Prof. Ji Chen

1. ECE 3317 Prof. Ji Chen Spring 2014 Notes 8 Transmission Lines(Bounce Diagram)

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

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

4. Bounce Diagram z T 2T 3T 4T 5T 6T Rg t = 0 + Z0 RL V0 [V] - z = L z = 0

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 [] V0 = 4 [V] - z = L z = 0 1 2 3 4 5 6 T = 1 [ns]

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.) L/4 The bounce diagram can also be used to get a “snapshot” of the line voltage at any point in time. 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. Current Voltage 1 2 3 4 5 6 Note: This diagram is for the normalized current, defined as Z0I (z,t).

12. Example (cont.) Note: We can also just change the signs of the reflection coefficients, as shown. Current 1 2 3 4 5 6 Note: These diagrams are for the normalized current, defined as Z0I (z,t).

13. Example (cont.) 1 2 3 4 5 6 Current 0.75 [ns] 1.25 [ns] 2.75 [ns] 3.25 [ns] (units are volts) Steady state current:

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

15. Example Rg = 225 [] t = 0 T = 1 [ns] T = 1 [ns] + RL = 50 [] Z0 = 75 [] Z0 = 150 [] V0 = 4 [V] - z = L z = 0 Reflection and Transmission Coefficient at Junction Between Two Lines Junction (since voltage must be continuous across the junction) KVL: TJ = 1 + J

16. Example (cont.) Bounce Diagram for Cascaded Lines Rg = 225 [] t = 0 T = 1 [ns] T = 1 [ns] + RL = 50 [] Z0 = 75 [] Z0 = 150 [] V0 = 4 [V] - z = L z = 0 1 2 3 0.2222 [V] 0.2222 [V] 0.4444 [V] -0.4444 [V] 0.0555 [V] -0.3888 [V] 4

17. Pulse Response Superposition can be used to get the response due to a pulse. Rg RL + + Z0 Vg (t) - - z = 0 z = L 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 [] Vg (t) T = 1 [ns] + - z = L z = 0 z = 0.75 L V0 = 4 [V] W = 0.25 [ns] t W

19. Example: Pulse (cont.) W 0.25 1 1.00 [ns] 1.25 1.50 [ns] 2 2.25 3 3.25 3.00[ns] 3.50[ns] 4 4.25 5 5.00 [ns] 5.25 6 5.50 [ns] 6.25 Subtract W = 0.25 [ns] 0.75 [ns] 1.25 [ns] 2.75 [ns] 3.25 [ns] 4.75 [ns] 5.25 [ns]

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

21. Example: Pulse (cont.) W 1 3L / 4 2 3 4 5 6 Snapshot subtract t = 1.5 [ns] W = 0.25 [ns] 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 [] Z0 = 75 [] Vg (t) - T = 1 [ns] z = L z = 0 Snapshot of voltage Pulse is moving to the left

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

25. Capacitive Load (cont.) Rg = Z0 t = 0 + Z0 CL V0 [V] - z = L z = 0 At t = T: Thecapacitor acts as a short circuit: At t = : Thecapacitor acts as an open circuit: Between t= Tandt =, there is an exponential time-constant behavior. General time-constant formula: Hence we have:

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

27. Inductive Load Rg = Z0 t = 0 + LL Z0 V0 [V] - z = L z = 0 At t = T: inductor as a open circuit: At t = : inductor acts as a short circuit: Between t= Tandt =, there is an exponential time-constant behavior.

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

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

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