What a meter multiplies by

The power field’s standing question is what a number means when the current is not a sinusoid. Power factor stops being cos φ. The neutral of a three-phase system stops being empty. This essay asks the same question of the most basic reading there is — how many amps — and gets the same kind of answer.

Two instruments both claim to report the root-mean-square value of a current. One of them has a shape assumption buried in a constant. The other has no shape assumption and a bandwidth instead. Neither is measuring the heating, and they fail in ways that have nothing to do with each other.

The constant, and where it comes from #

An average-responding meter is the cheap and old one: a bridge rectifier and a moving-coil movement, or the same arrangement in silicon. What it physically measures is the mean of the absolute value. That is not the root-mean-square of anything, so the scale is multiplied by a constant chosen to make the answer right for a sinusoid.

For a sine, the mean of $|\sin|$ is $2/\pi$ and the root-mean-square is $1/\sqrt2$, so the ratio — the form factor — is

$$\frac{\pi}{2\sqrt2} = 1.110\,72$$

and that number is printed on the scale. Feeding a sinusoid through the arithmetic returns the right answer to $-7.8\times10^{-5}$ per cent, which is the sampling and not the constant.

For any other waveform the meter reports $1.1107$ times the mean of the absolute value, and the truth is the form factor times that same mean. So the error is the ratio of two form factors, and what it contains is worth stating: not the amplitude, not the frequency, not the resistance — only the shape. No calibration can remove it, because there is nothing to calibrate against; the instrument is right about what it measured and wrong about what it was asked.

Sixty degrees is not an exotic case. It is roughly what a capacitor-input rectifier — the front end of nearly every appliance made — actually draws, and the power field’s distorted-power-factor essay measures the rest of what it does. An averaging meter clamped round that cable reads two thirds of the current that is heating it.

The waveform it is exactly right about, which is not a sine #

There is a second waveform for which the meter reads exactly right, and finding it is the sort of thing this site looks for.

The error vanishes when the form factor is 1.1107, which for a rectangular pulse train of duty $D$ means $1/\sqrt{D} = 1.1107$, so $D = 1/1.1107^2 = 0.8106$. Bisecting on the solved waveform puts the crossing at a conduction angle of 145.90° against a closed form of 145.90°.

So an averaging meter is exact for a sine and exact for a rectangular waveform conducting for 81% of the time, and wrong everywhere in between and outside. The zero is not a feature of sinusoids; it is a level crossing on a monotonic curve, and any calibration constant has two waveforms it is right about rather than one.

The same reading of the table gives the practical rule. Below 146° the meter reads low and above it high, so a meter used on rectifier currents under-reads and one used on a chopped or square waveform over-reads — and the two errors have opposite signs, which is why an instrument that is out by a third in one application is out by a tenth the other way in the next.

Three quantities, and which one each instrument has #

It helps to have the three ratios in one place, because they are constantly confused and only one of them is what a meter’s constant is about.

The form factor is the root-mean-square over the mean of the absolute value. It is what an averaging meter’s constant is trying to be, it is 1.1107 for a sine and 1 for a square wave, and it is what this essay’s first half is entirely about.

The crest factor is the peak over the root-mean-square. It is what an instrument’s input range has to accommodate, it is $\sqrt2$ for a sine and 1 for a square wave, and it appears on data sheets.

The distortion factor is the fundamental’s root-mean-square over the whole waveform’s. It is what decides how much of a current a phasor calculation can see, it is 1 for a sine, and it is the power field’s own quantity from the distorted-power-factor essay.

Three ratios of the same waveform, all equal to different constants, and no two of them determine the third in general. A rectangular pulse train is the one family where two of them coincide — its form factor and crest factor are both $1/\sqrt{D}$ — which is a coincidence of that shape and is the source of most of the confusion between them.

The other instrument, and the other failure #

A true-RMS meter squares its input, averages, and takes the root. There is no shape assumption anywhere and no constant to be wrong. What it has instead is a bandwidth.

A narrow current pulse has energy far up the harmonic series — a rectangular pulse of duty $D$ has its first spectral null at harmonic $1/D$, and appreciable content well past it. An instrument that passes N harmonics returns the root-sum-square of the lines it can see, which is always low, because the missing lines are missing power.

The first four rows are worth reading twice. A true-RMS instrument reaching fifteen harmonics — which for a fifty-hertz supply is 750 Hz, and is a perfectly ordinary specification for a clamp meter — is more than one per cent low at every conduction angle in the sweep, including a full square wave. The square wave case is arithmetic anybody can check: the odd harmonics of a square wave carry $8/\pi^2 h^2$ of the power each, and stopping at the fifteenth leaves 1.2% of the total behind.

So the two instruments are wrong in the same direction on a rectifier current, for entirely unrelated reasons, and the one that has no shape assumption in it is not therefore right.

What the crest factor is doing, which is less than expected #

Instruments carry a crest-factor rating — usually three, sometimes five — and it is the specification people reach for when a reading looks wrong on a peaky waveform.

For a rectangular pulse train the crest factor and the form factor are the same number, $1/\sqrt{D}$, so the two rise together and it is easy to think the crest factor is causing the error. It is not. At sixty degrees of conduction the crest factor is 1.733, comfortably inside every instrument’s rating, and the averaging meter is 35.9% low anyway.

The crest factor limits something else: it is the point at which the instrument’s own front end clips or its converter runs out of range, which is a hard failure rather than a shape error. The two are independent and only one of them is on the data sheet.

That is worth separating carefully because the diagnosis matters. A reading that is wrong because of crest factor gets better when the input is attenuated. A reading that is wrong because of form factor does not change at all — the error is a ratio and the attenuation divides both halves of it.

The last two settings bracket what an instrument on a bench actually reaches, and they are worth drawing together because the answer is not the meter’s accuracy — it is its bandwidth, expressed in harmonics of the thing being measured.

What a modern instrument actually does #

Almost nothing sold today has a moving coil in it, and it is worth being precise about what has and has not changed.

A digital multimeter’s “true RMS” front end is either an analogue computing block — a log-antilog squarer and an averaging capacitor — or a sampler with the squaring done in arithmetic. The first has a bandwidth set by its own circuitry and an averaging time constant that sets how slowly the reading settles. The second has a bandwidth set by its anti-alias filter and, underneath that, a sampling problem: a narrow current pulse that lands between samples is missed entirely, and the digital field’s essays on aliasing are the same argument in the same words.

So the harmonic count in the table is not a legacy specification. It is the honest way of writing down whatever the instrument’s front end does, whether that is a filter, a squarer’s bandwidth or a sample rate, and the number is usually in the tens rather than the hundreds.

The averaging meter, meanwhile, has not gone anywhere: it is what most low-cost clamp meters, most panel meters and most current-monitoring integrated circuits do, because it needs no squarer and its error is invisible on the waveform the instrument is demonstrated with.

Three places the difference is money #

Sizing a neutral conductor. The line currents are read with whatever is available, the neutral is sized from them, and the neutral is carrying triplen harmonics the reading did not include. Two under-estimates compound: the meter’s and the arithmetic’s.

Verifying a supply’s rating. A power supply specified to draw ten amps rms is measured with a clamp meter and reads seven. Nothing is wrong with the supply; the fuse, the connector and the wiring are all being asked for ten and were sized for seven.

Comparing before and after a fix. Adding power-factor correction to a rectifier changes the shape of the current, so it changes the meter’s error at the same time as it changes the current. The measured improvement is the true improvement times the ratio of two form factors, and can be in either direction.

What “measuring the heating” would take #

Neither instrument measures what the wire is doing, and it is worth saying what would.

The heating in a conductor is $\int i^2 R\,dt$ over a cycle, and the only instrument that measures it without a model is a thermal converter: a heater, a thermocouple, and a second heater driven by a direct current until the two temperatures match. Then the direct current is the root-mean-square value, by construction, with no shape assumption and no bandwidth beyond the heater’s own — which is into the megahertz, because a small wire has a very short thermal time constant compared with anything electrical.

Thermal converters exist and are what national standards laboratories use to calibrate everything else. They are slow, fragile and expensive, which is why almost nothing else uses one, and the consequence is that every ordinary reading of a current is a model’s answer rather than a measurement’s.

That is a fair description of the whole subject as this collection sees it. The site’s premise is that no model is drawn without the frequency, amplitude or size at which it stops being true, and an instrument is a model too — with a constant in it, or a bandwidth, and in either case a range of waveforms it is right about.

What this essay does not claim #

That averaging meters should not exist. They are cheap, robust, need no active circuitry and are exact on the waveform most of the world’s electricity is. The claim is about knowing which of the two kinds is in one’s hand, and the case where it matters is precisely the case — a rectifier load — that has become the default since the meter was designed.

