Microwave Engineering
Cheng-Hsing Hsu
Department of Electrical Engineering
National United University
SCOPES:
Passive microwave circuits design and analysis using transmission line
theory and microwave network theory.
TEXTBOOK AND REFERENCE BOOKS:
1. “Microwave Engineering”, by David M. Pozar, 3rd Edition (Textbook)
2. “
Foundations for Microwave Engineering”
, by R. E. Collin, 2ed Edition (Ref)
3. “
Microwave Engineering”
, by Peter A. Rizzi, (Out of Print) (Ref).
微波工程
Electronic Materials and Devices Applications Lab
Outline
1. Transmission Line Theory
2. Transmission Lines and Waveguides
General Solutions for TEM, TE, and TM waves ; Parallel Plate waveguide ; Rectangular Waveguide ; Coaxial
Line ; Stripline ; Microstrip
3. Microwave Network Analysis
Impedance and Equivalent Voltages and Currents ; Impedance and Admittance Matrices ; The Scattering
Matrix ; ABCD Matrix ; Signal Flow Graphs ; Discontinuties and Model Analysis
4. Impedance Matching and Tuning
Matching with Lumped Elements ; Single-Stub Tuning ; Double-Stub Tuning ; The Quarter-Wave Transformer ;
The Theory of Small Reflections
5. Microwave Resonators
Series and Parallel Resonant Circuits ; Transmission Line Resonators ; Rectangular Waveguide Cavities
Dielectric Resonators
6. Power Dividers and Directional Couplers
Basic Properties of Dividers and Couplers ; The T-Junction Power Divider ; The Wilkinson Power Divider ;
Coupled Line Directional Couplers ; 180o hybrid
7. Microwave Filters
Periodic Structure ; Filter Design by the Insertion Loss Method ; Filter Transformations ; Filter Implementation ;
微波工程
Electronic Materials and Devices Applications Lab
Introduction
*Definition
Microwave: designating or of that part of the electromagnetic spectrum between the
far infrared and some lower frequency limit: commonly regarded as extending from
300,000 to 300 megahertz. (from Webster’
s dictionary)
f : 300MHz - 300GHz
λ : 100cm - 0.1cm
electromagnetic spectrum
*Why use microwaves
(1) Antenna gain is proportional to the electric size of the antenna.
f , gain
miniature microwave system possible
(2) f
available bandwidth
e.g., TV BW=6MHz
10% BW of VHF @60MHz for 1channel
1% BW of U-band @60GHz for 100 channels
微波工程
Electronic Materials and Devices Applications Lab
The Electromagnetic Spectrum
微波工程
Electronic Materials and Devices Applications Lab
(3) Line of sight propagation and not effected by cloud, fog,…
frequency reuse in satellite and terrestrial communications
(4) Radar cross section (RCS) is proportional to the target electrical size.
frequency , RCS
radar application
(5) Molecular, atomic and nuclear resonances occur at microwave frequencies
astronomy, medical diagnostics and treatment, remote sensing and
industrial heating applications
*Biological effects and safety
non-ionized radiation
thermal effect
IEEE standard C95.1-1991
Excessive radiation may be dangerous to brain, eye, genital, …..
cataract, sterility, cancer,…..
微波工程
Electronic Materials and Devices Applications Lab
1. Transmission Line Theory
The Lumped-Element Circuit Model for a Transmission Line
The Terminated Lossless Transmission Line
Smith Chart
Quarter-Wave Transformer
Generator and Load Mismatched
Lossy Transmission Lines
微波工程
Electronic Materials and Devices Applications Lab
The Lumped-Element Circuit Model for A Transmission Line
A transmission line is a distributed-parameter network, where voltages and
currents can vary in magnitude and phase over its length.
A transmission line is often schematically represented as a two-wire line,
since transmission line => TEM wave propagation
: coaxial line, parallel line and stripline
Lumped-Element Circuit Model:
R = series resistance per unit length (both conductors)
L = series inductor per unit length (both conductors)
G = shunt conductance per unit length
C = shunt capacitance per unit length
微波工程
Electronic Materials and Devices Applications Lab
From Kirchhoff’
s voltage and Kirchhoff’
s current law
微波工程
Electronic Materials and Devices Applications Lab
where


