Transmission Line Theory
Smith Chart
1
Transmission
a s ss o Linee Theory
eo y
• Wire connections in Analog / Digital circuits:
V, I
No voltage drop on wire connections
• Transmission Lines in Microwave circuits:
V, I
V, I waves on transmission lines
• Field analysis:
E, H
EM waves
2
Transmission
a s ss o Linee Theory
eo y
R  series resistance per unit length,
for both conductors, in /m.
L  series inductance per unit length,
for both conductors, in H/m.
G  shunt conductance per unit length,
in S/m
C  shunt capacitance per unit length,
in F/m
Figure 2.1 (p. 50) Voltage and current definitions
and equivalent circuit for an incremental length of
transmission line. (a) Voltage and current definitions.
(b) Lumped-element equivalent circuit.
3
Transmission
a s ss o Linee Theory
eo y
From Kirchhoff' s voltage law :
 i( z, t )
v( z , t )  R z i ( z , t )  L z
 v( z  z , t )  0,
(2 1a)
(2.1a)
t
From Kirchhoff' s current law :
 v( z  z , t )
i ( z , t )  G z v( z  z , t )  C z
 i ( z  z , t )  0.
(2.1b)
t
z  0 :
 v z , t 
 iz, t 
  R iz, t   L
(2.2a)
z
t
 iz, t 
 v z , t 


 G v z , t  C
(2.2b)
z
t
(time - domain transmission line equations / telegrapher equations)
4
Transmission
a s ss o Linee Theory
eo y
For the sinusoidal steady state condition,
the phasor form :
d V z 
 R  jL  I  z ,
(2.3a)
dz
d I z 
 G  jC V  z .
(2.3b)
dz
Wave equations for V ( z ) and I ( z ) :
Traveling wave solutions :
V ( z )  V0 e  z  V0 e z ,
((2.6a))
I ( z )  I 0 e  z  I 0 e z .
(2.6b)
Applying
pp y g ((2.3a)) to ((2.6a)) :
I ( z) 


V
R  jL
  z
0
e
 V0 e z

 characteristic impedance
d V z  2
  V ( z )  0,
(2.4a)
2
R  jL
R  j L
dz
Z0 

(2 7)
(2.7)
2

G  jC
d I z  2
  I ( z )  0.
(2.4b)
2
V0
V0
dz
 Z0   

I0
I0
where     j  R  jL G  jC  (2.5)
2
is the complex propagation constant
5
Transmission
a s ss o Linee Theory
eo y
The voltage waveform in the time domain :


v z , t   V0 cos  t   z    e  z


 V0 cos  t   z    e z (2.9)
wavelength
g on the line :

2

    j  j LC
   LC
 0
(2.12a)
(2.12b)
The characteristic impedance :
(2.10)
the phase velocity :
vp 
The Lossless Line : ( R  G  0)

 f (2.11)
(2 11)

L
Z0 
(2 13)
(2.13)
C
The general solutions :
V ( z )  V0 e  j z  V0 e j z , (2.14a)
V0  j z V0 j z
I ( z) 
e

e . ((2.14b))
Z0
Z0

2



2
, vp  

 LC
1
.
LC
6
Field
e d Analysis
a ys s oof Transmission
a s ss o Lines
es


L
I0
2

S
H  H  ds H/m (2.17)
(2 17)
Similarly for stored electric energy :
Figure 2.2 (p. 53) Field lines on an
arbitrary TEM transmission line.
The time - average
g stored magnetic
g
energy
gy :
Wm 

H H

4
S

ds,
and circuit theory gives :
Wm  L I 0
2

4 S
andd circuit
i it theory gives
i :
We 
E  E  ds,
We  C V0
2
4

C

V0
2

S
E  E  ds F/m ((2.18))
4
7
Field
e d Analysis
a ys s oof Transmission
a s ss o Lines
es
Power loss of the conductor :
Rs

