ECE 375, Majedi, A.H.
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Transmission Line (TL) Theory
1-1) Overview of General Concept
Electromagnetic Transmission Lines (TLs) are structures and media that serve to transfer
electromagnetic (EM) energy between two points. Such transmission lines include telephone
wires, coaxial cables, and optical fibers carrying optical signals, it fundamentally covers the
whole EM spectrum shown in Fig 1-1.
Figure 1-1: Electromagnetic Spectrum
TLs are special class of more general EM waveguides. Normally the term “Transmission line” is
used for the EM waveguiding structure that is capable of guiding TEM (Transverse Electric and
Magnetic) waves. TEM waves can only exist in structures that contain two or more separate
conductors to carry EM signals. As you have seen before, the interchange of electric and
magnetic energy gives rise to the propagation of EM waves in space, as shown in Fig. 1-2.
The magnetic fields that change with time induce electric fields as explained by Faraday’s law,
and the time-varying electric fields induce magnetic fields, as explained by the generalized
Ampere’s law (We will deal with these two laws later on to develop the Maxwell’s equations).
This interchange of energy causes that EM wave travels with speed of light in free space which
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fundamentally is related to free space electric permittivity and magnetic permeability, namely
c
1
 0 0
.
Ex
x
Direction of
Propagation
z
k
z
y
By
Fig 1-2: Any TEM wave is a travelling wave which has time
varying electric and magnetic fields which are perpendicular to each
other and the direction of propagation.
[Analogy between TEM waves and waves on the surface of Water. Waves & TLs from simple
electrodynamic point of view.]
[Your notes about TEM wave.]
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This electric and magnetic interrelation also occurs along conducting or dielectric boundaries and
can give rise to wave propagation guided by these boundaries (Think of getting your wireless
signal over your cell phone in a building), but not necessarily as a TEM wave. This higher order
EM wave propagation has at least one significant field component in the direction of propagation
(it could be TE (Transverse Electric) or TM (Transverse magnetic) and their superposition).
Needless to say, to guide these waves, we need TLs (better to say waveguides) that are capable
of supporting these modes such as hollow conducting waveguides, dielectric rods and cylindrical
optical fibers, refer to Fig. 2-4 pp. 38 of your textbook for more examples.
Now we focus on TLs that supports TEM waves (from now on wherever we say TL means those
can support TEM waves), for their simplicity, huge application, and all very important reasons in
their own rights! and another important reason to come. In many ways TL theory bridges the gap
between EM field analysis and basic circuit theory. TLs are composed of two –conductor system
in which the EM field can be uniquely related to a voltage and current that we use in circuit
theory via following equations:

