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# Smith Chart

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### Smith Chart

1. 1. Smith Chart Microwave Engineering CHO, Yong Heui
2. 2. Microwave Engineering1. Smith chartPhoto of Smith chart   Graphical method  Essential diagram for microwave engineering P. Smith in 1939 2 EM Wave Lab
3. 3. Microwave Engineering1. Smith chartInduction 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. 4. Microwave Engineering1. Smith chartInduction 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. 5. Microwave Engineering1. Smith chartConstant resistance circles 2 2  r   1   Γr −  + Γi =  2    1+ r  1+ r  5 EM Wave Lab
6. 6. Microwave Engineering1. Smith chartConstant reactance circles 2 2  1 1  ( Γr − 1) +  Γi −  =   2  x x 6 EM Wave Lab
7. 7. Microwave Engineering1. Smith chartInput 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. 8. Microwave Engineering1. Smith chartInput 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. 9. Microwave Engineering1. Smith chartAdmittance 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 ) 9 EM Wave Lab
10. 10. Microwave Engineering2. Applications VNA (Vector Network Analyzer)   Measurement equipment  Reflection coefficients with frequency sweep 10 EM Wave Lab
11. 11. Microwave Engineering2. 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 11 EM Wave Lab