Transmission Lines &
Waveguides
Fall 2022
Prof. Muhammad Amin
TL Theory
1
Transmission-Line Theory
We need transmission-line theory whenever the length of a line
is significant compared to a wavelength.
L
2
Transmission Line
2 conductors
4 per-unit-length parameters:
C = capacitance/length [F/m]
L = inductance/length [H/m]
R = resistance/length [/m]
Dz

G = conductance/length [ /m or S/m]
3
Transmission Line (cont.)
i  z, t 
B
x x x
+++++++
----------
v  z, t 
Dz
RDz
i  z, t 
LDz
i  z  Dz, t 
+
+
v  z, t 
-
G Dz
C Dz
v  z  Dz, t 
-
z
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.)
4
Transmission Line (cont.)
RDz
i  z, t 
LDz
i  z  Dz, t 
+
+
v  z, t 
-
G Dz
C Dz
v  z  Dz, t 
-
z
i ( z , t )
v( z , t )  v( z  Dz , t )  i ( z , t ) RDz  LDz
t
v( z  Dz , t )
i ( z , t )  i ( z  Dz , t )  v( z  Dz , t ) G Dz  C Dz
t
5
TEM Transmission Line (cont.)
Hence
v( z  Dz , t )  v( z , t )
i ( z , t )
  Ri ( z , t )  L
Dz
t
i ( z  Dz , t )  i ( z , t )
v( z  Dz , t )
 Gv( z  Dz , t )  C
Dz
t
Now let Dz

0:
v
i
  Ri  L
z
t
i
v
  Gv  C
z
t
“Telegrapher’s
Equations”
6
TEM Transmission Line (cont.)
To combine these, take the derivative of the first one with
respect to z:
v
i
  i 
 R  L  
z
z
z  t 
Switch the order of the
derivatives.
i
  i 
 R  L  
z
t  z 
v 
v
 v


  R  Gv  C   L  G  C
t 
t
t 


2
2
2
2
i
v
  Gv  C
z
t
7
TEM Transmission Line (cont.)
v
v 
 v

 v
  R  Gv  C   L  G  C
z
t 
t
t 


2
2
2
2
Hence, we have:
v
v
 v
  RG  v  ( RC  LG)  LC 
0
z
t
 t 
2
2
2
2
The same differential equation also holds for i.
Note: There is no exact solution in the time domain, in the lossy case.
8
TEM Transmission Line (cont.)
Time-Harmonic Waves:

 j
t
v
v
 v
  RG  v  ( RC  LG)  LC 
0
z
t
 t 
2
2
2
2
2
dV
  RG V  ( RC  LG ) jV  LC ( )V  0
dz
2
2
9
TEM Transmission Line (cont.)
2
dV
  RG V  j ( RC  LG )V   LC V
dz
2
2
Note that
RG  j ( RC  LG)   LC  ( R  j L)(G  jC )
2
Z  R  j L = series impedance / unit length
Y  G  jC = parallel admittance / unit length
2
Then we can write:
dV
 ( ZY )V
dz
2
10
TEM Transmission Line (cont.)
Define
d V z
  V z
dz
2
  ZY
2
Then
2
2
Solution:
V ( z )  Ae  Be
 z
 z
 is called the “propagation constant”.
We have:
  ( R  jL)(G  jC )
1/2
Question: Which sign of the square root is correct?
11
TEM Transmission Line (cont.)
  ( R  jL)(G  jC )
1/2
We choose the principal square root.
Principal square root: z  re j
z  r e j /2
    
Re
(Note the square-root (“radical”) symbol here.)
 z0
Hence
  ( R  j L)(G  jC )
Examples :
41/ 2  2,
42
 1 j 
j1/ 2   
,
 2 
j
1 j
2
Re   0
12
TEM Transmission Line (cont.)
  ( R  j L)(G  jC )
Denote:
    j
  phase constant rad/m
  propagation constant 1/m
  attenuationcontant [np/m]
  Re   0
13
TEM Transmission Line (cont.)
Im
Im
R  j L
G  jC
Re
Re
  ( R  j L)(G  jC )
There are two possible locations for the complex square root:
Im

