transmission line

1. Prof. David R. Jackson
Dept. of ECE
Notes 2
ECE 5317-6351
Microwave Engineering
Fall 2019
Transmission Lines
Part 1: TL Theory
1
Adapted from notes by
Prof. Jeffery T. Williams

2. Transmission-Line Theory
We need transmission-line theory whenever the length of a line
is significant compared to a wavelength.
2
L

3. Transmission Line
2 conductors
4 per-unit-length parameters:
C = capacitance/length [F/m]
L = inductance/length [H/m]
R = resistance/length [/m]
G = conductance/length [ /m or S/m]

Dz
3

4. Transmission Line (cont.)
z
D
 
,
i z t
+ + + + + + +
- - - - - - - - - -
 
,
v z t
x x x
B
4
Note: There are equal and opposite currents on the two conductors.
(We only need to work with the current on the top conductor, since we have chosen to put all of the series elements there.)
+
-
+
-
 
,
i z z t
 D
 
,
i z t
 
,
v z z t
 D
 
,
v z t
R z
D L z
D
G z
D C z
D
z

5. ( , )
( , ) ( , ) ( , )
( , )
( , ) ( , ) ( , )
i z t
v z t v z z t i z t R z L z
t
v z z t
i z t i z z t v z z t G z C z
t

  D  D  D

  D
  D   D D  D

Transmission Line (cont.)
5
+
-
+
-
 
,
i z z t
 D
 
,
i z t
 
,
v z z t
 D
 
,
v z t
R z
D L z
D
G z
D C z
D
z

6. Hence
( , ) ( , ) ( , )
( , )
( , ) ( , ) ( , )
( , )
v z z t v z t i z t
Ri z t L
z t
i z z t i z t v z z t
Gv z z t C
z t
 D  
  
D 
 D    D
   D 
D 
Now let Dz  0:
v i
Ri L
z t
i v
Gv C
z t
 
  
 
 
  
 
“Telegrapher’s
Equations”
TEM Transmission Line (cont.)
6

7. To combine these, take the derivative of the first one with
respect to z:
2
2
2
2
v i i
R L
z z z t
i i
R L
z t z
v v v
R Gv C L G C
t t t
   
 
    
   
 
  
 
    
  
 
  
   
      
   
  
   
Switch the order of the
derivatives.
TEM Transmission Line (cont.)
7
i v
Gv C
z t
 
  
 

8.  
2 2
2 2
( ) 0
v v v
RG v RC LG LC
z t t
  
 
    
 
  
 
The same differential equation also holds for i.
Hence, we have:
2 2
2 2
v v v v
R Gv C L G C
z t t t
   
   
      
   
   
   
TEM Transmission Line (cont.)
8
Note: There is no exact solution in the time domain, in the lossy case.

9.  
2
2
2
( ) ( ) 0
d V
RG V RC LG j V LC V
dz
 
     
 
2 2
2 2
( ) 0
v v v
RG v RC LG LC
z t t
  
 
    
 
  
 
TEM Transmission Line (cont.)
Time-Harmonic Waves:
9
j
t





10. Note that
   
2
2
2
( )
d V
RG V j RC LG V LC V
dz
 
   
2
( ) ( )( )
RG j RC LG LC R j L G j C
   
     
=
=
Z R j L
Y G j C


 
 
series impedance / unit length
parallel admittance / unit length
Then we can write:
2
2
( )
d V
ZY V
dz

TEM Transmission Line (cont.)
10

11. Define
We have:
Solution:
2
ZY
 
( ) z z
V z Ae Be
 
 
 
 1/2
( )( )
R j L G j C
  
  
 
 
2
2
2
d V z
V z
dz


Then
TEM Transmission Line (cont.)
 is called the “propagation constant”.
11
Question: Which sign of the square root is correct?

12. Principal square root:
TEM Transmission Line (cont.)
/2
j
z r e 
   
  
12
 1/2
( )( )
R j L G j C
  
  
We choose the principal square root.
 
