The Smith Chart
Gilbert Ching
July 31, 2003
Antennas and Propagation Group Seminar
Outline
• History
• Introduction
• Derivation of Smith Chart
• Properties of Smith Chart
• Moving along the Smith Chart
• Matching
• Smith Chart in Network Analyzer
• Other Uses
• Summary and References
History
Before Smith Chart
--- used rectangular chart by J.A. Fleming to graph reflections
--- data went off top and bottom of chart
1936 --- Phillip Smith made circular chart
--- all data inside circular chart
Smith Chart
--- published in Electronics magazine in January of 1939
--- sold about 9 million copies by 1975
Introduction
Smith Chart can be regarded as 2 charts :
Polar plot of voltage reflection coefficient, gamma,
Γload, at the load.
and circular plot of impedance of the load, Zload
Γload = ( Zload – Zo ) / ( Zload + Zo )
Transmission line with
Characteristic impedance, Zo
Load
Zload
Γi
Polar plot
Γr
complex number, Γ = Γr +j Γi = | Γ |<θ
Voltage reflection coefficient at the load, Γload , by definition
= Reflected voltage, Vr / incident or forward voltage, Vf
Γload = Vr / Vf
Vf
Transmission line with
Characteristic impedance, Zo
Load
Vr
VSWR
Another way to characterize mismatch.
The voltage standing wave ratio along a
transmission line ,VSWR = Vmax / Vmin
Vmax = |Vf| + |Vr|
Vmin = |Vf| - |Vr|
VSWR = ( 1 + | Γload| )/ ( 1 - | Γload| )
Derivation of Smith Chart
Γload = ( Zload – Zo ) / ( Zload + Zo )
In general,
Γ = ( Z – Zo ) / ( Z + Zo )
= ( (Z/Zo) – 1 ) / ( (Z/Zo) + 1 )
but Γ = Γr +j Γi
and
(Z/Zo) = z = r +j x
normalized impedance
therefore,
Γr +j Γi = ( r +j x – 1 ) / ( r +j x + 1 )
Derivation of Smith Chart
Smith Chart
Circular plot of impedance, Z
x=1
z = 0.5 +j 1
x =0.5
z = 1 +j 1
short
Pure resistance
x=0
r=0
r = 0.5
r=1
matched
open
z = 1 –j 0.5
Pure capacitance
x = -0.5
x=-1
Admittance
Let z = 1 +j 1
y = 1/z = 0.5 –j 0.5
Equal distance from
center and passes
through center
Impedance Admittance Chart
Blue – impedance chart
Red – admittance chart
Input reflection coefficient
If lossless transmission line,
Zo
+d, towards
source/generator
Along transmission line,
Γin = Vr exp(-jβd)
/ Vf exp(+jβd)
= Γload exp(-2jβd)
Load
d=0
Γload =
( Zload – Zo ) / ( Zload + Zo )
= Vr / Vf
One revolution
Γin = Γload exp(-2jβd)
β is the phase coefficient = 2π / λ
Γin = Γload exp(-j4πd / λ)
= Γload exp(-j4π(d+ λ/2) / λ)
Half wavelength
Moving along the Smith Chart
Γin = Γload exp(-2jβd)
As d increases, the phase value -2βd decreases
Towards source/generator
Zo
+d, towards source/generator
Load
d=0
Moving along the Smith Chart
If move along a lossless transmission line
Zo
Load
+d, towards source/generator
Moving along the Smith Chart
If add series inductance
+jx
Load
Moving along the Smith Chart
If add parallel inductance
-jb
Load
Moving along the Smith Chart
If add parallel inductance
-jb
Load
Matching
Objective is to move one point to another point
usually the center
Yellow point – easy
Green point - ??
Matching
circuit
Load
Lumped Element Matching
Green point - ??
Cap
Ind
Load
Smith Chart in Network Analyzer
• Every point corresponds to a certain frequency
• Shows R +j X and corresponding L or C at that
frequency
• The graph in Smith Chart and Polar plot form are the
same.
Other uses
• Matching using transmission lines
• Can draw constant Q plots
• Stability and Gain circles drawn in Smith Chart to
design amplifiers
• Noise figure circles for Low Noise Amplifiers
Summary
• Smith Chart gives us an idea of how matched a
load is by having a simple graphical relation of
impedance and reflection.
• Conversion of impedance to reflection is through
Γ = ( Z – Zo ) / ( Z + Zo )
Thank You
REFERENCES
http://www.sss-mag.com/smith.html#tutor
http://web.mit.edu/6.013_book/www/chapter14/14.6.html
http://www.ieee-virtual-museum.org/
‘Microwave Engineering’ by David M. Pozar
MIMP – Motorola Impedance Matching Program