Lecture 08
The Smith Chart and Basic
Impedance-Matching Concepts
Sections: 6.8 and 6.9
Homework: From Section 6.13 Exercises: 12, 13, 14, 15,
16, 17, 18, 19, 20
Nikolova 2012 2
The Smith Chart: Γ plot in the Complex Plane
• Smith’s chart is a graphical representation in the complex Γ plane of
the input impedance, the load impedance, and the reflection
coefficient Γ of a loss-free TL
• it contains two families
of curves (circles) in
the complex Γ plane
• each circle corresponds
to a fixed normalized
resistance or reactance
Nikolova 2012 Lecture 08: The Smith Chart 3
The Smith Chart: Normalized Impedance and Γ
0
00
1
where and
1
=| | =
L
LL
LLL
LL
j
ri

(1 )
ri
L
ri
i
L
ri
r
x
 

 



 
22
2
22
2
1
11
11
(1)
L
ri
LL
ri
LL
r

2
11
(1)
ri
LL
x
x

  


let the abscissa be Γ
r
and the ordinate be Γ
i
(the Γ complex plane)
• resistance and reactance equations are circles in the Γ
complex plane
• resistance circles have centers lying on the Γ
r
axis (Γ
i
= 0 or
ordinate = 0)
• reactance circles have centers with abscissa coordinate = 1
• a complex normalized impedance z
L
= r
L
+ jx

1
0.2
0.25
L
r

1
short
open
Nikolova 2012 Lecture 08: The Smith Chart 6
The Smith Chart: Reactance Circles
inductive
loads
capacitive
loads
Nikolova 2012 Lecture 08: The Smith Chart 7
The Smith Chart: Nomographs
at the bottom of Smith’s chart, a nomograph is added to determine
• SWR and SWR in dB,
• return loss in dB,
• power reflection |Γ|
2
(P)
• reflection coefficient |Γ| (E or I), etc.
perfect match
10
20lo
g||



0.59


Nikolova 2012 Lecture 08: The Smith Chart 9
The Smith Chart: Plotting Impedance and Reading Out Γ
0.5 1.0
L
zj


0.5
L
r

1
L
x

||

(1 0.135 / 0.25) 0.46 83



   

||
0.62
What is Z
L

in z L
j
LjL
zL
V
Ve e
ZZ Z
I
Ve e














2
0
2
1
1
j
L

2
2
1
1
j
L
in
j
L
e
z
e








 on the Smith chart, the point corresponding to z
in
is rotated by
−2βL (decreasing angle, clockwise rotation) with respect to the
point corresponding to z
L
along an SWR circle
 one full circle on the Smith chart is 2βL
max
= 2π, i.e., L

11
in
zj


to
w
a
r
d

g
e
n
e
r
a
to
r
t
o
w
a
r
d

l
o
a
d


• known load Z
L
75 75
L
Zj
 
• known Z
0
0
50
Z

1.5 1.5
L
zj


A
• measured Z
in
23 34
in
Zj
 
B
0.46 0.68
in
zj


,
2.6
LB
SWR r


SWR circle
A
B
A
B
SWR SWR
,
,
1
1
LB
B
LB
r
r



,
1| |
1| |
B
B
B

2
1
1
j
L
in in
j
L
e
yz
e









• the relation between y
in
and y
L
is the same as that between z
in
and z
L
– one can get from load to input terminals (and vice versa) by
following a circle clockwise (counter-clockwise)




11
(/4)
1
1
1
1
j
in
j
L
e
zL
e
z







 







t
o
w
a
r
d

g
e
n
e
r
a
t
o
r
t
o
w
a
r
d

l
o
a
d
/4L





 for impedance match at the input terminals of the λ/4 TL, Z
in
=Z
G
*
0 GL
Z
ZZ


⇒
⇒
in
Z
0
/4
L


0
(, )Z

L
Z
G
Z
G

( ) , where
tan( ) 4 2
L
in
L
ZjZ L
f
Zf Z L
Zj
ZL
f






  



0
0
()
|()|
()
in
in
Z
fZ

in
V

in
I

G
V

 active (or average) power delivered to the loaded TL (this is also
the power delivered to the load Z
L
if the line is loss-free)
2
22
11 1 1
( ) Re{ } | | Re | | Re
22 2
in
in av in in in in G
Gin in
Z
PVIVYV
Z
ZZ



 


+ jX
G
is known and fixed
opt
max ( )
in
in in in
Z
Z
PZ
find the optimal R
in
and X
in
by obtaining the respective derivatives
22 2
0 ( ) 0
in
Gin in G
in
P
RR X X
R


 

0 ( ) 0
in
in in G