Re 0
z 
( )( )
R j L G j C
  
   Re 0
 
Hence
(Note the square-root (“radical”) symbol here.)
j
z re 

1/2
1/2
4 2, 4 2
1 1
,
2 2
j j
j j
  
 
 
  
 
 
Examples :

13. Denote:
TEM Transmission Line (cont.)
Re 0
 
 
13
( )( )
R j L G j C
  
  
[np/m]
  attenuationcontant
j
  
 
 
rad/m
  phaseconstant
 
1/m
  propagationconstant

14. TEM Transmission Line (cont.)
14
There are two possible locations for the complex square root:
0, 0
 
 
Hence:
Re
Im
 
 


R j L


Re
Im
G j C


Re
Im



Re
Im
( )( )
R j L G j C
  
  
The principal
square root must
be in the first
quadrant.

15. TEM Transmission Line (cont.)
15
  z z j z
V z Ae Ae e
  
  
 
Wave traveling in +z direction:
 
j
  
 
Wave is attenuating as it propagates.
  z z j z
V z Ae Ae e
  
  
 
Wave traveling in -z direction:
 
j
  
 
Wave is attenuating as it propagates.

16. TEM Transmission Line (cont.)
16
  
10
10
dB 20log
20log
20 0.4343
8.686
out
in
z
V
V
A e
A
z
z




 
   
 
 
 
   
 
 
  

Attenuation in
   
10
log 0.4343 ln
x x

Note:
Attenuation in dB/m:   z j z
V z Ae e
 
 

dB/m 8.686

Attenuationin

17. Wavenumber Notation
17
( )( )
R j L G j C
  
   "propagationconstant"
  z z j z
V z Ae Ae e
  
  
 
  z
jk z z j z
V z Ae Ae e
 
  
 
( )( )
z
k j R j L G j C
 
    "propagationwavenumber"
z
jk
 
j
  
 
z
k j
 
 

18. TEM Transmission Line (cont.)
0 0
( ) z z j z
V z V e V e e
  
     
 
Forward travelling wave (a wave traveling in the positive z direction):
 
 
 
 
 
 
0
0
0
( , ) Re
Re
cos
z j z j t
j z j z j t
z
v z t V e e e
V e e e e
V e t z
  
   

  
   
  
 


  
2
g




2
g
 

The wave “repeats” when:
Hence:
18
g
 0
t 
z
0
z
V e 
 
“snapshot” of wave

19. Phase Velocity
Let’s track the velocity of a fixed point on the wave (a point of constant
phase), e.g., the crest of the wave.
0
( , ) cos( )
z
v z t V e t z

  
  
  
19
z
(phase velocity)
p
v

20. Phase Velocity (cont.)
0
constant
 
 


 
 

t z
dz
dt
dz
dt
Set
Hence p
v



 
Im )(
p
v
R j L G j C

 

 
In expanded form:
20

21. Characteristic Impedance Z0
0
( )
( )
V z
Z
I z



0
0
( )
( )
z
z
V z V e
I z I e


  
  


so 0
0
0
V
Z
I



Assumption: A wave is traveling in the positive z direction.
(Note: Z0 is a number, not a function of z.)
21
 
I z

 
V z



z

22. Use first Telegrapher’s Equation:
v i
Ri L
z t
 
  
 
so
dV
RI j LI ZI
dz

    
Hence
0 0
z z
V e ZI e
 
    
  
Characteristic Impedance Z0 (cont.)
22
0
0
( )
( )
z
z
V z V e
I z I e


  
  


Recall:

23. From this we have:
Use:
0
0
0
V Z Z
Z
I ZY



  
Characteristic Impedance Z0 (cont.)
Z R j L
Y G j C


 
 
23
Both are in the first quadrant
0
0
/
Re 0
ZY
Z Z ZY
Z

  
  
 
first quadrant
right - half plane
0
Z
Z
Y
 (principal square root)

24. Hence, we have
Characteristic Impedance Z0 (cont.)
24
0
Z R j L
Z
Y G j C



 

0
Re 0
Z 

25.  
0
0
0 0
j z j j z
z z
z j z
V e e
V z V e V
V e e e
e
e
 
 

  


  

   


 
 
   
 
 
 
0
0 cos
c
, R
os
e j t
z
z
V e t
v z t V z
z
V z
e
e t



 
  

 
 









In the time domain:
Wave in +z
direction
Wave in -z
direction
General Case (Waves in Both Directions)
25

26. Backward-Traveling Wave
0
( )
( )
V z
Z
I z



 0
( )
( )
V z
Z
I z


 
so
A wave is traveling in the negative z direction.
Note:
The reference directions for voltage and current are chosen the
same as for the forward wave.
26
z
 
I z

 
V z




27. General Case
0 0
0 0
0
( )
1
( )
z z
z z
V z V e V e
I z V e V e
Z
 
 
   
   
 
 
 
 
A general superposition of forward and
backward traveling waves:
Most general case:
27
z
 
I z
 
V z



28.  
 
