f)
= —^±L-!—
(6.40)
K
}
l + Mf)
QA(f)
For a given frequency/, if pVl(/5 = -1> the closed-loop gain becomes infinite. This means that
even a very small input signal may cause an infinite amplifier response. Such an amplifier is
unstable and may produce oscillations with no input signal. The amplitude and phase conditions
associated with oscillations both are related to the complex product
L(f)=6A(f)
(6.41)
called the loop gain. The loop gain L 0 is the total gain of the feedback loop from the amplifier
input back to the point of signal subtraction, including all loading effects at input and output.
Figure 6.16 shows that JL0 is the gain VJ Vs when the switch is in position 2, that is with the
feedback loop open.
f U J
We now state the Nyquist stability criterion, a necessary and sufficient condition for feedback
amplifiers to be unstable. If there exists any frequency/ such that the loop gain
L(f0) = jfl4(/ 0 ) = MZl80°, and M > 1
(6.42)
then the amplifier is unstable, and there will be oscillations a t / . Otherwise, the amplifier is stable.
The justification for such a simple stability criterion is that random noise is present in every
circuit with its power distributed over all frequencies - an infinity of tiny signal sources. For this
reason, if there is any frequency whatsoever at which Equation (6.42) is satisfied, oscillations are
inevitable even with no external signal applied to the circuit With an external signal applied, the
oscillations simply superimpose upon i t
-203-
Gain Margin and Phase Margin. In design of feedback amplifiers, it is often helpful to
consider Bode plots of me magnitude and phase of the loop gain $A(fi. For real and positive (3,
its effect on the magnitude plot is simply to shift it vertically by 201og(P) and there is no effect on
the phase plot. Thus the Bode plots of the loop gain $A(fi are the same as the Bode plots of the
amplifier open-loop gain A(J) except for the vertical shift in the magnitude plot
An example of the Bode plots for a BJT amplifier is shown in Figure 6.17. In considering the
stability of an amplifier, we examine the Bode plot for the loop gain $A(fi to find the frequency
fGM fot which the phase shift is 180°. If the magnitude of the loop gain is less than unity at this
frequency, die amplifier is stable. On die other hand, if the loop gain magnitude is greater than
unity, the amplifier is unstable.
For a stable amplifier, the gain a t / ^ is less than unity in magnitude (it is negative when expressed
in decibels). The amount mat the gain magnitude is below 0 dB is called die gain margin. The
gain margin is illustrated in Figure 6.17. It can be shown that a gain margin of zero implies a pole
lies on they'll) axis on the .r-plane. As gain margin becomes larger, the pole moves back into die
left half of the .r-plane. In general, larger gain margin results in less ringing and faster decay of the
transient response.
Another measure of stability that can be obtained from the Bode plots is the phase margin.
Phase margin is determined at the frequency fm for which the loop gain $A(fPAj) is unity in
magnitude (i.e., 201og | fyAffpy) \ — 0 dB). The phase margin is the difference between me actual
phase and 180°. This is also illustrated in Figure 6.17.
As we noted earlier, we usually want to design feedback amplifiers to avoid ringing transient
response and gain peaks in the frequency response. A. generally accepted rule of thumb is to design for a
minimum gain margin qflOdB and a minimum phase margin of45 °
For the amplifier illustrated by Figure 6.17, the frequency fGM is approximately equal to 30.1
MHz, and the gain margin equals to 21.7 dB. The phase margin can be determined at frequency
fPM £ 3.1 MHz as being equal to 79°. Thus the example amplifier is stable and shows reasonable
gain and phase margins.
-204-
Often an amplifier is unstable for a given gain l / p \ which is especially true for multistage
nonfeedback amplifiers that have complicated frequency response. We can make such amplifier
stable by altering its open-loop gain curve A(J). Deliberately changing the frequency response of
the nonfeedback amplifier to make a feedback amplifier stable is called frequency
compensation. Some techniques have been developed for frequency compensation, such as pole
addition used for operational amplifier design. However, the topic of frequency compensation is
outside the scope of this textbook. The modern operational amplifiers, which are most often
used as nonfeedback amplifiers, are internally compensated by adding an on-chip capacitor to
introduce the dominant pole for compensation.
Exercise 6.8 Use PSpice to find gain margin and phase margin for the uA741 operational
amplifier. Hint Run the .ac analysis and use evailib library.
6.7 Sinusoidal Oscillators
Oscillators are intentionally unstable circuits that serve as sources of electrical waveforms. There
are two broad classes of oscillators: sinusoidal oscillators, which produce sinusoidal waveforms,
and relaxation oscillators, which produce triangular, or rectangular, waveforms. Both classes of
oscillators are widely used for time bases in test and measurement equipment, and for signal
processing in analogue and digital communication systems. Here we concentrate on sinusoidal
oscillators.
6.7.1 General Theory of Sinusoidal Oscillators
A sinusoidal oscillator has three functional parts, a phase shifter to establish the frequency of
oscillation, a gain circuit to compensate for energy losses in the phase shifter, and a limiter to
control the amplitude of the oscillations. The gain circuit might be an operational amplifier or a
transistor amplifier. The phase shifter is typically an RC or JJC circuit. The limiter might be a
diode, a thermistor, or a variable-gain amplifier. In some oscillators the basic functions are
combined rather than relegated to individual subcircuits. For example, the internal capacitances
of the transistor that provides gain might contribute to the phase shifter, and inherent transistor
nonlinearities often provide the limiting. Common to all oscillator circuits is instability, which is
best understood in terms of positive feedback theory.
Figure 6.18 General structure of a sinusoidal oscillator
The voltage-shunt feedback structure of Fig. 6.18 describes many sinusoidal oscillators. A
voltage amplifier with gain A((o) = VJV-, provides the gain, and a feedback network described
by P(co) = VjfV, is the phase shifter. An oscillator representation, such as Fig. 6.18, differs in
several ways from a feedback amplifier diagram: (3 ((D) is defined without the notion of an input
subtraction; the feedback network includes reactive elements to provide the phase shift required
-205-
for positive feedback; and, of course, there is no external signal source. Nonlinearities that limit
the signal amplitude invariably arise but do not appear in this linear model. The switch helps us
examine the loop gain of the circuit.
Barkhausen Criterion. Assume that A((&) and (3(co) are defined in such a way that no loading
occurs when the switch is closed. The Barkhausen criterion states that there will be sinusoidal
oscillations at frequency C0o when the switch is closed, provided that with the switch open, the
loop gain is
Vf I Vt = A{o)0)P{co0) = M{co0)Z<f>{co0) = 1
(6.43)
When this condition is satisfied, a hypothetical sinusoidal signal generator V; attached to the
input can be removed when the switch is closed, because the amplitude and phase of the signal
fed back to the input are exactly those needed to replace this source. Since the Barkhausen
criterion involves a complex-valued function, it implies two conditions for oscillations, a
magnitude condition and a phase condition. We use the magnitude condition. M(C0o) — 1, as a
test to determine whether oscillations can exist in a given circuit. This condition arises from the
physical requirement that the amplifier provide sufficient gain to make up exacdy for energy
losses in the circuit. If Af(C0o) exceeds one, the oscillator is self-starting, with the oscillation arising
spontaneously and increasing in amplitude until nonlinearities cause a reduction in M((00)- Some
oscillator circuits require- a signal generator for start-up; however, self-starting circuits are die
norm. Provided that oscillations can occur, the phase condition <j)(co0) = 0 determines die
frequency of oscillation co0 of the circuit. This condition physically means the signal arrives back
at the input exacdy in phase with itself.
6.7.2 RC Oscillators
Two important oscillators use RC phase-shifting circuits. Both are suitable for generating
oscillations at frequencies from a few hertz to hundreds of kilohertz.
Wien Bridge Oscillator. Figure 6.19a shows the Wien bridge oscillator with a noninverting op
amp circuit for its gain element. It is also called a bridge oscillator because the active element is
imbedded in the RC bridge circuit, as we see by redrawing the circuit as Fig. 6.19b.
-206-
1/coC
co„+Aco
(c)
Figure 6.19 Wien bridge oscillator: (a) schematic; (b) redrawn to sliow the bridge;
(c) illustration of frequency stability.
To find design equations for this or any oscillator, we apply the Barkhausen criterion. From Fig.
6.19a, P(ffl) is the transfer function
where Z, and Z2 are the impedances of the parallel RC and series RC circuits, respectively. Since
A(a>) is the gain of the noninverting amplifier, Eq. (6.43) requires
1+
A{G>0)P{a>0) =
*2
R \J 1 + j<»0CR
=1
1
+R+
1 + jo)0CR
"
jo)0C
where 00o is the frequency of oscillation. Simplifying leads to the expression
R
1+
Ri
j(o0RC
*V
A(a>0)/3(a>0) =
=1
(6.44)
l+
3j0)oRC-a>lR2C2
For the expression to be real, the real part of the denominator must be zero. This gives the phase
condition
o
1
ffl =
(6.45)
o -^2
Solving for frequency gives
1
(6.46)
Substituting this into Eq. (6.44) gives the amplitude condition
*I
Thus if R2 > 2R, the circuit will oscillate; the inequality implies oscillations of increasing
amplitude.
An important attribute of an oscillator is its frequency stability, its ability to maintain a frequency
close to the design value. From Eq. (6.46) it is easy to show that
,a>,
con
SC°=SR°=-\
-207-
not particularly low values. To give an intuitive idea of what is involved, Fig. 6.19c shows Eq.
(6.46) as the graphical solution of the equation 1/toC = R. The dashed lines show how C0o
changes with variations in C. (Like the sensitivity expression, this diagram assumes that the
resistor and capacitor pairs in die P circuit are matched.)
The next example demonstrates another useful way to investigate frequency stability of an
oscillator - using SPICE to plot the phase of A(a>)p((o).
Example 6.4 Plot the phase shift of the loop gain for a Wien bridge oscillator with R = 1591.5
Q and C = 1000 pF. Find the change in frequency if the value of the shunt capacitor increases by
10%.
Solution. Figure 6.20a shows the phase-shifting network - Fig. 6.20b the SPICE code. An
asterisk marks the statement that describes C2 for the second run. Since Eq. (6.46) indicates an
oscillation frequency of 100 kHz, the phase of P(<»), which is the phase of V(l), is plotted from
80 to 120 kHz.
The upper curve of Fig. 6.20c shows that the phase curve of the original oscillator is roughly
linear over the range of the plot and crosses zero at 100 kHz. Increasing C2 by 10% gives die
lower curve, showing that the frequency drops to about 95 kHz, a change of 5%. Thus
Sr° s -0.5
c
2
From the discussion of frequency stability and the example we conclude that die components
diat contribute to the phase-shifting circuit should be of high quality and relatively insensitive to
temperature and other environmental factors.
-208-
Phase-Shift Oscillator. Figure 6.21a shows the phase-shift oscillator, The phase shifter consists
of three RC sections, The gain element is represented as an ideal inverting amplifier with voltage
gain -K.
To find amplitude and phase conditions from the Barkhausen criterion, we analyze the phaseshifting circuit of Fig. 6.21b. The transfer characteristic' of such a ladder circuit is most easily
found by systematic analysis from output to input as suggested by the following equations.
V: +
'
-
*
= 1+
•
,VjcoRC) '
1
2
l
R
K
jaRC) R
R
Continuing in this fashion finally gives
v
i
2+
1
5
1+
jcoRC) R
1
jcoRC o2R2C2+Jco3R3C3
Thus our design must satisfy
-K
A{O)0){3{CD0) = ^ - =
=1
6
5
1
1+
jco0RC~ CO2OR2C2+J
o)30R3C3
To find the phase condition, we set the imaginary part of the denominator to zero. This gives an
oscillation frequency of
(6 47)
^-7SE
'
It is easy to show that the sensitivity to changes in R and C is the same as for the Wien bridge
circuit. Substituting co0 from Eq. (9,39) into the preceding equation gives the gain condition
-K
TTir1
Thus the circuit oscillates for any K > 29.
A(<o)
Vi
Figure 6.21 Phase-shift oscillator: (a) circuit; (b) notation for computing p(co) for the oscillator
Exercise 6.9 Work out the first missing step in the preceding derivation of P(co); that is, find the
expression for V2 in terms of Ve
-209-
Ans. Vj =
jaRC
(wRC)2
Vt
Amplitude Limitets. To make an oscillator self-starting and to allow for uncertainties in
parameter values, we usually design the circuit so that the gain condition is exceeded. The
amplitude of the oscillation then increases until some nonlinearity reduces the effective loop gain.
If the signal amplitude becomes too large, the signal traverses a large segment of the nonlinear
transfer characteristic of the active device causing the sinusoidal output waveform to be highly
distorted. There are three basic approaches to controlling the signal amplitude while keeping the
waveform reasonably sinusoidal.
When the gain condition is not gready exceeded, a design can rely on inherent transistor
nonlinearities to limit the signal amplitude to a value corresponding to a reasonably undistorted
sine wave. This does not produce very robust designs, however, because component values are
rather critically related to waveform purity.
The second approach is to insert a special nonlinear component into the loop so that loop gain
begins to diminish with signal amplitude while signal amplitude is still small. A thermistor with
resistance that decreases- by self-heating, or a strategically placed diode are examples of
components used for this purpose. An example is Fig. 6.22a, which shows the phase-shift
oscillator of Fig. 6.21a with two diodes added for limiting. As long as the voltage across resistor
Rj has peak amplitude less than 0.5 V the circuit is approximately the same as Fig. 6.22a;
however, when output amplitude exceeds the forward-biased diode voltage, the diodes introduce
a low resistance in parallel with l y This nonlinear resistance decreases with increasing amplitude,
lowering the gain of the inverting amplifier and thereby limiting the oscillation amplitude. The
following example investigates this limiter in some detail.
Example 6.5 Use PSpice to examine the output waveform of the phase-shift oscillator of Fig.
6.22a without and with the diode limiter. Values are R = 10 kQ, K = 50, and C = 3000 pF. Use a
dependent source with gain 105 and input impedance of 1 MQ for the op amp.
Solution. Figure 6.22b is the PSpice code with diodes omitted. The initial condition statement,
".IC V(3) = 0.1," changes the initial voltage at node 3 to a nonequilibrium value to help start
oscillations. Figure 6.22c shows the output without the diode statements. Since the linear circuit
model has no limiting whatsoever, the oscillation amplitude quickly increases to tens of volts in
the simulation. The output of a real opamp would be driven to its saturation limits, giving a
highly distorted output voltage, a statement readily verified using the more realistic operational
amplifier model, e.g. from the EVAL.LIB library.
Figure 6.22d shows the output with the diode statements present in the input code. The
waveform quickly converges to a sinusoidal waveshape, with amplitude limited by the diodes and
virtual ground to the ±0.5 V range.
+
Exercise 6.10 Calculate the expected oscillation frequency in Example 6.5.
Ans. About 1.7kHz with limiting diodes omitted; 2.17 kHz with limiting diodes present.
-210-
(c)
(d)
Figure 6.22 Phase-shift oscillator with diode limiter (a) schematicj-(b) PSpice code;
(c) output with limiting diodes omitted; (d) output with limiting diodes included.
