The Smith chart is a graphical method that is essential for microwave engineering. It allows microwave engineers to represent normalized impedances on a chart. Developed by Philip Smith in 1939, the chart maps the reflection coefficient Γ which relates the load and source impedances. It uses constant resistance and reactance circles to plot impedance points. Microwave engineers can use the Smith chart and vector network analyzers to measure reflection coefficients over frequency sweeps.

1. Smith Chart
Microwave Engineering
CHO, Yong Heui

2. Microwave Engineering
1. Smith chart
Photo of Smith chart
  Graphical method
 Essential diagram for
microwave engineering
P. Smith in 1939
2 EM Wave Lab

3. Microwave Engineering
1. Smith chart
Induction of Smith chart
 Γ =| Γ | e jφ where | Γ |< 1
Z − Z0
Γ= L
Z L + Z0
Normalized load impedance :
Z L RL + jX L
z L = r + jx = =
Z0 Z0
zL −1 1+ Γ
 Γ=
zL + 1
or z L =
1− Γ
Γ = Γr + jΓi
3 EM Wave Lab

4. Microwave Engineering
1. Smith chart
Induction of Smith chart
1 + Γr + jΓi
z L = r + jx =
1 − Γr − jΓi
1 − Γr2 − Γi2 2Γi
r= x=
(1 − Γr ) 2 + Γi2 (1 − Γr ) 2 + Γi2
 Circle equations
2 2 2 2
 r   1   1 1
 Γr −  + Γi = 
2
 , ( Γr − 1) +  Γi −  =  
2
 1+ r  1+ r   x  x
4 EM Wave Lab

5. Microwave Engineering
1. Smith chart
Constant resistance circles
2 2
 r   1 
 Γr −  + Γi = 
2
 
 1+ r  1+ r 
5 EM Wave Lab

6. Microwave Engineering
1. Smith chart
Constant reactance circles
2 2
 1 1
 ( Γr − 1) +  Γi −  =  
2
 x x
6 EM Wave Lab

7. Microwave Engineering
1. Smith chart
Input impedance
V0+ − jβz
Vs = Vo+ (e − jβz + Γe jβz ) and I s = (e − Γe jβz )
Z0
Z in Vs e − jβz + Γe jβz
zin = = = − jβz
Z0 Z0 I s e − Γe jβz
At z = −l
1 + Γe − j 2 βl 1+ | Γ | e j (φ − 2 βl )
 zin = − j 2 βl
= j (φ − 2 βl )
1 − Γe 1− | Γ | e
7 EM Wave Lab

8. Microwave Engineering
1. Smith chart
Input impedance
 Γ ⇒ Γe − j 2 βl
Clockwise angle of 2 βl on the Smith chart.
 βl = π
one lap around the chart (l = 0.5λ )
8 EM Wave Lab

9. Microwave Engineering
1. Smith chart
Admittance
1
yL =
 zL
 1 − Γe − j 2 βl 1 + Γe − j ( 2 βl +π )
= − j 2 βl
=
1 + Γe 1 − Γe − j ( 2 βl +π )
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10. Microwave Engineering
2. Applications
VNA (Vector Network Analyzer)
  Measurement equipment
 Reflection coefficients
with frequency sweep
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11. Microwave Engineering
2. Applications
Representation of Smith chart
 Z L = 25 + j 50Ω on a 50 - Ω line
z L = 0.5 + j1
r = 0.5 x = 1 ⇒ Point A
| Γ |≈ 0.62, φ ≈ 83
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