V    E.dle

I   H.dl h
[Example: Voltage and Current definition in Coaxial Cable]
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In any transverse plane, electric field lines passes from one conductor to the other, defining a
voltage between conductors for that plane. Magnetic field lines surround the conductors,
corresponding to the current flow in one conductor and an equal but oppositely directed current
flow in the others. Both voltages and current are functions of distance along the TL. Therefore,
by extension of lumped circuit theory to distributed circuit theory we are able to analyze TLs and
develop TL theory. The key difference between TL theory and lumped circuit theory (from now
on simply circuit theory) is electrical size. Circuit analysis simply assumes that physical
dimensions of the circuit or network are much smaller than the electrical wavelength, while TLs
might be a considerable fraction of a wavelength or many wavelengths in size.
Example 1-1: Consider an audio circuit with the highest frequency is 25 kHz. For
electromagnetic waves, this corresponds to approximately 12 km. So even if the circuit is spread
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across a football stadium, the size of the circuit is very small compared to the shortest
wavelength,  .
Hint: More generally if d   , the circuit can be considered as lumped.
Example 1-2: Consider a 2 cm long resistor (including leads) in comparison with wavelength at
three different frequencies, 100 MHz, 1GHz and 10 GHz.
Discussion:
Homework 1-1: Consider a chip whose extent is 1 mm; Let the shortest signal time of interest is
0.1 nsec, calculate propagation time of EM wave on chip. Can we consider a chip as a lumped
circuit or not?
Your solution:
Hint: If d  c.t , then the circuit can be considered as lumped.
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1-2) Lumped Circuit Model & Distributed Circuit Model:
Knowing the electrical size (physical length and operating frequency) of the circuit immediately
leads us to know if the lumped element circuit theory is sufficient or not. Let’s discuss the
characteristic of each model.
Lumped Circuit Model:
* R, L, C are discrete components.
* Circuit paths are “ideal” short circuits.
* There is zero-time delay for a voltage to go from one end of a circuit to the other.
Distributed Circuit Model:
* Circuit paths are not ideal short circuits.
* Circuit paths have capacitance-per-unit-length, inductance-per-unit-length and resistance-perunit length.
* Circuit paths have a propagation velocity and so have a time delay, according to their lengths.
* These circuits are often called “TL circuits or distributed circuits”
How can we model distributed or TL circuits?
Answer: Transmission line theory!!!
1-3) Transmission Line Theory:
Since TLs (supporting TEM wave) always have at least two conductors, they are schematically
represented as a two-wire line, regardless of their specific shape. Let’s start to physically model a
TL with infinite length composed of two conductors and an insulating medium in between.
* Electrons traveling inside the conductors’ experience resistivity due to the finite conductivity
of the conductor, so we have distributed resistivity along the TL, represented by R  as the
combined resistance of both conductors per unit length, in  / m.
* Electrons are moving together, they are creating an electrical current which produce magnetic
field (Ampere’s law), and therefore TL has a distributed inductive effect, represented by L  as
the combined inductance of both conductors per unit length, in H/m.
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* Consider the electrical potential difference between two conductors that have opposite signs;
thus TL shows a distributed capacitance between two conductors, represented by C  as the
capacitance of the two conductors per unit length, in F/m.
* Due to small conductivity of the insulator between two conductors, TL should have a
distributed conductance, represented by G as the conductance of the insulating medium per unit
length, in S/m (1/  m). Therefore, to effectively model an infinitely long TL, TL can be
subdivided into small differential sections, in which each section can be modeled by an
equivalent lumped circuit as shown in Fig 1-4 (2-6 pp. 40 your textbook).
Fig 1-4: taken from F.T.Ulaby, Fundamentals of Applied Electromagnetics
Note that z   , where  is the operating wavelength and also the separation distance
between two conductors in TL is much less than  .
In order to fully analyze the TL, we have to come up with two important issues:
-
Calculation of TL parameters, namely R, L, C , G  for any TL configuration. These
parameters depend on two sets of parameters:
1- Geometric parameters defining the cross-sectional dimensions of the given geometry
2- EM constitutive parameters of conductors and insulating material
-
TL equations, namely relating the voltage and current along the TL and their dependency
on the distance at any instantaneous time.
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In the following, we review both analyses, starting TL equations.
1-4) Transmission Line Equations:
Fig 1-4 (Fig 2-8 pp. 43 of the textbook) indicates the equivalent circuit of a differential
length of a TL. Writing down the Kirchhoff’s voltage and current laws, yield:
[Your note]
v( z , t )
i ( z , t )
  Ri ( z , t )  L
,
z
t
i ( z , t )
v( z , t )
 G v( z , t )  C 
.
z
t
[1-1]
We can derive a second order partial differential equations for voltage and current separately,
called “Telegraphist’s equation”.
[Your note]
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For harmonic time dependence the use of phasors simplifies the TL equations to ordinary
differential equations and gives the sinusoidal steady-state condition. Defining the voltage
and current with cosine reference notation as:
~
jt
v( z , t )  e{V ( z )e
},
~
jt
i( z, t )  e{I ( z )e
}.
[1-2]
Applying [1-2] into [1-1], transforms the second partial differential equation to ordinary
second order differential equations, and two equations can be solved simultaneously to give
~
~
TL equations for V ( z ) and I ( z ) , as follows:
[Your note]
ECE 375, Majedi, A.H.
d2 ~
~
V ( z )   2V ( z )  0,
2
dz
d2 ~
~
I ( z )   2 I ( z )  0.
2
dz
10
[1-3]
where
    j  ( R  jL)(G  jC )
[1-4]
Eqs. [1-3] are called wave equation for voltage and current, respectively, and  is called the
complex propagation constant of the TL, which is a function of frequency.  consists of a
real part,  , called the attenuation constant of the line in Np/m, and an imaginary part,  ,
called the phase constant of the TL in rad/m. We choose the square root value to give
positive values for  and  .
[Your note]
Traveling-wave solutions of eqs. [1-3] can be found as
~
 z
z
V ( z )  V0 e
 V0 e
~
 z
z
I ( z)  I 0 e
 I 0 e
[1-5]
z term represents wave propagation in the +z direction, so called forward wave
where the e
z
and the e term represents wave propagation in the –z direction, called backward wave. The
TL voltage and current are a superposition of forward and backward traveling waves.
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Like lumped element analysis, we can relate the voltage and current across the line as
follows:
[Your note]
Z0 
V0
I0