  
0 0
0 0
0 0
0
z z
z z
V z V e V e
V V
I z e e
Z
j R j L G j C
R j L
Z
G j
Z
C
 
 
    


   
 
 
    







 
2
m
g




[m/s]
p
v



Guided wavelength:
Phase velocity:
Summary of Basic TL formulas
28
dB/m 8.686

Attenuationin
 
I z
 
V z



29. Lossless Case
0, 0
R G
 
( )( )
j R j L G j C
j LC
    

    

so
0
LC

 


0
R j L
Z
G j C




 0
L
Z
C

1
p
v
LC

p
v



(independent of freq.)
(real and independent of freq.)
29

30. Lossless Case (cont.)
1
p
v
LC

If the medium between the two conductors is lossless and homogeneous
(uniform) and is characterized by (, ), then (proof given later):
LC 

The speed of light in a dielectric medium is
1
d
c


Hence, we have that: p d
v c

In the lossless case the phase velocity does not depend on the frequency, and it
is always equal to the speed of light (in the material).
(proof given later)
30
0 r r
LC k k
     
   
and

31.   0 0
z z
V z V e V e
 
   
 
Where do we assign z = 0 ?
The usual choice is at the load.
Amplitude of voltage wave
propagating in negative z
direction at z = 0.
Amplitude of voltage wave
propagating in positive z
direction at z = 0.
Terminated Transmission Line
31
 
 
0
0
0
0
V V
V V
 
 


 
V z

 
V z
 Terminating impedance (load)
 
V z
 
I z
0
z 
z
L
Z



32.   0
0 0
z
V V z e 

 
 
     
   
0 0
0 0
z z z z
V z V z e V z e
 
  
 
 
  0
0 0
z
V z V e 

 

  0
0 0
z
V V z e 

 
 
Terminated Transmission Line (cont.)
  0 0
z z
V z V e V e
 
   
 
Hence
Can we use z = z0 as
a reference plane?
32
  0
0 0
z
V z V e 

 

Terminating impedance (load)
 
V z
 
I z
0
z 
z
L
Z


0
z z


33. Terminated Transmission Line (cont.)
     
0 0
z z
V z V e V e
 
   
 
Compare:
This is simply a change of reference plane, from z = 0 to z = z0.
33
     
   
0 0
0 0
z z z z
V z V z e V z e
 
  
 
 
Terminating impedance (load)
 
V z
 
I z
0
z 
z
L
Z


0
z z


34.   0 0
z z
V z V e V e
 
   
 
What is V(-d) ?
  0 0
d d
V d V e V e
 
  
  
  0 0
0 0
d d
V V
I d e e
Z Z
 
 

  
Propagating
forwards
Propagating
backwards
Terminated Transmission Line (cont.)
d  distance away from load
The current at z = -d is then:
34
(This does not necessarily have to be the length of the entire line.)
Terminating impedance (load)
 
I d

z d
 


0
z 
z
L
Z
 
V d


35.    
2
0
0
1
d d
L
V
I d e e
Z
 


   
  2
0
0 0 0
0
1
d d d d
V
V d V e V e V e e
V
   

    

 
    
 
 
Similarly,
L  load reflection coefficient
Terminated Transmission Line (cont.)
35
   
2
0 1
d d
L
V d V e e
 
 
   
or
0
0
L
V
V


 
 
I d

z d
 


0
z 
z
L
Z
 
V d

(-d) = reflection coefficient at z = -d
  2 d
L
d e 

   

36.    
   