Re

The principal
square root must
be in the first
quadrant.
  Re 
  Im 
Hence:
  0,   0
14
TEM Transmission Line (cont.)
Wave traveling in +z direction:
V  z   Ae z  Ae z e j z
    j  
Wave is attenuating as it propagates.
Wave traveling in -z direction:
V  z   Ae z  Ae z e j z
    j  
Wave is attenuating as it propagates.
15
TEM Transmission Line (cont.)
Attenuation in dB/m:
V  z   Ae z e j z
 Vout 
Attenuation in dB  20 log10 
 V 
 in 
 A e  z 
 20 log10 
 A 


 20  0.4343  z 
 8.686 z
Note: log10  x   0.4343 ln  x 
Attenuation in dB/m  8.686
16
Wavenumber Notation
V  z   Ae z  Ae z e j z
    j
V  z   Ae jkz z  Ae z e j z
k z    j
  jkz
  ( R  j L)(G  jC )
kz   j ( R  jL)(G  jC )
"propagationconstant"
"propagation wavenumber"
17
TEM Transmission Line (cont.)
Forward travelling wave (a wave traveling in the positive z direction):
V  ( z )  V0 e z  V0 e  z e  j  z

v  ( z, t )  Re V0 e  z e  j z  e jt
 Re
 V

0


 
e j e  z e  j z e jt
 V0 e  z cos t   z   
The wave “repeats” when:
g  2
t 0
g
Hence:
V0 e  z
z

2
g
“snapshot” of wave
18
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.
vp
(phase velocity)
z
v  ( z , t )  V0  e  z cos(t   z   )
19
Phase Velocity (cont.)
Set
t   z  constant
dz
 0
dt
dz 

dt 
Hence

v 

p
In expanded form:
v 
p
Im


R  j L)(G  jC

20
Characteristic Impedance Z0
I   z


V  z
z
Assumption: A wave is traveling in the positive z direction.
V  ( z)
Z0  
I ( z)
V  ( z )  V0 e z
I  ( z )  I 0 e z
so
V0
Z0  
I0
(Note: Z0 is a number, not a function of z.)
21
Characteristic Impedance Z0 (cont.)
Use first Telegrapher’s Equation:
v
i
  Ri  L
z
t
so
dV
  RI  j LI   ZI
dz
Recall:
Hence
V  ( z )  V0 e z
I  ( z )  I 0 e z
 V0 e  z   ZI 0 e  z
22
Characteristic Impedance Z0 (cont.)

V
Z
From this we have: Z  0 

0

I0

Z
ZY
Use:
Z  R  j L
Y  G  jC
Both are in the first quadrant
   ZY  first quadrant
 Z 0  Z / ZY  right - half plane
 Re Z 0  0
Z0 
Z
Y
(principal square root)
23
Characteristic Impedance Z0 (cont.)
Hence, we have
Z
Z0 

Y
R  j L
G  jC
Re Z0  0
24
General Case (Waves in Both Directions)
V  z   V0 e z  V0 e z

0
j 
V e e
 z  j  z
e
Wave in +z
direction
j 
 V e e  z e  j  z

0
Wave in -z
direction
In the time domain:
v  z, t   Re V  z  e jt 
 V0 e z cos t   z    
 V0 e z cos t   z    
25
Backward-Traveling Wave
I   z
V

 z


z
A wave is traveling in the negative z direction.
V  ( z)
 Z0

I ( z)
so
V  ( z)
 Z0

I ( z)
Note:
The reference directions for voltage and current are chosen the
same as for the forward wave.
26
General Case
I z
V  z


z
Most general case:
A general superposition of forward and
backward traveling waves:
V ( z )  V0  e  z  V0  e   z
1
V0  e  z  V0  e   z 
I ( z) 
Z0
27
Summary of Basic TL formulas
I z
 V  z

V  z   V0 e  z  V0 e   z

0

0
Guided wavelength:
V  z V  z
I  z 
e 
e
Z0
Z0
    j 
Z0 
g 
2

 m
 R  j L  G  jC 
R  j L
G  jC
Phase velocity:

v p  [m/s]

Attenuation in dB/m  8.686
28
Lossless Case
R  0, G  0
    j   ( R  j L)(G  jC )
 j LC
so

vp 

 0
   LC
R  j L
Z0 
G  jC
L
Z0 
C
(real and independent of freq.)
vp 
1
LC
(independent of freq.)
29
Lossless Case (cont.)
vp 
1
LC
If the medium between the two conductors is lossless and homogeneous
(uniform) and is characterized by (, ), then (proof given later):
LC  
(proof given later)
The speed of light in a dielectric medium is
Hence, we have that:
and
cd 
1

v p  cd
   LC     k  k0  r r
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).
30
Terminated Transmission Line
V   z
V  z  V e
  z
0
V   z
Terminating impedance (load)
  z
0
V e
I z
 V z
 

