AC Network Analysis

Learning objectives
AC Network Analysis
1.
2.
3.
4.
5.
6.
1
Compute current, voltage and energy stored in
capasitors and inductors
Calculate the average and rms value of periodic
signals
Write the differential equation for circuits
containing inductors and capasitors
Convert time domain sinusoidal voltages and
currents to phasors and vice versa
Represent circuits using impedances
Apply known circuit analysis methods to AC circuit
in phasor form
2
Ideell Kondensator
(Capasitor)
Structure of parallel-plate
capacitor
Kretsparameter:
Capasitans C
Kan både ta opp og avgi elektrisk energi.
Lagrer energi i form av elektrisk felt
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4
Serie og Parallel kobling av
Kapasitanser
Viktig observasjon for Ideell
Kondensator (Capacitor)
i
+
v
C
• Tillater ikke sprang (diskontinuitet) i spenningen
• Tillater imidlertid sprang i strømmen gjennom kapasitansen
• Representeres som en åpen krets ved konstant spenning
5
6
1
Calculation of energy stored in a
capasitor
Iron-core inductor
Magnetic flux
lines
Iron core
inductor
i ( t)
L
+
di
v L (t ) = L dt
_
Circuit
symbol
7
8
Inductanse and practical inductor
Ideell Spole (Inductor)
Kretsparameter:
Induktans L
Kan både ta opp og avgi elektrisk energi.
Lagrer energi i form av magnetisk felt
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10
Serie og Parallel kobling av
Induktorer
Viktig observasjon for Ideell
Spole (Inductor)
1
+
v
+
C
L
_
2
• Tillater ikke sprang (diskontinuitet) i strømmen
• Tillater imidlertid sprang i spenningen over induktansen
• Representeres som en kortsluttning ved konstant strøm
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2
Energy stored in an Inductor
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14
Analogy between electrical and
fluid resistance
v1
R
Analogy between fluid capacitance
and electrical capacitance
v1
v2
i
p2
qf p2 p1
Rf
+
v
p
Cf
_
qf
p2
qf
qf
P1
_
15
v2
p1
16
Analogy between fluid inertance
and electrical inertance
v1
i
L
+
qf
17
gas
+
C
p1
p2
i
p2
Analogy between electrical and fluid
circuits
v2
v –
If
p1
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3
Time-dependent signal sources
v (t) +_
i (t)
Sinusoidal
waveform
__
v (t), i(t) +
Generalized time-dependent sources
Sinusoidal source
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20
Phase angle / phase shift
Periodic signal
waveforms
The sine wave Vmsin(ωt + θ) leads Vmsinωt by θ rad.
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22
RMS Value of Sinusoidal
Waveform
Average and RMS values
The rms, or effective , value of a current (or voltage) is the DC (or DC voltage)
that causes the same average power to be dissipated by the resistor.
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24
4
Analysis of circuits containing
dynamic elements
Exercise
Given the sinusoidal voltage:
v(t ) = 325cos(100π t + 30o ) V
1. What is the period of the voltage?
2. Calculate the frequency?
3. What is the value of the voltage at t=3.333ms?
4. Calculate the rms value of the voltage?
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26
The steady-state response of circuits
excited by sinusoidal sources
27
The steady-state response of circuits
excited by sinusoidal sources cont.
28
The steady-state sinusoidal
response
Relationship between polar and
rectangular coordinates of a
complex number
In a sinusoidally excited linear circuit, all branch voltages and currents
are sinusoids at the same frequency as the excitation signal.
The amplitudes of these voltages and currents are a scaled version of
the excitation amplitude, and the voltages and currents may be shifted
in phase with respect to the excitation signal.
ρ e jθ = ρ cosθ + j ρ sin θ
ρ e jθ = a + jb
θ = tg
29
b
, ρ = a 2 + b2
a
30
5
The steady state response
The steady state response
Complex Forcing Function
From Eulers identity and and the superposition theorem we find:
•A complex forcing function may be considered as the sum of a real and
an imaginary forcing function
•The real part of the complex response is produced by the real part of
the forcing function. The imaginary part of the response is produced by
the imaginary part of the complex forcing function
The sinusoidal forcing function Vm cos (ωt + θ) produces the
steady-state response Imcos (ωt + θ).
The complex forcing function Vme j(ωt + θ) produces the complex
response Ime j(ωt + θ).
The imaginary sinusoidal forcing function j Vmsin (ωt + θ)
produces the imaginary sinusoidal response j Imsin (ωt + θ).
