Notes 8 Transmission Lines (Bounce Diagram) ECE 3317 Prof. David R. Jackson Spring 2013 Notes 8 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 = t1 RL z = 0 z = L V0 [V] t = 0 + - Rg Z0 t = t1 t = t2 V + (from voltage divider)

Bounce Diagram z Rg t = 0 Z0 RL + 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

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

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!

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 Wavefront is moving to the left

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).

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).

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

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

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 z = 0 RL = 50 [] z = L V0 = 4 [V] t = 0 + - Rg = 225 [] Z0 = 75 [] Z0 = 150 [] T = 1 [ns] 1 2 3 4 -0.4444 [V] 0.0555 [V] -0.3888 [V] 0.2222 [V] 0.4444 [V]

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 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

Example: Pulse (cont.) Subtract W W = 0.25 [ns] 0.25 1 1.25 2 2.25 3 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] W = 0.25 [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.) 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

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

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.

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

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:

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

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.

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

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

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

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.