ZL
z0
z
Amplitude of voltage wave
propagating in positive z
direction at z = 0.
V   0   V0
V   0   V0
Amplitude of voltage wave
propagating in negative z
direction at z = 0.
Where do we assign z = 0 ?
The usual choice is at the load.
31
Terminated Transmission Line (cont.)
Terminating impedance (load)
V  z   V0e z  V0e z
Can we use z = z0 as
a reference plane?
I z
 V z
 

ZL
z  z0
z0
V   z0   V0e z0
V   z0   V0e z0
 V0  V   z0  e z0
 V0  V   z0  e z0
z
Hence
V  z   V   z0  e
  z  z0 
 V   z0  e
  z  z0 
32
Terminated Transmission Line (cont.)
Terminating impedance (load)
I z
 V z
 

ZL
z  z0
z
z0
Compare:
V  z   V   0  e z  V   0  e   z
V  z   V   z0  e
  z  z0 
 V   z0  e
  z  z0 
This is simply a change of reference plane, from z = 0 to z = z0.
33
Terminated Transmission Line (cont.)
Terminating impedance (load)
I  d 
What is V(-d) ?
V  z  V e
  z
0
  z
0
V e
V  d   V e  V e
 d
0
  d
0
Propagating
forwards

V  d 

z  d
ZL
z
z0
Propagating
backwards
The current at z = -d is then:
V0  d V0  d
I  d  
e 
e
Z0
Z0
d  distance away from load
(This does not necessarily have to be the length of the entire line.)
34
Terminated Transmission Line (cont.)
I  d 

V  d 

z  d
ZL
z
z0
V  d   V0 e d  V0e d
or
(-d) = reflection coefficient at z = -d


V
 d
2 d 
0
 V0 e 1   e 
 V0

V  d   V0 e d 1   Le 2 d 
Similarly,
V0  d
I  d  
e 1   Le 2 d 
Z0
L  load reflection coefficient
V0
L  
V0
  d    L e 2 d
35
Terminated Transmission Line (cont.)
Z(-d) = impedance seen “looking” towards load at z = -d.
I  d 

V  d 

Z  d 
z  d
ZL
z
z0
Note:
If we are at the
beginning of the line,
we will call this the
“input impedance”.
V  d   V0 e d 1   L e 2 d 
V0  d
I  d  
e 1   L e 2 d 
Z0
V  d 
 1   Le 2 d
Z  d  
 Z0 
2 d
I  d 
1


e
L

 1    d  


  Z0 

 1    d  
36
Terminated Transmission Line (cont.)
At the load (d = 0):
 1  L 
Z  0  Z 0 
  ZL
 1  L 

At any point on the line(d > 0):
Z L  Z0
L 
Z L  Z0
  d   L e2 d
37
Terminated Transmission Line (cont.)
Recall
 1    d  
 1   Le 2 d 
Z  d   Z 0 
  Z0 
2 d 
1



d
1


e


L




Thus,
  Z L  Z 0  2 d
e
1 

Z L  Z0 


Z  d   Z 0
  Z L  Z 0  2 d
e
1 

  Z L  Z0 






38
Terminated Transmission Line (cont.)
Simplifying, we have:
  Z L  Z 0  2 d
1 
e
Z

Z
0 
Z  d   Z 0   L
  Z L  Z 0  2  d
e
1 

  Z L  Z0 


2  d


Z

Z

Z

Z
e




L
0
L
0
Z 
0
2  d 

Z

Z

Z

Z
e




0
L
0
 L



  Z L  Z 0  e   d   Z L  Z 0  e  d 
 Z0 
 d
 d 
Z

Z
e

Z

Z
e




0
L
0
 L

 Z cosh  d   Z 0 sinh  d  
 Z0  L

Z
cosh

d

Z
sinh

d




L
 0

Hence, we have
 Z L  Z 0 tanh  d  
Z  d   Z 0 

Z

Z
tanh

d
 
L
 0
39
Terminated Lossless Transmission Line
Lossless:
    j  j
V  d   V0 e j  d 1   L e 2 j  d 
V0 j d
I  d  
e 1   L e2 j  d 
Z0
Impedance is periodic with period g/2:
The tan function repeats when
 1   L e2 j d 
Z  d   Z 0 
2 j  d 
1


e

L

 Z L  jZ 0 tan   d  
Z  d   Z 0 

 Z 0  jZ L tan   d  
  d 2  d1   
2
g
 d 2  d1   
 d 2  d1   g / 2
Note: tanh  d   tanh  j d   j tan   d 
40
Summary for Lossy Transmission Line
I  d 