Pc 
H

H
dl

C

C
1
2
2
(assuming H is tangential to S )
and circuit theory gives :
Pc  R I 0
2
G
Rs
I0
Pd  G V0
2

2
2

2

R
Power loss of the lossy dielectric :
 

Pd 
E

E
ds,

S
2
and circuit theory gives :

H  H d l /m (2.19)
 
V0
2

S
E  E  ds S/m (2.20)
C1  C 2
where Rs  1  s
8
Field
e d Analysis
a ys s oof Transmission
a s ss o Lines
es
• Table 2.1 Transmission Line Parameters for Some Common Lines
COAX
TWO-WIRE
PARALLEL PLATE
a
a
D
b
L
C
R
G
 b
ln
2 a
2  
ln b a
Rs  1 1 
  
2  a b 
2  
ln b a
w
d
a

1  D 
cosh  

 2a 
 
cosh 1 D 2a 
Rs
a
  
cosh 1 D 2a 
d
w
 w
d
2 Rs
w
 w
d
9
Terminated
e
ated Lossless
oss ess Transmission
a s ss o Linee
Figure 2.4 (p. 58)
  j z
0
V e

0

0
V
 Z0
I
A transmission line terminated in a load impedance ZL. (modified)
The total voltage/current on the line :
  j z
0
V ( z)  V e

0
V e
j z
, (2.34a)
V0  j z V0 j z
I ( z) 
e

e . (2.34b)
Z0
Z0
At z  0, we must have
V (0) V0  V0
ZL 
 
Z0.

I (0) V0  V0
V0 e  j z
V
 ZL
I
V0
 Z 0 If Z  Z

L
0
I0
V0 e j z
V0
   Z0
I0
10
Terminated
e
ated Lossless
oss ess Transmission
a s ss o Linee
Z L  Z0 
V 
V0
Z L  Z0
Z L  Z0 
V 
V0
Z L  Z0

0

0
Voltage reflection coefficient,  :
V0 Z L  Z 0
  
V0
Z L  Z0
(2.35)
Voltage reflection coefficient,  :
V0 Z L  Z 0
  
V0
Z L  Z0
(2.35)
The total voltage/current on the line : The time - average power flow along the line :

e

.
V ( z )  V0 e  j z  e j z , (2.36a)
V0
I ( z) 
Z0
 j z
  e j z
 standing waves
 No
N reflection
fl i :   0
Z L  Z 0  matched
((2.36b))
Pav 

1
Re V ( z ) I ( z )
2
 2
0

A  A  2 j Im( A)

1V
2
  2 j z
2 j z

Re 1   e
 e

2 Z0
 2
0
1V

2 Z0
1   .
2

  0

 1
11
Terminated
e
ated Lossless
oss ess Transmission
a s ss o Linee
Incident Power : Pin 
Reflected Power : Pr 

V
magnitude of the voltage on the line :
2Z 0
 2
0
V
2Z 0

 2
0
V

2Z 0

V ( z )  V0 1  e 2 j z
2
  Pin
2

Transmitted Power : Pt  Pin  Pr  1   Pin
2
 V0 1  e  2 j l
 V0 1   e j   2  l 
When the load is mismatched, not all of available
Vmax  V0 1   
power is delivered to the load, defined the
Vmin  V0 1   
return loss (RL) in dB as :
RL  20 log  dB.
dB

V ( z )  V0 e  j z  e j z ,
 2
0
  0   dB

   1  0 dB
voltage  st anding
wave ratio
((SWR / VSWR)) can be definde as
Vmax 1  
SWR 

Vmin 1  
12
Terminated
e
ated Lossless
oss ess Transmission
a s ss o Linee
The input impedance seen toward
the load at l   z :
Zin

