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An addendum to The Devil’s Dictionary

Buypartisan, adj. As of, or pertaining to, a situation in which the partisans have been bought. Commonly misspelled.

(Not really entirely fair? Well, neither was the original…)

Power Factor In The Digital Age

Over the years, I’ve seen entirely too much confusion surrounding the electrical quantity known as power factor. Even its definition is often confused. Roughly half the sources I’ve encountered define it to be the cosine of the phase difference between current and voltage — a definition that was adequate sixty years ago when waveforms were almost all sinusoids of the same frequency, but which is entirely inadequate now that both current and voltage are commonly chopped up using silicon. The “phase” of a non-sinusoidal signal can have many definitions, and probably none of those definitions yields a meaningful number for power factor. The old formula is still fine as a formula for the power factor in the case that one is dealing only with sine-wave power supplying old-fashioned devices, but fails as a general definition.

A real definition (and the one used by the other half of the sources I’ve encountered) is that power factor is equal to the true power divided by the “apparent power”. The true power is defined as physics dictates: the average of the instantaneous power consumed by the device (instantaneous power being instantaneous current times instantaneous voltage). That average is usually best taken over a single full cycle of the AC waveform, or multiple full cycles; but even if there are no recognizable cycles, it can be computed for any given interval of time. Apparent power (aka “VA”) is defined to be the voltage multiplied by the current, both voltage and current being measured in root-mean-square (RMS) fashion. It is, as per the name, what one might think the power was, if one just measured current and voltage with a true-RMS meter. The average (the “mean” in RMS) is again best taken over a single full cycle; but again, there don’t even have to be cycles at all, for apparent power (and thus power factor) to be a well-defined quantity, for any interval one chooses.

Whether or not that definition makes any sense in general is another question. For one thing, the power factor is supposed to always be between 0 and 1 (or -1 and 1, if the device is allowed to supply net power rather than consuming it). And while it’s obvious that the cosine of a phase difference has to be between -1 and 1, it’s not obvious that the same thing applies to the general definition of power factor. Or at least, it’s not obvious unless one recognizes it as a direct consequence of the Cauchy-Schwarz inequality. That inequality states (in the version that’s useful here; it can also be written much more generally) that for any two real functions f and g of a single variable,

$\int f(x)^2 dx \ \int g(x)^2 dx \ \ge \ \left( \int f(x)g(x) dx \right)^2,$

with equality occurring if and only if f is proportional to g — that is, if

$f(x)=cg(x),$

for all x and for some constant c. (This web page uses MathJax to render equations; if the above equations appear as LaTeX source, with lots of backslashes, it’s probably because Javascript is not enabled. It needs to be enabled for this website and for the website “mathjax.org”.)

In this case, let f be the voltage, and g be the current, both as a function of time. Then take the square root of both sides, and divide both by the length of time over which the integrals are taken. The right hand side is then the absolute value of the true power, and the left hand side is the apparent power, proving that power factor is between -1 and 1 — and, as a corollary, that a power factor equal to one occurs only in the case of a resistive load (in which case c is the resistance).

Power factor, defined this way, is thus a solid concept, not one of those poorly-defined notions that sort of works as long as you stay within its traditional applications but which breaks when you do something unusual. There are no strange voltage or current waveforms lurking anywhere for which the power factor might be greater than one.

