## Charge in Motion

Last month's letter dealt in part with the density of charge carriers in a metallic conductor, this month's note continues that theme, adding some cool animations.

## Charge in Motion

The sea of electrons in a metallic conductor flows like an almost perfectly incompressible fluid, almost perfect, but not quite. The slight degree of compressibility generates all the interesting electronic effects we exploit so successfully in high-speed digital design. Visualizing the motion of charged particles within your circuits may lead you to new insights about how your circuits behave.

I'll open this discussion with Figure 1, illustrating a simple electric circuit. The schematic comprises a battery, a switch, a resistor, and two wires labelled signal and return. Assume a uniform diameter and spacing for the wires and adjust the spacing to create a consistent characteristic impedance of fifty ohms. At rest, with the switch open, no current flows.

The pink dots represent charge carriers dispersed evenly throughout the metallic lattice. Obviously, I can't show all of them. If the conductor is made of copper, every atom contributes one moblie charge carrier. That would be an astromonical number. The figure shows just enough dots to make a good picture and get the idea across.

In a real situation, charge carriers move about madly under the influence of thermal agitation. Mentally ignore that effect by imagining a very low temperature, near absolute zero, so you can observe the graceful movement of charge carriers under the influence of an applied electric field without interference from thermal noise. With zero current, the charge carriers in your imaginary situation should be motionless.

Figure 1 does not show the atomic lattice throughout which the mobile charge carriers circulate. If you want to imagine the lattice, think of a white dot under every pink one. The white and pink dots carry opposite charges, so the net charge on each wire equals zero and no voltage potential exists between the wires at this time.

When you close the switch, current circulates through the resistor. Before I get to the high-speed details of how that works, we need to treat the issue of charge-carrier polarity. Electrons, as you know, carry a negative charge. For this fact we can thank Mr. Benjamin Franklin.

Old Ben did correctly conclude from his experiments with vitreous and resinous substances that all electric effects were due not to two different types of fluids within a conductor, as previously conjectured, but rather one type only, according to whether the conductor held within an excess or deficit of that one type of electrical "fire". Ben then established the convention by which we still label all electrical effects. Rubbing wool on amber, he called the charge acquired by the wool positive (excess) and that acquired by the amber negative (deficit).

More than a century later, when the structure of atomic matter became clear, it turned out that the mobile charge carriers, the electrons, carry a negative charge. Oops. In Figure 1, that means that when the battery sends positive current from left to right along the signal wire, the mobile charge carriers (the electrons) actually move in retrograde fashion from right to left. How unfortunate.

I wish I could just set the dial on my way-back machine for the year 1749, go back to visit Ben, and have him do it the other way around. That would give the mobile charge carriers a positive charge, aligning their direction of physical movement with our convention for current flow, making the diagrams and animations to come much more intuitive. For the purposes of this discussion, let's do just that. Assume the mobile charge carriers are positive. Later, if you want, you can imagine how everything would look with the charge-carrier polarity reversed and everything circulating backwards.

Assuming positive charge carriers, when the battery sends positive current clockwise around the circuit, the charge carriers move in the same direction. I'm not going to bother animating that situation because it is so simple I know you can imagine how it looks.

What I shall do instead is depict the situation a few hundred picoseconds after closing the switch. From time zero, when the switch closes, current immediately begins to flow from the battery, but it does not instantaneously reach the load. It cannot. No effect can proceed faster than the speed of light. There must at some point in time be additional charges forced onto the signal conductor that have not yet reached the load. Figure 2 illustrates that situation.

The top of the figure displays details relevant to the simulation, which is implemented in MathCad.

• The reflection coefficient equals zero. That implies a perfect end-terminating resistor at the right end of the network.
• The input step voltage from the battery is one volt.
• You are viewing frame 230 from the simulation.
• TPORCH and PFRAME don't matter for this discussion.

The plot of voltage versus position resembles a scope waveform, but it isn't. This is a snapshot in time (at FRAME 230) showing voltage versus position along the transmission structure. At this instant, the 1-Volt step has processed 2/3 of the way to the load.

The schematic and charge-carrier plot at the bottom of the diagram reveals an important detail about the charge-carrier distribution. The distribution is no longer uniform. In the section prior to the leading edge of the signal waveform, the charge carriers bunch more tightly than normal. They must, because we have jammed in current from the battery and it has to go somewhere. The new current squishes together the charges on the signal wire. The effect is like a shock wave—it creates a column of squished particles, each shoving the next forward.

Play full-motion simulation: 14_02-R.wmv

As shown in the full-motion simulation, the leading edge of the wave progresses forward quite rapidly, even though the individual particles drift at a relatively slow pace. When the switch closes, put your finger on the first pink dot and follow it. At FRAME=380, when the leading edge of the waveform arrives at the resistor, your finger should be about 1/4 of the way to the end. In other words, the ratio of wave propagation speed to average drift velocity in my simulation equals about 4:1. In a typical PCB trace, at a realistic current, the ratio is about ten billion to one. Electrons in our circuits drift very slowly compared to the wave propagation speed.

Now take a look at the bottom wire. At the same time that current exits the battery onto the signal wire, the battery draws current in from the return wire, resulting in a deficit of charge carriers on the return wire. Between the signal and return wires, and this is the cool part, the excess of charge carriers on the top and the deficit of charge carriers on the bottom creates an electric field. The electric field creates the voltage you measure with your scope when you probe the signal wire.

My next letter will show the effects of open-circuit and short-circuit loads on the flow of mobile charge carriers.

Best Regards,
Dr. Howard Johnson