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

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

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

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.

Steady-State Solution Adding all infinite number of bounces (t = ), 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 + - 1 2 3 4 5 6

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:

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

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).

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).

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

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)

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

Bounce Diagram for Cascaded Lines Example (cont.) Bounce Diagram for Cascaded Lines + - Junction 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. + - We thus subtract two bounce diagrams, with the second one being a shifted version of the first one.

Example: Pulse + - Oscilloscope trace

Example: Pulse (cont.) Subtract Oscilloscope trace W 0.25 1 1.25 2 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]

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

Example: Pulse (cont.) + - Snapshot

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

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

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.

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

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:

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

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.

Inductive Load (cont.) Assume 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 = zF The time indicates where the break is. Resistive load, RF > Z0 Resistive load, RF < Z0

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

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