2
0
2
0
0
1
1
d d
L
d d
L
V d V e e
V
I d e e
Z
 
 
 


   
  
Z(-d) = impedance seen “looking” towards load at z = -d.
Terminated Transmission Line (cont.)
36
 
 
 
 
 
2
0 0
2
1
1
1 1
d
L
d
L
V d d
e
Z d Z Z
I d e d




 
   
 
 
     
 
     
   
Note:
If we are at the
beginning of the line,
we will call this the
“input impedance”.
 
I d

z d
 


0
z 
z
L
Z
 
V d

 
Z d


37. At the load (d = 0):
  0
1
0
1
L
L
L
Z Z Z
 
 
 
 
 
 
Terminated Transmission Line (cont.)
0
0
L
L
L
Z Z
Z Z

 

37

  2 d
L
d e 

   
At any point on the line(d > 0):

38. Thus,
 
2
0
0
0
2
0
0
1
1
d
L
L
d
L
L
Z Z
e
Z Z
Z d Z
Z Z
e
Z Z




 
 


 
 

 
 
 
 
 


 
 

 
 
Terminated Transmission Line (cont.)
 
 
 
2
0 0 2
1 1
1 1
d
L
d
L
d e
Z d Z Z
d e




 
    
 
  
   
    
 
 
Recall
38

39. Simplifying, we have:
 
 
 
0
0
0
tanh
tanh
L
L
Z Z d
Z d Z
Z Z d


 

   

 
Terminated Transmission Line (cont.)
 
   
   
   
   
   
   
2
0
2
0 0 0
0 0 2
2 0 0
0
0
0 0
0
0 0
0
0
0
1
1
cosh sinh
cosh sinh
d
L
d
L L L
d
d L L
L
L
d d
L L
d d
L L
L
L
Z Z
e
Z Z Z Z Z Z e
Z d Z Z
Z Z Z Z e
Z Z
e
Z Z
Z Z e Z Z e
Z
Z Z e Z Z e
Z d Z d
Z
Z d Z d




 
 
 
 




 
 
 
 


 
   
   
 
 
    
    
 
  

 
 

 
 
 
  
  
  
 
 

  

 
Hence, we have
39

40.    
   
2
0
2
0
0
1
1
j d j d
L
j d j d
L
V d V e e
V
I d e e
Z
 
 
 


   
  
Impedance is periodic with period g/2:
 
 
2 1
2 1
2 1
2
/ 2
g
g
d d
d d
d d
 




 
 
  
Terminated Lossless Transmission Line
j j
   
  
Note:      
tanh tanh tan
d j d j d
  
 
The tan function repeats when
 
 
 
0
0
0
tan
tan
L
L
Z jZ d
Z d Z
Z jZ d


 

   

 
40
Lossless:
 
2
0 2
1
1
j d
L
j d
L
e
Z d Z
e




 

   

 

41.  
 
 
2
0 2
0
0
0
1
1
tanh
tanh
d
L
d
L
L
L
e
Z d Z
e
Z Z d
Z
Z Z d






 
 
   

 
 

  
 

 
0
0
2
L
L
L
g
p
Z Z
Z Z
v






 



Summary for Lossy Transmission Line
41
 
I d

z d
 


0
z 
z
L
Z
 
V d

 
Z d

( )( )
j R j L G j C
    
    

42.  
 
 
2
0 2
0
0
0
1
1
tan
tan
j d
L
j d
L
L
L
e
Z d Z
e
Z jZ d
Z
Z jZ d






 
 
   
 
 
 

  

 
0
0
2 2
L
L
L
g d
p d
Z Z
Z Z
k
v c
 
 




 

  
 
Summary for Lossless Transmission Line
42
 
I d

z d
 


0
z 
z
L
Z
 
V d

 
Z d

LC k
   
  

43. 0
0
0
L
L
L
Z Z
Z Z

  

No reflection from the load
Matched Load (ZL=Z0)
  0
Z d Z z
  for any
43
 
2
0 2
1
1
d
L
d
L
e
Z d Z
e




 

   

 
 
I d

z d
 


0
z 
z
L
Z
 
V d

 
Z d



44. 1
L
  
Always imaginary!
Short-Circuit Load (ZL=0)
44
 
 
 
0
0
0
tan
tan
L
L
Z jZ d
Z d Z
Z jZ d


 

   

 
   
0 tan
Z d jZ d

 
0
0
L
L
L
Z Z
Z Z

 

