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

Recall, Generalized reflection Coefficient: Generalized Reflection Coefficient 2

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

At this link 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