E.E. Essential Knowledge Sereies My Online Courses: https://www.byparams.com/courses Transmission Line Theory

1. Chapter 1-2
Transmission Line Theory
Chien-Jung Li
Department of Electronics Engineering
National Taipei University of Technology

2. Department of Electronic Engineering, NTUT
Common Types of Transmission Lines
Two-wire line
Coaxial
Microstrip
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3. Department of Electronic Engineering, NTUT
Transmission Line Theory
• At high frequencies, especially when the
wavelength is not longer than the dimension of
circuitry, conventional circuit theory no longer
holds.
• Circuitry perspective on electromagnetic waves.
• Transmission-line effects:
 Standing waves generated (related to position)
 Load impedance changes
 Departs from max power transmission
• Transmission-line effect is obvious as frequency
or line length increases.
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4. Department of Electronic Engineering, NTUT
Theory
Conventional
Circuit Theory
Microwave
Engineering
Optics
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5. Department of Electronic Engineering, NTUT
Electrical Model for Transmission Line
Source
Source
impedance
Load
impedance
• In conventional circuit theory, you can easily find the voltage
appears at load by: L
s
s L
Z
v v
Z Z


Transmission line
sv
sZ
LZ
l
• If the transmission line is not just an ideal interconnection?
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6. Department of Electronic Engineering, NTUT
Distributed Circuit Model
• Assume that the transmission line is uniform and
the length l is divided into many identical
sections Δx.
• R: Ω/m, L: H/m, C: F/m, G: S/m
G x
L xR x
C x
sZ
sv LZ
l
dx dx dx
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7. Department of Electronic Engineering, NTUT
A Section of the Transmission Line
• The voltages and currents along the trans-
mission line are functions of position and time.
• Input: ,
• Output: ,
• R: finite conductivity G: dielectric loss
 ,v x t
 ,i x t
R x L x
G x C x
   ,v x x t
   ,i x x t
 ,v x t  ,i x t
   ,v x x t    ,i x x t
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8. Department of Electronic Engineering, NTUT
Transmission Line Equations (I)
• Apply Kirchhoff’s voltage law (KVL)
• Apply Kirchhoff’s current law (KCL)
         
 
      

,
, , ,
i x t
v x t v x x t R x i x t L x
t
         
   
        

,
, , ,
v x x t
i x t i x x t G x v x x t C x
t
 ,v x t
 ,i x t
R x L x
G x C x
   ,v x x t
   ,i x x t
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9. Department of Electronic Engineering, NTUT
Transmission Line Equations (II)
 
 
  
  
 
, ,
,
v x t i x t
Ri x t L
x t
 
 
  
  
 
, ,
,
i x t v x t
Gv x t C
x t
   
 
    
 
 
, , ,
,
v x t v x x t i x t
Ri x t L
x t
   
 
      
   
 
, , ,
,
i x t i x x t v x x t
Gv x x t C
x t
• Rearrange Δx then we have:
• Assume that Δx is very small
 The partial differential equations describe the voltages and currents along
the transmissions, called transmission line, or telegrapher equation.
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10. Department of Electronic Engineering, NTUT
Lossless Transmission Line
 
 
  
  
 
, ,
,
v x t i x t
Ri x t L
x t
 
 
  
  
 
, ,
,
i x t v x t
Gv x t C
x t
    
 
 
, ,v x t i x t
L
x t
    
 
 
, ,i x t v x t
C
x t
 ,v x t
 ,i x t
   ,v x x t
   ,i x x tL x
C x
• Of particular interest in microwave electronics is the lossless
transmission line, i.e., R=G=0.
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11. Department of Electronic Engineering, NTUT
Sinusoidal Steady-state Analysis
f(x) and g(x) are the real functions of position, and , describe
the positional dependence of the phase.
           
