1. Notes 8 Transmission Lines (Bounce Diagram)
ECE 3317
Applied Electromagnetic Waves
Prof. David R. Jackson
Fall 2018
Notes Transmission Lines (Bounce Diagram)

2. Step Response
The concept of the bounce diagram is illustrated for a step response on a terminated line: + -
Generator voltage

3. Step Response (cont.) The wave is shown approaching the load.+ -
(from voltage divider)

4. The purple values give the total voltage in each region.
Bounce Diagram T 2T 3T 4T 5T 6T
The purple values give the total voltage in each region.

5. Steady-State Solution
Adding all infinite number of bounces (t = ), we have:
Note: We have used

6. Steady-State Solution (cont.)
Simplifying, we have:

7. Steady-State Solution (cont.)
Continuing with the simplification:
Hence we finally have:
Note:
The steady-state solution does not depend on the transmission line length or characteristic impedance!
This is the DC circuit-theory voltage divider equation!

8. Example + - 1 2 3 4 5 6

9. Example (cont.)
The bounce diagram can be used to get an “oscilloscope trace” of the voltage at any point on the line.
0.75 [ns]
1.25 [ns]
2.75 [ns]
3.25 [ns]
Steady state voltage:

10. Example (cont.)
The bounce diagram can also be used to get a “snapshot” of the line voltage
at any point in time.
L/4
Wavefront is moving to the left

11. This diagram is for the normalized current, defined as Z0 i (z,t).
Example (cont.)
To obtain a current bounce diagram from the voltage diagram, multiply forward-traveling voltages by 1/Z0, backward-traveling voltages by -1/Z0.
Voltage
Normalized Current 1 2 3 4 5 6
Note:
This diagram is for the normalized current, defined as Z0 i (z,t).

12. Example (cont.)
Note: We can also just change the signs of the reflection coefficients, as shown. 1 2 3 4 5 6
Normalized Current
Note: This diagram is for the normalized current, defined as Z0 i (z,t).

13. Example (cont.) Normalized Current Steady state current: 0.75 [ns] 12 3 4 5 6
Normalized Current
2.75 [ns]
3.25 [ns]
0.75 [ns]
1.25 [ns]
(units are volts)
Steady state current:

14. Wavefront is moving to the left
Example (cont.) 1 2 3 4 5 6
Normalized Current
L/4
Wavefront is moving to the left
(units are volts)

15. Reflection and Transmission Coefficient at Junction Between Two Lines
Example
Reflection and Transmission Coefficient at Junction Between Two Lines + -
Junction
(This follows from the fact that voltage must be continuous across the junction.)
KVL: TJ = 1 + J

16. Bounce Diagram for Cascaded Lines
Example (cont.)
Bounce Diagram for Cascaded Lines + -
Junction 1 2 3 4
[V]
[V]
[V]
[V]
[V]

17. Pulse Response
Superposition can be used to get the response due to a pulse. + -
We thus subtract two bounce diagrams, with the second one being a shifted version of the first one.

18. Example: Pulse + -
Oscilloscope trace

19. Example: Pulse (cont.) Subtract Oscilloscope trace W 0.25 1 1.25 23 4 5 6
0.75 [ns]
1.25 [ns]
2.75 [ns]
3.25 [ns]
4.75 [ns]
5.25 [ns]
1.25
2.25
3.25
4.25
5.25
6.25 W
0.25
1.00 [ns]
1.50 [ns]
3.00[ns]
3.50[ns]
5.00 [ns]
5.50 [ns]

20. Oscilloscope trace of voltage
Example: Pulse (cont.) + -
Oscilloscope trace
Oscilloscope trace of voltage

21. Example: Pulse (cont.) + -
Snapshot

22. Example: Pulse (cont.) subtract Snapshot W W = 0.25 [ns] 1 2 3 4 5 6
1.25
L / 2
2.25
3.25
4.25
5.25
6.25

23. Pulse is moving to the left
Example: Pulse (cont.) + -
Snapshot
Snapshot of voltage
Pulse is moving to the left

24. The reflection coefficient is now a function of time.
Capacitive Load + -
Note: The generator is assumed to be matched to the transmission line for convenience (we wish to focus on the effects of the capacitive load).
Hence
The reflection coefficient is now a function of time.

25. Capacitive Load (cont.)+ - T 2T 3T t z

26. Capacitive Load (cont.)+ -
At t = T : The capacitor acts as a short circuit:
At t = : The capacitor acts as an open circuit:
Between t = T and t = , there is an exponential time-constant behavior.
General time-constant formula:
Hence we have:

27. Capacitive Load (cont.)
Assume z = 0 + -
steady-state T 2T 3T t z

28. Inductive Load At t = T: inductor as a open circuit:+ -
At t = T: inductor as a open circuit:
At t = : inductor acts as a short circuit:
Between t = T and t = , there is an exponential time-constant behavior.

29. Inductive Load (cont.)
Assume z = 0 + -
steady-state T 2T 3T t z

30. Time-Domain Reflectometer (TDR)
This is a device that is used to look at reflections on a line, to look for potential problems such as breaks on the line. + -
z = zF
The time indicates where the break is.
Resistive load, RF > Z0
Resistive load, RF < Z0

31. Time-Domain Reflectometer (cont.)
The reflectometer can also tell us what kind of a load we have. + -
Resistive load, RL > Z0
Resistive load, RL < Z0

32. Time-Domain Reflectometer (cont.)
The reflectometer can also tell us what kind of a load we have. + -
Capacitive load
Inductive load

33. Time-Domain Reflectometer (cont.)
Example of a commercial product
“The 20/20 Step Time Domain Reflectometer (TDR) was designed to provide the clearest picture of coaxial or twisted pair cable lengths and to pin-point cable faults.”
AEA Technology, Inc.