V0
I0



R  jL
 R0  jX 0 [  ]
G   jC 
[1-6]
where Z 0 are called a characteristic impedance with real part R0 and imaginary part X 0 .
Note that the characteristic impedance is not equal to the ratio of total voltage to the total
current, unless one of the waves is absent. The current across the line can be written in the
following form:
V  z V0 z
~
I ( z)  0 e

e
Z0
Z0
[1-7]
* Note that V0 and V0 are complex quantity that can be achieved through boundary
conditions, which is simply applied in both ends of TL (source and load) or any other points
across the line. We will discuss this in more detail later.
Converting back to the time domain, the voltage waveform can be expressed as
v( z, t )  V0 e z cos(t  z    )  V0 ez cos(t  z    )
where   is the phase angle of complex voltage V0 .
[Your note for derivation [1-8]]:
[1-8]
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Based on your previous knowledge of the wave phenomenon, the voltage wave across the TL
has following characteristics:

Voltage (current) waves propagates across the TL with a phase velocity
u ph 


 f

[1-9]
The wavelength of the propagating waves is:

2

[1-10]

The factors e z account for attenuation constants for  z propagation.

The presence of two waves on the TL propagating in opposite directions
produces a standing wave.
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Special cases of TLs:
1-4-1) Lossless TL ( R  G  0 ):
[Your note for calculating propagation constant, characteristic impedance, and phase
velocity, voltage/current equations]:
1-4-2) Low-loss TL ( R  L , G  C ):
[Your note for calculating propagation constant, characteristic impedance, and phase
velocity]
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1-4-3) Dispersion-less (Distortion-less) TL (
R G 

):
L C 
What is dispersion?
Answer: If  is not a linear function of frequency, then the phase velocity, u ph 

, will be

different for different frequencies. The implication is that the various frequency components
of a wideband signal (like a short pulse) will travel with different phase velocities, and so
arrive at slightly different times. This will lead to dispersion, a distortion of a signal which is
generally an undesirable effect, as shown in Fig 1-6. (Note that the information of a pulse is
traveling with group velocity which is u g 
d
.)
d
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[Your note for calculating propagation constant, characteristic impedance, and phase
velocity]
Fig 1-6: Dispersion effect in TL
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1-5) Transmission Line Parameters:
To fully describe a TL whose supports TEM wave, we need to describe the TL parameters,
namely, R, L, C , G  . These physical parameters depend on the geometrical configuration of
TL in which the electric and magnetic field are stored in a cross section of the TL,
perpendicular to the direction of propagation. We will give a recipe for calculating these
parameters, and introduce some examples, more importantly coaxial cables, parallel plate and
microstrip line.
Consider a 1m section of a uniform TL, where S is the cross-sectional surface area of the TL,
as is depicted in Fig 1-5. Assume that the conductivity of the conductor be  c , with magnetic
permeability,  c and the dielectric material between two conductors have a magnetic
permeability  , dielectric permittivity  and conductivity  d .
Let the voltage and current between conductors be V0 e
z
and I 0 e
 z
, respectively. We
know there is an electric field and magnetic field associated with voltage and current,
respectively. Based on the time-averaged stored magnetic and electric field energies for this
1m long TL and their circuit theory counterparts, the parameters of the TL can be written as:
[Your Note]
H
C1
dl
E
C2
ECE 375, Majedi, A.H.
L 

I0
C 
2

G 

2
H/m
[1-11]
~2
E ds
F/m
[1-12]
S
Rs
I0
~2
H ds
S
V0
R 

17

2
[1-13]
C1C 2
d
V0
~2
H dl  /m
2
~2
E ds

S/m
[1-14]
S
where in [1-13], Rs stands for surface resistance of the conductor and can be calculated as:
Rs 
f c
c
[1-15]
Note that all TLs (whose supports TEM wave) share the following useful relations:
LC   
[1-16]
G  d