     
    
 

   , cos Re Re
j t x j x j t
v x t f x t x f x e f x e e
           
     
    
 

   , cos Re Re
j t x j x j t
i x t g x t x g x e g x e e
     

j x
I x g x e
     

j x
V x f x e     
, Re j t
v x t V x e
    
, Re j t
i x t I x e
• Time-domain representation
  x   x
• Phasor-domain representation
time-domain
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12. Department of Electronic Engineering, NTUT
Solve for the Voltage
   
 
 
   
 
V x I x
L j LI x
x t
   
 
 
   
 
I x V x
C j CV x
x t
 
  
dV x
j LI x
dx
 
  
dI x
j CV x
dx
   


V x dV x
x dx
   
     
2
2
2
d V x dI x
j L LCV x
dx dx
• Laplace equation
 
 
 
    
2 2
2 2
2 2
d V x d V x
LCV x V x
dx dx
  LC• Propagation constant
• General solution
   
 j x j x
V x Ae Be ,where A, B are complex constant
• Lossless transmission line equations
(Think that if approaches zero?)
( is also called the “phase constant” that represents the phase change per
meter for the wave traveling along the path)

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13. Department of Electronic Engineering, NTUT
Solve for the Current
 
 
    
 
 

      
1 1 j x j xdv x
I x A j e B j e
j L dt j L
• Solve for the current
  LC
• Define
 
 
  0
L L L
Z
CLC
  
 
  
   
 
j x j x
A e B e
L L
   
 j x j x
V x Ae Be
   
 
0 0
j x j xA B
I x e e
Z Z
 
  
dV x
j LI x
dx
With and
,where
Z0 is the characteristics impedance of the transmission line, for the lossless
condition Z0 is real.
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14. Department of Electronic Engineering, NTUT
Time-domain Results
        
       
  , Re Re
j x t j x tj t
v x t V x e Ae Be
               
   
 0 0
, Re Re
j x t j x tj t A B
i x t I x e e e
Z Z
         cos cosA x t B x t
         
0 0
cos cos
A B
x t x t
Z Z
   
 j x j x
V x Ae Be
   
 
0 0
j x j xA B
I x e e
Z Z
• Time-domain results can be easily drawn from the phasor.
 Voltage Wave
 Current Wave
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15. Department of Electronic Engineering, NTUT
Wave Wavelength
  
 j x
V x Ae
• Consider
 Take a look on the term , the phase means how many radians
change for the wave to travel through the distance of x. If distance x is
equal to a wavelength long, i.e., :
 j x
e x

  
 2 x
x




2
 0x x
 0t t T
distance
time
phase
 0x
x 
 
 2x
x
      
 
  1 , Re cosj x j t
v x t Ae e A t x
x
For simplification, assume the
wave starts from x=0 and t=0.
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16. Department of Electronic Engineering, NTUT
Wave Velocity
 In the vacuum,    
  7
0 4 10 Wb/A-mL
 0 8.85419 F/mC
   
 
    8
,
0 0
1
light speed 3 10 /p vacuumv c m s
 


   0
0
0
377
L
Z
C
  ,
0
p vaccumv
f
is the wavelength in vacuum
    
 
   
      
2 1
2 2
pv f
T LC
• Wave velocity can be found: ( the wave goes one wavelength long in a period of T)
is the intrinsic/characteristic
impedance of vacuum (or free-space)
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17. Department of Electronic Engineering, NTUT
Wave Propagation in Material
 0  0r
 
    

  
8
0 0
1 3 10
/p
r r r
c
v m s
 Propagation in material with relative dielectric constant r
(non-magnetic material)
 


   0p r
g
r
c
v
f f
 Take water for an example:
 

 
8
7
,
3 10
3.32 10 /
81.5
p waterv m s

  0
, 00.11
81.5
g water
The propagation speed is slower than that in vacuum, and the wavelength
is also shorter than that in vacuum.
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18. Department of Electronic Engineering, NTUT
Wave at a Certain Point
 
 2t T
t 


2
T
     1 , cosv x t A x t
 0x
  1 0, cosv t A t
 2
t
 
0
0t
t
A
A
  1 0, cosv t A t
 0x x l
 0x
x l
• Consider
 At position
We only pay attention to this point
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19. Department of Electronic Engineering, NTUT
Wave at a Certain Time