V oe z V oe z
R  j L
R  j L
Zo 

I 
Z 
where phase constant
 
phase velocity v
input impedance
Z
p
in

2



vp

 f

V 
z

z 
I 
z

微波工程
Electronic Materials and Devices Applications Lab
For Lossless Line
From previous drived  ( R jL )(G jC )
; Z0 
R jL
; R and G are loss
G jC
if let R and G are zero :
jj LC
(0  LC ); Z 0 

1
from v pn  to obtained v pn 

LC
L
C
if dielectric medium is air : v pn c
微波工程
Electronic Materials and Devices Applications Lab
The Terminated Lossless Transmission Line
Assume that an incident wave of the form Vo+e-jz is generated from a source at z < 0.  we have
seen that the ratio of voltage to current for such a traveling wave is Zo, characteristic impedance.
When the line is terminated in an arbitrary load ZL Zo, the ratio of voltage to current at the
load must be ZL; a reflected wave must be excited with the appropriate amplitude to satisfy this
condition.
Sum of incident and reflected waves
The total voltage and current at the load are related by the load impedance
V 
0 VoVo
ZL 

Zo
I 
0 VoVo
at z 0
Z Z o 
 Vo  L
Vo
Z L Z o
Vo Z L Z o
   
Vo
Z L Z o

V
I 
z  
e
Z
 V 
z Vo e jz e jz

o
o
jz


e jz
total voltage and current waves on the line
A transmission line terminated in a load impedance ZL.
微波工程
Electronic Materials and Devices Applications Lab
V (0)
1 
From ohmic ' s law : Z L 
Z O  O
I (0)
1 O
Z Z O
  L
o Z L Z O
when Z L Z O O 0

O
For arbitrary of z : V ( z ) V ( e
z
z
z
V (z)
e
O e
Z in ( z ) 
Z O
z
z
I (z)
e
O e
Z in ( z ) Z O
Z in Z O
( Z L Z O ) e
( Z L Z O ) e
z
z
O e
z
VO z
z
) ; I ( z )  (e
O e
)
ZO
Z Z O
; because O  L
Z L Z O
( Z L Z O ) e
( Z L Z O ) e
z
z
e x cosh( x ) sinh( x )


sinh( x )
tanh( x ) 
cosh( x )






Z L Z O tanh( z )
Z O Z L tanh( z )
From z l , then Z in ( l ) Z O
Z in (l ) Z O
Z L jZ O tan ( l )
Z O jZ L tan ( l )
Z L Z O tanh( l )
Z O Z L tanh( l )
Lossless transmissi on line j


tanh (l ) tanh ( jl ) j tan ( l )



微波工程
Electronic Materials and Devices Applications Lab
2



1 Vo
2
where Pav 
Re 1 *e 2 jz e 2 jz  ,
2 Zo
middle two terms are purely imaginary A A* 2 j Im
A
which shows that the average power flow is constant at any point on the line
total power delivered to load is constant = incident power –reflected power
= 0  maximum power is delivered to the load
 = 1  no power is delivered
微波工程
Electronic Materials and Devices Applications Lab
When the load is mismatched, not all of the available power from the generator is delivered to the
load  Loss is called return loss (RL) and is defined in dB
RL = -20log || dB
If the load is matched to the line  = 0 and the magnitude of the voltage on the line is |V(z)| =
|Vo+|  is a constant
If the load is mismatched the presence of a reflected wave leads to standing waves where the
magnitude of the voltage on the line is not constant


jβz
2 jβz
V(z) Vo (e jβz Γe
) Vo e jβz( 1 Γe
)
2 jβz
2 jz
V(z) Vo 1 Γe
Vo 1 Γe
Vo 1 e j 2 

where e j  z is the positive d measured from the load at z 0,
and  is the phase of the reflection cofficient
V(z) max Vo ( 1 Γ) when the phase term e j 2 1 ;

