Summary of Transmission Lines

TRANSMISSION LINES
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EE 4347 Applied Electromagnetics
Transmission Line Model
R  z
L  z
Line Parameters
Transmission Line Behavior
Characteristic Impedance
lim
z 0
G  z
C  z
General
V0
R  j L
Z0   
 R0  jX 0
I0
G  jC
z

R  distributed resistance (W/m)
L  distributed inductance (H/m)
G  distributed conductance (1/Wm)
C  distributed capacitance (F/m)

Transmission Line
Equations
RG   2 LC 
 RG   2 LC 
R
2
  2 L2  G 2   2C 2 
2
L
L
R0 
X0  0
C
C
Distortionless Lines (RC = LG)
  RG
   LC
Transmission Line
Wave Equations
R
L
Z0 

G
C
R0  Z 0
Solutions to TL Wave
Equations
  z
I  z   I0 e
V  z   V0 e  z  V0 e z
 z
0
I e
Lossy
Lossy
Z in  Z 0 tanh 
Z in  Z 0 coth 
Z in  Z 0
Lossless
Lossless
Lossless
Z in  jZ 0 tan 
Z in   jZ 0 cot 
Z in  Z 0
Vmax  V0 1    Vmax  2 V0
Vmax  2 V0
Vmax  V0
Vmin  V0 1    Vmin  0
Vmin  0
Vmin  V0
Z in 
 Z L  Z 0 tanh  

 Z 0  Z L tanh  
  Z0 
Pavg 
 Z L  jZ 0 tan  

 Z 0  jZ L tan  
  Z0 


I max  2 V0 Z 0
I max  2 V0 Z 0
I max  V0 Z 0


I min  0
I min  0
I min  V0 Z 0
VSWR 
1 
1 
VSWR  
VSWR  
VSWR  1
max  Z in   Z 0  VSWR
max  Z in   
max  Z in   
max  Z in   Z 0
min  Z in   Z 0 VSWR
min  Z in   0
min  Z in   0
min  Z in   Z 0
I min  V0 Z 0 1   
Z0 
Z in,short Z in,open  Z 02
Lossy
Lossy
I max  V0 Z 0 1   
   LC
Misc.
L  0
Z in 
V  z , t 
I  z , t 

 R  I  z, t   L
z
t
I  z , t 
V  z , t 

 G V  z, t   C
z
t
d 2V  z 
  R  j L  G  jC V  z   0
2
dz
d 2I  z
  R  j L  G  jC  I  z   0
2
dz
L  1
2
Lossless Lines (R = G = 0)
 0
 L  1
Z L  Z0
Z L  Z0
Lossless
2
V0
2Z 0
X0  0
1   
Impedance
Transformation
General Case (with Loss)
 Z L  Z 0 tanh  
Z in    Z 0 

 Z 0  Z L tanh  
Matched
Z L  Z0
  2 L2  G 2   2C 2 
R
Open
ZL  
L 
 R  j L  G  jC 
Short
ZL  0
ZL
Complex Propagation Constant
    j 
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Types of Transmission
Lines
open two-wire
coaxial
2
L
microstrip
coplanar
Lossless Case
 Z L  jZ 0 tan  
Z in    Z 0 

Z

jZ
tan

L
 0

stripline
slotline