 
 2x
x




2
     1 , cosv x t A x t
 0t
  1 ,0 cosv x A x
 2
x
 
0
0x
x
A
A
  1 ,0 cosv x A x
 0x x l
 0x
x l
• Consider
 At time
We now pay attention to the whole line
at any time instant (here, t=0)
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20. Department of Electronic Engineering, NTUT
Wave Propagation versus x and t
 2 
x
A
A
t
t T
 2t T
x
x
t
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21. Department of Electronic Engineering, NTUT
Wave at Point x=
 2 
x
A
A
t
t T
 2t T
t

 0x x l
We only pay attention to this point
x
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22. Department of Electronic Engineering, NTUT
Terminated Transmission Line
LZ LZ0Z 0Z
 j x
Ae
j x
Be j x
Be
 j x
Ae
 0x x l  0dd l
   
 j x j x
V x Ae Be
   
 
0 0
j x j xA B
I x e e
Z Z
 IN d
   
 1 1
j d j d
V d A e B e
   
 1 1
0 0
j d j dA B
I d e e
Z Z

1
j
A Ae l 
1
j
B Be l
where and
 d xl
incident wave
reflected wave
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23. Department of Electronic Engineering, NTUT
Reflection Coefficient
   
 1 1
j d j d
V d A e B e 
1
j
A Ae l 
1
j
B Be l
 

 


 
    2 21 1
0
1 1
j d
j d j d
IN j d
B e B
d e e
A e A
     1
0
1
0IN
B
A
where and
• Moves from the load (at d=0) toward the source (at d=l)
where is the load reflection coefficient, which is the value
of at d=0:
0
 IN d
incident wave reflected wave
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24. Department of Electronic Engineering, NTUT
Reflection Coefficient at Load
         
      2
1 0 1 01j d j d j d j d
V d A e e A e e
         
      21 1
0 0
0 0
1j d j d j d j dA A
I d e e e e
Z Z
 
 
 
 
 


 
 
 
0
0
0
j d j d
IN j d j d
V d e e
Z d Z
I d e e
 
 
 
 
0
0
0
1
0
1
IN LZ Z Z

 

0
0
0
L
L
Z Z
Z Z
   
 1 1
j d j d
V d Ae B e
   
 1 1
0 0
j d j dA B
I d e e
Z Z
 Input impedance of the transmission line at any position d
is defined as
 Use boundary condition at load (d=0)
 0 0 when  0LZ ZProperly terminated (matched line):
LZ V d
 0dd l
 IN d
 INZ d
 I d


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25. Department of Electronic Engineering, NTUT
Input Impedance of a Terminated Line
 
   
   
 
 


  

  
0 0
0
0 0
j d j d
L L
IN j d j d
L L
Z Z e Z Z e
Z d Z
Z Z e Z Z e
 
 



0
0
0
cos sin
cos sin
L
L
Z d jZ d
Z
Z d jZ d





0
0
0
tan
tan
L
L
Z jZ d
Z
Z jZ d

 

0
0
0
L
L
Z Z
Z Z
 
 
 
 
 


 
 
 
0
0
0
j d j d
IN j d j d
V d e e
Z d Z
I d e e
It gives the value of the input impedance at any position d along the
transmission line
At d=0   0IN LZ Z
At d=l  





0
0
0
tan
tan
L
IN
L
Z jZ
Z Z
Z jZ
l
l
l
A very important property of a transmission line is the ability to change
a load impedance to another value of impedance as its input.
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26. Department of Electronic Engineering, NTUT
Voltage Standing-wave Ratio (VSWR)
  
   2
1 01 j d
V d A e
     1 0max
1V d A      1 0min
1V d A
 
 
 
 
 
0max
0min
1
1
V d
VSWR
V d
 The addition of the two waves traveling in opposite directions in a
transmission line produces a standing-wave pattern – that is,
a sinusoidal function of time whose amplitude is a function of position.
         
