1. Transmission Lines

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1. Transmission Lines
1.1 Ideal Transmission Line Theory
:series resistance per unit length in
:series inductance per unit length in
:shunt conductance per unit length in
:shunt capacitance per unit length in
By Kirchhoff’s voltage law:
By Kirchhoff’s current law:
As
,
For time-harmonic(
) circuits
.
.
.
.
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Thus
where
:complex propagation constant.
We have the solutions
: positive z-direction propagation wave.
: negative z-direction propagation wave.
: constants.
Also
Define characteristic impedance
Then
and
For lossless line
Terminated Lossless Transmission Line
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Assume incident wave
At
, reflected wave
,
Define return loss:
Special case:
1.
(short):
2.
.
(open):
3.
Half wavelength line:
4.
Quarter wavelength line:
Two-transmission Line Junction
.
then
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At
,
: transmission coefficient.
Define Insertion loss:
Conservation of energy
Incident power:
Reflected power:
Transmitted power:
Voltage Standing Wave Ratio (VSWR)
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Define Standing Wave Ratio
1.2 Coaxial Lines
TEM mode
Let be the inner radius of the coaxial line and be the outer
radius of the coaxial line.
Let
be the potential function of the TEM mode, then
satisfies Laplace’s equation
. In polar coordinate
and the boundary condition
Due to symmetry,
, we have
Use the boundary condition to solve
and
, we have
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1.3 Microstrip Line
Formulas
,
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Or
where
Loss
, where
where
Operating frequency limits
The lower-order strong coupled TM mode:
The lowest-order transverse microstrip resonance:
Frequency Dependence
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where
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1.3 Strip Line
Formulas
where
.
Or
where
.
Loss
where
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1.4 Coplanar Waveguide (CPW)
Benefit:
1. Lower dispersion.
2. Convenient connecting lump circuit elements.
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1.5 Generator and Load Mismatches
1.
Load Matched to Line
2.
Generator Matched to Loaded Line
3.
Conjugate Matched
Note this result means maximum power delivered to the
load under fixed . In reality, our concern is efficiency or how
much portion of total power is delivered to the load which is
related to
.
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1.6 Calculation of Transmission Line Parameters
Note:
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1.7 Criteria of Ideal Transmission Lines
TEM Wave
Assume
and dependence of the form
.
Substitute to Maxwell’s equations, we have
where
1.
and
. These lead to
The propagation constant of any TEM wave is the intrinsic
propagation constant
of the media.
Also,
2.
. The z-directed wave impedance of any TEM wave is
the intrinsic wave impedance of the medium.
Let
, then from wave equation we have
.
Similarly,
The boundary conditions at perfect conductors are
3.
The boundary-value problem for and
is the same as
the 2-dimensional electrostatic and magnetostatic
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4.
5.
problem. Thus, static capacitances and inductances can
be used for transmission lines even though the field is
time-harmonic.
The conductor must be perfect, otherwise
will exist.
Voltage is uniquely defined on the cross-section of the
waveguide.
To sum up, two conditions must be satisfied to support ideal
transmission lines:
1. Homogeneous, i.e., or are independent of location.
(Why?)
2. Two conductors. (Why?)
Multiplying both
, we have
.
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Equating both
, we have
1.8 Not Ideal Transmission Lines
Introduce mode functions
,
and mode currents
according to
TM:
, mode voltages
TE:
We can choose for
TM:
TE:
Also all modes are normalized according to
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Then, the characteristic impedance is
These is also the wave impedance. Also
satisfy transmission-line equations
The power transmitted is
Since
Then for
TE:
TM:
and
will
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1.9 Composite Right-Left Hand (CRLH) Transmission Lines
Let
Then,
where
and
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Applications
1. Dual-band Components.
2. Bandwidth Enhancement.
3. Zeroth-order Resonator
2. Microwave Network Analysis
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2.1 Concept of Impedance
1.
Intrinsic impedance:
2.
Wave impedance (of a propagation mode):
3.
Characteristic impedance (of a transmission line):
2.2 Properties of One Port
Complex power (Poynting vector)
where
: real positive. The average power dissipated.
: real positive. The stored magnetic energy.
: real positive. The stored electric energy.
Define real transverse model fields
and
and
such that
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then,
Thus, the input impedance
Properties:
1.
is related to
2.
.
equals zero if lossless.
is related to
.
, inductive load.
, capacitive load.
Even and Odd Properties of
and
since
. Similarly,
.
Summary
1. Even functions:
2. Odd functions:
3. Even functions:
.
.
2.3 Properties of N-Port
2.3.1 Impedance and Admittance Matrix
Let the total voltage and current at each port be
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where + and - sign mean the voltage or current entering the
port and leaving the port, respectively.
Define impedance matrix
, such that
, i.e.,
where
and admittance matrix
, such that
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where
Obviously,
Reciprocal Networks:
1. No source in the network.
2. No ferrite or plasma.
Lossless networks:
Example
2.3.2 The Scattering Matrix
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Define scattering matrix
where
Relationship with
Let
be the matrix formed by the
characteristic impedance of each port.
Thus
If lossless
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Therefore,
. When all ports are equal,
unitary.
Since
Also
Therefore,
If reciprocal
, or
,
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Example
1.
Reciprocal?
2.
Lossless?
3.
Return loss at port 1 when port 2 is matched.
4.
Return loss at port 1 when port 2 is shorted.