For sinusoids of very high quality, a „linear amplifier" can provide limiting with gain that
decreases as amplitude increases. In Fig. 6.23, the voltage-controlled amplifier is the gain element
in a Wien bridge oscillator. A half-wave rectifier with a capacitive load (envelope detector)
produces a dc control signal, Vc, proportional to signal amplitude. The follower isolates the
control circuit from the RC bridge so amplitude and phase conditions are unchanged. As
oscillation amplitude increases, Vc becomes more negative, and gain automatically decreases. The
detector time constant RDCD should be several periods of the oscillator waveform.
Because it involves nonlinear operation, limiting always introduces "impurity" into the oscillator
output waveform, causing the output to differ from the desired single-frequency sinusoid. One
can describe any periodic waveform as a Fourier series
00
v ( 0 = T,A„cos(na)0t+ </>„)
n=0
For an oscillator, the ideal output is the n = 1 term alone; terms for which n>2 are undesired
impurity terms introduced by the limiter. The smaller these harmonic terms, the greater the
-211-
waveform purity. A useful measure of waveform impurity is the percent total harmonic distortion,
THD, where
rms
value
of
harmonic
components
THD =
xl00%
rms
value
of
fundamental
frequency
terms
If we know the fundamental frequency^. = C0o/27t of the oscillator output, we can use PSpice to
compute THD by adding a SPICE statement ".FOUR". This asks SPICE to compute amplitudes
of the first nine Fourier series components and to use them to estimate THD. Adding the
statement
.FOUR2.0E3V(1)
reveals that the output voltage waveform contains 6.22% distortion (look into the PSpice output
file). The advantage of more sophisticated limiters like Fig. 6.23 is improved waveform purity.
The oscillator designer is often required to meet a THD specification along with specifications of
frequency and amplitude.
Figure 6.23 A limiter that uses sip~-' amplitude to control loop gain
6.7.3 LC Oscillators
Two important oscillators use the thtee-element n structure of Fig. 6.24a for the phase shifter. In
Fig. 6.24b, a transistor biased for small-signal active operation provides the gain. From Fig. 6.24b,
00
(b)
Figure 6.24 Oscillator configuration with three-element ladder phase shifter:
(a) general structure; (b) oscillator with transistor active element
-212-
A(a>) =Vf
= -gm[R0\\Zl\\{Z2
+ Z 3 )]
and
z3
This gives the loop gain
l s
i 4 ( « ) f l » ) = Z? + Zi ~ ZJ
^ Z?
+
J
A
„
+ *
+1
*o
Zl
The Barkhausen criterion requires
r
v
Z1Z2+Z1Z3+i?0(Z3+Z2+Z1)
'
If each impedance is an LC element, then Z,- = jXj, where X, =fflJL,-for an inductor and Xs
//(oC for a capacitor. Then
MZd> =
SmKZfo
= lzoo
M
^
-(XlX2+XlX3)
+ jR0(Xl+X2
+ X3)
Because die numerator is real, the phase condition is satisfied only if
Xx + X2 + X3 = 0
When Eq. (6.50) is satisfied, the gain condition in Eq. (6.49) reduces to
- g m V ^ ^
X2+X3
We next use these conditions to study two specific oscillators.
(649)
K
>
(6.50)
(651)
Colpitts Oscillator. In the Colpitts oscillator, Z2 is an inductor and Z, and Z, axe capacitors as in
Fig. 6.25. Substituting Z2 -faL.^ Z, = 1/foCt and Z} = 1//oC} into Eq. (6.50) gives
1
,
1
n
Solving for Of) gives the phase condition
2
C
l+C3
ffl
(6 52)
»=Â
-
which establishes the frequency of osculation. A second design condition comes from Eq. (6.51),
If!^ = _s^ a l
(6.53)
0>oL3
Substituting C0o from Eq. (6.52) leads to the second design equation
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(6.54)
gmRo>
which specifies the gain required for sustained oscillations.
Exercise 6.11 Compute the sensitivity of the resonant frequency of a Colpitts oscillator to L?
Use your answer to estimate the actual resonant frequency of an oscillator designed to operate at
1 MHz if L2 is 12% high. Ans. -0.5, 940 kHz.
+12V
+12V
+12V
+12V
Figure 6.26 Colpitts oscillator realizations: (a) circuit diagram for a BJT oscillator;
(b) small-signal equivalent for BJT circuit; (c) MOSFET oscillator
Example 6.6 The Colpitts oscillator of Fig. 6.26a has a transistor biased at 0.5 mA; pV = 120; Cc
is a large coupling capacitor. Find values for L^ C, and C, so the circuit oscillates atf, = 1 MHz.
Ignore transistor capacitances.
Solution. Figure 6.26b shows the high-frequency equivalent circuit. At 0.5 mA,&, = 0.02 S and rn
- 6 kQ. The small-signal equivalent circuit is Fig. 6.26b, where
5.5*0 = ^ ||159*Q||112Afi
From Eq. (6.54),
(0.02)(6xl03) = 1 2 0 > § c
(6.55)
l
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and from Eq. (6.52),
LoCea =
7TT = 25.3 x 10" 1 5
1
** ( 2 / r l 0 6 ) 2
where Ctq = C,C3/(Ct + Q .
(6.56)
We now have two equations involving Ch Cs and L.2. In a FET design we would make an
arbitrary choice of one component value; however, in a BJT design we must minimize the effect
of r„, since our theory does not include a resistor in this location. Therefore, we select C3 such
that its reactance is much smaller than rx = 5.5 kQ. This gives the third design equation. Using
two orders of magnitude to assure rn has negligible effect gives
1
1
3
2*10 6 C 3 " 100
orCjS2894pF.
5.5 xlO
Since our design depends upon approximation, and since actual component values may not have
exacdy the values we expect, we satisfy Eq. (6.55) by using C,/Cf = 75 instead of the limiting
value of 120 to ensure self-starting oscillations. With C} previously selected, this gives
2894
Q = — = 38.6pF
and C,q = 38.1 pF. From Eq. (6.56)
25.3 x l O - 1 5 n,tre
TT
L) =
7T = 0.65 5mH
12
38.6 x 10"
The coupling capacitor Cc must have reactance < < (£>oL2 at 1 MHz. Since the latter is 4.11 kfi, Cc
- 10 nF will do.
In Colpitts oscillators, an inductor called an RF choke usually replaces the collector or drain
resistor. The choke coil is a short circuit for biasing purposes but presents an open circuit at RF
(radio frequency) oscillator frequencies. When a choke is used, R, in Fig. 6.25 becomes the
output resistance r„ of the transistor. The RF choke reduces power dissipation of the circuit and
improves the purity of the output waveform.
Exercise 6.12 Figure 6.26c shows a MOSFET oscillator with drain biased at 1 mA through an
RF choke. If transistor parameters are K = 4 mA/V 2 , V, = 1.2 V, and VA = 70 V, find the
condition for oscillation. Ans. 197 > C3/Ct.
We have seen that SPICE simulations are useful in verifying oscillator designs, examining
distortion, and exploring sensitivity limitations. There are some practical simulation difficulties
when large bypass and coupling capacitors are included in simulations along with components
having short time constants. Sometimes there are convergence problems in the numerical
algoriuims. Sometimes a very large number of oscillator cycles are required before the circuit
settles into steady-state operation. The following example shows a way to avoid time constant
problems in oscillator simulations. An initial dc analysis determines the voltages across the large
capacitors, which we then replace by dc voltage sources of appropriate polarity.
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Example 6.7 Use PSpice to show how the oscillations arise in die circuit designed in Example
6.6.
Solution. The SPICE code of Fig. 6.27a uses values determined in die design of Fig. 6.26a.
Following initial dc analysis, capacitors CE and Cc were replaced, respectively, by sources VEE
and VCC. Because the oscillation period is expected to be approximately 1 Lis, the initial .TRAN
statement runs the simulation for 30 periods to show die build-up of oscillations.
Figure 6.27b shows that die oscillation is superimposed upon a 9.2 V dc collector voltage. By
measuring die average period of the last nine cycles from the output data, we find that^ = 0.9414
MHz. The oscillations reach steady-state conditions after about 15 Lis; however, longer times
should be expected if die capacitors CE and Cc are not replaced by voltage sources.
This simulation demonstrates limiting performed by inherent transistor nonlinearities rather man
by a special limiter circuit Because we used C3/C, = 75 in our design, the loop gain is rather high
and die resulting oscillation has peak amplitude of about 3.5 V. Harmonic distortion is about
13%.
Hartley Oscillator. The Hartley oscillator, Fig. 6.28, is Fig. 6.24b with inductors for Z, and Z,
and a capacitor for Z? For diis circuit, Eq. (6.50) gives
(00L\~——
+ G)oL3=0
tf>oc2
which establishes the oscillator frequency
1
<°o = C (L
2 l
+ L3)
Substituting reactances into Eq. (6.51) gives
(6.57)
216-
-ZmKOok
<0OL3-
1
ZmRo<°oL7>C2
>1
\-<oz0I^C2
0>„C
o«-2
or
(gmR0 + l)^L3C2
>1
Substituting Eq. (6.57) into this expression and simplifying gives the gain condition
(6.58)
Figure 6.28 Hartley oscillator small-signal circuit
6.7.4 Quartz Crystal Oscillators
Frequency Stability. We know that frequency stability is an important consideration in an
oscillator. The Colpitts and Hartley design equations show that in these circuits the frequency
depends entirely upon the reactive components in the resonators. We are therefore concerned
with variations in these components due to aging, temperature, and tolerances, especially since
transistor capacitances are sometimes part of the phase shifter.
Reactance
1/(0 C.
Figure 6.29 Frequency stability of Colpitts oscillator
By defining Ctq = C,C}/(C, + C,) for the Colpitts oscillator, we can view <a0 in Eq. (6.52) as that
frequency where reactances (aL2 and //©C^ are equal, as illustrated by the bold curves in Fig.
6.29. The light curves show how variations in Cf1 and L2 affect the frequency of oscillation. Many
applications, such as radio transmitters or electronic watches, call for oscillators having long-term
frequency variations of the order of 1 part per million (1 ppm) or less. Even well designed RC or
LC oscillators typically have long-term variations of 100 to 1000 ppm. When greater frequency
stability is required than discrete capacitance and inductance can realize, quartz crystals are used
in the phase shifting circuit
Crystal Resonator. Certain materials, such as quartz, display the piezoelectric effect. If we apply
an electric field to these materials, forces on the ions in the crystal lattice deform the material. For
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example, consider a bar of quartz firmly held at the left-hand end but free to flex up and down at
the right-hand end as illustrated in Fig. 6.30. Conducting, e.g. silver, electrodes are plated to the
upper and lower faces of the bar. Under suitable conditions, voltage applied to the electrodes
forces the right-hand end of the bar to move upward. On the other hand, voltage of the opposite
polarity flexes the bar downward. The piezoelectric effect is reciprocal. In other words, if the
terminals are open circuited and a force is applied to flex the bar, a voltage appears across the
electrodes.
\
4
Figure 6.30 Simplified physical structure of piezoelectric crystal
In quartz crystals, slight; reversible, physical deformations result in an electrical voltage, and,
conversely, applied voltage produces physical deformations. The crystal is thus an
electromechanical device in which electrical excitation and mechanical deformation are tightly
coupled, a feature that makes the crystal highly useful as a transducer. Another unusual property
of a quartz crystal is that, once set in motion, its energy losses per cycle are very slight. Its
electrical equivalent circuit is an LC resonant circuit like Fig. 6.31a, in which the energy losses of
the crystal are embodied in the crystal parameter r, and C is associated with the external holder
that makes electrical contacts to the quartz. Representative component values for a 90 kHz
crystal are L = 137 H, C = 0.0235 pF, r- 15 kQ and C - 3.5 pF.
When used as a frequency-determining element, the crystal is mounted so that it can vibrate
freely at die desired frequency. The mechanical vibrations result in an ac current in the external
circuit. In an oscillator circuit, an amplifier maintains the vibrations. Because quartz is an
extremely stable material, frequency variations due to changes in power-supply voltage or
temperature are very small compared to those of UZ or RC oscillators.
C
(a)
(b)
Figure 6.31 Crystal resonator: (a) schematic symbol and electrical equivalent circuit;
(b) approximate reactance curve
We now derive an expression for the reactance of the quartz crystal that allows us to compare the
crystal resonator characteristics with the JJC reactance curves of Fig 6.29. Near resonance where
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me crystal is used, r « ©L. With resistance omitted to simplify die development, the impedance
of the crystal in Fig. 6.31a is
1
jd)L +
-o)2LC + l
j<uC) joaC
Z=
jcoL +
1
1
1
jooC
jaC
- ja>5LCC+ja>(C + C )
where the second expression follows from multiplying numerator and denominator by
{/<aQ{/0)C'). We next factor LC from the numerator and -jCoLCC from the denominator. This
gives
-LC
Z = jX(a>) =
-jcoLCC
a>2-
LC
2_C + C
LCC
-J
<Q*
0)r
2
-ml
aCCa
(6.59)
where
a
''7m and (Op =
I LCC
\c+c
define die series and parallel resonant frequencies of the crystal, respectively. Figure 6.31b is a
sketch of Eq. (6.59). Intuitively, as oo approaches G>s, JL and C go into series resonance, producing
zero reactance. Above this frequency, the inductive reactance dominates the L/C branch, and this
reactance then goes into parallel resonance with C at G)p. The series and parallel resonant
frequencies are very close together in a quartz crystal, 88700 and 88998 Hz, respectively, for the
nominal 90 kHz crystal described earlier. Thus the curve segment between series and parallel
resonance is nearly a vertical line. Furthermore, in a properly cut quartz crystal, Op varies by only
a few hundred ppm over a wide range of temperature. As we see next, this leads to a very stable
oscillator frequency of frequency C0o = (Dp.
X(<o)
Inductive
Capacitive
(b)
Figure 6.32 Pierce oscillator (a) equivalent circuit;
(b) sensitivity of oscillatorfrequencyto capacitor variation
Pierce Oscillator. The crystal-controlled Pierce oscillator is a Colpitts oscillator with inductor
replaced by a crystal as in Fig. 6.32a. Substituting the reactance of the crystal X(©) for the
reactance of Iîn Eq. (6.50) gives
The frequency of oscillation is thus the frequency (Do that satisfies
UJ
arêq
-219-
The graphical construction of Fig. 6.32b shows why the oscillation frequency of the Pierce
oscillator is virtually independent of variations in resonator capacitance. Another popular crystal
oscillator circuit is obtained by replacing one of the inductors in a Hartley oscillator with a crystal.
Usually, a quartz crystal can vibrate in many different ways called modes. For example, returning
to die bar of quartz fixed at one end, the bar could flex up and down. On the other hand, it could
flex sideways. If the widdi and height of the bar are different, the frequency for the sideways
motion is different from that of vertical flexure. Another possibility is for the bar to twist around
its axis.
End
stationary
A
Stationary
point
(c) Third overtone
Figure 6.33 Overtone vibrations
Commonly, there are overtone vibrations for each mode. For example, several overtone
vibrations are shown for vertical flexure of a bar in Fig. 6.33. The lowest frequency is called
fundamental. The «th overtone frequency is nearly — but not exactly — n times the frequency of
the fundamental vibration. {The amplitudes of vibrations in Fig. 6.33 are exaggerated for clarity.
Actual amplitudes of vibration in quartz crystal are much smaller.)
