# Point of resistance

How often is the derivative of a variable, versus a variable itself, the major driver for the class of equations we deal with on a day-to-day basis; moreover, what does HP's invention of the "fourth passive element," the memristor, hold for power electronics?

HP Labs' touted breakthrough in bringing us the fourth passive component, the "memristor", is certainly something to think about, although right now I'm a long way away from thinking about it as an advance that belongs in the Mendeleev or Newton or Maxwell class. The news, however, rather rejuvenates a related philosophical question on the power of The Calculus that I (and many) have pursued for years. And that is, how often is the derivative of a variable, versus a variable itself, the major driver for the class of equations we deal with on a day-to-day basis? And, can HP's finding now be further applied to power and power management in general? Perhaps their latest discovery adds meat to the view that the derivative is more often more fundamental in equations where we've taken something else for granted.

After all, Newton didn't really say *F = ma*. What he did say was that *F = d (mV)*. And when you work that out (with respect to time *t*), you get *F = ma + V dm/dt*. Amazingly, for a while in the 50s and 60s I actually had a number of instructors who appeared to somehow believe—at least put it in their students' minds—that the second term in Newton's second law indicates that Newton may well have understood that mass could change with time in the relativistic sense. I seriously doubt Newton thought anything more than some hay could possibly fall off the wagon that he was rolling down a hill, but that's another discussion.

Electronics isn't the only area where I've seen a different look at the derivative strike in a big way. The area of radiometeorology, for instance, involves the refractive index of the atmosphere and its role in extending radio wave propagation at the VHF/UHF frequencies. Most people you run into with experience in this area will observe, and conclude, that you don't generally get enhanced propagation or ducting conditions in the wintertime because the absolute humidity is usually low, and basically is a very small term in the grand scheme of things. They assume the arithmetical differences in absolute humidity with height are too small to be concerned with, and don't bother to think about it further. That's true to a good extent, but under special circumstances you can extend the radio range even when the humidity is very low. In those cases, it's closer to the *percentage change* in the absolute humidity that counts.

But most people wouldn't readily recognize that, because few if any apparently bother to think about a situation where relative humidity changes greatly for a small change in absolute humidity, and that it might make a difference to look at it in that way. And know well enough, just to check it out, to take the derivative of *relative* humidity with height, and then plot it out. While they still would have seen the same result if they went through an analysis using absolute humidity (except for the fact that they discounted it), maybe part of the reason for avoiding relative humidity is that it isn't a direct fundamental variable (it's a function of temperature, whereas absolute humidity is not). Either way, it's a compounding of errors, and all the while the "secret" is sort of hiding in the math, waiting to be found. How many such cases of that do we have in electrical engineering? How many have we missed thus far?

In any case, I see certain parallels here with HP's announcement. Their advance seems a step beyond the kind of mental gymnastics that, in passives, made such great devices as the swinging choke, the varistor, the varactor, and yes, the FET possible. What happened is that over the course of time the technology for many of these components caught up with the math. And so HP's advance is something that apparently will yield dividends in advanced applications, and hopefully will extend deeper into power electronics. Which is, in the last reckoning, what would make the advance really noteworthy. But not the hype—given the power of mathematics—that went with it.