Note 3 Transmission Lines (Bounce Diagram) ECE 3317 Prof. D. R. Wilton Note 3 Transmission Lines (Bounce Diagram)

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

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

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

Steady-State Solution Adding all infinite number of bounces, we have: Note: We have used the geometric series formula

Steady-State Solution Simplifying, we have:

Steady-State Solution 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 just the voltage divider equation!

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

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:

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

Example (cont.) To obtain current bounce diagram from 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).

Example (cont.) Note: We can also just change the signs of the reflection coefficients, as shown. current current 1 2 3 4 5 6 1 2 3 4 5 6 Note: These diagrams are for the normalized current, defined as Z0 I (z,t).

Example (cont.) current oscilloscope trace of current 1 2 3 4 5 6 current oscilloscope trace of current 0.75 [ns] 1.25 [ns] 2.75 [ns] 3.25 [ns] steady state current:

Example (cont.) 1 2 3 4 5 6 current L/4 snapshot of current

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

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

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.

Example: Pulse 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) + - W = 0.25 [ns] V0 = 4 [V] t W

Example: Pulse - W = 0.25 [ns] z = 0.75 L W 0.25 1 1.25 2 2.25 3 3.25 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] -

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

Example: Pulse (cont.) - t = 1.5 [ns] W 1 2 3 4 5 6 1.25 2.25 3.25 4.25 5.25 6.25 0.25 L / 2 W L / 4 -

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

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

Capacitive Load Z0 t = 0 + V0 [V] Z0 CL - z = L z = 0 T 2T 3T t z

Capacitive Load (cont.) Z0 t = 0 + V0 [V] Z0 CL - z = L z = 0 At t = 0: capacitor acts as a short circuit. (valid for t < T) At t = : capacitor acts as an open circuit. Between t = 0 and t = , there is an exponential time-constant behavior. Time-constant formula: Hence we have:

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

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

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

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 + - Z0 (matched source) t V (0, t) t V (0, t) resistive load, RL > Z0 resistive load, RL < Z0

Time-Domain Reflectometer (cont.) z = 0 load z = L V0 [V] t = 0 + - Z0 (matched source) t V (0, t) t V (0, t) capacitive load inductive load

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.