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ELEC 401 – Microwave Electronics
ELEC 401
MICROWAVE ELECTRONICS
Lecture on Smith Chart
Instructor: M. İrşadi Aksun
Acknowledgements:
1. Some Smith Chart figures were taken from the textbook “Fundamentals of
Applied Electromagnetics” by F. T. Ulaby.
M. I. Aksun
Koç University
1/18
ELEC 401 – Microwave Electronics
Outline
 Chapter 1: Motivation & Introduction
 Chapter 2: Review of EM Wave Theory
 Chapter 3: Plane Electromagnetic Waves
 Chapter 4: Transmission Lines (TL)
 Chapter 5: Smith Chart & Impedance Matching
 Chapter 6: Microwave Network Characterization
 Chapter 7: Passive Microwave Components
M. I. Aksun
Koç University
2/18
ELEC 401 – Microwave Electronics
Smith Chart - Introduction
 Smith chart is a graphical tool
that represents a mapping
between impedance and reflection
coefficient;
 It was introduced by Phil
Smith of RCA in 1936;
 It is nothing but the polar plot
of reflection coefficient with the
corresponding impedances
written on it;
 It is a very convenient tool for
presentation purpose.
M. I. Aksun
Koç University
3/18
ELEC 401 – Microwave Electronics
Smith Chart - Introduction
Polar plot of    e j r
Point A:
A  0.5e
j 53 0
 0.3  j 0.4
Point B:
B  0.54e
M. I. Aksun
Koç University
j 202 0
 0.5  j 0.2
4/18
ELEC 401 – Microwave Electronics
Smith Chart - Introduction
 To show how the chart is constructed, we first write the
relationship between input impedance and the reflection coefficient:
1  in
1  in
Z in  Z 0
 Z in 
1  in
1  in
R  jX 

1   e j
1   e j
1  u  jv
1  u  jv
By equating the real and imaginary parts
2
u  12   v  1   12
X

X
Reactance circles
2

R 
1
 u 
  v 2 
R 1

R  12
M. I. Aksun
Koç University
Resistance circles
5/18
ELEC 401 – Microwave Electronics
Smith Chart - Introduction
2

R 
1
 u 
  v 2 
R 1

R  12
(v )
Resistance circles
Equation of a circle with
radius R located at (x0,y0)
( x  x 0 ) 2  ( y  y0 ) 2  R 2
(u)
2
u  1   v  1   12
X

X
2
Reactance circles
M. I. Aksun
Koç University
6/18
ELEC 401 – Microwave Electronics
Smith Chart - Introduction
Unit circle
 Point P represents a
normalized impedance Z=2-j1.
The reflection coefficient has a
magnitude OP/OR, and an angle
-26.6 degrees.
R
O
-26.60
P
 Point R is an arbitrary point
on the R=0 circle, which is also
||=1.0 circle.
M. I. Aksun
Koç University
7/18
ELEC 401 – Microwave Electronics
Smith Chart - Application
 Show the locations of the
following impedances:
Z A  (1.0  j 0.0)
Z B  (1.0  j1.0)
YE
Z C  (  j 0.0) o.c
Z D  (0.0  j 0.0) s.c
Z E  (1.0  j1.0)
 Show the locations of the
following admittance:
Y A  (1.0  j 0.0)
ZD
ZB
ZA
ZC
YA
YD
YC
ZE
YB
YB  (0.5  j 0.5)
YC  (0.0  j 0.0) o.c
YD  (  j 0.0) s.c
YE  (0.5  j 0.5)
M. I. Aksun
Koç University
8/18
ELEC 401 – Microwave Electronics
Smith Chart - Applications
 Find the admittance
from a given impedance,
or vice versa:
Z
Constant || circle
1 
1 
1   1  e j 1  e j (    )
Y 


1   1  e j 1  e j (    )
Z  1.0  j1.0
 

Y  0.5  j 0.5
M. I. Aksun
Koç University
9/18
ELEC 401 – Microwave Electronics
Smith Chart - Applications
 Find the input impedance
of a TL terminated in a load
impedance ZL.
Z L  1.0  j1.0
Z0 , 
z=-l
Constant || circle
ZL
z=0
Z L  1.0  j1.0
Angle  2 l
 

l  0.148
1  in 1  Le  j 2 l
Z in

1  in 1  Le  j 2 l
Z in
Z in  1.4  j1.2
1  e j (   2 l )
1  e j (   2 l )
M. I. Aksun
Koç University
10/18
ELEC 401 – Microwave Electronics
Smith Chart - Applications
 Find the SWR, voltage
maxima and minima.
Constant || circle
Z L  1.0  j1.0
Z0 , 
ZL
z=-l

