1. Notes 8 Transmission Lines (Bounce Diagram)
ECE 3317
Prof. David R. Jackson
Spring 2013
Notes Transmission Lines (Bounce Diagram)

2. Step Response
The concept of the bounce diagram is illustrated for a unit step response on a terminated line. RL
z = 0
z = L
V0 [V]
t = 0 + - Rg Z0 t

3. Step Response (cont.) The wave is shown approaching the load. t = t1RL
z = 0
z = L
V0 [V]
t = 0 + - Rg Z0
t = t1
t = t2
V +
(from voltage divider)

4. Bounce Diagram z Rg t = 0 Z0 RL + V0 [V] - z = L z = 0 T 2T 3T 4T 5T

5. Steady-State Solution
Adding all infinite number of bounces, 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 Rg = 225 [] t = 0 RL = 25 [] Z0 = 75 [] T = 1 [ns] +
z = L
V0 = 4 [V]
t = 0 + -
Rg = 225 []
Z0 = 75 []
T = 1 [ns] 1 2 3 4 5 6

9. Example (cont.)
The bounce diagram can be used to get an “oscilloscope trace”
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. 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
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
Current
Note: These diagrams are for the normalized current, defined as Z0 I (z,t).

13. Example (cont.) Current Steady state current: 0.75 [ns] 1 1.25 [ns] 23 4 5 6
0.75 [ns]
1.25 [ns]
2.75 [ns]
3.25 [ns]
(units are volts)
Steady state current:

14. Example (cont.) Current 1 2 Wavefront is moving to the left 3 L/4 45 6
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
z = 0
RL = 50 []
z = L
V0 = 4 [V]
t = 0 + -
Rg = 225 []
Z0 = 75 []
Z0 = 150 []
T = 1 [ns]
(since voltage must be continuous across the junction)
KVL: TJ = 1 + J

16. Bounce Diagram for Cascaded Lines
Example (cont.)
Bounce Diagram for Cascaded Lines
z = 0
RL = 50 []
z = L
V0 = 4 [V]
t = 0 + -
Rg = 225 []
Z0 = 75 []
Z0 = 150 []
T = 1 [ns] 1 2 3 4
[V]
[V]
[V]
[V]
[V]

17. Pulse Response
Superposition can be used to get the response due to a pulse. RL
z = 0
z = L
Vg (t) + - Rg Z0 t W
We thus subtract two bounce diagrams, with the second one being a shifted version of the first one.

18. Example: Pulse Oscilloscope trace Rg = 225 [] RL = 25 [] Z0 = 75 []
z = 0.75 L
z = 0
z = L
Rg = 225 []
Z0 = 75 []
T = 1 [ns]
Vg (t) + -
W = 0.25 [ns]
V0 = 4 [V] t W

19. Example: Pulse (cont.) Subtract W W = 0.25 [ns] 0.25 1 1.25 2 2.25 34 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]
W = 0.25 [ns]

20. Example: Pulse (cont.) Rg = 225 [] RL = 25 [] Z0 = 75 [] T = 1 [ns]
z = 0.75 L
z = 0
z = L
Rg = 225 []
Z0 = 75 []
T = 1 [ns]
Vg (t) + -
Oscilloscope trace of voltage

21. Example: Pulse (cont.) subtract Snapshot t = 1.5 [ns] W W = 0.25 [ns]3 4 5 6 W
3L / 4
0.25
1.25
L / 2
2.25
3.25
4.25
5.25
6.25

22. Example: Pulse (cont.) t = 1.5 [ns] Rg = 225 [] RL = 25 []
z = 0
z = L
Rg = 225 []
Z0 = 75 []
T = 1 [ns]
Vg (t) + -
Snapshot of voltage
Pulse is moving to the left

23. Capacitive Load Rg = Z0 t = 0 C Z0
z = L
V0 [V]
t = 0 + -
Rg = Z0 Z0
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.

24. Capacitive Load (cont.)CL
z = 0
z = L
V0 [V]
t = 0 + - Z0
Rg = Z0 z T 2T 3T t z

25. Capacitive Load (cont.)CL
z = 0
z = L
V0 [V]
t = 0 + -
Rg = Z0 Z0
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:

26. Capacitive Load (cont.)
Oscilloscope trace CL
z = 0
z = L
V0 [V]
t = 0 + -
Rg = Z0 Z0
Assume z = 0 +
V(0,t) - t
V(0,t) T 2T
V0 / 2 V0
steady-state T 2T 3T t z

27. Inductive Load Rg = Z0 t = 0 Z0 LL
V0 [V]
t = 0 + -
Rg = Z0 Z0
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.

28. Inductive Load (cont.) Rg = Z0 t = 0 Z0 LL V(0,t) z t V(0,t) t
V0 [V]
t = 0 + -
Rg = Z0 Z0
Assume z = 0 +
V(0,t) - t
V(0,t) T 2T
V0 / 2 V0
steady-state T 2T 3T t z

29. 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 = 0
Load
z = L
V0 [V]
t = 0 + -
Rg = Z0 Z0
Fault
The fault is modeled as a load resistor at z = zF.
z = zF t
V (0, t) t
V (0, t)
The time indicates where the break is.
resistive load, RL > Z0
resistive load, RL < Z0

30. Time-Domain Reflectometer (cont.)
The reflectometer can also tell us what kind of a load we have.
z = 0
Load
z = L
V0 [V]
t = 0 + - Z0
(matched source) t
V (0, t) t
V (0, t)
Capacitive load
Inductive load

31. 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.