Z L  Z 0 e j l  Z L  Z 0 e  j l
Z in  Z 0
Z L  Z 0 e j l  Z L  Z 0 e  j l
generalized reflection coefficient :
V0 e  j l
(l )   j l  0 e  2 j l ,
V0 e
the input impedance seen toward
the load at l   z :
 Z0
Z L cos  l  jZ 0 sin  l
Z 0 cos  l  jZ L sin  l
Z L  jZ 0 tan  l
 Z0
Z 0  jZ L tan  l




1   e  2 j l
Z0

 2 j l
1  e
((2.43))
(2.44)
 transmission line impedance equation
V  l  V0 e j l  e  j l
Z in 
Z0
  j l
 j l
I  l  V0 e  e
13
Terminated
e
ated Lossless
oss ess Transmission
a s ss o Linee
• Special case: short terminated 
Figure 2.5 (p. 60) A transmission line
terminated in a short circuit.
The total voltage/current on the line :

e

2V
 Z
V ( z )  V0 e  j z  e j z  2 jV0 sin  z , (2.45a)
(2 45a)
V0
I ( z) 
Z0
 j z
e
The input impedance :
Z ini  jZ 0 tan  l.
j z

0
cos  z.
(2.45b)
0
Figure 2.6 (p. 61) (a) Voltage, (b) current,
and (c) impedance (Rin = 0 or ) variation
along a short-circuited transmission line.
14
Terminated
e
ated Lossless
oss ess Transmission
a s ss o Linee
• Special case: open terminated 
Figure 2.7 (p. 61) A transmission line
terminated in an open circuit
circuit.
The total voltage/current on the line :

e

 2 jV
 Z
V ( z )  V0 e  j z  e j z  2 V0 cos  z ,
V0
I ( z) 
Z0
 j z
e
The input impedance :
Z ini   jZ 0 cot  l.
j z

0
(2 46a)
(2.46a)
sin  z. (2.45b)
0
Figure 2.8 (p. 62) (a) Voltage, (b) current,
and (c) impedance (Rin = 0 or ) variation
along an open-circuited transmission line.
15
Terminated
e
ated Lossless
oss ess Transmission
a s ss o Linee
If l  n / 2 (n  1, 2, 3, ) :
Z in  Z L .
(2.47)
If l   / 4  n / 2 (n  1, 2, 3, ) :
Z 02
Z ini 
.
ZL
The loading line infinitely long :
Z1  Z 0

. (2.49)
Z1  Z 0

(2.48)
 quarter  wave transformer

V ( z )  V0 e  j z  e j z , z  0, (2.50a)
V ( z )  V0Te  j1 z , z  0. (2.50b)
 T  1  
2 Z1
.
Z1  Z 0
(2.51)
isertion loss :
IL  20 log T dB
Zin
Figure 2.9 (p. 63) Reflection and transmission
at the junction of two transmission lines with
different characteristic impedances
impedances.
16
Smith
S
t C
Chart
a t
Figure 2.10 (p. 65)
The Smith chart.
chart
17
Smith
S
t C
Chart
a t
The lossless line ( Z 0 ) with a load Z L :
The Smith chart :
a polar plot of  

   e j
magnitude
g
:   radius ( 1))
angle :  (180    180 )
o
o
zL 1
  e j
zL  1
where z L  Z L Z 0 .
zL 
1   e j
1   e j
Express z L and  in term of real and
reflection coefficient

imaginary parts :
normalized impedance (admittance)
  r  ji
z L  rL  j xL
normalized impedance :
z  Z Z0

1  r   ji
rL  j xL 
.
1  r   ji
18
Smith
S
t C
Chart
a t
1  r2  i2
rL 
1  r 2  i2
(2.55a)
2i
xL 
1  r 2  i2
(2.55b)
Constant resistance (rL) circles
+xL
0
1
3
rL
xL
0
2

