1. Prof. David R. Jackson Dept. of ECE Notes 2 ECE 5317-6351 Microwave Engineering Fall 2011 Smith Charts 1

2. Recall, Generalized reflection Coefficient: Generalized Reflection Coefficient 2

3. Lossless transmission line (  = 0 ) Generalized Reflection Coefficient (cont.) For Proof: 3

4. Complex  Plane Increasing l ( toward generator) l Decreasing l ( toward load) Lossless line 4

5. Define Substitute into above expression for Z n (- l ): Impedance ( Z ) Chart Next, multiply both sides by the RHS denominator term and equate real and imaginary parts. Then solve the resulting equations for  R and  I in terms of R n and X n. This gives two equations. 5

6. 1) Equation #1: Equation for a circle in the  plane Impedance ( Z ) Chart (cont.) 6 RR II

7. Equation for a circle in the  plane: Impedance ( Z ) Chart (cont.) 2) Equation #2: 7 RR II

8. Impedance ( Z ) Chart (cont.) Short-hand version R n = 1 X n = 1 X n = -1 8

9. Impedance ( Z ) Chart (cont.)  plane R n = 1 X n = 1 X n = -1 9

10. Note: Admittance ( Y ) Calculations Define: Same mathematical form as for Z n : Conclusion: The same Smith chart can be used as an admittance calculator. 10

11. Admittance ( Y ) Calculations (cont.)  plane B n = 1 B n = -1 G n = 1 11

12. Impedance or Admittance ( Z or Y ) Calculations The Smith chart can be used for either impedance or admittance calculations, as long as we are consistent. 12

13. As an alternative, we can continue to use the original  plane, and add admittance curves to the chart. Admittance ( Y ) Chart Compare with previous Smith chart derivation, which started with this equation: If ( R n X n ) = (a, b ) is some point on the Smith chart corresponding to  =  0, Then ( G n B n ) = (a, b ) corresponds to a point located at  = -  0 (180 o rotation). Side note: A 180 o rotation on a Smith chart makes a normalized impedance become its reciprocal. 13  R n = a circle, rotated 180 o, becomes G n = a circle. and X n = b circle, rotated 180 o, becomes B n = b circle.

14. Admittance ( Y ) Chart (cont.)  plane 14

15. Short-hand version G n = 1 B n = -1 B n = 1  plane 15 Admittance ( Y ) Chart (cont.)

16. Impedance and Admittance (ZY) Chart Short-hand version G n = 1 R n = 1 X n = 1 X n = -1 B n = -1 B n = 1  plane 16

17. The SWR is given by the value of R n on the positive real axis of the Smith chart. Standing Wave Ratio Proof: 17

18. At this link http://www.sss-mag.com/topten5.html Download the following.zip file smith_v191.zip Extract the following files smith.exesmith.hlpsmith.pdf This is the application file Electronic Smith Chart 18