The average a square root pulls low

What a meter multiplies by put two instruments on one current and found each wrong for a reason the other does not have: the averaging meter carries a constant that belongs to a sine, and the true-RMS meter carries a bandwidth. Its list of what it did not claim included a third contributor to a true-RMS meter’s error, named and left alone — the averaging time.

That one is worth taking seriously, because it is the reason a true-RMS instrument has a lowest frequency as well as a highest, and the reason is not the one the specification suggests. It is not that a slow signal makes the reading wobble, although it does. It is that a square root is applied to something that wobbles, and a square root is curved.

The square is averaged, and the root is not #

An explicit converter does three things in order. It squares the input, it averages the square through a one-pole filter of time constant τ, and it takes the square root of the average. Every figure here has τ = 100 ms, a common value for a meter, and every input has a root-mean-square value of exactly one, so that an error is a fraction of the truth.

On a sine the square is a constant plus a single line at twice the frequency: for a unit sine, 1 − cos 2ωt. The averager passes the constant exactly and the line reduced by its gain at 2ω, which far above its corner is 1/(2ωτ). So the averaged square has a mean of exactly one, whatever the frequency, and a ripple whose size is that gain. The converter’s reading is the root of it.

The averaged square is right on average and the reading is not, and the gap between them is Jensen’s inequality in its most ordinary form. The square root is concave, so the mean of the root of anything that varies is below the root of its mean. A true-RMS converter reads low whenever its averaged square has any ripple on it, and it never reads high.

The size of the error follows from two terms of a series. Writing the averaged square as $1 + e$, the root is $1 + e/2 - e^2/8$ and a little more. The first term averages to nothing, because $e$ is a ripple. The second does not, because $e^2$ is never negative. So the reading is low by an eighth of the averaged square’s ripple power, and on a sine, whose averaged square carries a single line of amplitude $k$, that is $k^2/16$, with $k$ the averager’s gain at twice the signal frequency.

It is the same arithmetic how small is small signal applies to an exponential. A smooth curve driven by a small wobble about a point shifts its average by half its curvature times the wobble’s variance, because the curvature is the part of the response that does not average away. There the curve was a junction’s exponential and the shift was a distortion product; here it is a square root, whose curvature at one is −¼, and the shift is a reading that is low. Nothing about the averager is nonlinear. The nonlinearity is the root, placed after a filter that could only ever leave some ripple behind.

The ripple falls as the frequency, and the bias as its square #

Those two quantities — how much the reading moves and how far its mean is low — are both set by k, and they depend on it at different powers. That difference is what makes the error hard to see.

The reading is one per cent low below 1.86 Hz and its ripple is within ±1 per cent only above 39.8 Hz, a factor of 21 higher. So in the whole band where the bias matters, the reading is visibly swinging — by about ±20 per cent at 1.86 Hz — and the bias sits underneath the swing where nobody looks for it. Above the band where the swing stops being visible, the bias is already negligible: at 5 Hz it is −0.155 per cent, at 100 Hz a few parts per million.

That ordering would be harmless if the swing were what a user saw. It is not what a user is allowed to see, for a reason with its own section below.

The two routes to that figure share nothing but the converter’s equation. The exact steady state joins the input’s samples with straight lines, solves each interval in closed form, and closes one period on itself; the march integrates the same equation forwards with a fourth-order rule, from the true reading, until the start-up is gone. The dashed curve is the small-ripple series, the eighth of the ripple power, which the measured bias approaches from below as the frequency rises and meets to half a per cent at 100 Hz.

A second filter steadies the reading and leaves it low #

A reading that swings by ±20 per cent is not presented to anybody. Meters smooth their display; data acquisition averages its readings; a converter feeding a slower loop is filtered by the loop. Each of those is a filter after the square root.

The second filter does what it is for: the ripple is cut by twenty-five times and the display settles. Its mean does not move at all, and it cannot. A linear filter with unit gain at direct current passes the mean of its input through untouched, so whatever was below the truth before it is below the truth by the same amount after it. The reading that was a wobbling 0.88 per cent low on average is now a steady reading 0.88 per cent low — which is the worst form the error can take, because nothing about it looks wrong.

The only filtering that reduces the bias is filtering before the root. A longer time constant in the averager itself shrinks k and the bias with its square; a filter after the root shrinks nothing but the ripple. The difference between the two is invisible in the reading’s steadiness and is the whole of its accuracy.