 1 
r 
 r  L   i2  

1  rL 

 1  rL 
2
2
2

1   1 

r  1   i      .
xL   xL 

1   e  2 j l
Z in  Z 0
1   e  2 j l
r

PLANE
CONSTANT RESISTANCE
LINES IN THE zL=rL+jx
j L
PLANE
Constant reactance (xL) circles
xL
i
1
05
0.5
3
3
+xL
1
1
-1
3
1
rearranging (2.55) :
2
i
xL
rL
-3
CONSTANT REACTANCE
LINES IN THE zL=rrL+jxL
PLANE
xL
0
r
-3
-0.5
-1

PLANE
19
Smith
S
t C
Chart
a t
 The constant r and the constant x loci
form two families of orthogonal circles
in the chart.
chart
 The constant r and constant x circles all
pass through the point (r = 1, i = 0).
 The upper half of the diagram
represents +jx.
 The lower half of the diagram
represents jx.
 For admittance the constant r circles
become constant g circles,
circles and the
constant x circles become constant
susceptance b circles.
 The distance once around the Smith
chart is one-half wavelength ( / 2)
20
Smith
S
t C
Chart
a t - Example
a pe
Locate in Smith Chart with
following normalized
impedances
1. z1=1+j1
2. z2=0.4+j0.5
3 z3=3-j3
3.
3 j3
 z7  1
 z6  
4. z4=0.2-j0.6
5. z5=0
6. z6= 
7. z7=1
21
Smith
S
t C
Chart
a t - Example
a pe
load impedance: 40  j 70 
Z 0  100 , l  0.3 , find Z in  ?
solution :
z L  Z L Z 0  0.4  j 0.7
  0.59
SWR  3.87
RL  4.6 dB
WTG: 0.106
 0.3 : 0.406  0.365  j 0.611
Z in  Z 0 zin  36.5  j 61.1 
Figure 2.11 (p. 67)
Smith chart for Example 2.2.
22
22
Smith
S
t C
Chart
a t – Z vs Y
z L  /4 long
l
t
transmiss
i ion
i line
li : zin  1 / z L  normalized
li d admittance
d itt
Z Smith chart
Y Smith chart
ZY Smith chart
23
ZY Smith chart
24
Smith
S
t C
Chart
a t - Example
a pe
 /4 long transmission line:
zin  1/ z L
 normalized admittance
load impedance: 100  j 50 
Z 0  50 , l  0.15 , find Yin  ?
solution :
z L  Z L Z 0  2  j1
yL  0.4  j 0.2
yL
YL  yLY0 
 0.008
0 008  j 0.004
0 004 S
Z0
WTG: 0.214
 0.15 : 0.364  y  0.61  j 0.66
Y  yY0 
y
 0.0122  j 0.0132 S
Z0
25
Slotted
S
otted Linee
z  0.2cm,
cm 2.2cm,
2 2cm 4.2cm
4 2cm
z  0.72cm,
cm 2.72cm,
2 72cm 4.72cm
4 72cm
Figure 2.13 An X-band waveguide slotted line.
minima
i i
repeat every  / 2
   4 cm
4
1.48  86.4o
  
4
lmin  4.2  2.72  1.48
1 48 cm  0.37
  0.2e
SWR - 1 1.5  1

 0.2

SWR  1 1.5  1
Z L  Z0
j 86.4 o
 0.0126  j 0.1996
1 
 47.3  j19.7
1 
26
Thee Quarter-Wave
Qua te Wave Transformer
a so e
Figure 2.16 (p. 73)
The quarter-wave matching transformer.
Z in  Z1
RL  jjZ1 tan  l
Z1  jRL tan  l
Z12
Z in  f 0  
(at  l   / 2)
RL
Z1  Z 0 RL
Figure 2.18 (p. 75) Multiple reflection
analysis of the quarter-wave
quarter wave transformer.
transformer
27
Thee Quarter-Wave
Qua te Wave Transformer
a so e
Example 2.5
• Consider a load resistance RL = 100, to be matched to a 50 line with
a quarter-wave transformer. Find the characteristic impedance of the
matching section and plot the magnitude of the reflection coefficient
versus normalized frequency, f / f0, where f0 is the frequency at which
the line is  / 4 long.
• Solution:
Z1  Z 0 RL  50 100  70.71
Z  Z0
  in
Z in  Z 0
Z in  Z1
 2  0   2 f
 l      
   4   v p
Z L  jZ1 tan  l
Z1  jZ L tan  l
 v p   f


