High-Speed Digital Design Online Newsletter: Vol. 14 Issue 04
This is the third (and last) issue concerning the motion of charge carriers in a metallic conductor. The programming effort required to move the little dots in the correct patterns was extraordinary—I'm glad to finally be finished.
I hope this video clip helps you better understand the behavior of a closed-circuited transmission-line endpoint. Even if you already know this material well, perhaps the animations will help you describe to others how the particles "know" what to do.
Charge carriers move within a conductor under the influence of local electrical fields. A uniform density of charge carries within a conductor gives the charges no impetus to move; they remain still. A non-uniform distribution creates local electric fields that accelerate the mobile charge carriers.
For example, Figure 1 illustrates a circuit at rest. The pink dots represent mobile charge carriers dispersed evenly throughout an invisible metallic lattice. The carriers are uniformly distributed, so there is no impetus to move. The carriers remain, except for thermal agitation, perfectly still.
When an excess of charge carriers accumulates within some region, the electric fields from that region act on the surrounding charges, pushing them away. Figure 2 depicts that situation, showing the distribution of charges just a few hundred picoseconds after closing the switch. A wave of compression has traveled partway down the line, but not yet reached the end.
In the region to the left of the traveling wavefront, the charge carriers are squeezed together, creating an excess of charges. The word "excess" here implies a very slight excess of conduction-band charge carriers, above and beyond the nominal density of their respective donor atoms within the metallic lattice.
The pink dots in Figure 2 represent hypothetical positive charge carriers. The actual charge carriers in a metallic lattice are electrons, which posses a negative charge, but I like to imagine positive carriers just so that an excess of carriers makes a positively-charged wire, and a deficit makes a negatively-charged wire. Also, with positive carriers, the direction of carrier drift aligns with the normal convention for the flow of current.
In the lower left portion of Figure 2 the return-current wire exhibits a deficit of charge carriers. This happens because the battery, which stuffs positive carriers onto the top wire, creating an excess, must draw those charges from the bottom wire, creating a deficit.
In the region to the left of the traveling wavefront, the excess of charges above, combined with the deficit of charges below, creates an electric field (E-field) that points straight down, from the top wire to the bottom.
Consider the spot on the top wire just underneath the traveling wavefront. To the left of that position the charge carriers appear pressed close together. To the right, they stand at a normal spacing. That subtle difference creates a small, local electric field pointing to the RIGHT, pushing on the charge carriers ahead of the wavefront, according to the rule that "like charges repel." That "push" is the thing that makes the subsequent particles move.
The rightward-pointing electric field does not extend very far. It exists only underneath the rising edge of the traveling wavefront. To the right of the wavefront, in the region marked "Nothing Has Happened Here Yet," the rightward-pointing field does not exist. It does not push on those charge carriers. The rightward-pointing field underneath the rising edge acts only on the charge carriers within its domain, accelerating them forward.
Stated in terms of the telegrapher's equations, the acceleration of charge carriers underneath the rising edge equals the spatial derivative, taken longitudinally along the conductor, of the local electric field potential. That's more than a mouthful. What it means is that the charge carriers move in reaction to non-uniform distributions of voltage potential.
The least uniform spot in Figure 2 is the point right underneath the traveling wavefront. Near that tiny spot, the voltage suddenly changes from +1V (tightly compressed charges to on the left side) to 0V (normally-distributed charges, in balance with the atoms of the surrounding metallic lattice, on the right). The electric field on the top wire right at that spot points strongly to the right, giving the charge carriers in that region a huge shove, forcing them forward at their drift velocity. On the bottom wire, the charge carriers are shoved in the opposite direction, peeling them away from the uniformly-distributed mass of particles to the right, accelerating them toward the battery.
In this discussion the charge carriers, once set in motion, drift freely along the conductor forever. That free-wheeling behavior occurs only if the conductors have no resistance. Any practical, slightly resistive conductor creates other small, longitudinal changes in voltage that do not appear in Figure 2. I assume here that the total transmission line loss is small, so you may ignore, for this discussion, the effects of conductor resistance.
Now, imagine what happens when, as shown in Figure 3, the wavefront slams into the end of the wires. At that point the top wire comprises a region of compressed charges. The bottom wire has a deficit of charges. The charge carriers making up the physical wire that creates the short-circuit experience a double whammy of electric effects: pushing from the top wire and pulling from the bottom. That double-size impetus unleashes a torrent of current, twice as much as exists elsewhere in the system. Once begun, charges flush through the endpoint, rapidly discharging the excess of charge on the top wire and re-populating the bottom.
The third animation in this series shows the torrent in great detail.
In the closed-endpoint simulation, at any time before the wavefront reaches the far (right-hand) end of the line, the charges on the top wire are pushed from the left into a sea of particle that exist at nominal density. When the wavefront reaches the end, the mitigating effect of the nominal charge carrier density to the right is lost. The charges at the end experience double acceleration, doubling the current, while the voltage at that point remains zero. This is the "current-doubling" effect that happens at a short-circuited endpoint. That is the exact opposite of the behavior that happens at an open-circuited endpoint, where the voltage doubles, but the endpoint current remains zero.
By the end of the animation, when the wavefront has reflected back to its source, the charge carrier density has once again become uniform (zero voltage everywhere), but the carriers are still drifting like mad (lots of current).
I'll be at the EMC 2011 symposium in Long Beach, CA, August 15-19 if you want to talk about any of these issues. My special session on connectors (a long diatribe of complaints) happens Wednesday, 1:30 - 5:00pm. I'll be signing books on the show floor before that, and that morning screening my latest movie, Mixed Signal Isolation (it's pretty amusing). Hope to see you there.
Dr. Howard Johnson