1. Notes 10 Transmission Lines (Reflection and Impedance)
ECE 3317
Applied Electromagnetic Waves
Prof. David R. Jackson
Fall 2018
Notes Transmission Lines (Reflection and Impedance)

2. Consider a transmission line that is terminated with a load:
Reflection
Consider a transmission line that is terminated with a load: + -
Voltage and current on the line:

3. Reflection (cont.) Important point:
The forward-traveling and backward-traveling wave amplitudes are the amplitudes that describe the two waves in sinusoidal steady-state (after all bounces have died down and we are in steady state). + -
A = amplitude of net forward-traveling wave
B = amplitude of net backward-traveling wave

4. Reflection (cont.) + -
At the load (z = 0):
Hence we have or

5. Reflection Coefficient
We define the load reflection coefficient:
Hence
We then have, form the last slide, or

6. Voltage and Current We can then write:
Note: The generator (source) will determine the unknown (complex) constant A.

7. Impedance We define the input impedance at any point z on the line: +-
The input impedance is the impedance “seen” looking to the right.

8. Impedance (cont.) + - + - Note:
The input impedance does not care what is to the left of the point z. We can remove everything to the left if we wish. (What is to the left of the point z only affects the amplitude level of the voltage and current.)

9. Impedance (cont.)
We then have
Canceling the constant A, we have or

10. Impedance (cont.)
Now let z = - d : + -

11. Impedance (cont.) At the beginning of the line we have:
l = length of line + -

12. Impedance (cont.)
Substituting for the load reflection coefficient, we have:

13. Impedance (cont.)
Rearranging the expression, we have:

14. Impedance (cont.)
Now let z = - d : + -
d = distance from load

15. Impedance (cont.) At the beginning of the line we have:
l = length of line + -

16. These limiting cases agree with circuit theory.
Impedance (cont.)
Limiting cases
1) General (lossy) line:
2) Lossless line:
These limiting cases agree with circuit theory.

17. Impedance (cont.) Limiting cases (cont.) 3) Lossy infinite line:
Proof:

18. Impedance (cont.)
Lossless Case
Use
We then have
where

19. Summary of final formula for a lossless line:
Impedance (cont.)
Summary of final formula for a lossless line:

20. Impedance (cont.) For a lossless line:
The input impedance repeats every one-half wavelength.
The voltage and current repeat every wavelength.
The magnitude of the voltage and current repeat every one-half wavelength.
The voltage and current become their negatives after one-half wavelength.

21. Special Cases of Lossless Line
Impedance (cont.)
Special Cases of Lossless Line
Short-circuit line or

22. Impedance (cont.) Short-circuit line or π/2 3π/2 inductive capacitive
5π/2 
Short circuit
The line is one-half of a wavelength long.

23. Impedance (cont.)
Low frequency:
Short-circuit line

24. Special Cases of Lossless Line (cont.)
Impedance (cont.)
Special Cases of Lossless Line (cont.)
Open-circuit line or

25. Impedance (cont.) Open-circuit line or π 2π 3π inductive capacitive
The line is one-half of a wavelength long.

26. Impedance (cont.)
Low frequency:
Open-circuit line

27. Microstrip filter (application of open-circuited “stubs”)

28. Appendix: Summary of Formulas