The flexure modes that are illustrated in Fig. 6.33 are not often used for crystals. (An exception is
32,768-Hz crystal used in electronic watches.) We have discussed this mode mainly because it is
easy to illustrate. Typical high-frequency crystals use shear modes. Crystals are practical as
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frequency determining elements for frequencies in die approximate range from 10 kHz to 200
MHz. Below about 30 MHz, the fundamental mode is used. At higher frequencies, overtones are
used.
6.8 Summary
Adding negative feedback to an amplifier reduces sensitivity to parameter variations, increases
bandwidth, and reduces nonlinear distortion. We can also use feedback to increase or decrease
midrange input resistance and (independendy) increase or decrease output resistance. All
improvements involve multiplication or division by the improvement factor 1 +Ap. The closedloop gain is simultaneously reduced by diis factor; however, diis counts as an improvement once
we realize that die new, lower, gain usually approximates 1/p. By using resistor ratios for P we
can control this closed-loop gain to ±1%.
With four classes of feedback, we can make our nonfeedback amplifier more closely resemble an
ideal voltage, current, transresistance, or transconductance amplifier at midrange frequencies. For
each class of feedback, the definition of gain, A, corresponds to that of the ideal amplifier. The
origin of the feedback signal defines the output circuit of the feedback amplifier-voltage (current)
feedback lowers (raises) output resistance, making the original amplifier more closely resemble a
dependent voltage (current) source. The way the feedback signal subtracts from die input signal
determines die nature of die input circuit. Series (shunt) feedback involves a voltage (current)
subtraction that increases (decreases) input resistance. This makes die amplifier better represent a
voltage- (current-) controlled dependent source.
The resistive P circuits we use in practical feedback amplifiers introduce loading at input and
output. To apply ideal feedback equations to this case, we represent die p circuit by a two-port
equivalent - the one that corresponds to the particular kind of feedback we employ.
To successfully implement feedback or evaluate die designs of others, we must be able to
recognize some standard practical feedback topologies. The difference amplifier structure is a
voltage-series feedback arrangement that bases the input subtraction on an amplifier's differential
input. In classical voltage-feedback circuits, as in the difference amplifier topology, the output
node of die nonfeedback amplifier connects direcdy to the P circuit. Current feedback features
indirect sensing of output current using an unbypassed emitter or source resistor. Classical series
feedback involves a connection from die P circuit to an unbypassed emitter or source resistor in
the input circuit of the nonfeedback amplifier; shunt feedback uses a direct connection from die
P circuit to an input node of the nonfeedback amplifier. The A circuit must provide a 180° phase
shift to facilitate the current subtraction needed for shunt feedback; zero phase shift is required
for series feedback.
To ensure that our feedback amplifier does not oscillate, we examine die magnitude and phase of
its loop-gain function, the product of the A circuit and P circuit gains. If diere exists no
frequency where the loop gain is negative and also greater than one in magnitude, the feedback
amplifier will be stable. Gain and phase margins are two measures of the degree of stability that
might be included in design specifications. If die feedback amplifier is destined to be unstable, we
must compensate it by modifying die open-loop gain curve to produce appropriate gain and
phase margins. This process, called frequency compensation, usually involves adding capacitance
to reduce die bandwiddi of die A circuit.
-221 -
Oscillators are circuits intentionally made unstable by positive feedback. Common to all
oscillators are phase shifters, which determine the frequency of oscillation, gain circuits to make
up for energy losses in the phase shifters, and limiters to control oscillation amplitude. The
Barkhausen stability criterion, based upon conditions for positive feedback, gives a complex
valued equation for any oscillator. The Barkhausen conditions give an equation for the frequency
of oscillation and an amplitude condition that must be satisfied for oscillations to occur. In this
chapter only oscillators with sinusoidal output waveforms were considered. Phase-shift and Wien
bridge oscillators use RC circuits for phase shifters; Colpitis and Hartley oscillators employ tuned
circuits.
Important considerations in sinusoidal oscillator design are the harmonic purity of the waveform
and frequency stability, the relative insensitivity of the oscillation frequency to variations in circuit
parameters, and environmental factors. Total harmonic distortion measures the impurity of the
output waveform in terms of the amplitudes of undesired harmonics present. The sensitivity
definitions are useful aids in oscillator analysis and design. SPICE simulations help us address
both distortion and sensitivity issues in a practical fashion. Quartz crystals introduced into
oscillator phase shifters result in great improvements in frequency stability.
Notes
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References
1.
2.
3.
4.
5.
6.
7.
8.
D. Christiansen (Ed.), Electronics Engineers'Handbook, McGraw-Hill, 1996.
A. R. Hambley, Electronics, Macmillan, 1994.
J. Keown, PSpice and Circuit Analysis, Maxwell Macmillan, 1991.
N. R. Malik, Electronic Circuits, Prentice Hall, 1995.
R Mauro, Engineering Electronics, Prentice Hall, 1989.
J. Millman and A. Grabel, Microelectronics, 2 nd edition, McGraw-Hill, 1987.
J. F. Morris, Introduction to PSpice, Houghton Mifflin, 1991.
C. J. Savant, M. £J. Roden and G. L. Carpenter, Electronic Design, The Benjamin/Cummings,
1991.
9. J. Scott, Analog Electronic Design, Prentice Hall, 1991.
10. A. S. Sedra and K. C. Smith, Microelectronic Circuits, Saunders College, 1992.
11. W.J. Tompkins (Ed), Biomedical Digital Signal Processing, Prentice-Hall, 1993.
12. P. W. Tuinenga, SPICE: A Guide to Circuit Simulation and Analysis Using PSpice, Prentice Hall,
1992.
-223-
Analog Electronics /Review Questions
Review Questions
Chapter 1: introduction
1.1 List five examples of electronic systems. Try to think of new examples that have not been
mentioned in this chapter.
1.2 Discuss briefly what the spectrum of a signal is and why it is important.
1.3 Give five examples of useful signals and their spectra.
1.4 List five types of functional blocks found in electronic systems.
1.5 Discuss how analog signals can be converted to digital form.
1.6 List the relative advantages of digital systems compared to analog systems and vice versa.
1.7 Explain why at certain frequencies a physical resistor can be perceived as a capacitor or
inductor.
Chapter 2: Amplifiers
2.1 How does an inverting amplifier differ from a noninverting one?
2.2 Draw the voltage amplifier model. Is the gain parameter measured under open-circuit or
short-circuit conditions? Repeat for current, transconductance and transresistance models.
2.3 What are "loading effects" in an amplifier circuit?
2.4 Draw the cascade connection of two amplifiers. What is the voltage gain of the cascade
connection in terms of the voltage gains of the individual stages?
2.5 Define the efficiency of a power amplifier. What form does the dissipated power take?
2.6 How is power gain converted to decibels? Voltage gain?
2.7 Give the input and output resistances of an ideal voltage amplifier. Repeat for other ideal
amplifier types.
2.8 Sketch the gain magnitude of a typical dc-coupled amplifier versus frequency. Repeat for an
ac-coupled amplifier.
2.9 How is a narrowband amplifier different from a wideband one?
2.10 Discuss the Miller theorem. Explain its significance to circuit analysis.
2.11 What are the requirements for the gain magnitude and phase of an amplifier so that linear
distortion does not occur?
2.12 Sketch the pulse response of an amplifier, showing the rise time, overshoot, ringing and tilt.
Give an approximate relationship between rise time and the upper half-power frequency of a
broadband amplifier. Give an approximate relationship between percentage tilt and the lower
half-power frequency.
2.13 What is the compensated oscilloscope probe? What for is it used in electronics?
2.14 What is harmonic distortion? What causes it? Is it a problem for narrowband amplifiers?
Explain.
2.15 Discuss briefly intermodulation and crossmodulation.
-224-
Chapter 3: Diode Circuits
3.1 Draw the circuit symbol for a diode. Label the anode and the cathode. Make a reference to
die p-type and »-type regions of a corresponding/^ junction.
3.2 Draw the volt-ampere characteristic of a typical diode and label the various regions of
operation.
3.3 What is a Zener diode? For what is it typically used? What are two other names for it?
3.4 Draw a circuit diagram of a simple voltage regulator.
3.5 Draw the volt-ampere characteristic of an ideal 5.8-V Zener diode.
3.6 Write the Schockley equation and define all the terms.
3.7 How does the forward voltage of a silicon diode change with temperature for a fixed value of
current? Explain.
3.8 What is an ideal diode? Draw its characteristic.
3.9 After solving a circuit with ideal diodes, what check is necessary for diodes initially assumed
to be on? Off?
3.10 If a nonlinear two-terminal device is modeled by die piecewise-linear approach, what is die
equivalent circuit of die device for each linear segment?
3.11 A resistor Ra is in series with a voltage source Va. Draw the circuit. Label die voltage across
die combination as v and the current as /. Draw and label the volt-ampere characteristic (I
versus v).
3.12 Draw the circuit diagram of a half-wave rectifier for producing a nearly steady dc voltage
from an ac source. Include a transformer to adjust the voltage level. Draw two different fullwave circuits.
3.13 What is a clipper circuit? Draw an example circuit diagram including component values, an
input waveform, and the corresponding output waveform.
3.14 Repeat Question 3.13 for a clamp circuit.
3.15 Draw die circuit diagram of a two-input diode OR gate. Repeat for an A N D gate.
3.16 Logic 1 voltages for a digital system are defined as being larger than 3V. Consider a number
k of two-input OR gates employing silicon diodes. Connect one input of each gate to the
ground (logic 0). Cascade the gates such diat the output of a gate is connected to the
remaining input of another gate that follows in the cascade. Apply the 5-V voltage to die input
and measure die output of the cascade. At what value of k die output voltage can not be
recopiized as logic 1?
3.17 Try to construct a logic inverter using diodes. Can you succeed? Based on results you
obtained answering Questions 3.16 and 3.17, describe two serious drawbacks of diode logic
circuits.
3.18 Of what does die small-signal equivalent circuit of a diode consist?
3.19 How is the dynamic resistance of a nonlinear circuit element determined at a given
operating point?
3.20 Sketch die voltage and current waveforms corresponding to large-signal diode switching.
Define and explain the delay times in die diode response.
3.21 What is die Schottky diode? What are its properties and applications?
3.22 Explain die difference between the photodiode and solar cell.
3.23 What is die construction of an optical isolator? List applications of diis device diat are
known to you.
Chapter 4: FET Circuits
4.1 Sketch the simplified physical structure of an »-channel JFET. Label the terminals and die
channel region. Draw the corresponding circuit symbol.
-225-
4.2 In normal operation, what bias condition exists between gate and channel of JFET?
4.3 Define the pinch-off voltage and IDSS of a JFET.
4.4 Write an equation for the drain current of a JFET in the saturation (pinch-off) in terms of
device voltages.
4.5 Sketch the characteristics of an «-channel JFET. Label the saturation, linear and cutoff
regions. Repeat for the/>-channel device.
4.6 Give the ranges of vGS and vGD in terms of the pinch-off voltage Vp for each region (cutoff,
saturation and linear) of an »-channel JFET and depletion MOSFET.
4.7 Sketch the physical structure of an ^-channel depletion MOSFET. Label the terminals and
the channel region. Draw the corresponding circuit symbol. Repeat for a ^-channel
enhancement MOSFET.
4.8 Explain the physical origin of the variations in FET parameters with temperature.
4.9 What is "gate protection" for a MOSFET? Why is it necessary?
4.10 Draw the load line for a simple FET amplifier.
4.11 Why does nonlinear distortion occur in FET amplifiers?
4.12 Draw the diagram of the fixed-bias circuit, the self-bias circuit and the fixed- plus self-bias
circuit for a FET. In general, which circuit maintains the most constant drain current from
device to device? Which shows the greatest variation? Which is used for enhancement
devices? Why?
4.13 Draw the small-signal equivalent circuit for the FET including r^
4.14 Give definitions of^n and rrf as partial derivatives.
4.15 Draw the circuit diagram of a resistance-capacitance coupled common-source amplifier.
Repeat for the source follower. Which amplifier would be used if a voltage gain magnitude
larger than unity is needed? Which would be used to obtain low output resistance?
4.16 Draw the circuit diagram of the FET amplifier most useful if extremely high input resistance
is required.
4.17 What is the function of coupling capacitors? With what are they replaced in a midband
small-signal equivalent circuit? In general, what effect do the coupling capacitors have on the
gain of the amplifier as a function of frequency?
4.18 Compare gain, bandwidth, input and output impedance values of basic FET amplifier
configurations.
4.19 What is the value of ^n for VDS=0? Draw the small-signal equivalent circuit at this bias
point. For what applications is the FET used at this bias point?
4.20 Draw the circuit diagram of a CMOS inverter. Repeat for a two-input NOR gate.
4.21 Of what does the input impedance of a CMOS inverter consist?
4.22 What is the static power consumption of CMOS gates?
4.23 Draw the circuit diagram of the CMOS transmission gate.
4.24 Draw the dynamic circuit model of the »-channel JFET. Repeat for/(-channel device and all
kinds of MOSFET transistors.
4.25 Explain how the channel-length modulation effect is incorporated into the SPICE model of
the FET.
Chapter 5: BJT Circuits
5.1 Draw the circuit symbol for an npn BJT. Label the terminals and the currents. Choose
reference directions that agree with the true current direction for operation in the active
region.
5.2 Repeat Question 5.1 for zpnp transistor.
5.3 In normal operation, which type of bias (forward or reverse) is applied to the emitter-base
junction? The collector-base junction?
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5.4 To forward-bias apn junction, which side of the junction should be connected to the positive
voltage?
5.5 Write the Shockley equation for the emitter current of an npn transistor.
5.6 Give the definition of a and p for a BJT. What bias conditions for each junction are assumed
in diese definitions?
5.7 Sketch the input characteristic curve for a typical small-signal silicon npn BJT at room
temperature. Sketch the output characteristic curves if P = 100. Isabel the cutoff, active and
saturation regions.
5.8 Draw the load lines on the input and output characteristic planes of the npn BJTfor a simple
amplifier. Repeat for thepnp device.
5.9 Why does distortion occur in BJT amplifiers?
5.10 Sketch the output characteristics of a BJT, illustrating the Early voltage and collector
breakdown.
5.11 Explain the base-width modulation effect.
5.12 Discuss the BJT parameter variation with temperature.
5.13 What is the typical extreme variation of P from unit to unit for a given type of BJT?
5.14 How does vBh vary with temperature for a fixed emitter current? Assume a small-signal
silicon transistor.
5.15 Draw the large-signal dc circuit model for a silicon npn transistor in the active region at
room temperature. Include the constraints of currents and/or voltages that guarantee
operation in the active region. Repeat for the saturation region. Repeat for the cutoff region.
5.16 Repeat Question 5.12 for &pnp transistor.
5.17 In the active region, how is the base-collector junction biased (forward or reverse)? How is
the base-emitter junction biased?
5.18 Repeat Question 5.14 for the saturation region.
5.19 Repeat Question 5.14 for the cutoff region.
5.20 Briefly discuss the procedure for dc analysis of a BJT circuit using the large-signal circuit
models.
5.21 Draw the fixed base bias circuit. What is the principal reason that this circuit is unsuitable
for mass production of amplifier circuit?
5.22 Draw the four-resistor bias circuit for the BJT. Give the rule-of-thumb design guidelines for
this circuit.
5.23 Whv are coupling capacitors often used to connect the signal source and the load to
amplifier circuits? Should coupling capacitors be used if it is necessary to amplify dc signals?