V  d 

Z  d 
z  d
 1   Le

Z  d   Z 0 
2 d 
1


e

L

2 d
 Z L  Z 0 tanh   d  
 Z 0 

Z

Z
tanh

d


L
 0

    j   ( R  j L)(G  jC )
ZL
z
z0
Z L  Z0
L 
Z L  Z0
2


vp 

g 
41
Summary for Lossless Transmission Line
I  d 

V  d 

Z  d 
z  d
2 j  d
 1   Le

Z  d   Z 0 
2 j  d 
1


e
L


 Z L  jZ 0 tan   d  
 Z0 

Z

jZ
tan

d


L
 0

   LC     k
ZL
z
z0
Z L  Z0
L 
Z L  Z0
2 2
g 

 d

k

v p   cd

42
Matched Load (ZL=Z0)
I  d 

V  d 

Z  d 
z  d
ZL
z
z0
Z L  Z0
L 
0
Z L  Z0
No reflection from the load
 1   L e2 d 
Z  d   Z 0 
2 d 
1


e

L


Z  d   Z0 for any z
43
Short-Circuit Load (ZL=0)
Lossless Case
L 
I  d 

V  d 

Z L  Z0
Z L  Z0
 L  1
Z  d 
z  d
z
z0
 Z  jZ 0 tan   d  
Z  d   Z 0  L

 Z 0  jZ L tan   d  
Z  d   jZ0 tan   d 
Always imaginary!
44
Short-Circuit Load (ZL=0)
Lossless Case
Z  d   jZ0 tan   d 
Z  d   jX sc
X sc  Z0 tan   d 
X sc
Note:  d  2
Inductive
0
1/ 4
1/ 2
3/ 4
d / g
d
g
g  d
lossless 
Capacitive
S.C. can become an O.C. with a
g/4 transmission line.
45
Open-Circuit Load (ZL=)
Lossless Case
I  d 
Z  Z0
L  L
Z L  Z0
L 
1  Z0 / Z L
1
1  Z0 / Z L

V  d 

Z  d 
z  d
z
z0
 L  1
 Z L  jZ 0 tan   d  
Z  d   Z 0 

Z

jZ
tan

d


L
 0

Z  d    jZ0 cot   d 
or
 1  j  Z 0 / Z L  tan   d  
Z  d   Z 0 

Z
/
Z

j
tan

d




 0 L

Always imaginary!
46
Open-Circuit Load (ZL=)
Lossless Case
Z  d    jZ0 cot   d 
Z  d   jX oc
X oc  Z0 cot   d 
X oc
Note:  d  2
Inductive
0
1/ 4
1/ 2
d / g
d
g
g  d
 lossless 
Capacitive
O.C. can become a S.C. with a
g/4 transmission line.
47
Using Transmission Lines to Synthesize Loads
We can obtain any reactance that we want from a short or open transmission line.
This is very useful in microwave engineering.
A microwave filter constructed from microstrip line.
48
Voltage on a Transmission Line
Find the voltage at any point on the line.
l
I z

VTh
ZTh
Z0 , 



V  z
ZL
z0
Z in
At the input:
 Zin 
V  l   VTh 

Z

Z
Th 
 in
 Z L  Z 0 tanh   l  
Z in  Z  l   Z 0 

Z

Z
tanh

l


L
 0


VTh

ZTh
V  l 


Z in
49
Voltage on a Transmission Line (cont.)
V  z  V e
  z
0
1   e 
2 z
L
L 
Z L  Z0
Z L  Z0
Incident (forward) wave (not the same as the initial wave from the source!)
At z = -l :
V  l   V e 1   L e
 l
0