C 
[1-17]
Example 1-3: Find the TL parameters of the coaxial cable, with inner and outer radii, a and b
respectively (Use cylindrical coordinate system ( r ,  , z ) ) with the following TEM fields
E
V0
e
z ˆ
r
b
r ln( )
a
I z
H  0 e ˆ
2r
[Your Note]
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Table 1-1: TL parameters for three different types of TL with following geometry
Parameters
coaxial
Two Wire
Parallel plate
1-6) Microstrip Transmission Line Parameters:
The microstrip TL is one of the most popular types of planar TL and the most used
microwave integrated circuit (MIC) TL. It has many advantages such as low cost, small size,
absence of critical matching and cutoff frequency, ease of active device integration, use of
photolithographic method for circuit production, good repeatability and reproducibility, ease
of mass production and compatibility with monolithic circuits. Monolithic MIC (MMIC) that
use microstrip lines as matching networks and passive circuits are fabricated on GaAs or Si
substrates with both active devices, such as diodes and transistors and passive components
such as filters and antennas on the same chip. Its disadvantages include higher loss, lower
power handling capability and temperature instability in comparison with Coaxial TL. The
geometry of the microstrip line is shown in Fig. 1.6. A conductor of width w, and thickness t
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is printed on a thin, grounded dielectric substrate of thickness d, and relative permittivity  r .
If the dielectric was not present (  r  1), we could think of the line as a two-wire line
composed of two flat strip conductors of width, w, separated by a distance 2d (according to
image theory). In this case we would have a simple TEM TL, with phase velocity equals to
speed of light.
The presence of the dielectric, and particularly that the dielectric does not fill the air region
complicates the behavior of microstrip line and makes that the line does not support a TEM
wave.
Fig 1.6: Microstrip TL and its E&H field lines (taken from Microwave engineering, D.
Pozar,3rd ed., 2005)
[Your note]
ECE 375, Majedi, A.H.
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In most practical applications, however, the substrate thickness is electrically very small
( d   ) and so the EM fields are quasi-TEM (remember our discussion in 1-1 and 1-2).
Thus, a good approximation would be to model a microstrip line with static or quasi-static
EM fields and obtain the phase velocity, propagation constant and characteristic impedance
based on the relative effective dielectric constant  e where 1   e   r (refer to Fig 1-7).
Fig 1-7: Equivalent geometry of quasi-TEM microstrip line.
[Your note]
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1-6-1) Effective Dielectric Constant, Characteristic impedance, and attenuation for
Microstrip:
a) Analysis Formulas:
For now, consider that the thickness of the conductor is negligible (t=0), then we have:
e 
up 
 r 1  r 1
2

2

1
12d
1
w
[1-18]
c
[1-19]
e
  k0  e
[1-20]
120


w
w

  e   1.393  0.677 ln(  1.444)
d
d

Z0  
 60  8d w 
ln 



  e  w 4d 
for
w
1
d
[1-21]
for
w
 1.
d
Considering microstrip as a quasi-TEM line, the attenuation could be mainly due to dielectric
loss and conductor loss (the other one is radiation loss), therefore:
  d  c 
k 0 r ( e  1)
R

 s (Np/m)
2  e ( r  1)  0 r Z 0 w
[1-22]
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B) Synthesis Formulas:
For given characteristic impedance and dielectric constant the
 8e A

w e2 A  2

d 2 
 r 1 
0.61
B

1

ln(
2
B

1
)

ln(
B

1
)

0
.
39





2 r 
 r 
 
w
ratio can be found as:
d

w
 2
d


w

for  2
d

for
[1-23]
where
A
Z0  r 1  r 1 
0.11 
 0.23 


60
2
 r 1
 r 
B
377
2Z 0  r
Example 1-4: calculate the width and a length of a microstrip line for 50-ohm characteristic
impedance and a 90 degree phase shift in 2.5 GHz. The substrate thickness is 0.127 cm with
relative dielectric constant 2.20.
Solution:
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1-6-2) Effect of Strip Thickness:
So far we assume that the thickness of the conducting strip is negligible (t=0). For a finite
thickness, the E field from the edge makes the line width appear larger. Hence the effect of the
thickness can be introduced to the effective width, weff which is larger than w. The effective
width is introduced as:
weff
t
2d

w

1
.
25
(
)[
1

ln(
)]


t

w  1.25( t )[1  ln( 4w )]


t
w 1

d 2
w 1
for 
d 2
for
Note that for a finite thickness, you should replace w in all the previous formulas by weff .
[1-24]
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1-7) Terminated Lossless Transmission Line:
[Write down the propagation constant, characteristic impedance, voltage/current equations,
wavelength, and phase velocity of lossless TL.]
Consider a TL of length l, connected to a generator at one end and a load, Z L at the other end as
shown in Fig. 1-8 (Fig 2-9 of your textbook).
Fig 1-8: TL connected to a source and a load. Note that
The load is placed at z=0 and a source is located at z=-l.
Fig 1-8 illustrates wave reflection on a finite TL, a fundamental property of the distributed
systems. Assume that an incident wave of the form V0 e
 jz
is generated from a source at z<0.
The ratio of voltage to current for such a wave is Z 0 . When the TL is terminated in an arbitrary
load Z L , the ratio between voltage to current at the location of the load must be Z L . Thus, a
reflected wave must be excited with proper amplitude to satisfy this condition. Now we can
define the voltage reflection coefficient as:
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[Your Note]

V0 Z L  Z 0

V0 Z L  Z 0
[1-25]
The total voltage and current waves on the line can then be written as
~
 jz
jz 
V ( z )  V0 e
 e


