1. Microwave Engineering
Adapted from notes by Prof. Jeffery T. Williams
ECE
Microwave Engineering
Fall 2018
Prof. David R. Jackson
Dept. of ECE
Notes 5
Smith Charts

2. Generalized Reflection Coefficient
Recall:
Generalized reflection Coefficient:

3. Generalized Reflection Coefficient (cont.)
Different forms for (z)
Magnitude property of (z)
Proof:
Lossless transmission line ( = 0)

4. Complex  Plane d Lossless line d = distance from load
Decreasing d (towards load)
Lossless line
d = distance from load
Increasing d
(towards generator) d
Note:
Going /2 on the line corresponds to going all the way around the Smith chart.

5. Impedance (Z) Chart Define Hence we have:
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 Rn or Xn. This gives two equations.

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

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

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

9. Impedance (Z) Chart (cont.)
 plane

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

11. Admittance (Y) Calculations with Smith Chart (cont.)
 plane

12. Impedance or Admittance (Z or Y) Calculations with Smith Chart
Normalized impedance or admittance coordinates
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.

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

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

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

16. All Four Possibilities for Smith Charts
The first two are the most common.
The third is sometimes convenient.
The fourth is almost never used.
Z Chart, impedance
 plane
Z Chart, admittance
 plane 1 2
 plane
Y Chart, admittance
 plane
Y Chart, impedance 3 4

17. Impedance and Admittance (ZY) Chart
Short-hand version
 plane
This is convenient for doing matching problems that involve both series and shunt elements.

18. Standing Wave Ratio  plane Proof:
The SWR is given by the value of Rn on the positive real axis of the Smith chart (Rnmax).
Proof:
 plane

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

20. Example 1
Impedance chart a
 plane

21. Example 1 (cont.) b
Impedance chart
 plane c

22. Example 2
Admittance chart a
 plane

23. Example 2 (cont.) b c
Admittance chart
 plane

24. ( = k0, g =0) to create an impedance of Zin = -j25  at f = 10 GHz.
Example 3
Use a short-circuited section of air-filled TEM, 50  transmission line
( = k0, g =0) to create an impedance of Zin = -j25  at f = 10 GHz.
Impedance chart
 plane

25. Example 4
Use an open-circuited section of 75  (Y0 = 1/75 S) air-filled transmission line at f = 10 GHz to create an admittance of:
Admittance chart
 plane

26. Example 5 Single-stub matching
In this example we will use the “usual” Smith chart (Z chart), but as an admittance calculator.

27. (We’ll use the first choice.)
Example 5 (cont.)
Admittance chart
(We’ll use the first choice.) X X X
 plane
Smith chart scale:
wavelengths toward load
wavelengths toward generator

28. Example 5 (cont.) From the Smith chart: Analytically: Admittance chart
 plane
Analytically: