ECE 451
Advanced Microwave Measurements
Transmission Lines
Jose E. Schutt-Aine
Electrical & Computer Engineering
University of Illinois
jschutt@emlab.uiuc.edu
ECE 451 – Jose Schutt-Aine
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Maxwell’s Equations
 E  
 H  J
B
t
Faraday’s Law of Induction

D
t
Ampère’s Law
Gauss’ Law for electric field
 D  
Gauss’ Law for magnetic field
B  0
Constitutive Relations
D  E
B  H
ECE 451 – Jose Schutt-Aine
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Why Transmission Line?
l
l
z
Wavelength : l
propagation velocity
l=
frequency
ECE 451 – Jose Schutt-Aine
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Why Transmission Line?
In Free Space
At 10 KHz : l = 30 km
At 10 GHz : l = 3 cm
Transmission line behavior is prevalent when the
structural dimensions of the circuits are comparable
to the wavelength.
ECE 451 – Jose Schutt-Aine
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Justification for Transmission Line
Let d be the largest dimension of a circuit
circuit
z
l
If d << l, a lumped model for the circuit can be used
ECE 451 – Jose Schutt-Aine
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Justification for Transmission Line
circuit
z
l
If d ≈ l, or d > l then use transmission line model
ECE 451 – Jose Schutt-Aine
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Telegraphers’ Equations
L
I
+
V
C
z
L: Inductance per unit length.
C: Capacitance per unit length.

V
I
L
z
t
I
V
 C
z
t
Assume
time-harmonic
dependence

V
 j LI
z
V , I ~ e jt

I
 jCV
z
ECE 451 – Jose Schutt-Aine
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TL Solutions
L
I
+
V
C
z
  V
 
z  z

 2V
I




j

L
  j LjCV

z
z

  I 
 I
V



j

C
  j LjCI
 
z  z 
z
z
2
ECE 451 – Jose Schutt-Aine
 2V
2



LCV
2
z
2 I
2



CLI
2
z
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TL Solutions
(Frequency Domain)
z
Zo

forward wave
backward wave
Forward Wave
Backward Wave
   LC
V ( z )  V e  j z  V e  j z
L
Zo 
C
V  j z V  j z
I ( z) 
e

e
Zo
Zo
Forward Wave
ECE 451 – Jose Schutt-Aine
Backward Wave
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TL Solutions
Propagation constant    LC Propagation velocity v 
Wavelength
L
Characteristic impedance Zo 
C
Forward Wave
1
LC
v
l
f
Backward Wave
V ( z, t )  V cos t   z   V cos t   z 
V
V
I ( z, t )  cos t   z   cos t   z 
Zo
Zo
Forward Wave
ECE 451 – Jose Schutt-Aine
Backward Wave
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Reflection Coefficient
At z=0, we have V(0)=ZRI(0)
But from the TL equations:
V (0)  V  V
ZR
V  V   V  V
Zo
V V
I (0) 

Zo Zo
Which gives V   RV
Z R  Zo
where  R 
is the load reflection coefficient
Z R  Zo
ECE 451 – Jose Schutt-Aine
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Reflection Coefficient
- If ZR = Zo, R=0, no reflection, the line is matched
- If ZR = 0, short circuit at the load, R=-1
- If ZR inf, open circuit at the load, R=+1
V and I can be written in terms of R
V ( z )  V e  j z   R e  j z 
V ( z )  V e  j z 1   R e 2 j z 
V  j z
e
I ( z) 
  R e  j z 
Zo
V e  j z
1   R e 2 j z 
I ( z) 
Zo
ECE 451 – Jose Schutt-Aine
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Generalized Impedance
 e  j z   R e  j z 
V ( z)
Z ( z) 
 Z o   j z
 j z 
I ( z)
e


e
R


 Z R  jZ o tan  l 
Z  l   Z o 

Z

jZ
tan

l
R
 o

ECE 451 – Jose Schutt-Aine
Impedance
transformation
equation
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Generalized Impedance
- Short circuit ZR=0, line appears inductive for 0 < l < l/2
Z (l )  jZ o tan  l
- Open circuit ZR  inf, line appears capacitive for 0 < l < l/2
Zo
Z (l ) 
j tan  l
- If l = l/4, the line is a quarter-wave transformer
Z o2
Z (l ) 
ZR
ECE 451 – Jose Schutt-Aine
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Generalized Reflection Coefficient
Backward traveling wave at z Vb ( z )
( z ) 