V
~
 jz
jz 
I ( z )  0 e
 e


Z0 
[1-26]
Homework 1-2) Calculate the voltage reflection coefficient for a 50-ohm lossless TL with load
impedance 30-j200 ohm.
Now consider the time-average power flow along the line at point z:
1
~ ~
Pav  e{V ( z ) I * ( z )}
2
Inserting [1-26] in [1-27] results in:
[1-27]
ECE 375, Majedi, A.H.
Pav  Pavi  Pavr 
V0
27
2
2Z 0
1   .
2
[1-28]
[1-28] shows that the average power flow is constant at any point on the line and the power
delivered to the load is the incident power minus reflected power, refer to Fig. 1-9 (Fig 2-19 of
textbook) .
Consider the case for   0,1 .
Fig 1-9) Incident and reflected power
When the load is mismatched , not all the power from the source is delivered to the load. This
loss is called “Return Loss (RL)” and is defined in dB as
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RL  20 log  (dB)
[1-29]
for matched load (so-called “flat” line), RL is infinity and for totally reflective TL RL=0.
1-7-1) Standing wave on Transmission Line:
The presence of both incident and reflected voltage and current on the TL leads to standing wave
along the line where the magnitude of the voltage on the line is not constant but the maximum
and minimum of the voltage do not vary with time. Thus from [1-26], we have:
[Your Note]


1/ 2
~
2
V ( z )  V0 1    2  cos(2z   r )
where    e
j r
[1-30]
. Therefore the maximum and minimum voltage can be written as:
[Your Note]
Note that when  increases the ratio between Vmax and Vmin increases, so the measure of the
mismatch of the line, called the Standing Wave Ratio (SWR), can be defined as
ECE 375, Majedi, A.H.
SWR 
29
Vmax 1  
.

Vmin 1  
[1-31]
This quantity is also called VSWR. Note that 1  SWR   .
From [1-30], it is seen that a distance between a maximum and a minimum voltage is quarter of
the wavelength, while the distance between two successive voltage maxima or minima is a half
the wavelength, if  r  0 . Refer to Fig 1.10 (Fig 2-11 textbook) for an example of standingwave pattern of voltage and current.
Homework 1-3: Consider a general case where the reflection coefficient is a complex value and
find the distance between two successive maxima or minima in a general case [Hint: Refer to pp.
53 of your textbook].
Fig 1-10) standing-wave pattern for a lossless TL with Z 0  50 terminated in a load with
  0.3e
j30
where V0  1 V. SWR=1.86.
Remember that the reflection coefficient was defined at the location of the load but this quantity
can be generalized to any point l on the line. From [1-26] , with z=-l, we have:
(l ) 
V0 e
 jl
V0 e
jl
 (0)e
 j 2 l
[1-32]
This form is useful when transforming the effect of load mismatch down the line. Note that the
magnitude of reflection coefficient does not change as we move along the line, only its phase
changes.
ECE 375, Majedi, A.H.
30
We have noted that the real power flow on the TL is constant but the voltage and current
amplitudes for a mismatched line, is oscillatory with position on the line (refer to [1-28], [1-30]).
Therefore it seems that the impedance seen looking into line must vary with position, and this is
true. At a distance z=-l, the input impedance seen looking toward the load is:
[Your Note]
~
Z  jZ 0 tan l
V ( z)
Z in (l )  ~
 Z0 L
Z 0  jZ L tan l
I ( z)
[1-33]
This is a very important result giving the input impedance of a length of TL with arbitrary load
impedance, so-called TL impedance equation.
1-7-2) Special Cases for Lossless TLs:
In this subsection we will examine four different cases, namely, short-circuited line, opencircuited line, quarter-wave transformer, and half-wavelength line. These four cases play a very
important role in the design of various microwave circuits and high-speed RF integrated circuits.
A) Short-Circuited Line:
Consider a TL that is terminated by a short circuit load impedance as shown in Fig 1-11 (Fig 215 textbook).
[Write down the voltage/current relations, calculate the input impedance, discussion of how the
short-circuited line can show both inductive and capacitive effect]
ECE 375, Majedi, A.H.
31
ECE 375, Majedi, A.H.
Fig 1-11) Short-Circuited TL with its normalized voltage, current and input impedance.
B) Open-Circuited Line
In this case the load impedance is open, as it is shown in Fig 1-12 (Fig 2-17 of the textbook).
[Write down the voltage/current relations, calculate the input impedance]
32
ECE 375, Majedi, A.H.
33
Fig 1-12) Open-Circuited Line
C) Quarter-wave Transformer:
Another case of interest is when the length of the line is a quarter wavelength or more generally
l