Explain.
5.24 Draw the small-signal equivalent circuit for the BJT.
5.25 Give the formula for determination of rx, assuming that P and the Q-point are known.
5.26 Draw the circuit diagram of a common-emitter amplifier circuit that uses the four-resistor
biasing network. Include a signal source and a load resistance.
5.27 Repeat Question 5.23 for an emitter follower. What resistance did you omit from the bias?
Why?
5.28 For a small-signal midband analysis of an amplifier, with what do we replace the coupling
capacitors? Dc voltage sources? Dc current sources? Very large inductors?
5.29 Outline the small-signal analysis procedure to find the output resistance of an amplifier.
5.30 What are three important features in the structure of a BJT for high P (i.e., for the base
current to be small compared to the collector current in the active region of operation)? ,
5.31 Sketch the common-emittet input characteristics of an npn transistor. Indicate the effect of
base width modulation.
5.32 Prepare a table showing the bias conditions (forward or reverse) for the collector junction in
each of the four regions of operation of a BJT.
-227-
5.33 Draw the common-emitter h-parameter equivalent circuit for the BJT, labeling each
parameter.
5.34 Draw the hybrid-7l model for the BJT. Characterize each element of the circuit.
5.35 Discuss the P dependence of frequency.
4.26 Compare gain, bandwidth, input and output impedance values of basic BJT amplifier
configurations.
5.36 What is the cascode amplifier? What are its properties?
5.37 Draw the Ebers Moll dynamic model for the BJT. Characterize each element of the model.
5.38 Draw the circuit diagram of an RTL inverter. Sketch a positive input pulse and the
corresponding output voltage. Label the delay time, the rise time, the storage time, and the fall
time.
5.39 For fast switching, do we want a BJT with a thin base region or a thick base region?
Explain.
5.40 Draw the circuit diagram of an RTL inverter including a speed up capacitor and a Schottky
clamp diode. Discuss how the Schottky clamp diode improves switching time.
5.41 What is the function of the rt buried layer under the collector region of an npn BJT on an
IC?
Chapter 6: Feedback Circuits
6.1 List four benefits that potentially result from the use of negative feedback.
6.2 What problems are associated with positive feedback in amplifiers?
6.3 Under what condition is feedback able to reduce nonlinear distortion? Draw the feedback
amplifier circuit and derive appropriate expression.
6.4 Define signal-to-noise ratio. Under what condition is feedback able to improve signal-tonoise ratio? Draw the feedback amplifier circuit and derive appropriate expression.
6.5 Define the following terms: voltage feedback, current feedback, series feedback and parallel
feedback.
6.6 Describe a way to test a circuit for the presence of voltage feedback. Draw a schematic
diagram of such a feedback circuit. Repeat for current feedback.
6.7 In a series feedback, we usually consider the input signal to be a voltage. Explain why. In a
parallel feedback we usually consider the input signal to be a current. Explain why.
6.8 Sketch the circuit diagram of the simple class B amplifier that was discussed in this chapter.
What causes crossover distortion in this circuit?
6.9 List four types of feedback and give the appropriate amplifier gain parameter for each type.
Also give the units of P for each type.
6.10 What type of negative feedback should be employed to increase input impedance? To
reduce input impedance?
6.11 What type of negative feedback should be employed to make the amplifier output behave
as a nearly ideal voltage source? As a nearly ideal current source?
6.12 What type of (negative) feedback should be used to obtain a nearly ideal current amplifier?
Transconductance amplifier? Voltage amplifier? Transresistance amplifier?
6.13 Draw the circuit diagram of a negative feedback amplifier, including a resistive feedback
network for series current feedback; for parallel current feedback; for series voltage feedback;
for parallel voltage feedback. In each case give the value of the feedback ratio in terms of the
resistor values. Assume an amplifier having a differential input.
6.14 In series feedback we usually try to select small values for the resistors in the feedback
network. Explain why.
6.15 In parallel feedback we usually try to select large values for the resistors in the feedback
network. Explain why.
-228-
6.16 In voltage feedback we usually try to select large values for the feedback resistors. Explain
why.
6.17 In current feedback we usually try to select small values for the feedback resistors. Explain
why.
6.18 Define gain margin and phase margin for a feedback amplifier.
6.19 What are the rule-of-thumb minimum values of gain margin and phase margin used in
design of feedback amplifiers?
6.20 Can a single-pole amplifier become unstable if negative feedback having constant feedback
ratio P is employed?
6.21 Explain the Barkhausen criterion of oscillations in a feedback circuit.
6.22 Draw the circuit diagram of a Wien-bridge oscillator. Explain its operation.
6.23 Draw the circuit diagram of a Wien-bridge oscillator. Explain its operation.
6.24 Explain the role of amplitude limiters in oscillator circuits. Draw schematic diagrams of
diode and FET limiters circuits used to stabilize the output waveform generated by a Wienbridge oscillator.
6.25 Draw a block diagram of LC oscillators. Modify it to obtain (a) Collpits, and (b) Hartley
oscillator.
6.26 Briefly describe the piezoelectric effect.
6.27 What is a crystal as the term used in relation to oscillator circuits?
6.28 Draw the equivalent circuit of a crystal and sketch its reactance versus frequency. Label the
series resonant frequency and the parallel resonant frequency.
6.29 A crystal has a fundamental mode at 10 MHz. What is the approximate frequency of the
second overtone? Third overtone?
6.30 Briefly discuss the way to use quartz crystal to stabilize the frequency of a conventional LC
oscillator.
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Analog Electronics /Problems
Problems
Chapter 1: Introduction
1.1 A sinusoidal signal source has an open-circuit voltage of 10 mV and a short-circuit current of
10 u A. What is the source resistance?
1.2 Give expressions for the sine-wave voltage signals having:
(a) 10-V peak amplitude and 10-kHz frequency,
(b) 120-V rms and 60-Hz frequency,
(c) 0.2-V peak-to-peak and 1000-rd/s frequency,
(d) 100-mV peak and 1-ms period.
1.3 Illustrate the composition of a square-wave signal by sketching the first four terms of its
Fourier series and then by performing graphical summations.
1.4 For a square-wave audio signal, what fraction of the available signal energy is perceived by an
average adult listener of age 40 whose hearing extends only to 16 kHz?
1.5 What fraction of the energy contained in a square wave of frequency / and peak-to-peak
amplitude U is contained in the harmonic at frequency 9ft
Chapter 2: Amplifiers
2.1 A signal source with an open-circuit voltage of Vs = 2mV rms and an internal resistance of
50kA is connected to the input terminals of an amplifier having an open-circuit voltage gain of
100, an input resistance of lOOkQ and an output resistance of 4Q. A 4-fi load is connected to the
output terminals. Find the voltage gains A„ = vjvt and Ar - vjvt Also find the current gain and
the power gain.
2.2. A certain amplifier has an open-circuit voltage gain of unity, an input resistance of 1MQ, and
an output resistance of 100Q. The signal source has an internal voltage of 5mV rms and an
internal resistance of lOOkfl. The load resistance is 50£X If the signal source is connected to the
amplifier-input terminals and the load is connected to the output terminals, find the voltage
across the load and the power delivered to the load. Next consider connecting the load direcdy
across the signal source without the amplifier and again find the voltage and power. Compare the
results. What do you conclude about the usefulness of a unity-gain amplifier in delivering power
to the load?
2.3. An amplifier has an open-circuit voltage gain of 100. With a 10-kQ load connected, die
voltage gain is found to be only 90. Find the output resistance of the amplifier.
2.4. The output voltage v, of the circuit of Fig. P2.4 is lOOmV with the switch closed. With the
switch open, the output voltage is 50mV. Find the input resistance of the amplifier.
4-r"T-4-(>
v,(0
— c
Figure P2.4
-230-
2.5. Two amplifiers have the characteristics shown in Table P2.5. If the amplifiers are cascaded in
order A-B, find the input impedance, output impedance and an open-circuit voltage gain of the
cascade. Repeat if die order is B-A.
2.6 A certain amplifier has an input voltage of lOOmV rms, an input resistance of lOOkfl, and
produces an output of lOVrms across an 8-Q resistance. The power supply has a voltage of 15V
and delivers an average current of 2A. Find the power dissipated in die amplifier and the
efficiency of the amplifier.
2.7. An amplifier has an input voltage of lOmV rms and an output voltage of 5V rms across a 10Q load. The input current is lîA rms. Find the input resistance. Find the voltage gain, current
gain and power gain as ratios and in decibels.
2.8. An amplifier has a voltage gain of 30dB and a current gain of 70dB. What is the power gain
in decibels? If the input resistance is 100k£2, what is the load resistance?
2.9. Find the voltage across a 50-fi resistance corresponding to (a) lOdBV, (b) -30dBV, (c) 10
dBmV, (d) 20dBW.
2.10. Find the power levels in watts corresponding to (a) 20dBm, (b) -60dBW, and (c) lOdBW.
2.11. An amplifier has an input resistance of lOOCt, an output resistance of 10Q, and a shortcircuit current gain of 500. Draw the current and voltage amplifier models for the amplifier,
including numerical values of all parameters. Repeat for transconductance and transtesistance
models.
2.12. Amplifier A has an input resistance of 1MQ, an output resistance of 200Q and an opencircuit transresistance gain of 100MQ. Amplifier B has an input resistance of 50fi, an output
resistance of 500k£2 and a short-circuit current gain of 100. Find the voltage amplifier model for
the cascade of A followed by B. Find the corresponding transconductance amplifier model.
2.13 Repeat Problem 2.12 if the order of the cascade is changed to B-A.
2.14. An ideal transconductance amplifier having a short-circuit transconductance gain of 0.1 S is
connected as shown in Fig. P2.14. Find the resistance JR,,= vjix seen at the input terminals.
Figure P2.14
-231-
2.15. Repeat Problem 2.14 if the amplifier has an input resistance of lkft, an output resistance of
20Q and an open-circuit transresistance gain of 1 Okfi.
2.16. An amplifier has an input resistance of 1Q, an output resistance of 1Q, and an open-circuit
voltage gain of 10. Classify this amplifier as an approximate ideal type and find the corresponding
gain parameter. In deciding of an amplifier classification, assume that the source and load
resistances are on the order of lkQ.
2.17. Repeat Problem 2.16 if the input resistance is 1MQ, the output resistance is lMft, and the
open-circuit voltage gain is 100.
2.18. In a certain application, an amplifier is needed to sense an open-circuit voltage of a source
and force current to flow through a load. The source and load resistances are variable. The
current delivered to the load is to be nearly independent of both the source resistance and load
resistance. What type of amplifier is needed? If the source resistance varies from 1 to 2kQ and
this causes a 1-% decrease in load current, what is the value of the input resistance? If the load
resistance varies from 100 to 300Q and this causes a 1-% decrease in load current, what is the
value of the output resistance?
2.19. The input signal to an amplifier is v,(t) = 0.01COS(2000TI/) + 0.02cos(40007t/). The gain of the
amplifier as a function of frequency is given by
100
A
~ \+
j(f/1000)
Find an expression for the output signal of the amplifier as a function of time.
2.20. The input signal to an amplifier is the same as in Problem 2.19. The complex gain of the
amplifier at lOOOHz is 100Z-45 0 . What complex value must the gain have at 2000Hz for
distortionless amplification?
2.21. Consider the simple low-pass filter shown in Fig. P.21.
(a) Find the complex gain A = V2I Fir as a function of frequency. What are the magnitudes of A
at dc and at very high frequencies? Find the half-power bandwidth B of the circuit in terms of R
and C.
(b) Consider the case for which the capacitor is initially uncharged and v,{l) is a unit step function.
Find v2(t) and an expression for its rise time tr in terms of R and C.
(c) Combine the results found in parts (a) and (b) to obtain a relationship between bandwidth and
rise time for the circuit. Compare the results to Equation (2.22).
R
—O
Figure P.21 Low-pass filter
2.22. Consider the simple high-pass filter shown in Fig. P.22a.
(a) Find the complex gain A = V2I Vt as a function of frequency.
(b) What is the magnitude of the gain at dc? At very high frequencies? Find the half-power
frequency in terms of R and C.
-232-
(c) Consider the input signal shown in Fig. P.22b. Assuming that the capacitor is initially
uncharged, find an expression for the output voltage v^t) for / between 0 and T. Assuming diat
RC is much greater than T, find an approximate expression for percentage tilt
(d) Combine the results of parts (b) and (c) to find relationship between percentage tilt and halfpower frequency.
v;(0
0
(a)
T
(b)
t
Figure P22 High-pass filter (a), its input signal (b)
2.23. An audio amplifier is specified to have half-power frequencies of 15Hz and 15kHz. The
amplifier is to be used to amplify the pulse shown in Fig. P.22b. Estimate the rise time and
percentage tilt of the amplifier output The pulse width T is 2ms.
2.24. The gain magnitudes of several amplifiers are shown versus frequency in Fig. P.24. If the
input to the amplifiers is the pulse shown in the figure, sketch the output of each amplifier versus
time. Give quantitative estimates of as many features of each waveform as you can.
A(/)
(a)
100kHz
Figure P24
2.25. The input signal and the corresponding output are shown for several amplifiers in Fig. P.25.
Sketch the gain magnitude of each amplifier versus frequency. Give quantitative estimates of as
many features on die gain sketches as you can.
2.26. (a) A 1-kHz sinusoid is applied to the input of a nonlinear amplifier. list the frequencies of
at least six frequency components that might be present at the amplifier output.
-233-
(b) Repeat if the input is the sum of a 1-kHz sinusoid and a 1.1-kHz sinusoid.
2.27. List the frequencies of all the components of [cos(27l£j)+cos(27l/J/)]2. Repeat for the cube
and fourth power. (Hint: Make use of trigonometric identities).
Chapter 3: Diode Circuits
Solve the following problems. Verify your results with SPICE where possible.
3.1. Recall that the forward voltages of low-current silicon diodes decrease about 2mV/K. Such a
diode has a voltage of 0.6 V with a current of 1 mA at a temperature of 25°C. Find the diode
voltage at 1 mA at a temperature of 175°C.
3.2. Sketch J versus v fot the circuits shown in Fig. P3.2. The diodes are typical low-current
silicon devices at 300 K. The reverse-breakdown voltages of the Zener diodes are shown.
Assume 0.6 V for all diodes in the forward-bias region.
-234-
Analog
Electronics/Problems
3.3. Use graphical load-line analysis to find the currents and voltages labeled in die circuits shown
in Fig. P3.3. The following relation describes the I-V characteristic ofthe nonlinear element X1
\
0,
V<0.5
I=where voltage is in volts and current is in milliamperes. The chai j-T*îstic of element X2 is given
as
1
1=4 -0.5
[l + cxp(-3V)
where voltage is in volts and current is in milliamperes.
/N2kO
-235-
3.6. Design a voltage regulator circuit to provide a constant voltage of 10 V to a load from a
variable supply voltage. The load current varies from 0.1 A to 0.5A and the source voltage varies
from 12 to 15 V. You may assume that ideal Zener diodes are available. Resistors should be
standard 5% values. Draw the circuit diagram of your regulator and specify the value of each
component. Also find the worst-case (maximum) power dissipated in each component in your
regulator.
