Prof. David R. Jackson Dept. of ECE Notes 2 ECE Microwave Engineering Fall 2015 Smith Charts 1

Recall: Generalized reflection Coefficient: Generalized Reflection Coefficient 2 I (z) + Z L - V (z) z=0 z

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

Complex  Plane Increasing d (toward generator) d Decreasing d (toward load) Lossless line 4 d = distance from load

Define Hence we have: 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 or X n. This gives two equations. 5

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

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

Short-hand version 8  plane Impedance ( Z ) Chart (cont.)

9  plane

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.) 11  plane

Impedance or Admittance ( Z or Y ) Calculations 12 The Smith chart can be used for either impedance or admittance calculations, as long as we are consistent. The complex plane is either the  plane or the  plane. Normalized impedance or admittance coordinates

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 circles, rotated 180 o, becomes G n circles. X n circles, rotated 180 o, becomes B n circles.

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

Short-hand version  plane 15 Admittance ( Y ) Chart (cont.)

Impedance and Admittance ( ZY ) Chart Short-hand version  plane 16

All Four Possibilities for Smith Charts 17  plane Y Chart  plane Z Chart Y Chart  plane Z Chart  plane The top two are the most common.

The SWR is given by the value of R n on the positive real axis of the Smith chart. Standing Wave Ratio Proof: 18  plane

At this link: Download the following zip file: smith_v191.zip Extract the following files: smith.exemith.hlpsmith.pdf This is the application file Electronic Smith Chart 19