4

n
n  1,2,3,...
2
[Calculate the input impedance]
ECE 375, Majedi, A.H.
34
An important application for quarter-wave transformer is to design a matching transformer that
matches the line to the load, as depicted in Fig 1-13. In order to have   0 for a load resistance
RL , the impedance of the quarter-wave transformer should be:
Z 1  Z 0 RL
[1-33]
Fig 1-13: The quarter-wave matching transformer (Taken from Microwave engineering, 3rd ed,
Pozar)
Example 1-5) Sketch a frequency response of the reflection coefficient of a quarter-wavelength
matching transformer:
Solution:
D) Half-wavelength Line:
ECE 375, Majedi, A.H.
When l 
35
n
n  1,2,3,... does not modify the load impedance.
2
[Calculate input impedance]
1-7-3) Junctions of Two TL:
Consider a TL with characteristic impedance Z 0 feeding a TL with characteristic impedance
Z 1 as shown in Fig 1-14. If the load line is infinitely long or terminated in its own characteristic
impedance, so there is no reflections from its end, so the reflection coefficient would be:

Z1  Z 0
Z1  Z 0
[Your Note]
[1-34]
ECE 375, Majedi, A.H.
36
The Transmission coefficient, T, can be defined as:
T  1  
2Z1
Z1  Z 0
[1-35]
The transmission coefficient between two points along TL is often expressed in dB as, insertion
loss, IL,
IL  20 log T dB
[1-36]
Fig 1-14) Reflection & Transmission at the junction of two different TLs.
1-8) Source and Load Mismatches
Consider a lossless TL with both mismatched source and load in which reflection/transmission
occurs at both ends of the TL as shown in Fig 1-15 (general case of Fig 2-14 of the textbook).
Fig 1-15: Mismatched source and load TL circuits and its circuit model
ECE 375, Majedi, A.H.
37
Because both the generator and load are mismatched, multiple reflections occur. We would like
to calculate the power delivered to the load and then investigate three different scenarios; load
matched to the TL, source matched to the TL and finally conjugate matching.
[Your note]
~
~
V0  V g
 j l
Z0
e
Z 0  Z g (1    e  j 2  l )
l g
[1-37]
ECE 375, Majedi, A.H.
38
If Z  R  jX ,
The power delivered to the load is:
1
1 ~
~~
Pav  e{Vin I in }  Vg
2
2
2
Rin
( Rin  Rg )  ( X in  X g ) 2
2
Consider a fixed generator load, now we have three scenarios:
a) Load matched to the TL:
b) Source matched to the TL:
[1-38]
ECE 375, Majedi, A.H.
39
c) Conjugate matching:
Assuming that Z g is fixed, we might vary the input impedance until we achieve the
maximum power delivered to the load. Knowing Z in , it is then easy to find the corresponding
load impedance Z L via an impedance transformation. To maximize power:
[Your note]
ECE 375, Majedi, A.H.
Z in  Z g

Conjugate matching
40
[1-39]
Homework 1-4) Calculate the power delivered to the load for conjugate matching condition.
If all three impedances of the line, source and load equal to each other, does it mean the
maximum power delivered to the load? How many percent of the power produced vy the
source is delivered to the load in this case?
ECE 375, Majedi, A.H.
41
Microwave Network Analysis
(Adapted from Chapter 4, “Microwave Engineering”, D.M.Pozar, 3rd ed., JW )
2-1) Introduction:
In this section we show how the familiar concepts of low-frequency circuit analysis can be
extended to characterize RF and microwave circuits. Since the physical size of the RF and
microwave circuits is comparable with wavelength, it is helpful to view voltages and currents
in terms of incident, reflected and transmitted waves. Our journey begins by discussing the
use of impedance and admittance matrices to describe the relationship between voltages and
currents defined at the terminal ports of an arbitrary N-port microwave network and show
how we can decompose them into sum of incident and reflected waves. This leads us to
introduction to scattering matrix, which is a better way of characterizing microwave devices,
since can be easily used for non-TEM TLs or in general any microwave networks with multimode operation. At the end, we briefly introduce the vector network analyzer (VNA) as a
measurement tool for scattering parameters, which is extensively used in microwave
characterization.
2-2) Impedance & Admittance Matrices
Consider the arbitrary N-port microwave network shown in Fig 2-1, where incident and
reflected voltages, ( Vn , Vn ), and currents, ( I n , I n ) are defined at each port. At specific point
on the nth port, a terminal plane, t n , is defined to provide a phase reference point for the
voltages and currents. If we set the terminal plane at z=0, the total voltage and current can be
written as:
Vn  Vn  Vn
I n  I n  I n
[2-1]
The impedance matrix [Z] of the microwave networks relates these voltages and currents:
[V ]  [ Z ][ I ]
[2-2]
Similarly we can define the admittance matrix [Y], as:
[ I ]  [Y ][V ]
[2-3]
ECE 375, Majedi, A.H.
42
Fig 2-1: An arbitrary N-port microwave network, taken from 4.5 “Microwave Engineering”, D.M.
Pozar, 3rd ed., 2005.
Note that both [Z] and [Y] matrices relate the total voltages and currents and they are the
inverse of each other.
[ Z ]  [Y ]1
[2-5]
The elements of each matrix can be found as follows:
Z ij 
Vi
Ij
Yij 
Ii
Vj
[2-6]
I k 0 for k  j
[2-7]
Vk 0 for k  j
Practically, Z ij , can be calculated by driving port j, with current I j , open circuiting all other
ports and measuring the open circuit voltage at port i. In analogy, Yij can be found by driving
port j with voltage V j , short circuiting all other ports and measuring the short circuit current
at port i. Thus, Z ii and Yii are input impedance and admittance seen looking into port i when
all other ports are open- and short- circuited, respectively.
Z ij and Yij is the transfer
impedance and admittance between port i and j when all other ports are open- and shortcircuited, respectively.
ECE 375, Majedi, A.H.