That a bandwidth specification is the whole of a true-RMS meter’s error. It is not: the crest factor rating, the converter’s own linearity and its averaging time all contribute. The bandwidth is the one that is a clean function of the waveform’s shape, and the only one computable from a spectrum.

That the harmonic count maps simply onto a bandwidth in hertz. It does for a fixed fundamental. The table above is at a fixed conduction angle in degrees, which for a fifty-hertz supply makes the fifteenth harmonic 750 Hz; on a four-hundred-hertz aircraft supply the same instrument reaches only the second.

That the 145.90° crossing is a useful place to work. It is not — it is a demonstration that the calibration constant has two roots rather than one, which is a fact about the arithmetic. Nobody designs a load to sit there.

The other readings this field checks #

A meter multiplying by 1.11 is one of several instruments in this field that answer a different question from the one asked. The neutral that carries more than a line is a current an ammeter reads correctly and a designer does not expect. The far end that rises is a voltage regulation figure that carries neither the load angle nor the line’s reactance. The capacitor that was right once is a correction computed from a power factor that a distorted current makes meaningless. Two terminals measure the leads as well is the instruments field’s version of the same complaint, and The first cycle, which no steady state contains is the waveform none of these readings is taken on.

The gate #

The constant is checked by feeding it a sinusoid, which is what it is for: the reading comes back exact to $10^{-6}$, so the 1.1107 is verified against its own definition rather than quoted.

And by feeding it a square wave, where it must be high by exactly $\pi/2\sqrt2 - 1$ — asserted to a part in $10^9$. A square wave’s own form factor is one, so the whole error is the constant, and it is the cleanest possible statement of what the error is.

The averaging meter’s error is asserted to recover monotonically across the whole conduction sweep, from 63% low at ten degrees to 11% high at a hundred and eighty, rather than merely to be large at the angle drawn.

And the band-limited meter’s one per cent point is bisected on the solved waveform at every setting of the slider, with the “never on this sweep” case reported as such rather than silently clamped — because at nine harmonics there is no such angle, and a figure that invented one would be the more comfortable and the less true.

Two instruments, two conditions, one waveform #

The average-responding meter’s condition is a shape and the true-RMS meter’s is a bandwidth, which is why neither is right about the load this field spends most of its time on.

The direct voltage that is a sawtooth is the source of the waveform in question: a rectifier drawing 157 milliamps of average current as a 2.098 ampere pulse over 28.8° of each half cycle, with a crest factor of 13.4 that gets worse as the reservoir is made larger. Against that, the average-responding meter’s 1.1107 is wrong by the ratio of two form factors and the nine-harmonic meter is short of the energy in the tenth harmonic and above — and the narrower the conduction interval, the further out the harmonics that carry the current, so both errors grow with the same design choice.

The consequence for the quantity the field is about is worse than either error alone. The current that does no work shows a displacement factor of 1.000000 beside a true power factor of 0.780 on that waveform, so an installation’s apparent power is being computed from currents that two different instruments read two different wrong values of, and combined with a phase angle that is exactly right about a component carrying most of nothing.

Which is why this essay’s second result — that the average meter is also exactly right at a 145.90 degree conduction angle — is worth more than it looks. It is the demonstration that “right for a sinusoid” is a coincidence of one form factor rather than a property of sinusoids, and that a correction factor derived for one waveform is a number about that waveform and no other.

Nine harmonics, and where the tenth is #

The true-RMS meter’s limitation is a bandwidth and the essay reports it as one — never within one per cent of anything narrower than a square wave — and it is worth converting that into the design question it actually is.

A narrow pulse’s energy is spread over harmonics reaching to about the reciprocal of the pulse width, so a conduction angle of sixty degrees puts significant content out to the sixth harmonic and one of twenty degrees out to the eighteenth. A meter that reaches nine is therefore adequate for the first and not for the second — and which of the two is present is decided by the reservoir capacitor in the load’s own supply, which is somebody else’s design decision made for a reason unrelated to metering.

That is the same dependence the direct voltage that is a sawtooth measures from the other side: the conduction interval shortens as the reservoir grows — 54.2° at 220 µF and 17.3° at 4700 — so the harmonic content that a meter has to reach grows with the same parameter. An instrument specified as adequate for the equipment of one decade is not specified as adequate for the next, without anything about the instrument having changed.