Z L  1.0  j1.0
1 
1 
Voltage Max.
Current Min.
M. I. Aksun
Koç University
s  2.8
Z in  2.8
l  0.148
VSWR  s 
Angle  2 l
 
z=0
Voltage Min.
Current Max.
Voltage Max.
Current Min.
11/18
ELEC 401 – Microwave Electronics
Smith Chart - Applications
0.13 0
Example: A 5.2 cm long, lossless
100 W line is terminated in a load
impedance ZL=30+j50 W.
a) Calculate |L|, fL, and VSWR:
30  j 50
ZL 
 0.3  j 0.5
100
Z 1
L  L
 0.62123.50
Z L 1
VSWR 
93.60
Z L  0.3  j 0.5
 

1  0.62
 4.2
1  0.62
b) Determine the impedance at
the input for the frequency of
750 MHz and 0:
c 3.0  1010 cm / s
0  
 40cm
6
f
750  10 / s
5 .2
2
l  5.2cm 
 0  0.13 0  2 l  2
l  93.60
40
0
M. I. Aksun
Koç University
VSWR=4.2
Constant || circle
12/18
ELEC 401 – Microwave Electronics
Smith Chart - Applications
Example: A 50W coaxial cable, filled with a dielectric material of e2.25,
is connected to a generator with 50W internal impedance and 200 MHz
signal frequency. The TL is terminated in an unknown impedance at a
distance of 10 cm from the generator.
a) Knowing that the input impedance at the source terminal was
measured and noted as 25+j5 W, what is the load impedance?
b) Find the VSWR;
c) What should the length of the line be in order to have a real input
impedance at the source terminal?
0
c 3.0 1010 cm / s
0  

150
cm
;


 100cm
f
200 106 / s
er
Z in  25  j5
Z 0  50W
z= -10cm
M. I. Aksun
Koç University
ZL  ?
z=0
l  10cm 
Z in 
10
2
  0.1  2l  2 l  720
100

25  j 5
 0.5  j 0.1
50
13/18
ELEC 401 – Microwave Electronics
Smith Chart - Applications
1. Locate Zin on the S.C.;
2. Draw the constant || circle;
3. Starting from Zin move
toward load by 0.1 on
constant || circle;
Constant || circle
Z in  0.5  j 0.1
Z L  0.6  j 0.4
s  2.0
Z L  Z L  50W  (30  j 20)W
Z in  0.5  j 0.1
Z L  0.6  j 0.4
Z 0  50W
z= -0.1
M. I. Aksun
Koç University
ZL  ?
z=0
14/18
ELEC 401 – Microwave Electronics
Smith Chart - Applications
Example: A length of TL with 50W characteristic impedance is connected
to a generator with 50W internal impedance and to an unknown load
impedance ZL. The VSWR and the locations of the maximum and
minimum of standing wave are measured and the following information
is obtained:
i) First voltage minimum occurs at a distance of /5 from the load
terminals;
ii) VSWR=Vmax/Vmin = 2.0
Find the load impedance by using the above information.
Standing wave
Vmax
 /5
Vmin
Z 0  50W
z= -l
M. I. Aksun
Koç University
ZL  ?
z=0
15/18
ELEC 401 – Microwave Electronics
Draw the SWR=2.0 circle
1. Find the s=2.0 point
on the Smith Chart.
2. Draw a circle that
is centered at s=1.0
and passes through
s=2.0 point
M. I. Aksun
Koç University
SWR=2
circle
s=2.0
16/18
ELEC 401 – Microwave Electronics
On the SWR=2.0 circle,
start from Vmin point
and go 0.2λ toward load.
Normalized load
impedance is:
1.55 - j0.65
Vmin , Imax
Vmax , Imin
Z L  1.55  j 0.65
Load impedance is:
77.5 – j32.5
0.2λ
M. I. Aksun
Koç University
17/18