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Phasor
Transformation
Definition of a Phasor
En prosess hvor en sinusformet
strøm eller spenninger blir konvertert
fra en størrelser i tidsplanet til en
kompleks størrelse i frekvensplanet
Vi merker oss at frekvensplan
representasjonen ikke eksplisitt
inneholder informasjon om den
aktuelle frekvensen til sinussignalet.
Frekvensen er kjent på forhånd og
er derfor unødvendig i
representasjonen
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v (t ) = Vm cos(ω t + φ )
v ( t ) = R e {V m e j ( ω t + φ ) }
V = Vm e jφ
V = Vm ∠ φ
Frequency domain
34
Complex
eksponetial function Vejωt
Phasor diagram
35
Time domain
36
6
A graphical representation of
two sinusoids v1 and v2
Summasjon
av visere
The magnitude of each sine
function is represented by the
length of the corresponding
arrow, and the phase angle by
the orientation with respect to
the positive x axis. In this
diagram, v1 leads v2 by 100o +
30o = 130o, although it could
also be argued that v2 leads v1
by 230o.
It is customary, however, to
express the phase difference
by an angle less than or equal
to 180o in magnitude.
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Spole representasjon i frekvensplanet
Motstand representasjon i frekvensplanet
i(t)
Diagrammet viser at summasjon av sinus størrelser
kan illustreres geometrisk ved hjelp av visere.
+ v(t) R
Spenningen over resistansen blir
i frekvensplanet :
V = RI
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Definisjon
Kondensator representasjon i frekvensplanet
41
Impedans
Resistans og Reaktans
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7
Definisjon
Resistor , Capasitor, and inductor
in time and in the phasor domain
Admittans
Konduktans og Suseptans
Enkelte ganger (f.eks i forbindelse med
paralellkobling av impedanser) kan det være
hensiktsmessig å innføre størrelsen admittans
(a)
(b)
Admittansen til et element defineres som den
inverse av impedansen:
(c)
Måleenheten for admittans (konduktans og suseptans) er
Siemens (S)
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44
In the phasor domain, (a) a
resistor R is represented by
an impedance of the same
value; (b) a capacitor C is
represented by an impedance
1/jωC; (c) an inductor L is
represented by an impedance
jωL.
Bestemmelse av impedansen
Impedances
R, L and C in
the complex
plain
For å bestemme impedansen til et
ukjent kretselement eller en toport må
vi først transformere støm og spenning
til frekvensplanet. Dvs.utrykke støm og
spenning som komplekse visere
Impedansen beregnes som forholdet
mellom spenningsviseren og
strømviseren.
Impedansen blir generelt en kompleks
størrelse. Måleenheten er ohm
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46
Numerical example
Den imaginære
delen til
impedansen
kalles reaktansen
Merk: Impedansen er ikke en viser
Kirchhoffs laws in the frequency
domain
Calculation of impedans
This circuit is operating in the sinusoidal steady state
with v(t) = 50 cos(500t) V and i(t) = 4 cos(500t < 60°) A.
Find the impedance of the elements in the box.
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8
Kirchhoffs laws in the
frequency domain
Example
49
Flow
diagram for
Phasor
circuit
Analysis.
50
Numerical example
Series connection of impedances
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52
Numerical example
53
Series connection of
impedances
Series connection of
impedances
54
Design the voltage divider so that an input
vS=15cos2000t V produces a steady-state
output v0(t) = 2sin2000t V.
9
Parallel connection of impedances
Figure 15-15 (p. 682)
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Parallell connection of two
impedances
Numerical Example
Steady-state currents
Find the steady-state currents i(t), iC(t), and
iR(t) for vS = 100 cos 2000t V,
L = 250 mH, C = 0.5 µF, and R = 3 kΩ.
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58
Numerical Example
steady-state currents
Find the i(t), iC(t), and iR(t)
for vS = 100 cos2000t V,
L = 250 mH, C = 0.5 µF, and
R = 3 kΩ .
59
Figure 4.37
60
10
Phasor diagram
(a) A phasor diagram showing the sum of
V1 = 6 + j8 V and V2 = 3 – j4 V, V1 +
V2 = 9 + j4 V = 9.85∠24.0o V.
(b) The phasor diagram shows V1 and I1,
where I1 = YV1 and Y = 1 + j S =
1.4∠45o S. The current and voltage
amplitude scales are different.
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An AC circuit
Figure
4.41
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