ECE 3317
Prof. David R. Jackson
Spring 2013
Notes 12
Transmission Lines
(Smith Chart)
1
Smith Chart
The Smith chart is a very convenient graphical tool for analyzing transmission
lines and studying their behavior.
A network analyzer (Agilent N5245A PNA-X) showing a Smith chart.
2
Smith Chart (cont.)
From Wikipedia:
Phillip Hagar Smith (April 29, 1905–
August 29, 1987) was an electrical
engineer, who became famous for his
invention of the Smith chart.
Smith graduated from Tufts College in
1928. While working for RCA, he
invented his eponymous Smith chart. He
retired from Bell Labs in 1970.
Phillip Smith invented the Smith Chart in 1939 while he was working for The Bell Telephone
Laboratories. When asked why he invented this chart, Smith explained: “From the time I could operate
a slide rule, I've been interested in graphical representations of mathematical relationships.”
In 1969 he published the book Electronic Applications of the Smith Chart in Waveguide, Circuit, and
Component Analysis, a comprehensive work on the subject.
3
Smith Chart (cont.)
  z    L e j 2 z
 1   z 
Z  z   

 1   z 
N
in
  L e j 2  l
z  l
The Smith chart is really a complex plane:
ΓL 
Im   z 
ZL  Z 0
ZL  Z 0
1
 z
2 l
L
Re   z 
4
Smith Chart (cont.)
R  jX
N
in
Denote
  z   x  jy
R  jX
N
in
so
N
in
N
in
 1   z  
 

1


z
 

(complex variable)
 1  x   jy 
 

 1  x   jy 
1  x   jy   RinN  jX inN   1  x   jy 
Real part:
Imaginary part:
1  x  RinN  yX inN  1  x
1  x  X inN  yRinN  y
5
Smith Chart (cont.)
1  x  RinN  yX inN  1  x
1  x  X inN  yRinN  y
From the second one we have
 y 
N
X 
1

R

in 

 1 x 
N
in
Substituting into the first one, and multiplying by (1-x), we have
 y 
N 
1  x  R  y 
 1  Rin   1  x
 1  x 

N
in
1  x 
2
RinN  y 2 1  RinN   1  x 1  x 
6
Smith Chart (cont.)
Algebraic simplification:
1  x 
2
RinN  y 2 1  RinN   1  x 1  x 
N
2
N
2
1

x
R

y
1

R

1

x
  in
 in 
2
x 2 1  RinN   2 xRinN   RinN  1  y 2 1  RinN   0
x 2 1  RinN   2 xRinN  y 2 1  RinN   1  RinN
N

R
x 2  2 x  in N
 1  Rin
 2 1  RinN
 y 
N
1

R
in

7
Smith Chart (cont.)
N

R
x 2  2 x  in N
 1  Rin
 2 1  RinN
 y 
N
1

R
in

2

 R 
R 
1 R
2
y 

x
N 
N 
1

R
1

R
1

R
in 
in 


N
in
N
in
N
in
1 R


R 
2
y 
x
N 
1  Rin 

N
in
2
N
in
N
in
2
1  R    R 
1  R 
N
in
N
in
N
in
2
2
2

R 
1
2
y 
x
N 
N 2
1

R
in 

1  Rin 
N
in
8
Smith Chart (cont.)
2

RinN 
1
2
x


y


N 
N 2
1

R
in 

1  Rin 
This defines the equation of a circle:
 R

x
,
y

,
0
center:  c c  

N
1

R
in


N
in
Note:
radius: R 
Im   z 
RinN  0
1
1  RinN
RinN  1
xc  R  1
RinN  3
Re   z 
RinN  0.2
RinN  
9
Smith Chart (cont.)
1  x  RinN  yX inN  1  x
1  x  X inN  yRinN  y
Now we eliminate the resistance from the two equations.
From the second one we have:
N
1

x
X


in  y
N
R 
in
y
Substituting into the first one, we have
 1  x  X inN  y 
N

yX
1  x  

in  1  x
y


10
Smith Chart (cont.)
Algebraic simplification:
 1  x  X inN  y 
N
1  x  
  yX in  1  x
y


1  x  X inN  y 1  x   y 2 X inN  y 1  x   0
2
1  x  X inN  2 y  y 2 X inN  0
2
 2 
 x  1   N  y  y 2  0
 X in 
2
11
Smith Chart (cont.)
 2 
2
x

1

   N  y  y2  0
 X in 
2

1   1 
 x  1   y  N    N 
X in   X in 

2
2
This defines the equation of a circle:
center:

1 
N 
X
in 

 xc , yc   1,
Note:
radius:
R
1
X inN
yc   R
12
Smith Chart (cont.)
This defines the equation of a circle:
X  0.5
N
in
 1 
x
,
y

 c c  1, N 
 X in 
Im   z 
R
1
X inN
X inN  1
X inN  3
X inN  0
Re   z 
X inN  
X inN  3
X inN  0.5
X inN  1
13
Smith Chart (cont.)
Im   z 
Actual Smith chart:
 z
Re   z 
14
Smith Chart (cont.)
Important points:
Z inN  z   1
z  N
Z in  z   1
RinN = 0
(   z   1)
1  jX
circle
S/C
Perfect Match
(   z   0)
O/C
15
Smith Chart (cont.)
j 2 z
2 j l
1


z


1


e
1


e
L
L
ZinN ( z ) 


