1. Notes 11 Transmission Lines
ECE 3317
Applied Electromagnetic Waves
Prof. David R. Jackson
Fall 2018
Notes Transmission Lines
(Standing Wave Ratio (SWR) and Generalized Reflection Coefficient)

2. Consider a lossless transmission line that is terminated with a load:
Standing Wave Ratio
Consider a lossless transmission line that is terminated with a load: + -

3. Standing Wave Ratio (cont.)
Consider a lossless transmission line that is terminated with a load: + -

4. Standing Wave Ratio (cont.)
Denote
Then we have
The magnitude is
Maximum voltage:
Maximum voltage:

5. Standing Wave Ratio (cont.)
The voltage standing wave ratio is the ratio of Vmax to Vmin .
We then have:
Perfect match: L = 0

6. Standing Wave Ratio (cont.)
For the current we have
Hence we have:
The current standing wave ratio is thus
Hence

7. Note: V+ is the net wave going in the +z direction.
Standing Wave Pattern
Note: V+ is the net wave going in the +z direction.

8. Standing Wave Ratio: Real Load
Special case of a real load impedance
Case a:

9. Standing Wave Ratio: Real Load (cont.)
Hence
Case b:
Hence

10. Standing Wave Ratio: Real Load (cont.)
Hence, for a real load impedance we have:

11. Example (6.6, Shen and Kong)
Given:
Find: + -

12. Example (6.6, Shen and Kong) (cont.)

13. Example (6.6, Shen and Kong) (cont.)
This problem has practical significance: often we are interested in figuring out what an unknown load is.
Reverse problem:
Given:
What is the unknown load impedance? so
(Any multiple of 2 can be added to .)
Solve for .

14. Example (6.6, Shen and Kong) (cont.)
Hence, we have
The calculation yields:

15. Generalized Reflection Coefficient
Define the “generalized reflection coefficient” at a point z0 on the line: + -

16. Generalized Reflection Coefficient
Rearranging, we have:
Solve for L
We can then write
where

17. Generalized Reflection Coefficient (cont.)
where + -
We identify (z0) as the reflection coefficient at the point z0, with Zin acting as the load impedance.
Hence

18. Generalized Reflection Coefficient (cont.)
Hence

19. Generalized Reflection Coefficient (cont.)
Define a normalized input impedance at point z0:
We then have
Note:
This can be used as an alternative formula to the tangent formula for calculating input impedance at z0 = -d.
(This is the starting point for the Smith chart discussion.)

20. Crank Diagram where Note:1
Moving from load
(angle change 2 z)
Note: We go all the way around the crank diagram when z changes by  / 2.

21. Example
Given:
Calculate the reflection coefficient and the input impedance at z0 =  so so

22. Appendix: Summary of Formulas