V(z) min V0 ( 1 Γ0 ) when the phase term e j 2 1

**Standing Wave Ratio (Voltage Standing Wave Ratio)
As  increases, the ratio of Vmax to Vmin increases
V(z) max 1 Γ
VSWR (or SWR ) :VSWR 

V(z) min 1 Γ
A measure of mismatch of a line
VSWR 1
Γ 
VSWR 1
微波工程
Electronic Materials and Devices Applications Lab
This is an important result giving the input impedance of a length of
transmission line with an arbitrary load impedance transmission line
impedance equation





V

VO e je j
1 e 2 j
Z L jZ O tan ( l )
Z in (
)
  j
Z

Z

Z
 VO e e j O 1 e 2 j O O Z O jZ L tan ( l )
I

微波工程
Electronic Materials and Devices Applications Lab
Special Case of Lossless Terminated Lines
For a line is terminated in a short circuit -> ZL = 0 -> = -1
(a) Voltage, (b) current, and (c) impedance (Rin = 0 or ) variation along a
short-circuited transmission line.
微波工程
Electronic Materials and Devices Applications Lab
For a line is terminated in a open circuit -> ZL =  -> = 1
(a) Voltage, (b) current, and (c) impedance (Rin = 0 or ) variation along an open-circuited transmission line.
微波工程
Electronic Materials and Devices Applications Lab
From consider terminated transmission lines with some special lengths
微波工程
Electronic Materials and Devices Applications Lab
Consider a transmission line of characteristic impedance Zo feeding a line of
different characteristic impedance Z1
If the load line is infinitely long, or if it is terminated in its own characteristic
impedance, so that there are no reflections from its end, then the input impedance
seen by the feed line is Z1, then the reflection coefficient is
Z Z 0
 1
Z 1 Z 0
A transmissi on coefficien t T
the voltage for z 0 is

V 
z Vo e jz e jz

where Vois the amplitude of the incident voltage wave on the feed line
V 
z VoTe jz
for z 0
Equating these voltage at z 0 gives the transmisio n coefficien t T
Z Z 0
2 Z1
T 1 1  1

Z 1 Z 0 Z 1 Z 0
IL (insertion loss )
IL 10 log T
dB
微波工程
Electronic Materials and Devices Applications Lab
Often the ration of two power levels, P1 and P2, in a microwave system is expressed in
decibels (dB) as

10 log(P1/P2) dB
 Using power ratios in dB makes it easy to calculate power loss or gain through a series of
components. For ex. : A signal passing through a 6 dB attenuator followed by a 23 dB
amplifier will have an overall gain of 23 –6 = 17 dB.
If P1= V12 / R1 and P2 = V22 / R2, then the resulting power ratio in terms of voltage ratios is
V12 R2
V R2
10 log 2 20 log 1
dB
V2 R1
V2 R1
And if the load resistance are equal => 20 log(V1/V2) dB
On the other hand, the ratio of voltages across equal load resistances can also be
expressed in terms of nepers (Np)
 ln(V1/V2) Np  1/2[ ln(P1/P2)] Np since voltage is proportional to the square root of power
 10 Np = 10 log e2 = 8.686 dB
If a reference power level is assumed, then absolute powers can also be expressed notation
 If we let P2 = 1mW, then the power P1 can be expressed in dBm as
10 log (P1/1mW) dBm  a power of 1mW is 0dBm, while a power of 1W is 30dBm
微波工程
Electronic Materials and Devices Applications Lab
Smith Chart
Developed in 1939 by P. Smith at the Bell Tel. Lab. -> impedance matching problem and
transmission line issue
It is essentially a polar plot of the voltage reflection coefficient, 
 let the reflection coefficient be expressed in magnitude and phase (polar) form as =||ej  then the
magnitude || is plotted as a radius (||1) from the center of the chart, and the angle (-180o 180o) is
measured from the right-hand side of the horizontal diameter
The real utility of the smith chart, it can be used to convert
from reflection coefficients to normalized impedances (or
admittance)
When dealing with impedances on a Smith chart,
normalized quantities are generally used  z = Z / Zo
 The normalization constant is usually the characteristic
impedance of the line
微波工程
Electronic Materials and Devices Applications Lab
If a lossless line of characteristic impedance Zo is terminated with a load impedance ZL
 =
j