43
For reciprocal networks (not containing any active devices and nonreciprocal
media), impedance and admittance matrices are symmetric, so that Z ij  Z ji ,
and Yij  Y ji .

If the network is lossless all elements of impedance and admittance matrices
are purely imaginary.

The average power can be calculated as:
1
1
Pav  [V ]T [ I ]  [ I ]T [ Z ][ I ]
2
2
Example 2-1: Find the Z parameters of the two-port network as shown in Fig 2-2.
Solution:
[2-8]
ECE 375, Majedi, A.H.
44
2-3) Scattering Matrix:
Equivalent voltages and currents, and the related impedance and admittance matrices become
somewhat an abstraction when dealing with high-frequency networks. A better representation
more in harmony with measurements, and with the ideas of incident, reflected and
transmitted waves, is given by the scattering matrix. Scattering matrix relates the voltage
waves incident on the ports to those reflected from other ports. Scattering parameters can be
directly measured by a vector network analyzer (VNA).
Consider again the N-port network shown in Fig. 2-1, where V n is the amplitude of the
voltage wave incident on port n, and V n is the amplitude of the voltage wave reflected from
port n. The scattering matrix, or [S] matrix is defined as:
V1   S11 S12  S1N  V1 
  
 
S 2 N  V2 
V2    S 21
  
  
  
 
V N   S N 1  S NN  V N 
[2-9]
or [V  ]  [ S ][V  ]
Each element of the [S] matrix can be determined as:
Sij 
Vi
[2-10]

V j
Vk 0 for k  j
S ij is found by driving port j, with an incident wave of voltage V j and measuring the
reflected voltage wave coming out of port i, while all other ports are matched to avoid
reflection.
* S ii is the reflection coefficient seen looking into port i when all other ports are terminated
in matched loads.
* S ij is the transmission coefficient from port j to port i, when all other ports are matched.
Example 2-2) Compute the S parameters of the 3dB attenuator circuit shown in Fig 2-2.
Fig 2-2: A matched 3 dB attenuator with a 50  characteristic impedance.
ECE 375, Majedi, A.H.
45
Solution:
Homework 2-1: Find the S parameters of a matched lossless TL with length l and characteristic
impedance Z 0 , terminated at one end to Z s and another end to Z L .
2-3-1) Relationship between [S] and [Z]:
ECE 375, Majedi, A.H.
46
We now look for a systematic way to convert [S] to [Z] and [Y] and vice versa. First, we
assume that the characteristic impedance of all ports is identical and for more convenience is
1 (we will relax this constraint when we talk about generalized scattering parameters).
Writing the voltage and current of the nth port:
[ S ]  [ Z ]  [U ] [ Z ]  [U ]
[2-11]
[Z ]  [U ]  [ S ] [U ]  [ S ]
[2-12]
1
1
2-3-2) Reciprocal & Lossless Networks:
* For reciprocal network the [Z] matrix is symmetric, therefore [ S ]  [ S ]T
Proof:
[2-13]
ECE 375, Majedi, A.H.
47
* For lossless network, the real power delivered to the network is zero, thus [S] is a unitary
matrix, so that
In this case,
Proof:
S
ki
1
S kj   ij  
0
for i  j
for i  j
[2-14]
ECE 375, Majedi, A.H.
48
Example 2-3: A two-port network is known to have the following scattering matrix:
0.150 
[S ]  

0.8545
0.85 - 45  

0.20 

Determine if the network is reciprocal, and lossless. If the port two is terminated to the matched
load, what is the return loss at port 1? If port two is terminated with a short circuit, what is the
return loss seen at port 1?
Solution:
ECE 375, Majedi, A.H.

49
Note that the reflection coefficient looking into port n is not equal to S nn ,
unless all other ports are matched.

Note that transmission coefficient from port m to port n is not equal to S nm
unless all other ports are matched.

Note that changes in the terminations or excitations of a network do not
change the [S] but might change the reflection and transmission coefficients.
2-3-3) A Shift in Reference Plane:
Consider the N-port microwave network as shown in Fig 2-3, where the original terminal
planes are assumed to be at z n  0 . The scattering matrix measured at this set of terminals is [S].
Now consider the reference planes are shifted to z n  l for the nth port and now the [ S  ] is the
new scattering matrix, if  n   n l n is the electrical length of the outward shift of the reference
plane of port n, then we have:
Fig 2-3: Shifting reference planes for an N-port network. taken from 4.5 “Microwave Engineering”,
D.M. Pozar, 3rd ed., 2005.
ECE 375, Majedi, A.H.
50
 j1
e  j1
0  e
0 

 

[ S ]   0

0 [ S ] 0

0 
 j N  
 j N 

e
e
 0
  0

 e
so that S nn
 j 2n
[2-15]
S nn
2-3-4) Generalized Scattering Parameters:
Up to now, the scattering matrix is considered for networks with the same characteristic
impedance for all ports. Although this holds for most practical cases where characteristic
impedance is 50 ohms, however the characteristic impedance might be different, which needs a
generalization of our definition of [S] matrix.
Consider the N-port network depicted in Fig 2-4, where Z 0 n is the real characteristic impedance
of the nth port.
Fig 2-4: An N-port network with different characteristic impedance, taken from 4.5 “Microwave
Engineering”, D.M. Pozar, 3rd ed., 2005.
ECE 375, Majedi, A.H.
51
We define a new set of wave amplitudes as:
Vn
an 
Z 0n
[2-16]
Vn
bn 
Z 0n
Voltage and current at port n can be written as:
Now the average power delivered to the nth port is:
Pn 
1
1
2
a n  bn
2
2
2
[2-17]
Now a generalized scattering matrix can be defined as:
[b]  [ S ][ a]
[2-18]
where:
S ij 
bi
aj

ak 0 for k  j
Vi 
V j
Z0 j
Z 0i
Vk 0 for k  j
[2-19]
which shows how the S parameters of a network with equal characteristic impedance can be
converted to a network connected to TLs with unequal characteristic impedance.
ECE 375, Majedi, A.H.
52
2-4: The Transmission (ABCD) Matrix:
The Z, Y, and S matrices represent a very convenient way to characterize multi-port microwave
networks, but in practice many microwave networks are cascaded connection of two-port
networks. In this case ABCD matrix is useful representation. We will see that for the cascade
connection of many two-port networks the transmission matrix (ABCD) can be easily found by
multiplying the ABCD matrices of each individual two-ports. Based on Fig 2.5, the ABCD
matrix can be defined as follows:
Fig 2.5: a) A two-port network b) Connection of two-port networks, taken from 4.5 “Microwave
Engineering”, D.M. Pozar, 3rd ed., 2005.
V1  AV2  BI 2
I 1  CV2  DI 2
or in a matrix form:
V1   A B  V2 
 I   C D  I 
 2 
 1 
[2-20]
where each element can be defined as:
A
V1
V
I
I
, B 1
,C 1
, D 1
V2 I 0
I 2 V 0
V2 I 0
I 2 V 0
2
2
2
2
Note the sign change of I 2 from our previous convention for Z and Y matrices.
For the cascaded two-port network shown in Fig 2.5 b)
[2-21]
ECE 375, Majedi, A.H.
53
V1   A1
 I   C
 1  1
B1  V2 
, and
D1   I 2 
V1   A1
 I   C
 1  1
B1   A2
D1  C 2
V2   A2
 I   C
 2  2
B2  V3 
D2   I 3 
B2  V3 
D2   I 3 
[2-22]
Which shows that the ABCD matrix of the cascaded connection of two networks is equal to the
product of the ABCD of each individual. This can be generalized to any number of two-port
networks, especially when we have a library of different two-port networks, as a building block
of any microwave circuits. Table 2.1 shows such a library.
Homework 2-2: For a reciprocal network, shows that AD  BC  1.
Table 2.2 shows the conversion between two-port network parameters, Z, Y, S, and ABCDs.