 
Z d
 z d
  0
z 
z
 
I d

 
V d



Lossless Case

45. Note: 2
g
d
d
 


  sc
Z d jX
 
S.C. can become an O.C. with a
g/4 transmission line.
Short-Circuit Load (ZL=0)
 
0 tan
sc
X Z d


45
   
0 tan
Z d jZ d

 
sc
X
Inductive
Capacitive
/ g
d 
0 1/ 4 1/ 2 3/ 4
Lossless Case
 
g d
 

lossless

46. 0
0
1 /
1
1 /
L
L
L
Z Z
Z Z

  

Always imaginary!
Open-Circuit Load (ZL=)
46
1
L
  
0
0
L
L
L
Z Z
Z Z

 

 
 
 
0
0
0
tan
tan
L
L
Z jZ d
Z d Z
Z jZ d


 

   

 
   
0 cot
Z d jZ d

  
 
   
   
0
0
0
1 / tan
/ tan
L
L
j Z Z d
Z d Z
Z Z j d


 

   

 
or
 
Z d
 z d
  0
z 
z
 
I d

 
V d



Lossless Case

47. 2
g
d
d
 


O.C. can become a S.C. with a
g/4 transmission line.
Open-Circuit Load (ZL=)
47
  oc
Z d jX
 
 
0 cot
oc
X Z d

 
   
0 cot
Z d jZ d

  
Note:
Inductive
Capacitive
oc
X
/ g
d 
0 1/ 4 1/ 2
Lossless Case
 
g d
 

lossless

48. Using Transmission Lines to Synthesize Loads
A microwave filter constructed from microstrip line.
This is very useful in microwave engineering.
48
We can obtain any reactance that we want from a short or open transmission line.

49. 49
Find the voltage at any point on the line.
 
 
 
0
0
0
tanh
tanh
L
in
L
Z Z l
Z Z l Z
Z Z l


 

    
 

 
  in
Th
in Th
Z
V l V
Z Z
 
   

 
At the input:
Voltage on a Transmission Line
0 ,
Z   
V z
 
I z


0
z 
L
Z
l
Th
Z
Th
V


in
Z
Th
V
Th
Z
in
Z
 
V l






50.    
0
2
1
z z
L
V z V e
e  
  
 
 0
0
L
L
L
Z Z
Z Z

 

   
2
0 1
l l in
L Th
in Th
Z
V l V e e V
Z Z
 
   
      

 
   
2
2
1
1
z
l z
in L
Th l
in Th L
Z e
V z V e
Z Z e




 

   
 
    
 
 
 
At z = -l:
Hence
0 2
1
1
l
in
Th l
in Th L
Z
V V e
Z Z e


 

   
    
 
 
 
50
Voltage on a Transmission Line (cont.)

Incident (forward) wave (not the same as the initial wave from the source!)

51. Let’s derive an alternative form of the previous result.
 
2
0 2
1
1
l
L
in l
L
e
Z Z l Z
e




 
 
    

 
 
   
 
   
 
2
2
0 2
0
2 2
2
0
0 2
2
0
2
0 0
2
0
2
0 0
0
0
0
1
1
1
1 1
1
1
1
1
1
l
L
l
l
L
L
l l
l
L Th L
L
Th
l
L
l
L
l
Th L Th
l
L
l
Th Th
L
in
in T
Th
h
h
T
e
Z
Z e
e
Z e Z e
e
Z Z
e
Z e
Z Z e Z Z
e
Z
Z Z Z Z
e
Z Z
Z
Z Z
Z
Z Z



 









 






 
 
   
 
 
  
      
 

 
 
 
 

   
 
 
  
  

     


 
 
 


 
2
2 0
0
1
1
l
L
l Th
L
Th
e
Z Z
e
Z Z




 

 

    

 
51
Start with:
Voltage on a Transmission Line (cont.)

52.    
2
0
2
0
1
1
z
l z L
Th l
Th s L
Z e
V z V e
Z Z e




 

   

    
  
   
2
0
2
0
1
1
l
in L
l
in Th L
s
Th
Z Z e
Z Z Z Z e




  
 
   
   
  
 
where 0
0
Th
s
Th
Z Z
Z Z

 

Therefore, we have the following alternative form for the result:
Hence, we have
52
(source reflection coefficient)
Voltage on a Transmission Line (cont.)
   