(zL-1)/(zL+1) = || e ; where zL=ZL/Zo  this relation can be solved for zL in terms of to give  zL =
(1+||ej) / (1-||ej) where Zin = [(1+e-2jl)/(1-e-2jl)]Zo , l = 0
This complex equation can be reduced to two real equations by writing and zL in terms of their real and
imaginary parts.
Let = r + ji and zL = rL + jxL
 rL + jxL = [(1+r)+ji]/[(1-r)-ji]
The real and imaginary parts of this equation can be found by multiplying the numerator and
denominator by the complex conjugate of the denominator to give
rL = [1-r2-i2]/[(1-r)2+i2]
xL = [2i]/[(1-r)2+i2]
 {r-[rL/(1+rL)]}2+i2 = [1/(1+rL)]2
(r-1)2+[i-(1/xL)] 2 = (1/xL)2
------- resistance circles
------- reactance circles
which are seen to represent two families of the circles in the r and i
For ex., the rL = 1 circles has its center at r = 0.5 , i = 0 -------- has a radius of 0.5, and so passes
through the center of the Smith chart
 All of the resistance circles have centers on the horizontal i = 0 axis, and pass through the = 1
point on the right-hand side of the chart.
The centers of all the reactance circles lie on the vertical r =1 line (off the chart), and these circles
also pass through the =1 point
The resistance and reactance circles are orthogonal.
微波工程
Electronic Materials and Devices Applications Lab
Since terms of the generalized reflection coefficient as
1 e 2 j
Z L jZ O tan ( l )
Z in 
Z