Forward traveling wave at z V f ( z )
V e  j z V 2 j z
2 j  z
( z ) 

e


e
R
V e  j z V
Reflection coefficient
transformation equation
1  ( z )
Z ( z)  Zo
1  ( z )
(l )   R e2 j l
( z ) 
ECE 451 – Jose Schutt-Aine
Z ( z)  Zo
Z ( z)  Zo
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Voltage Standing Wave Ratio (VSWR)
V ( z )  V e  j z 1   R e 2 j z 
We follow the magnitude of the
voltage along the TL
V ( z )  V e  j z 1   R e 2 j z  V 1   R e 2 j z
Maximum and minimum
magnitudes given by
Vmax  1   R 
Vmin  1   R 
ECE 451 – Jose Schutt-Aine
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Voltage Standing Wave Ratio (VSWR)
Define Voltage Standing Wave Ratio as:
Vmax 1   R
VSWR 

Vmin 1   R
It is a measure of the interaction between
forward and backward waves
ECE 451 – Jose Schutt-Aine
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VSWR – Arbitrary Load
Shows variation of amplitude along line
ECE 451 – Jose Schutt-Aine
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VSWR – For Short Circuit Load
Voltage minimum is reached at load
ECE 451 – Jose Schutt-Aine
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VSWR – For Open Circuit Load
Voltage maximum is reached at load
ECE 451 – Jose Schutt-Aine
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VSWR – For Open Matched Load
No variation in amplitude along line
ECE 451 – Jose Schutt-Aine
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Application: Slotted-Line Measurement
- Measure VSWR = Vmax/Vmin
- Measure location of first minimum
ECE 451 – Jose Schutt-Aine
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Application: Slotted-Line Measurement
At minimum, ( z )  pure real    R
Therefore, (dmin )   R e2 j dmin    R
So,  R    R e2 j dmin
VSWR  1
Since  R 
VSWR  1
 VSWR  1  2 j dmin
then  R   
e
 VSWR  1 
 1 R 
and Z R  Z o 

1


R 

ECE 451 – Jose Schutt-Aine
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Summary of TL Equations
Voltage
V ( z )  V e
 j z
1   R e
Current
2 j  z

V  j z
1   R e 2 j z 
I ( z) 
e
Zo
 Z R  jZ o tan  l 
Impedance Transformation  Z  l   Z o 

Z

jZ
tan

l
R
 o

2 j  l

(

l
)


e
Reflection Coefficient Transformation 
R
Reflection Coefficient – to Impedance  Z ( z )  Z o 1  ( z )
1  ( z )
Impedance to Reflection Coefficient 
ECE 451 – Jose Schutt-Aine
( z ) 
Z ( z)  Zo
Z ( z)  Zo
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Example 1
Consider the transmission line system shown in the
figure above.
(a) Find the input impedance Zin.
Z R  Z o (30  j 40)  50
R 

 0.590
Z R  Z o (30  j 40)  50
(d  l )   R e
2 j  l
 0.590e
j
4
l
.0.725 l
ECE 451 – Jose Schutt-Aine
 0.5  72
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Example 1 – Cont’
1  (l ) 
1  0.5  72 
Zin  Z o 
 50 



1  0.5  72 
1  (l ) 
Z in  64.361  51.738    39.86  j50.54  
(b) Find the current drawn from the generator
1000
Ig 

 1.55239.11 A
Z g  Z in (10  j10)(39.86  j 50.54)
Vg
I g  1.207  j 0.981 A
ECE 451 – Jose Schutt-Aine
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Example 1 – Cont’
c) Find the time-average power delivered to the load
1
V (l )  Z in I g  64.36151.731.556239.11  100.159  12.62
2
1
1
*


 P  Real V (l ) I g   Real 100.159  12.621.5562  39.11
2
2
1
 P  Real V (l ) I g*   48.26 W
2
ECE 451 – Jose Schutt-Aine
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Example 1 – Cont’
In the system shown above, a line of characteristic
impedance 50  is terminated by a series R, L, C circuit
having the values R = 50 , L = 1 H, and C = 100 pF.
(d) Find the source frequency fo for which there are no
standing waves on the line.
ECE 451 – Jose Schutt-Aine
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Example 1 – Cont’
No standing wave on the line if ZR = R = Zo which
occurs at the resonant frequency of the RLC circuit.
Thus,
fo 
1
2 LC

1
2 106 1010
108

Hz  15.92 MHz
2
ECE 451 – Jose Schutt-Aine
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