2
2
1
1
z
l z
in L
Th l
in Th L
Z e
V z V e
Z Z e




 

   
 
    
 
 
 
Recall:
Substitute

53.    
 
2
0
2
0
1
1
z
z l L
Th l
Th s L
Z e
V z V e
Z Z e




  

   

    
  
   
The “initial” voltage wave that would exist if there were no reflections from the load
(we have a semi-infinite transmission line or a matched load).
53
Voltage on a Transmission Line (cont.)
This term accounts for the multiple (infinite) bounces.
0 ,
Z   
V z
 
I z


0
z 
L
Z
l
Th
Z
Th
V


in
Z

54.  
 
       
2 2
2 2 2 2
0
0
1 l l
L L s
l l l l
Th L s L L s L s
Th
e e
Z
V l V e e e e
Z Z
 
   
 
   
 
    
 
  
   
          
     
 

 
 

 
Wave-bounce method (illustrated for z = -l):
54
Voltage on a Transmission Line (cont.)
We add up all of the bouncing waves.
0 ,
Z 


0
z 
L
Z
l
Th
Z
Th
V


in
Z
 
V l


55.  
   
   
2
2 2
2
2 2 2
0
0
1
1
l l
L S L S
l l l
Th L L S L S
Th
e e
Z
V l V e e e
Z Z
 
  
 
  
 
      
 
  
 
          
  
 
 

  

 
 
Geometric series:
2
0
1
1 , 1
1
n
n
z z z z
z


     

 2 l
L s
z e 

  
55
Group together alternating terms:
Voltage on a Transmission Line (cont.)
 
 
       
2 2
2 2 2 2
0
0
1 l l
L L s
l l l l
Th L s L L s L s
Th
e e
Z
V l V e e e e
Z Z
 
   
 
   
 
    
 
  
   
          
     
 

 
 

 

56. or
 
2
0
2
0
2
1
1
1
1
l
L s
Th
d
Th
L l
L s
e
Z
V l V
Z Z
e
e






 
 
  
  
    
 

   
 
 
  
 
 
 
 
2
0
2
0
1
1
l
L
Th l
Th L s
Z e
V l V
Z Z e




  

    
  
  
This agrees with the previous result (setting z = -l).
The wave-bounce method is a very tedious method – not recommended.
Hence
56
Voltage on a Transmission Line (cont.)
   
2
0
2
0
1
1
z
l z L
Th l
Th s L
Z e
V z V e
Z Z e




 

   
 
    
  
   
Previous (alternative)result :

57. At a distance d from the load:
     
 
  
   
*
*
*
2
0 2 2 * 2
*
0
2 2
2
0 0
2 4 2 2 * 2
* *
0 0
1
Re
2
1
Re 1 1
2
1 1
Re 1 Re
2 2
z z z
L L
z z z z z
L L L
P z V z I z
V
e e e
Z
V V
e e e e e
Z Z
  
    

  
 
    

 
 
   
 
 
   
   
     
   
   
   
2
2
0 2 4
0
1
1
2
z z
L
V
P z e e
Z
 

 
  
If Z0  real (low-loss transmission line):
Time-Average Power Flow
   
   
2
0
2
0
0
1
1
z z
L
z z
L
V z V e e
V
I z e e
Z
j
 
 
  
  

 
  
 
 
 
*
2 * 2
*
2 2
z z
L L
z z
L L
e e
e e
 
 
 
 
 
   
 pure imaginary
Note:
57
 
I z
z


0
z 
z
L
Z
 
V z
P(z) = power flowing in + z direction
(please see the note)

58. Low-loss line
   
2
2
0 2 4
0
2 2
2
0 0
2 2
0 0
1
1
2
1 1
2 2
z z
L
z z
L
V
P z e e
Z
V V
e e
Z Z
 
 

 
 
 
  
  
Power in forward-traveling wave Power in backward-traveling wave
   
2
2
0
0
1
1
2
L
V
P z
Z

  
Lossless line ( = 0)
Time-Average Power Flow (cont.)
58
 
I z
z


0
z 
z
L
Z
 
V z
Note:
For a very lossy
line, the total
power is not the
difference of the
two individual
powers.