But there’s another way in which one could doubt whether power factor makes sense to compute: if one objects to root-mean-square as the appropriate way to measure current and/or voltage. The square root of the sum of squares is a mathematically convenient entity, which makes a lot of formulas simpler than they would otherwise be. But mathematical convenience shouldn’t take priority over usefulness in applications. Fortunately, in this case, the two pretty much coincide. By Ohm’s law, heating in a conductor, at any instant, is proportional to the square of current. So total heating is proportional to the integral of the square of current; the RMS current is the square root of that, and thus tells you how much your wires are heating up in the process of carrying the current. An RMS current of 15 amps will yield about the same heating whatever the waveform; if it is 15 amps DC, the heating will be about the same as if it is 15 amps AC RMS — the latter being, by convention, a sinusoidal waveform with a maximum of 15$$\sqrt{2}$$ = 21.2 amps. (The reason for the qualifier “about”, in the preceding sentence, is skin effect; but the frequencies of interest here are too low for skin effect to play a big role.) Heating represents wasted energy, lost in transmission. Also, the amount of heating is usually what sets the limit on how much current a wire can carry. Heating in motors, transformers, and inductors is largely resistive heating, proportional to the square of current. On the other hand, if the current is through a diode, the situation changes: the diode’s voltage drop is nearly constant, rather than being proportional to the current. So instead of the square of current, the heating at any instant is proportional just to the current. But for power MOSFETs operated in saturation, the situation is again that they look like a resistance, with voltage drop proportional to the current. BJTs, however, are more like diodes. So, in power transmission and handling, RMS is a pretty decent measure of current, although it’s not as perfect as it was before silicon devices. As for the appropriate measure for voltage, if one is going to measure current in RMS terms, one pretty much has to measure voltage that way, too, so that Ohm’s law works for AC current.

So power factor, under the proper definition, is in all circumstances a good measure of how efficiently a device is sucking down current, as compared to the best it could do: not, of course, a measure of internal efficiency, but rather of how efficiently it loads down power production and distribution networks.

But there are some notions which one has to let go of, when using the general definition of power factor. One is the idea of measuring a phase difference and using that measurement to correct power factor. Oh, the old formulas still work in the old circumstances — those being when one is dealing only with sine-wave power and with linear devices such as motors, generators, transformers, and capacitors. But they don’t extend to the general situation. I’ve seen talk of patching them up by having two numbers for power factor, the one being the cosine of the phase difference and the other being due to harmonics. But there seems little point in this. For one thing, it could only apply to sine-wave power in the first place: if some other voltage waveform is being used, the best power factor is from a current waveform proportional to it, which has the same harmonics, which in this case are making power factor better rather than worse. Besides, unless one is going to try to correct the power factor, as has traditionally been done for motors by adding capacitors, there seems little point in computing any number for phase difference. And the power factor of nonlinear devices is not easily corrected: it is not a traditional “leading” or “lagging” power factor, where the current is a sinusoid that either leads or lags the voltage. Instead the pattern is commonly that power is drawn from the line near the peaks of the voltage waveform, and not near the zero crossings. The following are oscilloscope shots of such behavior, as displayed by an old computer power supply; the first shot is with it running, the second with it quiescent (plugged in, but only drawing enough power to keep its internal circuitry alive). The white line is voltage, and the purple line current (which is on a different scale in the second shot than in the first):

To correct power factor for a device like this by adding an external device across the line, the way capacitors have traditionally been used to correct power factor for motors, would mean fielding a device that drew power near the zero crossings, and fed it back into the line near the peaks. Such a device could be built, but would be much more complicated, expensive, inefficient, and unreliable than a capacitor. It is probably easier to demand that the devices being powered be power factor corrected, as are many modern computer power supplies, such as the one that produced the following scope traces — the first, again, when running, and the second when quiescent:

(As can be seen, the power factor correction only applies when the power supply is on; when it is quiescent, the small current it draws looks a lot like the current a capacitor would draw: about 90 degree phase lead. Indeed, that current is likely being drawn by a filtering capacitor inside the unit, such as is often placed across the input for the purpose of blunting power surges and suppressing RF emissions.)