Z
O
O
1 e 2 j
Z O jZ L tan ( l )
where is the reflection at the load, and l is the (positive) length of transmission line.  If we have
plotted the reflection coefficient ||ej at the load, the normalized input impedance seen looking into a
length l of transmission line terminated with zL can be found by rotating the point clockwise an amount 2l
(- 2l)  the radius stays the same , since the magnitude of does not change with the position along the
line
The smith chart has scales around its periphery calibrated in the electrical wavelengths, toward
and away from the “generator”(the direction away from the load)  these scales are relative, so
only the difference in the wavelength between two points on the Smith chart is meaningful.
=> The scales cover a range of 0 to 0.5 wavelengths => a line of length /2 requires a rotation
of 2l = 2around the center of the chart, bring the point back to its original position=>
showing that the input impedance of a load seen through a /2 line is unchanged.
微波工程
Electronic Materials and Devices Applications Lab
Ex. A load impedance of 40+j70terminated a 100 transmission line that is 0.3long. Find
the reflection coefficient at the load, the reflection coefficient at the input to the line, the input
impedance, the SWR on the line, and the return loss.
<Sol>
The normalized load impedance is zL = ZL / Zo = 0.4 + j 0.7
 using a compass and the voltage and the voltage coefficient scale below the chart, the
reflection coefficient magnitude at the load can be read as || = 0.59 -> SWR = 3.87, and to the
return loss (RL) = 4.6dB  Now draw a radial line through the load impedance point, the read
the angle of the reflection coefficient at the load from the outer scale of the chart as 104o
On the other hand, drawing an SWR circle through the load impedance point.
Reading the reference position of the load on the
wavelengths-toward-generator (WTG) scale gives a value of
0.106 moving down the line 0.3toward the generator
bring to 0.406
Zin = Zozin = 100 (0.365 –j 0.611) = 36.5 –j 61.1
 the reflection coefficient at the input still has a magnitude
of || = 0.59 ; phase = 248o
微波工程
Electronic Materials and Devices Applications Lab
Combined impedance-Admittance Smith Chart
The Smith chart can be used for normalized admittance in the same way that it is used for
normalized impedances  it can be used to covert between impedance and admittance
Z jZ O tan ( l )
From Z in Z O L
Z O jZ L tan ( l )
the input impedance of load zL connected to a /4 line is zin = 1/zL which has the effect of
converting a normalized impedance to a normalized admittance.
Since a complete revolution around the Smith chart
corresponds to a length of  /2, a /4
transformation is equivalent to rotating the chart
by 180o ; this is also equivalent to imaging a given
impedance (or admittance) point across the center
of the chart to obtain the corresponding
admittance (or impedance) point.
微波工程
Electronic Materials and Devices Applications Lab
Ex. Smith Chart Operations Using Admittances
A load of ZL = 100 + j 50terminated a 50line. What are the load admittance
and the input admittance if the line is 0.15long ?
<Sol>
Normalized load impedance is zL = 2 + j 1  plotted the zL and SWR circle
Conversion to admittance can be accomplished with a /4 rotation of zL (or drawing a
straight line through zL and the center of the chart to intersect the SWR circle) ; The chart
can now be considered as an admittance chart, and the input impedance can be rotating
0.15from yL.
Plotting zL on the impedances scales and reading the admittance scales at this same give
yL = 0.4 –j 0.2 => the actual load admittance is then
YL = yLYo=yL / Zo = 0.008 –j 0.004 S
Then , on the WTG scale, the load admittance is seen to have a reference position of 0.214
. Moving 0.15 0.364
A radial line at this point on the WTG scale intersects the SWR circle at an admittance
of y = 0.61 + j 0.66
 actual input admittance is then Y = 0.0122 + j 0.0132 S
微波工程
Electronic Materials and Devices Applications Lab
Slotted Line
A slotted line is a transmission line configuration (usually waveguide or coax) that allows the
sampling of the electrical field amplitude of a standing wave on a terminated line.  with this
device the SWR and the distance of the first voltage minimum from the load can be measured,
and from this data the load impedance can be determined  due to the load impedance is in
general a complex number (with two degrees of freedom), two distinct quantities must be
measured with the slotted line to uniquely determine this impedance
 Measured impedance
Slotted Line (previous)  Vector Network Analyzer (now)
Assume that, for a certain terminated line, we have measured the SWR on the line
and lmin, the distance from the load to the first voltage minimum on the line. The
load impedance ZL can be determined as follows.
|| = (SWR-1)/(SWR+1) ; a voltage minimum occurs when ej(- 2l) = -1, when is
the phase angle of the reflection coefficient, = || ej
=> = + 2lmin where lmin is the distance from the load to the first voltage
An X-band waveguide slotted line.
minimum
Since the voltage minimums repeat every /2, where is the wavelength on the line,
and multiple of /2 can be added to lmin without changing the result in = + 2
lmin, because this just amounts to adding 2n/2 = 2n to , which not change 
 the complex reflection coefficient at the load can be find by SWR and lmin
To find the load impedance form with l = 0 : ZL = Zo [(1+)/(1-)]
微波工程
Electronic Materials and Devices Applications Lab
The Quarter-Wave Transformer
The quarter-wave transformer is a useful and practical circuit for impedance matching and also
provides a simple transmission line circuit that further illustrates the properties of standing waves
on a mismatched line.
For Impedance Viewpoint
These two components are connected with a lossless piece of transmission line of characteristic
impedance Z1 and length /4  It is desired to match the load to the Zo line, by using the /4 piece
of line, and so make = 0 looking into the /4 matching section.

Z in Z1
RL jZ1 tan 
Z1 jRL tan 
=> to evaluate this for l = (2
) (/4) = /2
 we can divide the numerator and denominator by tan l and take the limit as l  /2 to get
Zin = Z12 / RL
In order for = 0, we must have Zin = Zo, which yields the characteristic impedance Z1 as
Z1  Z o RL
the geometric mean of the load and source impedances
When the length of the matching section is λ/ 4 ,
or an odd multiple (2n+1) of λ/ 4 long,
so that a perfect match may be achieved at one frequency,
but mismatch will occur at other frequencies.
微波工程
Electronic Materials and Devices Applications Lab
Ex. Consider a load resistance RL = 100, to be matched to a 50 line with a quarterwave transformer. Find the characteristic impedance of the matching section and plot the
magnitude of the reflection coefficient versus normalized frequency, f/fo, where fo is the
frequency at which the line is /4 long.
<Sol>
z1  
50 