Shift in Reference Planes
If at port n, the reference plane is shifted out by a length of
voltage at the reference plane will be
where
. Let
, the
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We have
2.3.3 Generalized Scattering Parameters
Define the scattering parameters based on the amplitude of the incident
and reflected wave normalized to power.
Let
thus
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The generalized scattering matrix is defined as
where
or
If lossless,
If reciprocal,
or
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2.3.4 The Transmission (ABCD) Matrix
Define a transmission matrix of a two port network as
or in matrix form
Relationship to impedance matrix
If reciprocal,
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Cascading of ABCD matrix:
2.3.5 Two-Port Circuits
Example: prove the first entry of Table 4.1.
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Homework #2: 2.14, 2.17., 2.20.
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3. Impedance Matching
3.1 Quarter-Wave Transformer
Match a real load
to
by a section of transmission line
with characteristic impedance
and length .
The reflection coefficient becomes
for a given
, solve for
Assume TEM mode,
, we have
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The bandwidth becomes
`
Example: Match a 10
load to a 50
line at
Determine the percent bandwidth for SWR
=3 GHz.
1.5.
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3.2 Matching Using L-Sections
jX
Z0
jX
jB
(a)
ZL
Z0
jB
ZL
(b)
Analytic Solutions
(a)
(b)
Smith Chart Solutions
1.
. Use (a)
a.
b.
c.
d.
2.
a.
b.
c.
Convert to admittance plot.
Move along constant conductance curve until
intercept with the constant resistance curve equal to
1.
Convert back to impedance plot.
Find the required reactance.
. Use (b)
Move along constant resistance curve until intercept
with the constant admittance curve equal to 1.
Convert to admittance plot.
Find the required susceptance.
Example 2.5
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Smith chart
Lumped Elements
3.3 Single-Stub Tuning
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Analytic Solutions
1. Shunt Stubs
Open stub:
Short stub:
Where
2. Series Stubs
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Open stub:
Short stub:
where
Smith Chart Solutions
Shunt (Series) Stubs
1. Use admittance (impedance) plot.
2. Rotate clockwise along constant
curve until intercept
with the constant conductance (resistance) curve of value
1.
3. Compensate the remaining susceptance (reactance) by a
suitable length of open or short stub.
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Example
Smith Chart
3.4 Double-Stub Tuning
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Analytical Solution
where
Open stub:
Short stub:
where
or
Smith Chart Solutions
Smith Chart
1. Use admittance plot.
2. Rotate the constant conductance circle of value 1
counterclockwise by a distance d.
3. Move
along the constant conductance curve until
4.
intercepting the rotated circle in 2. The difference of the
susceptance determines the length of the stub 2.
Rotate the intercepting point back to constant
conductance circle of value 1. The susceptance value
determine the length of stub 1.
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Example
Smith Chart
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4. Power Dividers and Directional Couplers
4.1 The T-Junction Power Divider
Lossless Divider
1.
2.
3.
4.
Lossless
Match at the input port.
Mismatch at the output ports.
No isolation at the output ports.
4.2 Resistive Divider
1.
2.
3.
Lossy.
Match at all ports.
No isolation.
From the figure
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4.3 The Wilkinson Power Divider
1.
Matched at all ports.
2.
Isolation between output ports.
3.
No power loss from input to output ports.
4.
Half power loss from output to input ports.
Analysis
1.
Excite port 1.
Symmetry  equal voltages at port 2 and 3  no current flows
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through the resistor  open. The circuit becomes
From the figure
To compute
, let
and
denote the voltages of
the forward and backward propagating modes in one of the two
lines. Assume the reference plane is located at port 1. Let
the voltage of the incident wave at port 1 be
and port 3
We have at port 1
At port 3,
2.
Even and odd mode excitation at port 2 and 3
Rearranging the circuit as follow
.
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a.
Even mode: Symmetry  equal voltages at port 2
and 3  no current flows through the resistor  open.
The circuit becomes
b.
Odd mode: Anti-symmetry  opposite voltages at
port 2 and 3  short at the middle of the resistor. The
circuit becomes
Since
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From
, we have
Unequal Power Division
If power ration between ports 2 and 3 is
N-way, equal-split, Wilkinson power divider
,
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4.4 Basic Properties of a Three Port Device
Impossible scenario: reciprocal, matching at all ports, lossless.
Reciprocal and matching at all ports give the following S matrix
If lossless, the matrix is unitary, that is,
Two of
must be zero to satisfy the last 3 equations.
However, then, the first 3 equations will not be satisfied.
Possible scenario:
1. Nonreciprocal, matching at all ports, lossless.
Lossless
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Two possible solutions
and
Example: Circulators
2.
Reciprocal, lossless, matching only two ports.
Lossless
Possible solution
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3.
Lossy, matching at all ports, reciprocal.
4.5 Basic Properties of a Four Port Device
Reciprocal, matched at all ports.
If lossless, the following conditions are required.
if
If
and
, the following conditions can be derived:
, then
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real.
1.
Symmetrical:
2.
Anti-symmetrical:
,
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4.6 The Quadrature (90) Hybrid
Even-Odd Mode Analysis
Even Mode
Using ABCD matrix, we have
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Odd mode
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4.7 Coupled Line Directional Coupler
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Assume
and
, we have
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where
.
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If
For
,
and
. Also
, choose the mid-band frequency such that
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Example Design a 20 dB singlesection coupler.