 4 f  2 f
0
 0 
Figure 2.17 (p. 74)
Reflection coefficient versus normalized frequency
f the
for
th quarter-wave
t
t
transformer
f
off Example
E
l 2.5.
25
28
Generator
Ge
e ato and
a d Load
oad Mismatches
s atc es
Figure 2.19 (p. 77) Transmission line
circuit for mismatched load and generator.
The power delivered to the load :


Generator Matched to Loaded Line :
2
2
 1 
Z in
1
1
Re 
Re Vin I in  Vg
Z in  Z g
2
2
 Z in 
2
Rin
1
 Vg
2
Rin  Rg 2  X in  X g 2
P
Load Matched to Line :
2
Z0
1
P  Vg
2
Z 0  Rg 2  X g2

Z in  Z g
0
Z in  Z g
Rg
2
1
P  Vg
2
4 Rg2  X g2


29
Generator
Ge
e ato and
a d Load
oad Mismatches
s atc es
Conjugate Matching :
Z g fixed, to maximize P

P
2
 0  Rg2  Rin2  X g  X in   0
 Rin
or Z in  Z g
P
 0  X in X g  X in   0
 X in
Rin  Rg ,
Pin, max
X in   X g
2 1
1
 Vg
2
4 Rg
 maxmum available p
power from
the generator
30
Lossy
ossy Transmission
a s ss o Lines
es
   j 

R  j L G  j C 
The Low - Loss Line :
G 
 R



L

C


  j LC 1  j 
 j R
G 
 j LC 1  


 2   L  C 

1
C
L  1 R


G
  R
   GZ 0 

2
L
C  2  Z0

The Distortionless Line :
R G

L C
C
R
 constant
L
   LC
 vp   /   1
LC
 constant
Z0 
L
C
   LC
Z0 
L
C
31
Lossy
ossy Transmission
a s ss o Lines
es
 2
0


V
1
2

1  l  e 2 l
Pin  Re V  l I  l  
2
2Z 0


 2
0

V
1
2

1 
PL  Re V 0I 0 
A lossy transmission
2
2Z 0

Figure 2.20
2 20 (p.
(p 82)
line terminated in the impedance ZL.
The Terminated Lossy Line :

e
V ( z )  V0 e  z  e z
V0
I ( z) 
Z0
 z
 e  z


l   e  2 l
Z in  Z 0
Z L  Z 0 tanh  l
Z 0  Z L tanh  l
Ploss  Pin  PL 

V0
2
2Z 0
e
2 l



 1   1  e  2 l
2

The Perturbation Method for 
P( z )  P0 e  2 z
 P
 2 P0 e  2 z  2 P( z )
z
Pl  z  Pl  z  0


2 P( z )
2 P0
Pl 
32
Lossy
ossy Transmission
a s ss o Lines
es
The Wheeler Incremental inductance Rule
2
R
Pl  s  H t d l W/m
2 C
L 
0 s
2I
2

C
Rs I L
2
Ht d l
2
Pl 
αc 
 0 s
I L
2

0 s2

Pl
L

2 P0 2Z 0
L
Z0 

C
Z 0
αc 
Z0
I L
2
L
 Lv p
LC
2

 
Z0  s   Z0  s
2
2
 s d Z0
Z 0 
2 dl
Z 0  s
αc 

Z0
4Z 0
d Z0
dl
d Z0
dl
Rs d Z 0

2Z 0 d l
roughness of conductor surface :
2
 2


 
1
αc  αc 1  tan 1.4  
 
  s  
33