1    z  1   L e j 2  z 1   L e2 j l

Movement in negative z direction
Clockwise motion on circle of constant L
(toward generator )
Im   z 
Generator
Zg
Transmission Line
L
Load
S
Z0
ZL
ΓL
z = -l
z=0
 z
To generator
Re   z 
z
angle change = 2l
16
Smith Chart (cont.)
Im   z 
j 2 z


1


e
N
L
Z in  z   
j 2 z 
 1 L e

ΓL
We go completely around
the Smith chart when
Re   z 
l /2
 2   
2 z  2 l  2 
   2
   2 
17
Smith Chart (cont.)
In general, the angle change
on the Smith chart as we go
towards the generator is:
Im   z 
  2    z 
 2 
 2
  z 



 z 
 4  
  
 l 
 4  
 
ΓL

Re   z 
 l 
  4  
 
The angle change is twice
the electrical length change
on the line:  = -2( l).
  z   L e j 2 z  L e j e j 2 z  L e j
18
Smith Chart (cont.)
Note:
The Smith chart already
has wavelength scales on
the perimeter for your
convenience (so you don’t
need to measure angles).
The “wavelengths towards
generator” scale is
measured clockwise,
starting (arbitrarily) here.
The “wavelengths towards
load” scale is measured
counterclockwise, starting
(arbitrarily) here.
19
Reciprocal Property
Im   z 
Normalized
impedances become
normalized
admittances.
j 2 z


1


e
N
L
Z in  z   
j 2 z 
1


e

L

ΓL
Go half-way around the
Smith chart:
A
l /4
 2   
2 z  2 
    


 4 
Re   z 
B
1
Z inN  z 
 1  L 
ZinN  0   

1



L 
 1 L 
Z inN  l   

1



L 
ZinN ( B)
1
 N
Zin ( A)

1
Z in  z  / Z 0

1/ Z in  z 
1/ Z 0
Yin  z 

Y0
 YinN  z 
20
Normalized Voltage
Normalized voltage
Assume V+  z   1
Γ z
V(z) = V+  z  1+ΓL e j 2 z  1+Γ  z 
V( z )
Vmin
ΓL
Vmax
Vmax  V  z  max  1   L
Vmin  V  z  min  1-  L
We can use the Smith chart as a crank diagram.
21
SWR
As we move along the transmission line,
we stay on a circle of constant radius.
Γ  z   ΓL e j 2 z
Γ z
On the positive real axis:
ZinN

RinN
 real  Z0
SWR 
Rin
 RinN
Z0
ΓL
Positive real axis
(from the previous property proved about a real load)
The SWR is read off from the normalized
resistance value on the positive real axis.
22
Example
Given:
L =0.707
 L =0.70745
1.707
load
Use the Smith chart to plot
the magnitude of the
normalized voltage, find the
SWR, and find the
normalized load admittance.
ZLN 
V(z)
1  L
 1  j2
1  L
V (z)  1
0.293
z

5
16
16
V z
Z LN
45
45   / 4  rad 
Set

 2 
2 l  2 
l =
4
  
l

Vmin
ΓL
Vmax
1   L =1.707
16
23
Example (cont.)
 L =0.70745
Z LN  1  j 2
RinN  1
 /16  0.1875
45
X inN  2
YLN  0.2  j 0.4
SWR = 5.8
24
Smith Chart as an Admittance Chart
The Smith chart can also be used as an admittance calculator instead of an
impedance calculator.
 1   z 
Z inN  z   

1


z
 

YinN  z  
Yin  z  1/ Zin  z 
1
1


 N
Y0
1/ Z0
Zin  z  / Z0 Zin  z 
Hence
 1   z  
Y  z   

1


z




or
 1    z  
Y  z   


1


z




N
in
N
in
where
  z     z 
25
Admittance Chart (cont.)
 Im   z 
Bin / Y0  0.5
Gin / Y0  0.5
  z 
 Re   z 
26
Comparison of Charts
Impedance chart
Inductive region
X 0
N
in
Im   z 
Capacitive region
X inN  0
27
Comparison of Charts (cont.)
Admittance chart
Capacitive region
BinN  0
 Im   z 
Inductive region
BinN  0
28
Using the Smith chart for Impedance
and Admittance Calculations
 We can use the same Smith chart for both impedance and admittance
calculations.
 The Smith chart is then either the  plane or the -  plane, depending
on which type of calculation we are doing.
For example:
 We can convert from normalized impedance to normalized admittance,
using the reciprocal property (go half-way around the smith chart).
 We can then continue to use the Smith chart on an admittance basis.
29
Example
Z LN  1  j 2
Find the normalized admittance /8 away from the load.
30
Admittance Chart (cont.)
Im   z  or Im    z  
YinN  0.23  j 0.48
Z LN  1  j 2
l
2  l  4  
 
1
 4  
8


2
   90o
YLN  0.2  j 0.4
90 o
Re   z  or Re    z  
Answer:
YinN  0.23  j 0.48
31