3.7. Consider the circuit shown in Fig. P3.7. The diodes are identical and have »=1. The
temperature of each diode is 300 K. Before the switch is closed, the voltage v is 600mV. Find v
after the switch is closed. Repeat for »=2. Neglect the effect of device self-heating.
1mA
Figure P3.7
3.8. A junction diode has »=1 at 300K with a current of 1 mA and a voltage of 600 mV. By how
much must the voltage be increased to (a) double the current? (b) increase the current by one
order of magnitude? Repeat parts (a) and (b) if «=2.
3.9. Consider the diodes shown in Fig. P3.9. The diodes are identical and have »=1. For each
diode a forward current of 100 mA results in a voltage of 700 mV at a temperature of 300K. (a) if
both diodes are at 300K, what are the values of IA and 7B? (b) If diode A is at 300 K and diode B
is at 305 K, again find IA and JB. Assume that I, doubles in value for a 5-K increase in
temperature. (Hint: For part (b), a transcendental equation for the voltage across the diodes can
be found. Solve, it by trial and error. An important observation to be made from this problem is
that, starting at the same temperature, the diodes should theoretically each conduct half of the
total current. However, if one diode conducts slightly more, it becomes warmer, resulting in even
more current Eventually, one of die diodes "steals" most of the current. This is particularly
noticeable with large currents for which significant heating occurs. One cannot straight connect
diodes in parallel to increase maximum available current.)
-236-
3.10 A certain diode has n - 1 and Rt= 0. At 300 K, vD - 650 mV when iD = 1 mA. Plot iD versus
% for this diode at 300 K using linear axis for the voltage and logarithmic axis for the current
Allow the current to range from 0.1 to 100 mA. Repeat if the diode has a series resistance of R, =
ion.
3.11. Find the values of I and V for the circuits of Fig. P3.11 assuming the diodes are ideal.
Figure P3.ll
3.12. Power is available from a 220-V 50-Hz ac source. Design a half-wave rectifier power supply
to deliver an average voltage of 9 V with a peak-to-peak ripple of 2 V to a load. The average load
current is 100 mA. Assume that ideal diodes are and transformers are available. Draw the circuit
diagram for your design. Specify the values of all components used. Be sure to give the turns
ratio for the transformer.
3.13. Repeat the Problem 3.12 using a mil-wave bridge rectifier.
3.14. Repeat Problem 3.12 using two diodes and a center-tapped secondary winding to form a
full-wave rectifier.
3.15, Repeat Problem 3.12 assuming diodes having forward drops of 0.8 V.
3.16. Consider the circuit of Fig. P3.16 that contains ideal diodes. The capacitors are very large,
so they discharge only a very small amount per cycle. (Thus no ac voltage appears across the
capacitors and the ac input plus the dc voltage of C, must appear at point A) Sketch the voltage
at point A versus time. Find the voltage across the load. Why this is called a voltage doubler?
What is the peak inverse voltage across each diode?
Figure P3.16
3.17. Design a clipper circuit to clip off the portions of an input voltage that fall above 3 V or
below -5 V. The input voltage ranges from -10 V to 10 V. Assume that diodes having a constant
voltage drop of 0.7 V are available. Ideal Zener diodes of any breakdown voltage required are
available. Use standard 5% resistor values and design for a peak current of about 1 mA in the
diodes. The only supply voltages available are ±15 V, if needed
-237-
3.18. Repeat Problem 3.17 if the clipping levels are +2 V and +5 V (Le. every part of the input
waveform below 2 V and above 5 V is clipped off).
3.19. Design circuits that have the transfer characteristics shown in Fig. P3.22. Assume that v-m
ranges from -10 to +10 V. Use diodes, Zener diodes and standard 5% resistor values. Assume a
0.6-V forward drop for all diodes and that the Zener diodes have ideal characteristics in the
breakdown region. Power supply voltages of ±15 V are available.
v0(V)
v,(V)
(b)
Figure P3.22
3.20. Design a clamp circuit to clamp the negative extreme of a periodic input waveform to - 5 V.
Use diodes, Zener diodes and standard 5% resistor values. Assume a 0.6-V forward drop for all
diodes and that the Zener diodes have ideal characteristics in the breakdown region. Power
supply voltages of ±15 V are available.
3.21. Repeat Problem 3.22 for a clamp voltage of +5 V.
3.22. Current-voltage relationship for a certain breakdown diode is described as follows:
iD
=
-10"
[mA],
for - 5 V < vD < 0
(••#
Plot iD versus vD in the reverse-bias region. Find the dynamic resistance of this diode at ID = 1
mA and at ID = -10 mA.
3.22. Consider the voltage regulator circuit shown in Fig. P3.26. The ac ripple voltage is 1 V
peak-to-peak. The dc load voltage is 8 V. What is the jg-point current in the Zener diode? What
is the maximum dynamic resistance allowed for the Zener diode if the output ripple is to be less
than 10 mV peak-to-peak?
-238-
Analog Electronics / Problems
25fi
rtîoon
Figure P3.22
Chapter 4: Field-Effect Transistor Circuits
4.1. An //-channel JFET has Vf = -3 V and IDSS = 9 mA. Assuming operation in the saturation
region, what value of vcs is required for iD — 4 mA?
4.2. The FET of Problem 4.1 has y G f = - l V . For what range of vDS is the device in die saturation
region? Repeat for vcs = - 2 V.
4.3. The FET of Problem 4.1 has vGS = - 1 V and vDS = 1 V. Find the drain current. Repeat for v^
= - 1 V and vDS = 5 V.
4.4. For what range of vGS is the FET of Problem 4.1 in cutoff? Assume that vDS > 0.
4.5. Consider the circuit shown in Figure P4.5. It is found that as VDD is increased, the voltmeter
reading increases until VDD reaches 16 V, after which the reading is constant at 13 V. What are
die values of IDSS and Vp for the FET?
Voltmeter
Figure P4.5
4.6. An n-channel enhancement MOSFET has Vlt = 3 V and K = 0.5 mA/V 2 . If vGS = 5 V, for
what range of vDS is the device in die saturation region? In the triode region? Plot iD versus va in
die saturation region.
4.7. An //-channel depletion MOSFET has VP = -4 V and K = 0.25 mA/V 2 . Find the value of
IDSS. Plot die boundary between the triode region and die saturation region on die iD-vm plane.
4.8. A p-channel JFET has Vp = 4 V and 1DSS = -16 mA. Sketch die drain characteristics to scale
for vDS ranging from 0 to - 10 V. Show the curves for vGS = 0,1, 2, 3, and 4 V.
4.9. The voltmeter shown in Figure P4.9 has very high impedance. Approximately what value
does die*meter read?
-239-
15V
+
1 Voltmeter
Figure P4.9
4.10. A />-channel enhancement MOSFET has Vth - -6 V and K = -2 mA/V 2 . Assuming
operation in the saturation region, what value of vGS is requited for iD — 8 mA?
4.11. For the circuits shown in Figure P 4 . l l , find the currents and voltages labeled. For each
FET, | Vp\ - 2 V and \IDSS\ = 8 mA.
R
15V
i
15V
R
30V
(b)
Figure P4.11
4.12. Consider the amplifier shown in Figure P4.12. (a) Find vcs(i) assuming that the coupling
capacitor is a short circuit for the ac signal, (b) If the FET has Vth = 5 V and K = 0.5 mA/V 2 ,
sketch its drain characteristics to scale for %y = 5,6, 7, and 8 V. (c) Draw the load line for the
amplifier on the characteristics, (d) Find the values oi.VDS, vDSmill, and vDSmM.
Figure P4.12
4.13. What is the largest value of RD allowed in the circuit of Problem 4.12 if the instantaneous
operating point is required to remain in the saturation region at all times?
4.14. Find the values of ID and VDS for each of the circuits shown in Figure P4.14. Assume that
Vp= .4 V and IDSS = 8 mA for all FETs.
-240-
15V
Figure P4.14
4.15. (a) Find the value of ID for the circuit shown in Figure P4.15. Assume that Vlh = 4 V and K
= ImA/V 2 . (b) Repeat for Vlh - 2 V and K = 2 mA/V 2 .
1MQ
lkQ
20V
lkfl
IM«
Figure P4.15
4.16. Find the value of Rs if ID = 4 mA in the circuit of Figure P4.16. Assume that Vp = -3 V, IDSS
= 18 mA, and operation is in saturation. What is the largest value o f Rp allowed if the operating
point must remain in the saturation region?
20V
lOOkQ
Figure P4.16
4.17. The F E T o f Figure P4.17 has Vp = -2V and IDSS = 4 mA. If ID = 9 mA, find the value of
R2, assuming operation in the saturation region. What is the largest value of R 0 allowed if the
operating point must remain in the saturation region?
-241-
Figure P4.17
4.18. Repeat Problem 4.17 if the depletion MOSFET is replaced with an «-channel enhancement
MOSFET having Vlh = 4V and K = 1 mA/V 2 .
4.19. Two identical JFETs are connected in parallel: gate-to-gate, source-to-source, and drain-todrain. Each FET has the parameters g*, IDSS, and Vp. Find the parameters g'^ I'DSS, and V'p of a
single JFET that is equivalent to the parallel combination.
4.20. Find the value of the input resistance of the amplifier shown in Figure P4.12. Assume that
the coupling capacitor is a short circuit for the frequencies of interest
4.21. Find midband values of the voltage gain, input resistance, and output resistance for the
common-source amplifier shown in Fig. P4.21. The transistor has Vp = -3 V and IDSS = 9 mA.
Assume that rd - «>.
h
1.2kO
4
v -20V
Figure P4.21
4.22. Repeat Problem 4.21 if VP = -1 V and IDSS = 1 2 mA. Compare the results.
RD
fi
2.2kO
20V
Figure P4.23
-242
Analog Electronics/Problems
4.23. Find VDS and ID for the FET shown in Figure P4.23 given Krt '= 3 V and K = 0.5 mA/V 2 .
Find the value of £, at the operating point. Draw the small-signal equivalent circuit assuming that
rd = oo. Derive an expression for the resistance R„ in terms of Rp and &,. Evaluate the expression
for the values given.
4.24. Consider the amplifier shown in Figure P4.24.
(a) Draw the small-signal midband equivalent circuit.
(b) Assume that 0 = 0 0 and derive expressions for the voltage gain, input resistance, and output
resistance.
(c) Find ID if R = 100 kQ, Rf = 100 kQ, Rp = 3 kQ, RL = 10 kQ, VDD = 20 V, Vlb = 5 V, and K
- 1 mA/V 2 . Determine the value of g, at the j2-point.
(d) Evaluate the expressions found in part (b).
(e) Find v.(t) & VW = 0.2 sin(2000n/).
(f) Is this amplifier inverting or noninverting? Would you classify-the input resistance as high,
moderate, or low compared to other FET amplifier types?
20V
Figure P4.24
4.25. Consider the common-gate amplifier shown in Figure P4.25.
(a) Draw die small-signal midband equivalent circuit.
(b) Assume that rd = 00 and derive expressions for the voltage gain, input resistance, and output
resistance.
(c) Fin* ID if R = 100 Q, Rs - 1 kQ, RD = 6,8 kQ, RL = 10 kfl, VD£ = 20 V, VP = -2 V, and IDSS
= 8 mA. Determine the value of g, at the Q-point.
(d) Evaluate the expressions found in part (b).
(e) Find »,(/) if v(/) = 0.1 sin(20007i/).
(f) Is this amplifier inverting or noninverting? Would you classify the input resistance as high,
moderate, or low compared to the common source amplifier?
R
Cl
C
2
I
VD
to
O——'
Figure P4.25
4.26. A depletion MOSFET is to be used as a voltage-controlled resistor with VDS = 0. The
device has Vp = -2 V and IDSS - 8 mA. Find rd for VGS = -3, 2, -1, 0, and +0.5 V.
-243-
4.27. Consider the circuit shown in Figure P4.27. The FET has K P = - 3 V and IDSS = 9 mA.
(a) Draw die small-signal equivalent circuit. D o not assume tiiat the capacitor is a short circuit
(b) Derive an expression for die voltage gain as a function of frequency, r^ and C.
(c) Find the values of rd for v^ = -3, - 1 , and + 1 V
(d) If C - 0,01 J4.F, sketch die magnitude of die voltage gain versus frequency for each of the
values found in part (c).
C
Figure P4.27
4.28. Consider the CMOS inverter of Figure 4.36. Assume diat the FETs have \Vlb\ — 3 V and
| K\ - 0.1 mA/V 2 . Find die current drawn from die source if Vin = VDD/2 for VDD = 5,10, and
15 V. Repeat for Vfc = 0.
4.29. Consider die CMOS inverter shown in Figure 4.36. The transistors have \Vlb\ = 2 V and
\K\ =0.1 mA/V 2 . The load capacitance is 100 pF. Prior to t = 0 the input voltage is zero.
(a) Assuming that steady-state conditions have been reached, what is die output voltage prior to t
= 0?
(b) If die input voltage switches to VDD at / = 0, what is die output voltage waveform?
(c) How much current is flowing out of die capacitor immediately after / = 0? At what time does
die output voltage reach 8 V?
Chapter 5: Bipolar Transistor Circuits
5.1. An npn transistor is operating widi die base-emitter junction forward-biased and die basecollector junction reverse-biased. If ic= 9mA for iB = 300|iA, find iE, a and p.
5.2. A transistor has P = 50. What is the value of a?
5.3. Consider an npn transistor at room temperature mat has 1^= 10"13A, P = 100, v^ ~ 10V and
iE = 10mA. Find vBE, iB, ic and a. (Assume VT = 26mV at room temperature.)
Figure P5.4
-244-
5.4. Consider the circuit shown in Fig. P5.4 The transistors Qt and Q^ are identical, both having
IES = lOfA and P = 100. Find VBE and ZQ. Assume a temperature of 300K for both transistors.
5.5. Repeat Problem 5.4 ifQ, has Imi = lOfA and P = 100, whereas Q has 7 ^ = lOOfA and P =
100.
5.6. Two transistors J2, and Qj, connected in parallel are equivalent to a single transistor as
indicated in Fig. 5.6. If the individual transistors have I&, = IBS2 = 10"13A and Pi = P2 = 100, find
igj and P for the equivalent transistor. Assume the same temperature for both transistors.
{Comment: Sometimes we may be tempted to parallel transistors to obtain an equivalent transistor
with higher current ratings. However, unless the transistors are mounted on the same heat sink to
maintain nearly equal temperature, one of the transistors "steals" most of the current Sometimes
we add resistors in series with the emitters to ensure more nearly -equal current division. See
Problem 3.9.)
Figure P5.6
5.7. Find the value of |3 fot the transistors of Fig. P5.7.
10kO
r C£ =7V
V=5V
(b)
Figure P5.7
5.8. An npn transistor has VBE = 0.7V and IE — 10mA. Find VBB if IE = 1mA. Repeat for IE =
lpA. Assume a temperature of 300K.
5.9. Design a "P-meter" for the measurement of P of small-signal npn silicon transistors at room
temperature. Assume that vBE = 0.7V for the transistors to be measured. The following parts are
available:
• A 1-mA-full-scale meter having a resistance of 150 Q.
• Standard 5%-tolerance resistors.
• Potentiometers of 100ft, lkQ, lOkQ, lOOkQ and 1MQ.
• A 4.7-V Zener diode.
• Switches and mechanical components as required.
• A 9-V battery.