      2
1 0 1 01j d j d j d j d
V d A e e Ae e
 Magnitude of the voltage along the line:
 The voltage standing-wave ratio (VSWR):
Next we will discuss 4 important cases of the terminated line, which are
matched line, short-circuited line, open-circuited line, and quarter-wave line.
and
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27. Department of Electronic Engineering, NTUT
Matched Line (Properly Terminated)
• Matched line: the characteristic impedance is equal to the
load impedance, i.e.,   IN LZ d Z
 0LZ Z
 0dd l
 l  0INZ Z
   0INZ d Z
0Z
 There is no reflection wave, the input impedance is at any location d,
VSWR has its minimum value of 1, i.e.,  0 0  1VSWRand
0Z
 No matter how long the line is, there is no transmission line effects .
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28. Department of Electronic Engineering, NTUT
Short-Circuited Line
• For , it follows that , , and the
input impedance at a distance d from the load, , is
given by    0 tanscZ d jZ d
 The amplitude of the incident and reflected waves are
the same since (total reflection from the load), VSWR attains
its largest value of infinity.
 0LZ
 0dd l
 l l 0 tanINZ jZ
   0 tanscZ d jZ d
0Z
 0LZ   0 1  VSWR
 scZ d
 When
Short-circuited load is transformed to
open-circuited.
 0 1
l


4
 l  scZ
 When
Short-circuited load is still short-
circuited seen from the input.
l


2
 l  0scZ
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29. Department of Electronic Engineering, NTUT
Open-Circuited Line
• For , it follows that , , and the
input impedance at a distance d from the load, , is
given by
 The amplitude of the incident and reflected waves are
the same since (total reflection from the load), VSWR attains
its largest value of infinity.
 LZ  0 1  VSWR
 ocZ d
 When
Open-circuited load is transformed to
short-circuited.
 0 1
l


4
 l  0ocZ
 When
Open-circuited load is still open-
circuited seen from the input.
l


2
 l  ocZ
    0 cotocZ d jZ d
 LZ
 0dd l
 l l  0 cotINZ jZ
    0 cotocZ d jZ d
0Z
29/40

30. Department of Electronic Engineering, NTUT
Quarter-wave Line
• Quarter-wave transformer:
  
2
0
4IN
L
Z
Z
Z
  0 4IN LZ Z Z
 In order to transform a real impedance to another real impedance
given by , a quarter-wave line with real characteristic impedance
of value
can be used.
LZ
 0d

4
d
 
 
 
2
0
4
IN
L
Z
Z
Z
0Z
l

 
4
d
LZ
  4INZ
 Example: , how to trans-
form it to at certain frequency?
 75LZ
50
      0 4 50 75 61.2IN LZ Z Z
You can simply use a transmission line
with 61.2 Ohm characteristic impedance!
 4
30/40

31. Department of Electronic Engineering, NTUT
Half-wave Line
• Half-wave line:
  2IN LZ Z
 No matter what the line impedance is, when a half-wave line is used
and it does not affect the impedance seen from the input.
LZ
 0d

2
d
 
 
 2
IN LZ Z
0Z
  2IN LZ Z
l

 
2
d
31/40

32. Department of Electronic Engineering, NTUT
Voltage on the Shorted-circuited Line
    

  1 12 sinj d j d
V d A e e j A d
    




 
 
 
  
   
  
2
1, Re Re 2 sin
j t
j t
v d t V d e A d e
 When
     1 14 2 sin 2 2V j A j A
     12 2 sin 0V j A


4
d
 When


2
d
• Voltage on the short-circuited line:
incident wave reflected wave
32/40

33. Department of Electronic Engineering, NTUT
Standing-wave Pattern
 

 
 
   
 
1, 2 sin cos
2
v d t A d t
 
2
3
2
2
 
 
  max
min
V d
VSWR
V d
 V d
 1 max
2A V d
  min
0 V dd
d

2

4
3
4

 In order to proceed we need to know the value of the complex constant A1.
For simplicity let us assume that A1 is real. Hence we obtain
 
2
3
2
2
 ,v d t
12A
  min
0 V dd
d

2

4
3
4

12A
  3
2
t  5
4
t
   3,
4 4
t  
2
t
  0,t
33/40

34. Department of Electronic Engineering, NTUT
Example(I)
  100 50sZ j
  50 50LZ j

 10 0sv
 
   
50 50
10 0 3.92 11.31
50 50 100 50
L
L s
L s
jZ
V V
Z Z j j

    
   
 Consider the circuit shown below that there is no transmission line
between the source and load, find the voltage at the terminal of the
load.
34/40

35. Department of Electronic Engineering, NTUT
Example(II)
 Find the load reflection coefficient, the input impedance, and the
VSWR in the transmission line shown, the length of the
transmission line is and its characteristic impedance is 50
Ohm.

