Last month's letter dealt with the motion of charge carriers in a metallic conductor. This note adds more cool animations showing the behavior of those particles at an open-circuited transmission-line endpoint.
Figure 1 illlustrates the circuit from my previous article, Charge in Motion, with one important difference. Previously, the circuit included a resistive termination at the right-hand side of the transmission line. This time, the end of the line leads to nowhere. The endpoint is left open. That's called an "open-circuit" boundary condition.
As before, the pink dots represent mobile charge carriers dispersed evenly throughout an invisible metallic lattice. The figure assumes positive charge carriers, even though we know electrons are really negative, because that assumption aligns the direction of carrier drift with the direction of current flow. Further assume that there is no thermal agitation, and that the characteristic impedance for signals traveling along the signal and return wires equals fifty ohms.
When you close the switch, a wave of compression travels, just as before, along the signal wire from the battery heading from left to right, approaching the open endpoint. Coincident with that activity, the battery draws in charge from the return wire, creating a relative scarcity of pink dots on that wire.
Figure 2 illustrates the situation a few hundred picoseconds after closing the switch. At that moment in time, a compression wave has traveled partway down the line, but not yet reached the end. In that condition, the battery has no knowledge of what lies ahead. It simply supplies a constant electrostatic pressure, causing the mobile charge carriers on the signal wire to move and forming the rapidly-advancing traveling wave front.
The battery (driver) in this example puts out only a half-volt step, in contrast to the full-volt step I used in the previous article. I've done that because I anticipate something happening that would, were I to use a full-sized step, cause the waveform to shoot offscreen. You will see that shortly.
Because the battery voltage is only half the amount used previously, the degree of charge carrier compression is only half as great as before. Figure 3 shows the amount of carrier compression experienced by the signal wire under conditions of full, half, or no voltage.
On the return wire, the paucity of charge carriers reacts in a manner opposite the signal wire. The larger the applied voltage, the further apart the carriers spread. The effect is subtle. Look closely on the left side of Figure 2 at the carrier spacing on the bottom wire to see the effect. To the left of the wavefront the charges are in motion—signal charges drifting to the right and return-wire charges drifting to the left. Ahead of the wavefront, nothing has yet happened; the charges remain stationary at their nominal spacing.
Consider what must happen when the wavefront slams into the end of the wires. At that instant, you have a column of compressed charges on the top wire moving to the right, carrying with them a certain degree of momentum. When that column encounters the bitter end of the signal wire it can no longer drift to the right—there is nowhere left to go!
The charges at the right end of the wire will have to stop, but the entire column cannot stop instantaneously. There is no way to notify the battery, without violating speed-of-light restrictions, that anything untoward has occurred. The battery will continue pouring charges into the line, piling them up at the end, for a considerable period.
The following full-motion animations show that piling-up process in great detail.
In the open-endpoint simulation, when the wavefront reaches the far (right-hand) end of the line, the charges at the end accumulate to a level twice as high as the incoming signal. That is the precise level of compression required to arrest the momentum of further incoming particles. Does that make sense to you? We used a voltage of 1/2 volt to get the particles moving, so it takes an additional 1/2 volt, working in the opposite direction, to stop them.
This is the so-called "doubling effect" that happens at the end of an open-circuited transmission line. It is closely related to the Water Hammer effect.
Dr. Howard Johnson