The meter is to have switch-selectable full-scale values of PFSÎO, 100 and 1000. Adjustments are
to be provided which allow calibration of the meter. The meter is to provide accurate readings
-245-
for battery voltages ranging from 7 to 9V. Under reasonable battery conditions (including shortcircuited test terminals), the battery current should not exceed 5mA.
5.10. A certain npn transistor has vBE = 0.7V for iB = 0.1mA at a temperature of 30°C. Sketch the
input characteristic to scale at 30°C. What is the approximate value of vBB for iB = 0.1mA at
180°C? (Use the rule of thumbs that vBE is reduced in magnitude by 2mV per I K increase in
temperature.) Sketch the input characteristic to scale at 180°C.
5.11. A certain npn silicon transistor has |5 = 100 and iB — 0.1mA. Sketch ic versus v^ for JÊ
ranging from 0 to 5V. Repeat for P = 300. Ignore second-order effects.
5.12. Repeat Problem 5.11 for znpnp transistor if v^ ranges from 0 to -5V.
5.13. At a temperature of 30°C, a particular/>«p transistor has vBE = -0.7V for % = 2mA. Estimate
vBE for iE — 0.1mA at a temperature of 180°C.
5.14. An npn transistor has P = 100 and iB = 0.1mA. The collector-to-emitter breakdown voltage
is 20V and the Early voltage is VA = 100V. Sketch ic versus v^ for the voltage range from 0 to
25 V.
5.15. Determine the region of operation for a room-temperature silicon npn transistor that has p
= 100 if (a) V^ =10V and IB = 20\iA; (b) Ic = IB = 0; (c) Va = 3V and VBE = 0.4V; Ic - 1mA
and IB = 50uA.
5.16 Determine the region of operation for a room-temperature siliconpnp transistor that has P =
100 if (a) V^ =-5V and VBE = -0.3V; (b) Ic = 10mA and IB = 1mA; (c) IB = 0.05mA and V^ -5V.
5.17. Use the large-signal models for the transistors to find Ic and V^ for the circuits of Fig.
P5.17. Assume that P = 100. Repeat for p = 300 and compare the results for both values.
+15V
+15V
+15V
6.8kO
a
iMn
+15V
|6.8kn
H
-15V
\kCl
-15V
(b)
(c)
Figure P5.17
5.18. Find J and K i n the circuits shown in Fig. P5.18. For all transistors assume that P = 100 and
I VBE I = 0.7V in both the active and the saturation regions. Repeat for P = 300.
-246-
+ 10V
+ 10V
-•5V.
Figure P5.18
5.19. Consider the circuit shown in Fig. P5.19. A jg-point value for Ic between a minimum < •£
4mA and a maximum of 5mA is required. Assume constant resistor values and that p ranj
from 100 to 300. It is desired for Rg to have the largest possible value while meeting the otb t
constraints. Find the values of Rg and RE. The resistors in this problem are not required to be
nominal values.
+15V
lkn
Figure P5.19
5.20. Consider the four-resistor bias network of Fig. 5.20a with Vcc = 15V, R, = lOOkQ, R^ =
47kfl, Re = 4.7kfi and RE = 4.7kQ. Suppose that P varies from 50 to 200, VBE = 0.7V and the
resistors have the tolerance of ±5%. Find the maximum and minimum values of Ic
5.21. Consider the circuit shown on Fig. P5.21. Find R, and Re if a bias point V^ - 5V and Ic =
2mA is required. What are the closest 5%-tolerance nominal values for R, and R^
-247-
+ 15V
•15V
Figure P5.21
5.22. Find Ic and V^ in the circuit of Fig. P5.22.
+ 15V
-15V
Figure P5.22
5.23. A certain npn silicon transistor at room temperature has P = 100. Find the corresponding
values of rKi£Ic= 1mA, 0.1mA and l|iA. Assume operation on the active region.
5.24. Consider the common-emitter amplifier of Fig. P5.24. Draw the dc circuit and find Ic. Find
the value of r*. Then calculate the values of A„ A^ Z& Ah G and Z,. Assume operation in the
midband region for which the coupling and bypass capacitors are short circuits.
lkfi
Figure P5.24
5.25. Repeat Problem 5.24 if all resistance values, including Rs and R^ are increased in value by a
factor of 100. Prepare a table comparing the results for the low-impedance amplifier of Problem
5.24 with those for the high-impedance amplifier. (Comment: When we consider the high-
-248-
frequency response of these circuits, we will find that die gain of the high-impedance circuit falls
off at lower frequencies than the gain of the low-impedance circuit does. Thus if we want
constant gain to extend to very high frequencies, we should use die low-impedance circuit.)
5.26. Consider the emitter-follower amplifier of Fig. P5.26. Draw the dc circuit and find Ic Find
the value of r* Then calculate midband values of A„ A^ Zit> Ab G and Z,.
+15V
cj
h
+15V
iokn
j l k Q f "
| 47kQ
Figure P5.26
5.27. Repeat Problem 5.26 if all resistance values, including Rs and R^ are increased in value by a
factor of 100. Prepare a table comparing the results for the low-impedance amplifier of Problem
5.26 with those for the high-impedance amplifier.
5.28. Draw the small-signal equivalent circuit for die amplifier shown in Fig. P5.28. Derive
expressions for the voltage gain and input impedance in terms of the resistor values, r„ and p\
Assume that the capacitors are short circuits for the signal.
vc
vc
Figure P5.28
5.29. Find the values of Ia r„, A„ and Zu for the circuit of Problem 5.28 if Vcc = 15V, p = 100,
VBE = 0.7V, RB = 270kQ, r^ = lkfl, r^ = lOOfi and R^ = lkfi. Repeat for RE = 0 and prepare a
table comparing the results.
5.30. Find an expression for the output impedance of the amplifier shown in Fig. P5.28.
5.31. Draw the small-signal equivalent circuit for the amplifier shown in Fig. P5.31 and derive
expressions for the input impedance and die voltage gain. Assume that die capacitors are shortcircuits for the signal.
-249-
Figure P5. 31
5.32. Consider the circuit of Problem 5.31 with V^ = 15V, R, = lOkfl, R, = lOkQ, RB = lOOkfl,
RE = lOkQ and B^ = 4.7kQ. Assume a transistor having P = 200, VBE = 0.7V. Evaluate the
expressions found in Problem 5.31 for input impedance and voltage gain.
5.33. A certain npn BJT has an Early voltage of V = 100 V. Find the value of r„ - 1/hK for Ic = 1
mA. Repeat for Ic = 0.1 mAand Ic = 10 mA.
5.34. A certain transistor has hp = 200. Find the approximate value of hk for lc — 0.1 mA, 1 mA
and 10 mA. Assume a temperature of 300 K.
5.35. The transistor shown in Figure P5.35 has hn = 10-4. Find the reading of the voltmeter in the
ac mode. Assume that the ac base current is negligible (because of the high impedance of RB and
the voltmeter).
15V
Ac or dc
voltmeter
J
Figure P5.35
5.36. The transistor shown in Figure P5.35 has hK = 10"4 S. Find the rms value of the ac collector
current. Assume that the ac base current is negligible.
5.37. Consider the circuit shown in Figure P5.35. In the dc mode, the voltmeter gives a reading
of 0.65 V. In the ac mode, the voltmeter gives a reading of 1 mV rms. The dc collector current is
known to be Ic = 5 mA. The ac collector current is 0.1-mA rms. Find approximate values of h„,
hJa hK, and h?
5.38. Derive an exact expression for bk in terms of the parameters of the hybrid-7t model at low
frequencies. Evaluate the exact expression to find hk from the 2N2222A equivalent circuit shown
in Figure 5.31. What percentage error results if die approximation hk = r„is used? Is this error
significant considering the unit to unit variations of these devices? Consider the hybrid-7i model
for the BJT with a short connected from the collector to the emitter.
-250-
5.39. Derive an expression for the ratio of the collector current phasor to the base current phasor
as a function of frequency. To simplify the analysis, replace rM by an open circuit and use the
approximation lc = g^y* (In other words, neglect the current through Cf, when computing the
collector current.)
5.40. The transition frequency fT is the frequency at which the short circuit common-emitter
current gain has a magnitude of unity. Use the expression found in 5.39 for the current gain to
obtain an expression for fT in terms of the hybrid-7t parameters. Show that your result is
equivalent to Equation (5.71).
5.41. The data sheet for a certain transistor gives the following data for a j2-point of V^ — 10 V
and Ic = 1 mA:
A„ =1x10-5, fr = 400MHz, hft = 500,
Q = 2pF,
h„= 2x10-5 S
Furthermore the collector-base time constant is 20 ps. Find values for the parameters of die
hybrid-n equivalent circuit.
5.42. A certain npn transistor has 7 ^ = l x l O 1 3 A, PF = 200, and (3R = 0.5. Use the Ebers-Moll
equations to find ic, iE, and iB. Also identify the region of operation. Assume that VT = 26 mV.
(a) vBE = 0.65 V and v^ = -10 V.
(b) vBE = 0.65 V and vK - 0.6 V.
(c) vBE = -1 V and v^ = -10 V.
(d)r BH = - 5 V a n d ^ = 0 . 7 V .
5.43. Consider die RTL inverter of Figure 5.37.
(a) The value of 13 for the 2N2222A is approximately 150. Find the maximum value of
RB allowed if the transistor is to be in saturation for viH = 3 V.
(b) Use a PSpice program to perform a transient analysis with a 200-ns 3-V input pulse
for RB = 10 kfi and for RB = 5 kQ.
Prepare a table comparing the delay, rise, storage, and fall times of the output for the two
values of RB Give a brief explanation of the effect that the value of RB has on each of
these time intervals.
5.44. Consider the RTL inverter of Figure 5.37.
(a) Use a PSpice program to perform a transient analysis with a 200-ns 3-V input pulse.
Plot the output pulse. (The result should be substantially the same as Figure 5.38a.)
(b) Increase the resistor values by a factor of 10 (i.e., R$ = 50 kQ and E^ = 20 kQ) and
repeat die analysis for an input pulse duration of 2 j*s.
(c) Prepare a table comparing the delay, rise, storage, and fall times of the output for the
two sets of resistor values.
Chapter 6: Feedback Circuits
6.1. A certain negative feedback amplifier has P = 0.1. Plot the closed-loop gain Aj versus openloop gain A. (Assume that A is a pure real number.) Also plot ^ v e r s u s A on the same set of
axes for p = 0.01.
6.2. A certain negative feedback amplifier (as shown in Figure 6.1) has xs = cos(fi«), A = 103, and
P = 0.1. Find x^ x;. and xf Repeat for A - 104. What does x(- approach as A approaches infinity?
251-
6.3. An amplifier has a nominal open-loop gain of A = 104. It is found that^4 varies by ±3% with
changes in ambient temperature. Negative feedback is to be used with the amplifier. What value
of P should be used so that the variations in A{ with temperature are no more than ±0.1%? What
is the nominal value of Aj for this value of P? Assume that P is constant with temperature.
6.4. For the feedback amplifier configurations shown in Figure P6.4, determine the overall gain
Af = xjxf
*-x.
Figure P6.4
6.5. (a) Derive an expression for the closed-loop gain Af = xjx, of the feedback amplifier shown
in Figure P6.5. Under what conditions is the closed-loop gain magnitude \Aj\ less than the open
loop gain magnitude \A\i In other words, under what conditions is the feedback negative?
Assume that A and P are real but can assume negative values.
(b) Suppose that A is negative and P is positive. To have negative feedback, should the summer
add the signals as in Figure P6.5 or subtract them as in Figure 6.1?
(c) Repeat part (b) if A is negative and P is negative.
>x0
Figure P6.5
6.6. Consider the transfer characteristic of a certain nonlinear amplifier as shown in Figure P6.6.
i
Plot the output voltage of the amplifier assuming a sinewave x,- = sin(cot) at its input. Find the
voltage gain A^xjxf of the feedback amplifier in Fig. P6.6 for the positive and negative input x>
Plot the output voltage x, of the feedback amplifier for xs = sia(o)t) .
-252-
6.7. Consider using positive feedback with the nonlinear amplifier of Figure P6.6. In other
words, the feedback signal is added to the source signal as shown in Figure P6.5. Assume that P
= 0.09 and xs = 0.1sin((0^. Find xc(t) and sketch it to scale versus time. Find the ratio of the
positive peak to the negative peak for xt(t). Compare to the ratio for the waveform without
feedback (as plotted in Problem 6.6). What effect does positive feedback have on distortion?
6.8. Consider interchanging the order of the amplifiers in Figure 6.8. This is shown in Figure
P6.8. Find the signal-to-noise ratio at the load. Compare the result to that given in Equation
(6.17). What do you conclude concerning the best order of cascading a noisy amplifier with a
low-noise amplifier?
Figure P6.8
6.9. A certain power amplifier has a voltage gain of 100. Because of poor power-supply filtering,
a "hum" of 2-V peak appears in the amplifier output It is required to reduce the output hum to
0,1 V peak. It is not practical to change the internal design of either the power supply or die
power amplifier. However, it is practical to cascade an additional amplifier and employ negative
feedback. Design die block diagram of a feedback system to achieve the hum reduction. Give die
gain of each amplifier and the feedback ratio for the feedback block. The overall voltage gain is
required to remain 100.
6.10. What is the minimum phase margin possible for a single-pole amplifier? As usual, assume
that P is constant - not a function of frequency.
6.11. A ceicain amplifier has an open-loop dc gain of 5000. The open-loop poles are located at s
= 2071 and s = 20071. Find the allowed range for P if a minimum phase margin of 60° is required.
What is the gain margin for die maximum allowed value of P? Verify your results with Spice.
6.12. Repeat Problem 6.11 if a third open-loop pole is added at s = 400071.
6.13. For die circuit of Fig. P6.13 find the product V4(CD)P((O), die frequency of zero loop-phase
and R2/R, for .oscillation.
-253-
A(a>)
6.14. Design a Wien-bridge oscillator with frequency of 5MHz.
6.15. (a) Design a phase-shift oscillator for a frequency of 20 kHz. Use SPICE to plot the phase
curve of the phase-shifting circuit.
(b) Find the percent change in frequency if one of the resistors increases by 10%.
(c) Repeat part (b) for a 10% decrease in one capacitor.
6.16. (a) Design a 2 MHz oscillator like Fig. 6.26c using a MOSFET with k = 4xl0" 3 A/V 2 , VT =
1.2 V and Cp — C^ = 2 pF. Begin with the high-frequency equivalent circuit, assuming that your
transistor is biased at 1 mA and that the Early voltage is 70 V. Include transistor capacitances as
part of your design.
(b) Design a suitable biasing circuit for your oscillator transistor, including all necessary biasing
and coupling capacitor values. Draw the final equivalent circuit.
6.17. A Hartley oscillator uses L,+L} = 25 ^ H and C2 = 40 pF. Use SPICE to plot phase versus
frequency for the phase shifting network driven by an independent ac current source, both for
the original design and when C2 is 10% high. One expects the circuit to oscillate at that frequency
where the phase shift is 180°. From your curves, determine the nominal design frequency and the
frequency when C2 is 10% high.
6.18. Use SPICE to plot the phase versus frequency curve for the phase-shifting circuit of Fig.
P6.18. The input signal should be an ac current. Use the crystal equivalent of Fig. 6.30a. Crystal
parameters are L = 137 H, C = 0.0235 pF, r = 15 kQ, and C - 3.5 pF. Try to determine the
percent frequency change when the 32.2 pF capacitor increases by 10%.