2 l
 Z in 
  VTh  Z  Z 
Th 
 in
 Zin   l 

1
V  VTh 
e 
2 l 
Z

Z
1


e
Th 

L

 in

0
Hence
 Zin   l  z   1   L e2 z 
V  z   VTh 
e

2 l 
 1   Le 
 Zin  ZTh 
50
Voltage on a Transmission Line (cont.)
Let’s derive an alternative form of the previous result.
Start with:


Z in
Z in  ZTh
 1   L e2 l 
Zin  Z  l   Z 0 
2 l 
1


e

L

 1   L e 2 l 
Z0 

Z 0 1   L e 2 l 
1   L e 2 l 



2 l
 1   Le 
Z 0 1   L e 2 l   ZTh 1   L e 2 l 
Z0 
 ZTh
2 l 
1


e

L

Z 0 1   L e 2 l 
 ZTh  Z 0    L e2 l  Z 0  ZTh 
1   L e 2 l 

 Z0 


Z

Z
 Z  ZTh 
0 
 Th
1   L e 2 l  0

 ZTh  Z 0 
1   L e 2 l 

 Z0 


 ZTh  Z 0  1   e 2 l  ZTh  Z 0 


L
 ZTh  Z 0 
51
Voltage on a Transmission Line (cont.)
Hence, we have
 Z0  1   L e2 l 
Zin


2 l 
Zin  ZTh  Z0  ZTh   1   s  L e 
Substitute
Recall:
where
Z Th  Z 0
s 
Z Th  Z 0
(source reflection coefficient)
 Zin   l  z   1   L e2 z 
V  z   VTh 
e

2 l 
Z

Z
1


e
Th 

L

 in
Therefore, we have the following alternative form for the result:
 Z0   l  z   1   L e2 z 
V  z   VTh 
e

2 l 
Z

Z
1



e
Th 
s L
 0


52
Voltage on a Transmission Line (cont.)
l
I z

VTh
ZTh
Z0 , 

Z in


V  z
ZL
z0
 Z0    z  l   1   L e2 z 
V  z   VTh 
e

2 l 
Z

Z
1



e
Th 
s L
 0


This term accounts for the multiple (infinite) bounces.
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.)
l

VTh
ZTh

V  l 


Z0 , 
ZL
z0
Z in
Wave-bounce method (illustrated for z = -l):
We add up all of the bouncing waves.
1   L e2 l    L e2 l   s


 Z 0  
2 l
2 l
2 l
2 l
V  l   VTh 
    L e   s    L e     L e   s   L e    s 

 Z 0  ZTh  





54
Voltage on a Transmission Line (cont.)
1   L e2 l    L e2 l   s


 Z 0  
2 l
2 l
2 l
2 l
V  l   VTh 
    L e   s    L e     L e   s   L e    s 

 Z 0  ZTh  





Group together alternating terms:
1     e 2 l      e 2 l 2 
L S
L S

 Z0  
2 l 
2 l
2 l 2
V  l   VTh 


e
1



e



e






L
L S
L S


 Z 0  ZTh  






 



Geometric series:

 zn  1 z  z2 
n 0

1
,
1 z
z 1
z   L  s e2 l
55
Voltage on a Transmission Line (cont.)
Hence
1

2 l

 Z 0  1   L  s e
V  l   VTh 

Z

Z
1
2 d 
Th 
 0
   Le 
2 l
1



e

L s






 
or
 Z0   1   L e2 l 
V  l   VTh 

2 l 
Z

Z
1



e
Th  
L s
 0

This agrees with the previous result (setting z = -l).
 Z 0   l  z   1   L e 2 z 
Previous (alternative) result : V  z   VTh 
e

2 l 
Z

Z
Th 
 0
 1  s Le 
The wave-bounce method is a very tedious method – not recommended.
56
Time-Average Power Flow
I z
P(z) = power flowing in + z direction
At a distance d from the load:

V  z

ZL
z
z0
z
1
V  z   V0 e z 1   L e2 z 
P  z   Re V  z  I *  z 
2

V
0
I  z 
e z 1   L e2 z 
V 2

*
Z0
1
0
 Re  * e 2 z 1   L e 2 z  1  *L e 2 z 

2  Z0
    j


V 2

V 2

*
1  0
1
2 4 z
0
2 z

2

z

2

z
*

2

z
  Re 

 Re
e
1 L e
e
 Le   Le
*
*
 2  Z0

2  Z0









If Z0  real (low-loss transmission line):
 2
0

1V
2
P z 
e2 z 1   L e4 z
2 Z0


Note:
 L e 2 z  *L e 2
*
z
  L e 2 z    L e 2 z 
*
(please see the note)
 pure imaginary
57
Time-Average Power Flow (cont.)
I z
Low-loss line
 2
0