59. 0
0
0
tan
tan
L T
in T
T L
Z jZ d
Z Z
Z jZ d


 

  

 
2
4 4 2
g g
g
d
 
 
 

  
0
0
T
in T
L
jZ
Z Z
jZ
 
   
 
0
2
0
0
0
in in
T
L
Z Z
Z
Z
Z
   
 
Quarter-Wave Transformer
2
0T
in
L
Z
Z
Z

so
0 0
T L
Z Z Z

Hence
(This requires ZL to be real.)
59
Matching condition
0T
Z L
Z
0
Z
in
Z / 4
g
d 

Lossless line

60. Match a 100  load to a 50  transmission line at a given frequency.
0 100 50
70.7
T
Z  

0
0
2 2 2
g d
r r
k k

  
 
  
    
0 70.7
T
Z  
60
50
/ 4
g

 
100
L
Z  
0T
Z
0
c
f
 
Lossless line
Quarter-Wave Transformer (cont.)
LC k
   
  
Note:
Example

61.   2
0 1 L
j j z
L
V z V e e
 
 
  
   
 
2
0
2
0
1
1 L
j z j z
L
j
j z j z
L
V z V e e
V e e e
 

 
  
  
  
  
 
 
max 0
min 0
1
1
L
L
V V
V V


  
  
Voltage Standing Wave
61
z
1+ L

1
1- L

0
( )
V z
V 
/ 2
g
z 
D 
0
z 
2 2 / 2
g
z z
  
D   D 
 
I z
z


0
z 
z
L
Z
 
V z
Lossless Case
Note: The voltage repeats every g. The magnitude repeats every g /2.
Note: The voltage changes by a minus sign after g /2.

62.   max
min
V
V

Voltage Standing Wave Ratio VSWR
Voltage Standing Wave Ratio
1
1
L
L
 

 
VSWR z
1+ L

1
1- L

0
( )
V z
V 
/ 2
g
z 
D 
0
z 
62
 
 
max 0
min 0
1
1
L
L
V V
V V


  
  
 
I z
z


0
z 
z
L
Z
 
V z

63. At high frequency, discontinuity effects can become important.
Limitations of Transmission-Line Theory
Bend
Incident
Reflected
Transmitted
The simple TL model does not account for the bend.
63
L
Z
0
Z
Th
Z



64. At high frequency, radiation effects can also become important.
When will radiation occur?
We want energy to travel from the generator to the load, without radiating.
Limitations of Transmission-Line Theory (cont.)
64
This is explored next.
L
Z
0
Z
Th
Z



65. The coaxial cable is a perfectly shielded system – there is never any radiation at
any frequency, as long as the metal thickness is large compared with a skin depth.
The fields are confined
to the region between
the two conductors.
Limitations of Transmission-Line Theory (cont.)
65
Coaxial Cable
r
 a
b
z

66. The twin lead is an open type of transmission
line – the fields extend out to infinity.
The extended fields may cause interference
with nearby objects.
(This may be improved by using “twisted pair”.)
Limitations of Transmission-Line Theory (cont.)
Having fields that extend to infinity is not the same thing as having radiation, however!
66
 

67. The infinite twin lead will not radiate by itself, regardless of how far
apart the lines are (this is true for any transmission line).
The incident and reflected waves represent an exact solution to
Maxwell’s equations on the infinite line, at any frequency.
 
*
1
ˆ
Re 0
2
t
S
P E H dS

 
   
 
 

S
+ -
Limitations of Transmission-Line Theory (cont.)
67
Incident
Reflected
No attenuation on an infinite lossless line
h

68. A discontinuity on the twin lead will cause radiation to occur.
Note:
Radiation effects usually increase as
the frequency increases.
Limitations of Transmission-Line Theory (cont.)
68
Incident wave
Pipe
Obstacle
Reflected wave
h
Bend
Incident wave
Bend
Reflected wave
h

69. To reduce radiation effects of the twin lead at discontinuities:
1) Reduce the separation distance h (keep h << ).
2) Twist the lines (twisted pair).
Limitations of Transmission-Line Theory (cont.)
CAT 5 cable
(twisted pair)
69
h