-254-
6.19. Figure P6.19 shows two oscillator circuits of the Colpitts type, complete with bias details.
For each circuit derive an equation governing circuit operation, and find the frequency of
oscillation and the gain condition that ensures the oscillation start.
Notes
-255-
Analog Electronics /Appendix
Appendix
A. Nominal Values and the Color Code for Resistors
Several types of resistors are available for use in electronic circuits. Carbon-film and carboncomposition resistors with tolerances of 5%, 10% or 20% are available with various power
ratings (such as 0.125, 0.25 and 0.5W). These resistors are used where greater precision is
required. For example, we often choose metal-film resistors in applications such as feedback
resistor of an op amp or as the frequency-determining elements of an oscillator.
Second digit
First digit
(closest to the end)
Optional band
Multiplier
Digit
0
1
2
3
4
5
6
7
8
9
Examples:
Color
Black
Brown
Red
Orange
Yellow
Green
Blue
Violet
Gray
White
Tolerance
Color
2%
5%
10%
20%
Gold
Silver
No fourth band
Yellow Violet Black
Yellow Violet Red
Brown
Black
Yellow
Red
47xl0°=47O
47xl02«4700Q
10xl0 4 =100kO
Figure A1
Wire-wound resistors are available with high-power dissipation ratings. Wire-wound resistors
often have significant series inductance because they consist of resistance wire that is wound on a
form, such as ceramic. Thus they are often not suitable for use as resistance at high frequencies.
The value and tolerance are marked on 5%-, 10%- and 20%-tolerance resistors by color bands as
shown in Figure A l . The first band is closest to one end of the resistor. The first and second
bands give the significant digits of the resistor value. The third band gives the exponent of the
-256-
multiplier. The fourth band indicates the tolerance. The fifth band is optional and indicates
whether the resistor meets certain military reliability specifications.
Table Al shows the combinations of significant figures available as nominal values for 5%-, 10%and 20%-tolerance resistors. Table A2 shows the standard nominal significant digits for 1-%tolerance resistors.
TABLE Al
STANDARD NOMINAL VALUES FOR 5%-TOLERANCE RESISTORS
10
11
12
13
15
16
18
20
22
24
27
30
33
36
39
43
47
51
56
62
68
75
82
91
TABLE A2
STANDARD VALUES FOR 1%-TOLERANCE METAL-FILM RESISTORS
100
102
105
107
110
113
115
118
121
124
127
130
133
137
140
143
147
150
154
158
162
165
169
174
178
182
187
191
196
200
205
210
215
221
226
232
237
243
249
255
261
267 -..
274
280
287
294
301
309
316
324
332
340
348
357
365
374
257
383
392
402
412
422
432
442
453
464
475
487
499
511
523
536
549
562
576
590
604
619
634
649
665
681
698
715
732
750
768
787
806
825
845
866
887
909
931
953
976
B. Introduction to PSpice
PSpice is a computer program that helps to simulate electronic circuits' design before building
them.
PSpice is a member of the Spice "family" of simulators:
Simulation
Program with
Integrated
Circuits
Emphasis
developed at the University of California, Berkeley.
Prescribed textbook:
P. W. Tuinenga, Spice: A guide to circuit simulation and analysis using PSpice. Second edition.
Prentice-Hall, 1992.
The student version 5.0^ of PSpice is available from Prof. A. Materka, Instytut Elektroniki,
Politechnika Lodzka, uL StefanowsJtiego 18/22, 80-924 Lodz, materka@ck-sg.p.lodz.pl (come
with two 1.44 MB floppies). Obviously, the faster your PC machine, the shorter simulation times;
however, no coprocessor is required to run PSpice on a PC.
Example 1. Use PSpice to calculate the voltage at node 2 and all the branch currents in the
following dc circuit;
Figure Bl
Introduction. All the respective files of the PSpice simulator are stored in a directory
..\PSEVAL5(L Call PS program from that directory. The PSpice control shell loads in a few
seconds. You can see a horizontal menu bar across the top of the screen.
In PSpice, any simulation is run accordingly to circuit description and analysis commands that are
included in the input file. Then before running a simulation you have to edit the appropriate
input file for your circuit Suppose your input file will be stored in the A: drive, insert a floppy
disk into that drive.
You have to specify a name of the file that you ate going to create. To do this, either use the
arrow keys to move the highlight to Files and press <ENTBR>, or simply press <F>. The dropdown menu appears below Files. Select the Current File option. Type in your input file name
AiMYFIRST. An extension .CIR will be added automatically to the file name, by default The
full file name appears at the bottom of the screen, along with the notation New. The Files dropdown menu disappears. Now you are ready to call the text editor to create your input file.
-258-
Creating a circuit file: the text editor. Press F or <ENTER> to bring down the Files menu
again. This time all the items are intensified, meaning that all the functions are available. Select
Edit. A new window opens which is a blank text field of the input file.
The line number and column number tell you the current cursor location. The notation [Insert]
indicates that die editor will insert your keyboard entry at the cursor location and move right
whatever text is to the right of the cursor. If you press <INSERT> on the computer keyboard
diis mode will change from Insert to Overstrike. Overstrike means that your keyboard entry will
overstrike or replace whatever is at the cursor location.
When typing in the PSpice circuit file, the number of spaces between entries is not important
You may use <TAB> or <SPACE BAR> to separate the data. A space is equivalent to a
comma, in terms of separation of data entries.
Use <ENTER>, <PGUP>, <PGDN>, <HOME>, <END>, and the arrow keys to move
around die file. The <BACKSPACE> key deletes any character to the left of the cursor location.
You can delete blank lines with the <DEL> key when the cursor is at the beginning of a blank
line. To see Help Screen of the editor commands, press < F / > . Press <ESC> to return to the
screen you were using.
Now type in your first PSpice program. Please note mat the nodes of your circuit in Figure Bl
were numbered, and the node with a number '0' is a reference node.
Direct current circuit example
VI
1
11
2
V2
3
Rl
1
R2
2
R3
2
.DC VI
. PRIN
.END
0
0
0
2
0
3
5
DC
DC
DC
DC
2
2
4
5
V(2)
5
3
10
1
I (VI)
I(V2) I(R1)
I(R2)
I(R"3)
The first line in the program above is the program tide line. You must
even if it is blank.
The voltage source V I has its positive terminal connected to node 1, its negative terminal at the
reference node 0, and a dc voltage of 5 V.
PSpice uses a convention in which all currents are assumed to pass through the device from die
first listed node to the second listed node. Following this convention, the current source I I has
its reference direction passing through the source from node 2 to node 0.
The lines with the resistance names are the circuit device statement lines. (A name that starts with
"R" or "r" is a name of a resistance.) Each device is described widi a name, the nodes to which it
is attached in the circuit, and its value.
-259-
The . DC statement is a command statement Begin all command statements with a dot. In this
case, the . DC command calls for the source voltage V I to be calculated at a single value of 5
volts.
The . PRINT command saves the dc analysis values of the listed voltages and currents to the
output file. You can view the output file on the monitor screen and/or print it out on the
printer.
The . END is to terminate the PSpice file.
Now, press <ESC> to indicate that you have finished file editing. You will be asked whether or
not to save your edits. Press < E N T E R > to accept the default (Save).
If there are any syntax errors in your file, the Files menu displays with the Edit selection
highlighted. The word Errors appears at the bottom of the screen in the Current File line. In this
case, press <F6> to see a listing of the line number and a description of the error. Press <ESC>
to close the error screen, and press < E N T E R > to re-enter the editor and correct the error. If
there are no errors in the file, the Files menu selection closes and PSpice is ready to accept your
next command. The word Loaded appears in the Current File line.
Running a PSpice circuit analysis. To select Analysis, either press A or use the arrow keys to
highlight Analysis and press <ENTER>. The drop-down menu appears in which the Run PSpice
selection is highlighted. The » arrows to the left of DC Sweep indicate that the dc analysis has
been enabled. The other entries are not yet functional with this file, because you have not use the
appropriate commands. This will be explored later.
Press <ENTER> to accept Run PSpice option. The status screen displays a continuous update of
the actions performed by PSpice.
When the analysis is complete you return to the Control Shell menu screen. To view the data
output file you can select Files and then Browse Output function.
If you select Browse Output, the output file appears in a window similar to the text editor window.
You cannot edit the output file. You can use the arrow, <PGUP>, and <PGDOWN> keys to
move around the output file. For this program, the second page has the output of the dc analysis,
labeled DC TRANSFER CURVES. The voltage V (2) and currents I ( V I ) , I ( R l ) , I ( R 2 ) ,
and I (R3) are listed across the screen.
To exit the Browser, press <ESC>.
+
In addition to the printed text output, PSpice can produce graphics output. The PROBE graphics
post-processor is used for this purpose. As an example, we consider an analysis of the RLC filter
circuit shown in Figure B2.
Example 2. Write the appropriate PSpice code and run the transient analysis for the circuit
shown in Figure B2. Then create a CRT screen graph, similar to an oscilloscope screen that will
show transient time-varying voltages across each of the devices in the circuit. The input voltage
to the circuit will be a step of 5 volts.
-260-
Figure B2
To get started, select Current File under the Files menu heading. Enter the file name, e.g.
RLCTRAN. Select Files again. Press E to select the editor.
Type the following file for me RLC circuit:
RLC circuit transient analysis
* Analyze the RLC circuit for the transient time
* varying voltage across the capacitance when
* input voltags is a 5-volt pulse.
5
0.1ms
O.lus)
VI
1
0
PULSE (0
Rl
1
2
250
LI
2
3
15.9mH
CI
3
0
15.9nF ;This line relates to the capacitor
.TRAN
12US
600US
0
1US
.PLOT
TRAN
V(3)
.PROBE
.END
The first line in the file is the required tide line. The lines beginning with an asterisk (*) are
comment lines and are ignored by PSpice. An asterisk in the first column or a semicolon (;) in
any column identifies everything to the right of it as a comment also.
The source V I is described as having a 5-volt pulse waveform that is zero at t=0, jumps to 5 V at
t=0.1 ms and has a time of rise equal to 0.1 fj,s.
The .TRAN command tells PSpice to perform transient analysis. The first value (12US = 12 JJ.S)
is the time interval between data points on the printer-plot output called for by the . PLOT
command. The second value (600US s 600 |is) specifies the time interval for the analysis.
The . PLOT command specifies that the values of the node voltage V(3) calculated in the
transient analysis are to be saved in the output file.
The . PROBE command tells PSpice to create an RLCTRAN. DAT file containing all of the data
points calculated in the transient analysis. Those data can be used by the PROBE graphics postprocessor to construct graphs of any voltage or current waveforms in the circuit. Various
functions of voltages and currents can be plotted against the horizontal axis variable, which is
time in this case.
After entering and checking your file, press <ESO to leave the editor. Select Analysis. The
Transient line in the menu has an indicator showing that the transient analysis has been enabled by
the .TRAN command in the file. Press <ENTER> to choose the highlighted Run PSpice
selection.
-261-
When the analysis run is complete, select Probe from the Control Shell menu bar. The drop-down
menu appears with the Run Probe selection highlighted. Press <ENTER>. After a few seconds an
empty graph screen appears, with the available selections listed across the bottom line. Use arrow
keys to move highlight to the Add_trace selection and press <ENTER>. When asked Enter
Variables or Expressions type: V ( 3 ) <ENTER>. After a brief time a graph of the capacitance
voltage waveform will be displayed.
Select Add_trace and add the variable V ( 2 , 3 ) . Probe re-scales the graph (automatically) and
displays both the capacitance voltage and the graph of the voltage across the inductance due to
pulse source.
Select Addjtrace again and add the voltage V ( 1 , 2 ) . This is the graph of the voltage across the
resistance.
16U-
-5U+-
6s
166us
269us
3Wus
D o(3) o u(l,2) A 11(2,3) 11+12*13
Tine
466us
566us
6Mus
Experiment According to Kirchhoffs voltage law (KVL), the sum of the voltages across the
circuit loop is equal to zero, or the sum of the voltages across the R, L, and C of this circuit is
equal to the source voltage. You can add the traces together by simply adding their trace
numbers.
First, select Add_trace. Then, type the expression:
#1 + #2 + #3
<ENTER>
A waveform equal to die sum of the three voltages appears. This is equal to the 5 V input
waveform, of course.
-262-
Quick Reference for PSpice Programmers
1. The circuit reference node must be numbered zero.
2. The remaining nodes may be named with unique positive integers or alphanumeric character
strings.
3. A dc path to the reference node from each node is required.
4. Two or more components must be connected to each node.
SUFFIXES FOR POWERS OF TEN
3t
4BltJtdfi
You can use either upper- or lower-case letters for all suffixes, statements, commands, etc. in
PSpice.
Program Statements
1. Thefirstline of the program is the tide line. It may be left blank. There may be only one tide
line in the input file.
2. An asterisk (*) in the first column identifies a comment.
3. Comments may also be written to the right of a semicolon (;).
4. A plus sign (+) at the beginning of a line indicates previous line continuation.
PSpice Device Statements
1. Thefirstcharacter of the name identifies the element type.
2. The next two fields identify node connections: N+ and N - . It is assumed that a positive
current flows through the element from node N+ to node N - .
3. Thefieldsafter the node connections are parameter fields.
4. The circuit device names and the syntax required by PSpice are listed below. Please note that
the parameter in square brackets [ ] are optional.
Resistance:
Rname
N+
NCapacitance: Cname
N+
NInductance:
Lname
N+
N(Where IC are initial conditions.)
Value
Value
Value
Magnetic Coupling Coefficient:
Kname
Ll
L2
Value
-263-
[IC = v o l t s ]
[IC = amps]
Dependent Sources:
VCVS: Enamo
N+
NNC+ NCValue
VCCS: Gnamo
N+
NNC+ NCValue
[Where the voltage control is V(NC+,NC-).]
CCVS: Hname
N+
NVX
Value
CCVS: Fname
N+
NVX
Value
(Where VX is the voltage source through which the control current flows.)
Independent sources (X=V for voltage. X=I for current sources):
DC sources:
Xname
N+
N-
[DC]
[Value]
Piecewise linear sources:
Xname
N+
NPWL(tl v l t 2 v 2
(Where ti = point in time where voltage is vi.)
Pulse sources: Xname N+ N - PULSE (VS VP [TD] [TR]
Default
Parameter
VS
Initial value
None
VP
Pulsed value
None
TD
Delay time
0
TR
TSTEP
Rise time
Fall time
TSTEP
TF
TFINAL
Pulse width
PW
TFINAL
Pulse period
PER
.)
[TF]
[PW]
Damped sinusoidal source:
Xname N+ N - SIN(VO VA [FREQ] [TD] [ALPHA]
Meaning
Default
Parameter
None
VO
Offset value
Peak amplitudeNone
VA
Frequency in Hz
1 /TFIN AL
FREQ
Delay time
0
TD
Damping coefficient 0
ALPHA
Phase in degrees
0
PHASE
-264-
[PER])
[PHASE])
Ambt Electronics /Appendix
PSpice Command and Control Statements
Operating P ^ i f ( f t i m ) AlWlyftift
.OP
1. Replaces all capacitances with open circuits and all inductances with short circuits.
2. Performs a dc analysis and prints out all node voltages.
3. Calculates currents of all independent voltage sources and the power delivered by
these sources.
