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Fields and Waves
Lesson 1.3
TRANSMISSION LINES - INPUT IMPEDANCE
Darryl Michael/GE CRD
Input Impedance
Do Experiment 1
• After Experiment: Cable+Load can look very different
than Load
• Characterize V/I as Impedance
Input Impedance
From previous class:
V̂ ( z )  V  ( z )  V  ( z )  Vm  e  j  z  ( 1   ( z ))
What about I?
V  ( z ) V  ( z ) Vm  j    z
Î ( z ) 


e
 ( 1   ( z ))
Rc
Rc
Rc
Also,
 ( z )   L  e 2 j ( L z )
Input Impedance
Form the Ratio:
V̂ ( z )
1  ( z )
 Rc 
Î ( z )
1  ( z )
Book calls this Zin(z)
We are primarily interested in z=0 value
• treat connection to rest of circuit as 2 port with,
Z in ( z  0 )  Rc 
1  ( z  0 )
1  ( z  0 )
Input Impedance
After lots of algebra, one can show:
Z L  j  Rc  tan(   L )
Z in ( z  0 )  Rc 
Rc  j  Z L  tan(   L )
Special Case example: ZL=0 (short circuit)
0  j  Rc  tan(   L )
Z in ( z  0 )  Rc 
 j  Rc  tan(   L )
Rc  j  0  tan(   L )
Input Impedance - SHORT CIRCUIT
Z in( z  0 )  j  Rc  tan(   L )
Can change Zin by
changing these two
parameters
• Fix , vary L - different effects
• Vary , fix L - get same effects
Note that L is the
length of the
Transmission Line
Input Impedance - TL
Setup for Problem 1a & 1b
1a. Open Circuit Case
ZL  
small
Z L  j  Rc  tan(   L )
Z L  Rc
Rc  j  Z L  tan(   L )
small
1b. ZL = 93W - lots of complex algebra
Problem 1a and 1b
We know Zin (z=0) - treat as 2-PORT
80 W
1V
Power 
Zin

1
Re Vin  I in
2

Vin
Voltage
Divider
2



V
V

V
1
1 

 Re in  in   Re in 
2  Z in  2 
 Z in 

Do problem 1b - parts 2 &3
Problem 1a and 1b
In a Lossless Transmission Line, Pin flows into the
Transmission Line and it is dissipated at the
2
LOAD
1 VL
Pin  
2 ZL
Do Problem 1b - part 4
V̂ ( z )  Vm  e  j   z  ( 1   ( z ))
V̂ ( z  0 )  Vin  Vm  e  j  z  ( 1   ( z  0 ))
 Vm 
Vin
Can then plug back and get the full phasor
1  (0 )
Do Problem 1b - part 5