1. Notes 7 Transmission Lines (Pulse Propagation and Reflection)
ECE 3317
Applied Electromagnetic Waves
Prof. David R. Jackson
Fall 2018
Notes Transmission Lines (Pulse Propagation and Reflection)

2. Pulse on Transmission Line
A voltage signal is applied at the input of a
semi-infinite transmission line. + -
Example:
Sawtooth wave

3. Pulse on Transmission Line (cont.)+ -
Goal: determine the function f
At z = 0:
Hence

4. Pulse on Transmission Line (cont.)
At any position z, the pulse that is measured is the same as the input pulse, except that it is delayed by a time td = z / cd.

5. Pulse on Transmission Line (cont.)
Note the delay in the trace on this oscilloscope.
Here is what oscilloscopes will show.

6. Pulse on Transmission Line (cont.)
“Snapshot” of voltage wave at one fixed time. + -
Pulse
Note that the shape of the pulse as a function of z is a scaled mirror image of the pulse shape as a function of t.

7. Pulse on Transmission Line (cont.)
This wave moves past the oscilloscope to create the trace.
(The oscilloscope trace is the mirror image of the snapshot shape.) + -
Pulse

8. Pulse on Transmission Line (cont.)
The pulse is shown emerging from the source end of the line.
A series of “snapshots” is shown. + -

9. Another example (battery and switch)
Step Function Source
Another example (battery and switch) + -
Unit step function u(t)

10. Step Function Source (cont.)
The step function is shown propagating down the line. + -
u(t) = unit step function

11. Step Function Source (cont.)
Steady-state solution (t = )
The steady-state voltage is just the battery voltage. + -

12. At z = L, the two right-hand side terms are the same since RL = Z0.
Matched Load + -
On line:
At load:
At z = L, the two right-hand side terms are the same since RL = Z0.
Hence the total voltage at the load is the same as the incident voltage. There is thus no reflection. When the waveform hits the load, it “sees” a continuation of the line.

13. This shows the sawtooth waveform propagating on a matched line.
Absorption by Load
This shows the sawtooth waveform propagating on a matched line. - +
The pulse is shown emerging from the source end of the line, traveling down the line, and then being absorbed by the matched load.

14. Absorption by Load (cont.)
This shows the step function waveform propagating on a matched line. + -
Time to reach the load end:
For t > T we have reached steady state: V(z,t) = V0 everywhere on the line.

15. Load Reflection
On line:
where
(known function)
At load:

16. Load Reflection (cont.)
At load:
Hence we have:

17. Load Reflection (cont.)

18. Load Reflection (cont.)
Define
voltage of forward-traveling wave at the load
voltage of backward-traveling wave at the load
Then
Define
load reflection coefficient
We then have

19. Load Reflection (cont.)
Summary for load reflection
load reflection coefficient
Note:
There is no reflection for a matched system (RL = Z0).
Note:

20. Load Reflection (cont.)
We now proceed to obtain
(for arbitrary z)
At the load:
This is a known function:
Use
Hence we have
Hence
The reflected wave is delayed from the load due to the distance L - z.

21. Reflection Picture Also, we have Hence Reflected wave Incident wave
Notes:
The reflected wave is the mirror image of the incident wave, reduced in amplitude.
The reflected wave is delayed from the load after traveling a distance L-z.
Also, we have
Hence

22. Reflection Picture (cont.)
Incident wave
Reflected wave
Physical interpretation:
Time it takes to go from load to observation point z
Time it takes to reach load

23. Reflection Picture (cont.)
Including Thévenin Resistance
Voltage divider:
Incident wave
Reflected wave
Incident wave:
Reflected wave:

24. Reflections at Source End
After the reflected wave hits the source end, there may be another reflection.
Here we allow the source to have a Thévenin resistance Rg.
Reflected wave
Re-reflected wave
Note:
In calculating the reflection from the source, we turn off the source voltage.

25. We'll stop here…but we could do more reflections.
Complete Wave Picture
Reflected wave
Incident wave
Incident:
Voltage divider:
Reflected:
Re-reflected:
Re-re-reflected:
We'll stop here…but we could do more reflections.

26. Comments
The higher-order reflected waves get smaller, due to the reflection coefficients.
The bounce diagram (discussed in the next set of notes) gives us a convenient way or tracking all of the waves and determining the waveform observed at any point on the line, when the generator voltage is a step function or a pulse.

27. Example
Incident wave
Reflected wave

28. Example (cont.) Z0 = 75 [], r = 2.1 Incident wave Reflected wave
Re-reflected:
Re-re-reflected:

29. Example (cont.) Z0 = 75 [], r = 2.1 Total voltage: Incident wave
Reflected wave
Total voltage:
L = 100 [m]