It’s not just when a device consumes power in a nontraditional way that it is difficult to correct power factor; it also is difficult when a device provides power in a nontraditional way — that is, as something other than a sine wave. The usual nonsinusoidal power waveform is what marketing people have decided to call a “modified sine wave”, which really would be better termed a modified square wave. Whereas a square wave of 115VAC would alternate between 115V and -115V, the modified square wave alternates zero, 162V, zero, -162V:

That peak voltage is chosen to be the same as the peak voltage of a sine wave with RMS voltage 115V; the time spent at zero volts is chosen so that the signal as a whole is 115V RMS. This is the waveform output by most inverters — inverters being devices for converting DC, usually at 12V or 24V, into alternating current at line voltage. Most uninterruptible power supplies dish out the same sort of modified square wave when running off battery power. If a motor is driven by such a voltage source, it will have a lagging power factor; but if one were to try to correct it by adding a capacitor, at the sudden transitions between voltages the capacitor would try to draw enormous currents. Rather than correcting the power factor, that would dramatically worsen it — if, indeed, the inverter didn’t shut itself off instantly, as it probably would in self-defense when it detected those enormous currents.

Nevertheless, modified square waves have their upsides, as regards power factor. The old computer power supply that yielded the first graphs above, I measured drawing 219W and 313VA on AC power from the utility (a power factor of 0.70). On an inverter, with the same load, it drew 207W and 240 VA (a power factor of 0.86); the current waveform looked like this:

The fact that this power supply draws current only near the peaks makes for a better power factor on the inverter, since its voltage waveform has wider peaks. Also, it improves the power supply’s internal efficiency a bit, so it draws less real power.

On the other hand, the same power supply also has a small filtering capacitor across its input, to help deal with power surges and to suppress RF emissions. With the power supply quiescent, it draws 2 watts at 8 VA from the AC line, but draws something like 75 VA from the inverter (though that measurement is quite imprecise, since the wattmeter I was using couldn’t really resolve the current spikes). The power-factor-corrected power supply graphed above behaved even worse on the inverter when quiescent, drawing about 90 VA; even when switched off using the switch on the back of the power supply (which turns off the current to every part of it that is at all active, leaving only a filtering capacitor or two drawing power), it drew about 60 VA. Yet under load, its behavior was again good: 264 W at 287 VA, a power factor of 0.90, although it still shows current spikes:

So although power factor is usually specified as just a single number (or as a function of load), really those numbers apply only to sinusoidal voltage waveforms, and can’t be extrapolated to other waveforms.

Runaway Starter

“Hmm, I shouldn’t be going anywhere with the car making a noise like that.”

I pulled back into the driveway, and turned off the ignition.

The engine went off, but the growling noise that had disturbed me continued.

I got out of the car, opened the hood, and looked. The engine wasn’t vibrating the way it does when it’s running; so, as my ears had already told me, whatever was making the noise didn’t involve pistons and cylinders and such. So was the noise electrically powered, or was fuel leaking someplace and burning? No smoke was coming out of anywhere I could see. Should I disconnect the battery? Go get a voltmeter and see whether the battery is draining? Or perhaps go get a fire extinguisher? Could combustion from a leak really be that regular and uniform, even inside some hidden space from which no smoke could escape? (In retrospect: no; and disconnecting the battery would be the right thing to do even if there were a fire.)

I stood there trying to figure out what to do for a minute or two, until the noise stopped with a bang, accompanied by a bit of smoke blowing out from behind the engine.

When, after some pondering, I tried turning on the car again, it wouldn’t start. After jacking up one side of the car, and going under it to fasten an alligator clip to a starter terminal (the starter being down behind the engine, in roughly the same area the puff of smoke had come from), I found that the starter solenoid seemed to be working: the voltage on its output was zero until I turned the key, then went to eleven volts and some. But by the same measurement, the starter motor was broken — since it didn’t run, and since it couldn’t just be jammed: the current draw of a starter is large enough that the voltage would be lower than that.

I finally got around to considering the possibility that the starter had run away, something I don’t recall ever hearing about, but of which Google quickly found me many examples. Of course how the starter would have run away was not entirely clear, since the starter solenoid was now working properly, as were all the circuits feeding it. But if it had stuck on, the bang at the end might have unstuck it. In any case, the starter plus solenoid being a complete assembly, it was clearly time to pull that assembly off and order a new one.

