Notes 7 Transmission Lines (Pulse Propagation and Reflection) ECE 3317 Applied Electromagnetic Waves Prof. David R. Jackson Fall 2018 Notes 7 Transmission Lines (Pulse Propagation and Reflection)

Pulse on Transmission Line A voltage signal is applied at the input of a semi-infinite transmission line. + - Example: Sawtooth wave

Pulse on Transmission Line (cont.) + - Goal: determine the function f At z = 0: Hence

Pulse on Transmission Line (cont.) At any position z, the pulse that is measured is the same as the input pulse, except that it is delayed by a time td = z / cd.

Pulse on Transmission Line (cont.) Note the delay in the trace on this oscilloscope. Here is what oscilloscopes will show.

Pulse on Transmission Line (cont.) “Snapshot” of voltage wave at one fixed time. + - Pulse Note that the shape of the pulse as a function of z is a scaled mirror image of the pulse shape as a function of t.

Pulse on Transmission Line (cont.) This wave moves past the oscilloscope to create the trace. (The oscilloscope trace is the mirror image of the snapshot shape.) + - Pulse

Pulse on Transmission Line (cont.) The pulse is shown emerging from the source end of the line. A series of “snapshots” is shown. + -

Another example (battery and switch) Step Function Source Another example (battery and switch) + - Unit step function u(t)

Step Function Source (cont.) The step function is shown propagating down the line. + - u(t) = unit step function

Step Function Source (cont.) Steady-state solution (t = ) The steady-state voltage is just the battery voltage. + -

At z = L, the two right-hand side terms are the same since RL = Z0. Matched Load + - On line: At load: At z = L, the two right-hand side terms are the same since RL = Z0. Hence the total voltage at the load is the same as the incident voltage. There is thus no reflection. When the waveform hits the load, it “sees” a continuation of the line.

This shows the sawtooth waveform propagating on a matched line. Absorption by Load This shows the sawtooth waveform propagating on a matched line. - + The pulse is shown emerging from the source end of the line, traveling down the line, and then being absorbed by the matched load.

Absorption by Load (cont.) This shows the step function waveform propagating on a matched line. + - Time to reach the load end: For t > T we have reached steady state: V(z,t) = V0 everywhere on the line.

Load Reflection On line: where (known function) At load:

Load Reflection (cont.) At load: Hence we have:

Load Reflection (cont.)

Load Reflection (cont.) Define voltage of forward-traveling wave at the load voltage of backward-traveling wave at the load Then Define load reflection coefficient We then have

Load Reflection (cont.) Summary for load reflection load reflection coefficient Note: There is no reflection for a matched system (RL = Z0). Note:

Load Reflection (cont.) We now proceed to obtain (for arbitrary z) At the load: This is a known function: Use Hence we have Hence The reflected wave is delayed from the load due to the distance L - z.

Reflection Picture Also, we have Hence Reflected wave Incident wave Notes: The reflected wave is the mirror image of the incident wave, reduced in amplitude. The reflected wave is delayed from the load after traveling a distance L-z. Also, we have Hence

Reflection Picture (cont.) Incident wave Reflected wave Physical interpretation: Time it takes to go from load to observation point z Time it takes to reach load

Reflection Picture (cont.) Including Thévenin Resistance Voltage divider: Incident wave Reflected wave Incident wave: Reflected wave:

Reflections at Source End After the reflected wave hits the source end, there may be another reflection. Here we allow the source to have a Thévenin resistance Rg. Reflected wave Re-reflected wave Note: In calculating the reflection from the source, we turn off the source voltage.

We'll stop here…but we could do more reflections. Complete Wave Picture Reflected wave Incident wave Incident: Voltage divider: Reflected: Re-reflected: Re-re-reflected: We'll stop here…but we could do more reflections.

Comments The higher-order reflected waves get smaller, due to the reflection coefficients. The bounce diagram (discussed in the next set of notes) gives us a convenient way or tracking all of the waves and determining the waveform observed at any point on the line, when the generator voltage is a step function or a pulse.

Example Incident wave Reflected wave

Example (cont.) Z0 = 75 [], r = 2.1 Incident wave Reflected wave Re-reflected: Re-re-reflected:

Example (cont.) Z0 = 75 [], r = 2.1 Total voltage: Incident wave Reflected wave Total voltage: L = 100 [m]