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Smith Chart: Introduction
Introduction– A graphical tool used to solve
transmission-line problems.– Today, a presentation medium in
computer-aided design (CAD) softwaresand measuring equipment for displaying the performance of microwave circuits.
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°=Γ
+=Γ535.0or
4.03.0j
A
A
ej
Smith Chart and Γ
°=Γ−−=Γ 20254.0or 2.05.0 jBB ej
The Smith chart lies in the complex plane of Γ
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Smith Chart: Refection Coefficient at z
In a lossless transmission line, there is no attenuation and a wave traveling along the line will only have a phase shift. So the reflection coefficient Γ(z) at a point of distance l from the load at the end of the line is related to the load reflection coefficient ΓL by:
It means the reflection coefficient has same magnitude but only a phase shift of 2βz if we move a length z along the line
ZL
z=-l z=0 z
)(zΓ LΓ ( )0 :Note =Γ=Γ zL
zjLzj
o
zjo eeVeVz β
β
β2)( Γ==Γ −+
−
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Smith Chart: Refection Coefficient at z
Reflection coefficient has same magnitude but only a phase shift of 2βl if we move a length l along the line
( Γ rotates clockwise on the Smith Chart when moving away from the load and anti-clockwise when moving towards the load).
Im
Γ(z)
ΓL
Re
2βz
ZL
z=-l z=0 z
)(zΓ LΓ
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The unit circle corresponds to |Γ|=1
** Load impedances can berepresented by points on a Smith chart **
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=Γ−=Γ
OC
SC
Example: Special cases
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Special curves (circles)
Impedance on a Smith chart is represented by normalized impedance:
From this equation, we can obtain expressions for r and x in terms of Γ:
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)(
+−
=+−
=Γ⇒
++
==
zz
ZZZZ
jxrZ
jXRZ
zZz
o
o
oo
( )22
2
22
2
111
11
1
=
−Γ+−Γ
+=Γ+
+−Γ
xx
rrr
ir
ir
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Families of circles
These equations define family of circles on the ( Γr , Γi ) plane corresponding to constant resistance r, and constant reactance x. The reflection coefficient at a point on the line with normalized input impedance z = r+jx is then the intersection point between the constant r and x circles.
Note: The equation
is a circle in the x-y plane with center at (xo,yo) and radius a
( )22
2
22
2
111
11
1
=
−Γ+−Γ
+=Γ+
+−Γ
xx
rrr
ir
ir
( ) ( ) 222 ayyxx oo =−+−
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Examples
For r=1, we have
For r=2, we have
For x=+1, we have
For x=-1, we have
22
2
21
21
=Γ+
−Γ ir
( ) ( ) 111 22 =−Γ+−Γ ir
22
2
31
32
=Γ+
−Γ ir
( ) ( ) 111 22 =+Γ+−Γ ir
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Example
(a)
(b)
(c)
(e)
(g)
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Example
0125.0;01
)(
jzzljz
a
Lin
L
+===+= λ
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115.0;11
)(
jzzljz
b
Lin
L
+===+= λ
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84.076.03.0;11
)(
jzljz
c
in
L
+==−= λ
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66.059.02.1;5.05.0
)(
jzljz
d
in
L
+==−= λ
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73.001.0;0
)(
jzlz
e
in
L
+=== λ
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72.004.0;3
)(
jzljz
f
in
L
+=== λ
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32.002.0;
)(
jzlz
g
in
L
−==∞= λ
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