## A Transmission Line is Always a Transmission Line

Toss one end of a stout rope to a circus strongman. Then back up, pulling the rope taut as you go. When you are standing about 50 ft apart, flick the rope with a rapid up-and-down motion. If the man at the other end holds the line taut, you will observe a familiar pattern of wave propagation. Your up-and-down stroke first passes quickly from you to the strongman. At his end, the waveform reflects, sending an inverted copy of the original pulse back towards you. One roundtrip delay after the initial flick, you feel the echo of your (attenuated and inverted) original excitation. Then, the residual signal bounces back and forth many times with an exponentially decaying amplitude.

This simple physical analogy reveals much about the behavior of pc-board transmission lines. It shows propagation of the input signal, reflection at the far end, and residual ringing.

It also reveals a temporal disconnection between the ends of a long transmission line. In the example, your strongman stands so far away that the propagation delay across the taut rope easily exceeds the rise-and-fall time of your input signal. Under these conditions, when you first flick the rope, you feel only the mass and tautness of the rope, not the strongman.

Your interaction with the strongman proceeds in three stages. First, you interact with the rope. Then, the rope conveys your inputs of force and velocity to the load. Finally, your signal (now delayed and possibly attenuated) interacts with the load. This sequence corresponds precisely to the behavior of an electrical source, a transmission line, and its load-provided that the delay of the line exceeds the rise (or fall) time of the source.

To express the temporal disconnection in more interesting terms, suppose I drape a black curtain halfway between you and the strongman. With the curtain in place, as long as I don't change the tautness of the rope, you can't tell whether the rope is anchored to a man, a block of wood, or another long section of rope. Obviously, you can infer from the size and timing of the echo the characteristics of the far-end load, but before the echo returns, in the first moment after you create an outgoing waveform, you feel only the mass and tautness of the rope, not the anchor.

Electrical transmission lines exhibit precisely the same effect. The input impedance of a long transmission line, in the brief interval of time before the echo returns, depends only on the characteristics of the line itself, not the load.

"But what," asks a student, "about a short transmission line? In that case, doesn't the driver see the load directly? Does the input impedance thus behave one way on a long transmission line but differently when the load is adjacent to the driver? How does it know what to do?"

To answer this question, I want you to walk over to your strongman and clench the rope right next to his hands. Pull hard. What you feel now is the strength of his grip, not the rope. At a short distance, no matter what kind of rope you use, thick or thin, the same result applies: You feel the strongman, not the rope.

Keep in mind that in both cases, the rope remains a rope. It doesn't suddenly change character. It still conveys your forces to the strongman, only it does so with such speed that the returning signal influences you instantaneously. Before you even begin to create part of a rising edge, the returning (and opposing) force holds the rope back down. The instantaneously returning forces, in contrast to the temporally disconnected reflections of the previous case, are responsible for the change in behavior.

Similarly, in the world of high-speed digital design, a pc trace of any length always remains a transmission line. It supports two modes of propagation, going out to the load and back. When the line is short, these two modes of propagation still exist, only their temporal superposition creates the illusion of a direct connection between source and load.