10070.71 
The reflection coefficient magnitude is given as

Z in Z o
Z in Z o
where the input impedance Z in is a function of frequency
The frequency dependence in comes from the term, which can be written in terms of f / f o as
v p  f
2f 
2
o  



 


  




4
v
4
f
 
 p 
 o  2 fo
For higher frequencies the line looks electrically longer,
and for lower frequencies it looks shorter.
The magnitude of the reflection coefficient is plotted versus f / fo
微波工程
Electronic Materials and Devices Applications Lab
The Multiple Reflection Viewpoint
= overall, or total, reflection coefficient of a wave incident on the /4 transformer
1 = partial reflection coefficient of a wave incident on a load Z1, from the Zo line
2 = partial reflection coefficient of a wave incident on a load Zo, from the Z1 line
3 = partial reflection coefficient of a wave incident on a load RL, from the Z1 line
T1 = partial transmission coefficient of a wave from the Z0 line into the Z1 line
T2 = partial transmission coefficient of a wave from the Z1 line into the Z0 line
1 = (Z1-Zo) / (Z1+Zo)
2 = (Zo-Z1) / (Zo+Z1) = - 1
3 = (RL-Z1) / (RL+Z1)
T1 = 2Z1 / (Z1+Zo)
T2 = 2Zo / (Z1+Zo)
Clearly, this process continues with an infinite number of bouncing waves,
And the total reflection coefficient is the sum of all of these partial
reflections. Since each round trip path up and down the /4 transformer
Section results in a 180o phase shift, the total reflection coefficient
can be expressed as
2
2 3


T
T


T
T



T
T

......
1
1 2 3
1 2 2 3
1 2 2
3 


T1T2 3 
2 3 
1
n
n 0
微波工程
Electronic Materials and Devices Applications Lab
when 3 1 and 2 1, the infinte series can be using the geometric series result that

1

for x 1
x 1
n 0
to give
T1T23
 T T 

1 1 2 3 1 2 3
1
1 2 3
1 2 3
x
n
The number of this expression can be simplified to give
2


Z1 Z o 
4 Z1 Z o 

3  T1T2 
3 


3
1
1
1
2
2




Z

Z
Z

Z
o
1
o
1




Z1 Z o 
RL Z1 

RL Z1 
Z1 Z o  2 Z12 Z o RL


Z1 Z o 
RL Z1 
Z1 Z o 
RL Z1 

2
1



which is seen to vanish if we choose Z1  Z o RL
Then is zero, and the line is matched
微波工程
Electronic Materials and Devices Applications Lab
Generator and Load Mismatches
In general, both generator and load may present mismatched impedances to the transmission line. We
will study this case, and also see that the condition for the maximum power transfer from the generator
to the load, in some situations, require a standing wave on line.
 Figure shows a transmission line circuit with arbitrary generator and load impedance, Zg and Zl, which may be
complex. transmission line is assumed lossless with a length l and characteristic impedance Zo => Due to
mismatched  multiple reflections can occur on the line  problem of the quarter-wave transformer
The input impedance looking into the terminated transmission line from the generator end is
Z in Z o
1 e 2 j
Z jZ o tan 
Z o 
2 j
1 e
Z o jZ tan 
Z Z
where is the reflection coefficient of the load   o
Z Z o

The voltage on the line can be written as V 
z Vo e jz e jz


o
and we can find V from the voltage at the generator end of the line , where z - 



V

Vg

Vo Vg

Zo
e j
V Vg
Z o Z g 1 g e 2 j
Z in
Vo e je j
Z in Z g
Z in
1
j
Z in Z g e e j


o



where g is the reflection coefficient seen looking into the generator
1 
SWR 
1 - 
微波工程
Electronic Materials and Devices Applications Lab
The power delivered to the load is
2
2
1  1
1 
1
1
Z in
2
P  Re Vin I in*  Vin Re   Vg
Re  
2
2
Z in Z g
Z in  2
Z in 
Now let Z in Rin jX in and Z g Rg jX g
 
