Transients on Transmission Lines

The phasor analysis of Chapters 24 and 25 answers "what does the line do to a continuous sine wave?" It says nothing about the moment that wave is switched on. For a step or a pulse — a digital bit, a surge, the leading edge of any signal — we need the transient response in the time domain. The key physical fact is that information travels at finite speed \(u\), so a line of length \(\ell\) has a one-way transit time:

Until a full round trip \(2T\) has elapsed, the source has received no news from the load. We restrict attention to lossless lines with purely resistive terminations \(R_g\) and \(R_L\), driven by a DC source switched on at \(t=0\) — the canonical transient problem.

At the instant the switch closes, the source looks into the line and sees… not \(R_L\), which is a transit time away and still unknown, but the line's own characteristic impedance \(Z_0\). The line accepts current exactly as a resistor \(Z_0\) would. So the launched voltage is a simple resistive divider between \(R_g\) and \(Z_0\):

This step propagates toward the load at speed \(u\), arriving at \(t=T\). Behind its leading edge the line is charged to \(V_1^{+}\); ahead of it the line is still quiet. This is the seed from which the entire transient grows.

When the step reaches the load at \(t=T\), the load demands its own ratio of voltage to current, \(R_L\) — which \(V_1^{+}\) and \(I_1^{+}\) generally do not satisfy. The line resolves the mismatch by sending back a reflected wave, scaled by the familiar load reflection coefficient. That reflected wave reaches the source at \(t=2T\), where \(R_g\) imposes its ratio, so a second reflection is launched, scaled by the source reflection coefficient:

From then on the pattern is mechanical. Every bounce off the load multiplies the travelling wave by \(\Gamma_L\); every bounce off the source multiplies it by \(\Gamma_g\). One complete round trip — load then source — therefore scales the wave by the product \(\Gamma_g\Gamma_L\), and the successive waves form a geometric sequence \(V_1^{+},\,\Gamma_g\Gamma_L V_1^{+},\,(\Gamma_g\Gamma_L)^2 V_1^{+},\dots\) Since both coefficients have magnitude \(\le1\) for resistive terminations, the bounces shrink and the line settles.

The bookkeeping is made visual by the bounce diagram (or lattice diagram): position runs across, time runs down, and each travelling wave is a diagonal line crossing the line in one transit time \(T\). At each end the diagonal reflects, picking up a factor of \(\Gamma_L\) or \(\Gamma_g\). To find the voltage at any place and time, draw a vertical line at that position and add up every wave whose diagonal it has already crossed.

At the load, the voltage updates every time a new wave arrives — once per round trip, at \(t=T, 3T, 5T,\dots\) Each arrival adds a forward wave and its instantly-reflected partner, a contribution \((1+\Gamma_L)\) times the incoming wave. The result is a staircase that steps toward the final value, overshooting and correcting if the steps alternate in sign:

Summing the geometric series of all contributions gives a closed form, and it collapses — reassuringly — to the plain DC voltage divider, exactly what circuit theory predicts once the wave nature has died away:

The behaviour ranges from a single clean step to endless ringing, depending on the terminations. Three cases are worth committing to memory:

This is the whole rationale for terminating high-speed digital lines. An unterminated trace (\(R_L\) large, \(R_g\) small) rings violently as edges reflect back and forth, producing overshoot and false logic transitions. Adding a series resistor at the driver to make \(R_g=Z_0\), or a parallel resistor at the receiver to make \(R_L=Z_0\), kills the reflections and lets the signal settle in one or two transit times. Signal integrity at gigahertz edge rates is, at bottom, this chapter applied to circuit-board traces.

Run the logic backwards and the transient becomes a measuring instrument. Send a fast step down an unknown cable and watch the echoes: every impedance discontinuity — a break, a crush, a bad connector, a branch — sends a reflection back, arriving after a round-trip delay that pinpoints its distance. This is time-domain reflectometry (TDR):

The sign of the echo even tells you the fault type — a positive reflection means a higher impedance (an open or break), a negative one means a lower impedance (a short or crush) — directly from \(\Gamma=(Z_{\text{fault}}-Z_0)/(Z_{\text{fault}}+Z_0)\). Cable technicians, board designers, and chip engineers all use TDR to find discontinuities they can never see.

For each item, compute \(V_1^{+}\), \(\Gamma_L\), and \(\Gamma_g\) first — they generate the whole bounce sequence. Remember the load voltage updates every \(2T\) starting at \(t=T\). Difficulty rises down the list.