4. Calculates the currents and voltages for controlled sources.
***• A^fywfft
.DC [SWEEP] NVAR START STOP STEP
Parameter
SWEEP
NVAR
START
STOP
STEP
Mânjng
Default
LIN for linear sweep
LIN
OCT for log sweep by octave
DEC for log sweep by decade
Name of independent source to be swept
Starting value for NVAR
Maximum value for NVAR
Increment in values of NVAR
Tirana wm Analysis
.TRAN TSTEP TFINAL [TNOPRINT] [MAXSTEP] [UIC]
Default
Parameter
Meaning
TSTEPTime step for plots or print out
None
TFINAL
Final time for analysis
None
TNOPRINT Output suppressed until TNOPRINT 0
MAXSTEP
Maximum time step used in analysis see textbooks
UIC
Specifies use of initial conditions in capacitance
and inductance statements
AC Analysis
.AC SWEEP NPOIMTS FSTART FSTOP
Parameter
Mining
SWEEP
LIN for linear sweep
OCT for log sweep by octave
DEC for log sweep by decade
Total # of points for LIN
Points per decade for DEC
Points for octave for OCT
Starting frequency for sweep
Maximum frequency for sweep
NPOINTS
FSTART
FSTOP
-265-
C. English-Polish Dictionary of Saiected Terms
A
abrupt junction - zla_cze skokowe
ac circuit - obwod prajdu przemiennego
acceptor — akceptor
A / D converter (ADC) — przetwornik analogowo-cyfrowy
admittance — admitancja
AFC (automatic frequency control) - automatyczna regulacja czestotliwosci
AGC (automatic gain control) - automatyczna regulacja wzmocnienia
AM (amplitude modulation) — modulacja amplitudy
ambient temperature — temperatura otoczenia
amplifier — wzmacniacz
analog - analogowy
angular frequency — pulsacja
anode — anoda
approximation — przyblizenie, aproksymacja
attenuator - dzielnik, tlumik
avalanche breakdown - przebicie lawinowe
average — sredni
B
bandpass — pasmowoprzepustowy
bandstop — pasmowozaporowy
bandwidth — pasmo
base — baza
bias — polaryzacja
bipolar - bipolarny
block diagram — schemat blokowy
breakdown - przebicie
bridge - mostek
buffer — bufor
c
capacitance - pojemnosc
capacitor — kondensator
cascade connection - pola.czenie kaskadowe
cathode — katoda
cellular — komorkowy
channel — kanal
characteristic — charakterystyka, charakterystyczny
charge — ladunek
circuit - obwod, uklad
clamp - uklad przylegania
clipper - ogranicznik
closed-loop — w p?tli zamkni^tej
CMOS (complementary MOS) - CMOS
-266-
coil - cewka
collector - kolektor
common — wspolny
compensated - skompensowany
complex - zespolony, zlozony, skomplikowany
component - cz?sc skladowa, element
concentration — koncentracja
conductance — przewodnosc
conduction angle — ka_t przeplywu
conductivity — przewodnictwo
converter — przetwomik
crossmodulation - modulacja skrosna
crystal - krysztal
current - pra_d
current gain — wspolczynnik wzmocnienia pra_dowego
cutoff - odciecie
D
D/A converter (DAC) - przetwomik cyfrowo-analogowy
dc (direct coupled, direct current) — bezposrednio sprz^zony, stalopra^dowy
decibel - decybel
delay - opoznienie
depletion layer — wartswa zubozona
detector — detektor, demodulator
differential — roznicowy
diffusion - dyfuzja
digital — cyfrowy
diode - dioda
distortion — znieksztalcenia
donor — donor
drain - dren
drift - unoszenie
dynamic - dynamiczny
E
discrete - dyskretny
Early effect - zjawisko Early'ego
efficiency - sprawnosc
emitter - emiter
equivalent circuit - obwod (schemat) zastepczy
F
fall time - czas opadania
feedback - sprz?zenie zwrotne
FET - tranzystor polowy
field-effect transistor - tranzystor polowy
filter - filtr
first harmonic - pierwsza harmoniczna
FM (frequency modulation) - modulacja cz^stotliwosci
follower - wtornik
-267-
forward bias - polaryzacja w kierunku przewodzenia
frequency - czestodrwoic
full-w^ve — dwupotowkowy
G
gain - wspolczynnik wzmocnienia, wzmocnosc
gate — bramka
ground — masa, ziemia
H
half-wave - jednopolowkowy
harmonic - harmoniczny
higher cutoff frequency - gorna cz^stotliwosc graniczna
high frequency - duza czcstotliwosc
highpass — gomoprzepustowy
high voltage — wysokie napiecie
hybrid - hybrydowy
I
IC -> integrated circuit
impedance - impedancja
incremental - przyrostowy
inductance — indukcyjnosc
input - wejscie
integrated circuit — uklad scalony
interference — interferencja
intermediate frequency - cze^totliwosc posrednia
intermodulation - intermodulacja
inverse - inwersyjny
inverter - inwerter
inverting amplifier — wzmacniacz odwracajajcy
J
JFET - tranzystor polowy zlajczowy
junction - zlajcze
K
KCL (Kirchhoff s Current Law) - pierwsze prawo Kirchhoffa
knee current, voltage - prad, napiecie "kolana" (punktu zagiccia krzywej)
KVL (Kirchhoff s Voltage Law) - drugie prawo Kirchhoffa
L
large-signal - wielkosygnalowy
layer - warstwa
lead - odprowadzenie
lifetime - czas zycia (nosnikow mniejszosciowych)
line regulation - wspolczynnik stabilizacji napieciowej
-268-
load - obciazenie
load regulation — wspolczynnik stabilizacji prajdowej
logic - logiczny
loop - petla
loudspeaker - glosnik
lower cutoff frequency — dolna czestotliwosc graniczna
low frequency - mala czestotliwosc
low voltage - niskie napiecie
lowpass - dolnoprzepustowy
M
microphone - mikrofon
midband range - zakres srednich czestotliwosci
mixer - mieszacz
MOSFET - tranzystor polowy z izolowana. bramka.
multistage — wielostopniowy
mutual — wzajemny
N
narrowband — wa.skopasmowy
noise — szum, zaklocenie szumowe
noninverting amplifier - wzmacniacz nieodwracajajcy
o
opamp, operational amplifier - wzmacniacz operacyjny
open circuit — rozwarcie
operating point — punkt pracy
oscillator - generator
output — wyjscie
output stage - stopien wyjsciowy
overshoot - maksymalna amplituda oscylacji, "przerost" odpowiedzi jednostkowej
P
parasitic - pasozytniczy
peak value — wartosc szczytowa
peak-to-peak value — wartosc miedzyszczytowa
phase angle — ka_t fazowy
phase shift - przesuni^cie fazowe
photodiode — fotodioda
piecewise linear - odcinkami liniowy
pin — koncowka
PM (phase modulation) - modulacja fazy
pole - biegun
power — moc
power gain - wspolczynnik wzmocnienia mocy
power electronics - energoelektronika
power supply — zasilacz
primary winding — uzwojenie pierwotne
probe - sonda
-269-
pulse — impuls
PWM (pulse width modulation) — modulacja szerokosci impulsow
Q
Q-point (quiescent point) - punkt pracy (punkt spoczynkowy)
quantization - kwantowanie
quartz crystal — krysztal kwarcowy
R
reactance — reaktancja
receiver — odbiornik
recording head — glowica zapisuja.ca
rectifier — prostownik
regulator - stabilizator
reverse bias — polaryzacja zaporowa
resistance — rezystancja
resistor - opornik, rezystor
reverse recovery time — czas wyla.czania diody
ringing — oscylacje
ripple - t^tnienia
rise time — czas narastania
rms value (root mean square value) - wartosc skuteczna
s
sample — probka, (to s.) probkowac
sampling — probkowanie
saturation - nasycenie
secondary — wtorny
secondary winding — uzwojenie wtorne
second harmonic — druga harmoniczna
self-bias - samopokryzacja
semiconductor — polprzewodnik
sensitivity — wrazliwosc
schematic diagram - schemat ideowy
short circuit - zwarcie
short circuit gain — zwarciowy wspolczynnik wzmocnienia
shunt — zwierac, bocznikowac
slope - nachylenie
small-signal - malosygnalowy
smoothing capacitor - kondensator wygladzaja.cy
source — zrodlo
speaker - glosnik
spectrum — widmo
storage interval - czas magazynowania, czas pierwszej fazy wyla_czania diody
stored charge — ladunek zmagazynowany
stray — rozproszony
switch - przel%cznik
system - system, uklad
-270-
T
tangent — styczna
terminal — zacisk, koncowka
diermal voltage (KT) - potencjal elektrokinetyczny
threshold — prog
direshold voltage - napiecie progowe
tilt - zwis (grzbietu impulsu)
time constant - stala czasowa
total harmonic distortion - wspolczynnik zawartosb harmonicznych
ttansconductance — transkonduktancja, przewodnosc wzajemna
transducer - przetwomik
transfer function — transmitancja
transit rime - czas przelotu
transition time - czas drugiej fazy wyla.czania diody
transresistance - transrezystancja, opornosc wzajemna
transmitter — nadajnik
turns ratio — przekladnia transformatora
u
unilateral - unilateralny
unit - jednostka
V
variable-capacitance diode (varicap) - dioda pojenmosciowa
voltage — napi?cie
voltage-controlled — sterowany napieciem
voltage gain — wspolczynnik wzmocnienia napi?ciowego
w
waveform — przebieg
wavelength — dlugosc fali
wideband — szerokopasmowy
winding — uzwojenie
z
Zener diode — dioda Zenera
-271-
Symbol
MBit
2N2222
2N2222A
CoHector-Emitter Voltage
VCEO
30
40
Vdc
Collector-Base Voltage
VCBO
60
75
Vdc
Emtoar-Baaa Voltage
VEBO
&0
&0
Vdc
t
•00
HO
mAdc
Rating
Coaartcr Currant—Continuous
Unit
2M22MA
M B B >
Total Davica Wtaliiatlon
#T*-25X
Derate above 2S*C
*>
Total Device DMpanon
:
9TC.25*C
Darata abova 2S*C
*D
Opafatiiiy and Ssorage Junction
Tamparatura Ranga
T
OJ)
*5!
0L4
Z2»
34
17.1
1.2
•JJS
-•510+200
J. T stg
Watt
mWC
Wans
mwrc
•c
THERMAL C H A H A C I t W S I I C S
2N221SA
Symbol 2N2219.A
Ck-aâria*
Tharmal Resistance. Junction to Ambiant
Thannal floi'mema. Junction to Caaa
2M222ZA
Unit
B«JA
219
145.8
«*JC
58
437.5
•c/w
•ow
a£CTWCALCHAWACIH«IICS{TA-25Xunla«aothaiwi»ar>otad.>
OFF CHARM. 11MB IM. 8
CoRactor-EmJttar Breakdown V o t a g a
OC - 10 m A d c *B - 0 )
Coaector-Base Breakdown Vokaga
0 c - l O j i A d c l E - 0)
30
40
—
80
75
—
5.0
6.0
—
—
10
Vdc
V<BRICBO
Non-A Suffix
A-Suffix
EmMer-Bese Braakdown Vokaga
flE - 1 0 / i A d c . l c - 0 )
'
NorvA Suffix
A-Suffix
Collector Cutoff Currant
(VCE - 60Vdc.VEB(off) - l O V d e )
A-Suffix
Collactor
(VCB (VC8 (VCB WcB -
Non-A Suffix
A-Suffix
Non-A Suffix
A-Suffix
V(BR)EBO
<CEX
Vdc
•CBO
1 11 1
Cutoff Currant
S o V d c l E - 0)
O0 Vdc. I E - 0)
* 0 V d c . l E - 0 . T A - 150*C)
MVdc;»E-0.TA-150*C)
Vdc
V(BfttCEO
Non-A Suffix
A-Suffix
aoi
nAdc
MAdc
0.01
10
10
'EBO
—
10
nAdc
'BL
—
20
nAdc
2N2218A
2N2219A 2N2222.A
20
35
—
Clc - 10 mAdc. VCE = W Vdc)
2N2218A
2N2219A. 2N2222.A
25
50
—
lie - 10 mAdc VCE " lOVdeMI)
2N2218A
2N2219.A. 2N2222.A
35
75
—
OC - 10 mAdc V C E = 10 Vdc,
T A - -55*CMD
2N2218A
2N2219A 2N2222.A
15
35
—
( l c = 150 mAdc. VCE = 10 Vdc)(1)
2N2218A
2N2219.A. 2N2222.A
40
100
120
300
Emitter Cutoff Currant
( V E B - 3.0 V d c <C - ° )
A-Suffix
Base Cutoff Currant
(VcE - » V d c V E B(off) - 3.0 Vdc)
A-Suffix
ON CHARACTERISTICS
0C Current Gain
flC - 0.1 mAdc VCE - 10 Vdc)
"FE
MOTOROLA SMALL-SIGNAL TRANSISTORS. FETs AND DIODES
"
2N2218A/19/19A/22/22A
MOTOROLA SMAI-l S'GNAl. TRANSISTORS, FETs AND DIODES
2N2218A/19/19A/22/22A
MOTOROLA SMALL-SIGNAL TRANSISTORS. FETs AND DIODES
2N2218A/19/19A/22/22A
SWITCHING TIME CHARACTERISTICS
MOTOROLA SMALL-SIGNAL TRANSISTORS, FETs AND DIODES
Q'TQBM- s
- .
7
'
•
.
'
,
•
'
.
-
'
This textbook, describes principle of operation arid
properties of basic analog electronic circuits that
210000076947
comprise diodes, field-effect and bipolar transistors,
including, negative and positive feedback circuits.
It is arnply illustrated by computer sitnulationT*
using PSpice program. N u m e r o u s probk
exercises to be solved by the reader are ]
Review questions are included to turn the r
elements of the theory' and to help sorting tjwi
E n g l i s h - P o l i s h d i c t i o n a r y of s e l e c
T h e b o o k is a p r e s c r i b e d , t e x t
I n t e r n a t i o n a l ^ JFaculty of E n g i n e e r i n
of Lodz. It is' also r e C o m m e n d p to
s t u d y b a s i c e l e c t r o n i c cir.c/uits
t e r m i n o l o g y . A m o d e r n , applicâtio
s t u d y i n g e l e c t r o n i c s is refleVc^tfe^r ifi^
m a y b e a v a l u a b l f r e f e r e n c e T f o r p Dlitechnikatodzkaôdz-biblioteka
* * * *
•
Professor Andrzej «&fmterka (Senior Member, IEEE) was graduated in
radio engineering From Warsaw University of Technology in 1972. He has
authored and co-authored 3 monographs and more than 100 technical papers,
most of ^them published in international journals and conference proceedings.
In 1980-82 he worked on microwave MESFET oscillators in Shizuoka University,
Japan. In 1991-94 he was with Monashy University, Australia lecturing analog'
of Electronics and Head of Medical Electronics Division, Technical University
of Lodz. His current research interests include computer vision and applications
of a r t i f i c i a l i n t e l l i g e n c e ( h t t p i Z / w w w ^ e l e t e L p J o d z ^ p l V -

1 - Andrzej Materka

Transcription

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