1
P  Vg
2
2
Rin
Rin Rg 2 X in X g 2
Load Matched to Line
=> Zl = Zo  ℓ= 0 and SWR = 1 => the input impedance is Zin = Zo
1
2
=> the power delivered to the load is P  Vg
2
Zo
Z o Rg 2 X g2
Generator Matched to Loaded Line
The load impedance Zℓand/or the transmission line parameters ℓ, Zo are chosen to make the
input impedance Zin = Zg, so that the generator is matched to the load presented by the terminated
transmission line => the overall reflection coefficient, , is zero => =(Zin-Zg) / (Zin+Zg) = 0
 However, a standing wave on the line since ℓmay not be zero
The power delivered to the load is P 1 Vg
2
2

Rg
4 Rg2 X g2

微波工程
Electronic Materials and Devices Applications Lab
Conjugate Matching
Assuming that the generator series impedance, Zg, is fixed, we may vary the input impedance Zin until we achieve
the maximum power delivered to the load.
=> Knowing Zin  easy to find Zℓvia an impedance transformation along the line
To maximum P, we differentiate with respect to the real and imaginary parts of Zin
2 Rin 
Rin Rg 
P
1
0 

0
Rin
Rin Rg 2 X in X g 2 Rin Rg 2 X in X g 2 2
Rg2 Rin2 
X in X g 0


2
or ,
2 Rin 
X in X g 
P
0 
0
X in
Rin Rg 2 X in X g 2 2

X in 
X in X g 
0
or ,

solving simulatenously for Rin and X in gives
Rin Rg ,
or ,
X in X g
Z in Z g*
This condition is known as conjugate matching, and results in maximum power transfer to the load, for a fixed generator impedance
The power delivered is
2 1
1
P  Vg
2
4 Rg
微波工程
Electronic Materials and Devices Applications Lab
Lossy Transmission Lines
In practice, all transmission lines have loss due to finite conductivity and/or lossy dielectric.  we
will study the effect of loss on transmission line behavior and show how the attenuation constant can
be calculated.
For low loss line  R L G C
The general experssion for he complex propagation constant
 R 
 G 





jC 
 
R jL 
G jC  jL 
1

1
 jL 


j

C




G  RG
R
j LC 1 j   2
L C   LC
with RG 2 LC
G 
R
j LC 1 j   
L C 
If we were to ignore the ( R/ω/G/ω/) and using Taylor series expression

 j R

G 
1
C
L
1 R




j LC 
1   
,
so
that


R

G


GZ
o

Z


2

L

C
2
L
C
2

 



 o
 LC
 Zo 
R jL
L

G jC
C
微波工程
Electronic Materials and Devices Applications Lab
where Po is the power at the z=0 plane
微波工程
Electronic Materials and Devices Applications Lab
The Terminated Lossy Line
V 
z Vo
e z ez 

Vo z
I 
z  e ez
Zo

where is the reflection coefficient of the load and
Vois the incident voltage amplitude reference at z 0
 
e 2 je 2e 2 the reflection coefficient at a distance from the load
The input impedance Zin at a distance from the load
V

Z Z L Z o tanh 
 Z in 
I

 o Z o Z L tanh 
 the power delivered to the input of the terminated line at z -as
2


2


Vo 2
Vo
1
2 2
2



Pin  Re
V

I 
e  e

1 
 e 2
2
2Z o
2Z o
The power actually delivered to the load is
2
 
Vo
1
2
*
PL  Re V 
0 I 
0 
1 
2
2Z o


The difference in thee powers corresponds to the power lost in the line
2
Ploss


Vo
2
Pin PL 
e 21  1 e 2
2Z o
 
 The first term accounts for the power loss of the incident wave,
while the second term accounts for the power loss of the reflected wave
note : that both terms increase as increases
微波工程
Electronic Materials and Devices Applications Lab