1V
2
P z 
e2 z 1   L e4 z
2 Z0

 2
0
1V
e2 z
2 Z0

Power in forward-traveling wave
2

ZL
z
z0


z
 2
0
1V
2 2 z
L e
2 Z0
Power in backward-traveling wave
Lossless line ( = 0)

1 V0
2
P z 
1  L
2 Z0

V  z

Note:
For a very lossy
line, the total
power is not the
difference of the
two individual
powers.
58
Quarter-Wave Transformer
Lossless line
 Z L  jZ 0T tan  d 
Zin  Z 0T 

Z

jZ
tan

d
L
 0T

Z in
d  
g
4

Z 0T
Z0
ZL
d  g / 4
2 g 

g 4 2
Matching condition
 jZ 
 Zin  Z 0T  0T 
 jZ L 
in  0
 Zin  Z 0
Z 02T
 Z0 
ZL
(This requires ZL to be real.)
so
2
Z 0T
Zin 
ZL
Hence
Z 0T  Z 0 Z L
59
Quarter-Wave Transformer (cont.)
Example
Match a 100  load to a 50  transmission line at a given frequency.
g 
2



2
2

 0  d
k k0  r
r
Lossless line
50 
Z 0T
0 
Z L  100 
c
f
g / 4
Note :    LC     k
Z 0T  100  50
 70.7
Z0T  70.7 
60
Voltage Standing Wave
I z
Lossless Case
V  z  V e
  j z
0
 V0 e  j  z
1   e 
1   e e 
2 j  z
L
jL 2 j  z
L

V  z

ZL
z
z0
V  z   V0 1   L e jL e  j 2  z
z
1+  L
V ( z)
Vmax  V0 1   L

0
Vmin  V
1  
L


V0
1
1-  L
Dz   / 2
g
z
z0
2 Dz  2  Dz  g / 2
Note: The voltage repeats every g. The magnitude repeats every g /2.
Note: The voltage changes by a minus sign after g /2.
61
Voltage Standing Wave Ratio
Vmax
Voltage Standing Wave Ratio  VSWR  
Vmin
I z
Vmax  V0 1   L
Vmin  V0 1   L



V  z

ZL
z
z0
z
1+  L
V ( z)
V0
1
1  L
VSWR 
1  L
1-  L
Dz  g / 2
z
z0
62
Limitations of Transmission-Line Theory
At high frequency, discontinuity effects can become important.
Transmitted
Incident
Bend
Reflected
The simple TL model does not account for the bend.
ZTh


Z0
ZL
63
Limitations of Transmission-Line Theory (cont.)
At high frequency, radiation effects can also become important.
We want energy to travel from the generator to the load, without radiating.
ZTh


Z0
ZL
When will radiation occur?
This is explored next.
64
Limitations of Transmission-Line Theory (cont.)
Coaxial Cable
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.
r
a
b
z
65
Limitations of Transmission-Line Theory (cont.)
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”.)
Having fields that extend to infinity is not the same thing as having radiation, however!
66
Limitations of Transmission-Line Theory (cont.)
The infinite twin lead will not radiate by itself, regardless of how far
apart the lines are (this is true for any transmission line).
1

Pt   Re   E  H *    ˆ dS  0
2

S
Reflected
S
Incident
h
+
-
No attenuation on an infinite lossless line
The incident and reflected waves represent an exact solution to
Maxwell’s equations on the infinite line, at any frequency.
67
Limitations of Transmission-Line Theory (cont.)
A discontinuity on the twin lead will cause radiation to occur.
Incident wave
Pipe
Obstacle
h
Reflected wave
Note:
Radiation effects usually increase as
the frequency increases.
Bend
Incident wave
h
Reflected wave
Bend
68
Limitations of Transmission-Line Theory (cont.)
To reduce radiation effects of the twin lead at discontinuities:
1) Reduce the separation distance h (keep h << ).
2) Twist the lines (twisted pair).
h
CAT 5 cable
(twisted pair)
69