 10 0sv
  100 50sZ j
  50 50LZ j
l
 
 
 
    
  
0
0
0
50 50 50
0.447 63.44
50 50 50
L
L
jZ Z
Z Z j
 
 
 
 


  
    
   
50 50 50tan45
8 50 100 50
50 50 50 tan45
IN
j j
Z j
j j
  
  
  
0
0
1 1 0.447
2.62
1 1 0.447
VSWR
 8
 0 50Z
 8
  8INZ
LV
35/40

36. Department of Electronic Engineering, NTUT
Example(III)
 
 
 
 
   




     
   
8 100 50
8 10 0 5.59 26.57
8 100 50 100 50
IN
s
IN s
Z j
V V
Z Z j j
    
   2
1 01j d j d
V d A e e
 
 


 
       
 
4 2
18 5.59 26.57 1 0.447 63.44
j j
V A e e

  1 3.95 63.44A
   
      0 3.95 63.44 1.77 5 45LV V
  100 50sZ j

 10 0sv
    8 100 50INZ j
Equivalent circuit
Transmission-line effect makes things different! It is not always bad, since it
cab be used for matching and makes maximum power transfer possible.
36/40

37. Department of Electronic Engineering, NTUT
 V x
 I x
R x  j L x
G x  j C x
  V x x
  I x x
Lossy Transmission Line
 
     
dV x
R j L I x
dx
 
     
dI x
G j C V x
dx
 
  
2
2
2
d V x
V x
dx
          j R j L G j C
 Complex propagation constant
is the attenuation constant in
nepers/m and propagation constant
is in rads/m


37/40

38. Department of Electronic Engineering, NTUT
Reflection Coefficient and Input Impedance
        
        
  , Re Re
j x t j x tj t x x
v x t V x e Ae e Be e
                
   
 0 0
, Re Re
j x t j x tj t x xA B
i x t I x e e e e e
Z Z
 With x=l - d
   
 1 1
d d
V d A e B e
   
 1 1
0 0
d dA B
I d e e
Z Z
l
1A Ae l
1B Be
 The reflection coefficient of the terminated lossy line
 





    21
0
1
d
d
IN d
B e
d e
A e
where  

    

01
0
1 0
0 L
IN
L
Z ZB
A Z Z
 





0
0
0
tanh
tanh
L
IN
L
Z Z d
Z d Z
Z Z d
where and
 The impedance of the terminated lossy line
38/40

39. Department of Electronic Engineering, NTUT
Summary (I)
 





0
0
0
tanh
tanh
L
IN
L
Z Z d
Z d Z
Z Z d
  
   2
0
d
IN d e

 

0
0
0
L
L
Z Z
Z Z
 

 
0
0
1
1
VSWR
 





0
0
0
tan
tan
L
IN
L
Z jZ d
Z d Z
Z jZ d
  
   2
0
j d
IN d e


pv




2
  LC• Propagation constant
• Reflection coefficient at load
• Reflection coefficient at any position
lossless
• Input impedance
lossless
    j where
• Phase velocity
• Wavelength
• Voltage standing wave ratio
39/40

40. Department of Electronic Engineering, NTUT
Summary (II)
• Transmission line affects the input impedance, and you
may not get the voltage you want at the load. On the other
hand, this property is useful when you need impedance
matching to get maximum power transfer.
• Both the wave velocity and wavelength decreases when the
wave travels from vacuum into the material that has a
relative dielectric constant greater than 1.
• Generally speaking, when the circuit dimension is
under , the transmission effects can be considered
negligible.
 20
40/40