A variety of places on the net sell starters; I chose one from Amazon — about a hundred dollars for a new starter, NSA brand. Curiously, the refurbished starters on offer mostly went for more than that. The one that had failed was, to judge from the part number on a sticker on it, itself a refurbished unit, from the “Quality-Built” corporation. On their website, they advertise that the solenoids on their rebuilt starters have “100% new contacts”, among other things.

With a new one on its way, it was time for the fun part: failure analysis. The starter was held together by two bolts that ran the whole length of the motor. Removing them was a bit difficult; they came out a bit bent in one place. Then the front of the starter came off easily, revealing a planetary gear set, a one-way clutch attached to the output gear (and loosely mounted on a helical spline), a lot of grease, and nothing at all wrong.

Taking off the other end of the motor, though, revealed a scene reminiscent of Chernobyl. The whole volume was packed full of black-ish, somewhat fluffy debris, including bits of copper and of graphite brush. This explained the bang: something had gotten loose, slammed into something else moving fast, broken it, and a chain reaction of destruction had ensued. Here is a photograph of the recognizable pieces that remained; the photo can be clicked on to bring up a larger version:

That area of destruction was also the place in which the aforementioned bolts were a bit bent. But the real place of interest was the solenoid contacts. Being crimped in place, that portion could not be disassembled nondestructively; I got it apart by filing off the crimp. Here is a photo of the contacts (again, click for a larger version):

This is a decent contact design. The contacts are copper. The copper washer that closes the contact is loosely mounted, so that it can rotate, evening out its own wear. It can also swivel a bit to make good contact even if one of the two fixed contacts wears down (as one has). It lasted through several years and thousands of starts, so I can’t complain too much. But in the end it wasn’t enough to reliably switch the hundreds of amps that a starter draws. The contacts, which started out smooth, roughened from sparks upon opening. Eventually they roughened to the point where the resistance was high enough to generate serious heat, and to weld them together. The places where they welded are clearly visible in the photo, as fresh copper which was exposed when the weld was cracked away.

My impression is that the way to make contacts like this reliable is to make them out of silver rather than copper. But silver costs money. A thin layer of silver won’t do it, because silver gets eroded too, in this duty. And even a thick layer of silver eventually fails.

The drive gear off the old starter wasn’t much chewed up by the runaway. Nor, getting under the car and looking at it, was the ring gear. Properly hardened steel seems to have been used throughout.

The replacement starter looks good, and has worked fine so far. The only curiosity about it is that it comes with a two-year roadside assistance plan. This is rather odd, since the cost of two years of AAA roadside assistance is about the same as the whole cost of the starter. Of course the plan that comes with the starter is not provided by AAA, but rather by another company, “Auto Road Services Inc.”. That company advertises on their website that:

For just pennies per unit, you too can give your customers the added value of FREE emergency roadside assistance, and your company the marketing edge over the competition.

which of course tells me, the customer, the most that their plan could be expected to be worth to me: “pennies”. After their profit and overhead is taken out, it might even have a negative expected value. According to the plan description that came with the starter, one has to call their 800 number, use the roadside assistance provider that they dispatch, pay him his full fee, then send in lots of paperwork to them to get reimbursed. The ways in which they could sleazeball this are too numerous to mention. Not that I care; I’m just glad the starter’s manufacturer didn’t spend too much on this marketing gimmick.

One thing I noted when Googling for “runaway starter” is that some people advise against doing the obvious, and disconnecting the battery. Now, it’s true that in general, disconnecting the battery while the engine is running is a bad idea: it’s called a “load dump”, and can cause the voltage to rise excessively, damaging the car’s electronics. But in this case there’s zero danger: the alternator, which is what produces the excess voltage in the load dump scenario, is not running in the first place. Even if it were running, the runaway starter is sucking down so many amps that the voltage could hardly rise. The “load” that gets “dumped” in a load dump is the current going into the battery; but here current is coming out of the battery.

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