There is a second way, and it is exact. An average of the square taken over a whole number of periods is the mean square exactly, with no ripple to take a root of, which is what the filter an average is calls a null: a rectangle of whole periods rejects the square’s line at twice the frequency completely. A sampling converter that computes the sum of squares over whole cycles has no bias of this kind at any frequency, and trades it for needing to know how long a cycle is.

The shape decides how much averaging is enough #

A sine’s square has one line. Other waveforms’ squares have more, and the error is set by how much a waveform’s square moves rather than by how much the waveform does.

The square wave is the instructive case. Its value is ±1 and its square is 1 at every instant, so the averager has nothing to smooth, the root is taken of a constant, and the converter is exact at any time constant down to none. The waveform with the largest possible swing between its extremes is the one the converter has no trouble with at all.

The rectifier current is the opposite case, and the same one what a meter multiplies by found the averaging meter misreading by a form factor. A current drawn in two narrow pulses a cycle has a square that is nearly all ripple — zero for most of the cycle and large in between — so the averaged square moves a great deal for a given averager, and the converter needs more averaging to read it. At 60° of conduction the one per cent edge is 1.80 times a sine’s; at 20° it is 2.41.

Which of those a converter meets is decided by somebody else’s reservoir capacitor. The direct voltage that is a sawtooth finds a rectifier’s conduction interval shortening as its reservoir grows, so a supply redesigned with a larger capacitor, for less ripple on its own output, draws a current that a true-RMS converter needs more averaging to read. The averaging meter’s form factor and the true-RMS meter’s bandwidth both depended on the same design choice, and this is the third instrument error to arrive from it.

A lowest-frequency specification is quoted for a sine. On the currents that the current that sizes the transformer measures — root-mean-square values three times the direct current, crest factors of thirteen — it is optimistic by more than a factor of two.

The narrow pulse keeps the shape the sine had — ripple falling as the frequency, bias as its square — with both edges pushed up together: the bias edge by 2.41 times and the ripple edge from 39.8 to 111 Hz, 2.79 times. So the distance between the band where a reading visibly swings and the band where it is biased stays about the same for every waveform, between twenty-one and twenty-five times, and the place the bias hides under the swing is the same place. What moves is where the whole picture sits on the frequency axis, and it moves up for exactly the currents a rectifier draws.

A heater switched in whole seconds #

The frequency that matters to the averager is not always the one printed on the supply. A burst-fire heater controller switches the mains on for some number of whole cycles and off for some number, so that the element sees 50 Hz and the averager sees the bursts. The square of that current is a 50 Hz sine squared and gated, and its slow part is a square wave in power — the power itself switching between nothing and twice its mean — which is the case the averager has the most trouble with rather than, like a square wave in current, none.

A 100 ms averager reading a heater run at half power in bursts of five cycles on and five off reads it 1.00 per cent low. With bursts of 25 cycles it reads 11.97 per cent low, with 50 cycles 19.60 per cent, and with 250 cycles 27.33 per cent, on its way to the limit a reading that followed each burst exactly would give: $\sqrt 2$ times the truth for half the time and nothing for the other half, which averages to 0.707 of it and is 29.3 per cent low. At that extreme a display’s second filter produces a steady reading of seven tenths of the heating current, from a heater doing nothing unusual, and the averaging time needed to read it properly is set by the controller’s burst period rather than by anything about the mains.

Two converters that are one converter #

The other common form of true-RMS converter does not square and then root. It divides: its output $y$ is fed back, the input’s square is divided by $y$, and the quotient is averaged, so that in steady state the average of $x^2/y$ equals $y$ and $y$ is the root-mean-square value. The reason usually given for it is range, since the quotient spans the input’s range rather than the square of it.

Its averaging is not a different kind of averaging. The equation is $y' = (x^2/y - y)/\tau$, and writing the output as the root of $z$ turns it into $z' = 2(x^2 - z)/\tau$ — the explicit converter’s averager with the time constant halved. The implicit converter is the explicit one at $\tau/2$, exactly, for every input.

So at the same capacitor the implicit form is four times as far low far above its corner and one per cent low below twice the frequency, and it settles in half the time, which is the other face of the same factor of two. Nothing about the division helps or hurts the bias beyond that; two converters of the two kinds compared at equal averaging have identical low-frequency errors. A comparison of their specifications that does not say which of them divides is a comparison of two capacitors.

The march is worth a sentence, because the identity is exactly the kind of thing that ought to be measured rather than accepted. The implicit equation has the output in a denominator, which is a different equation to integrate from the explicit one — stiffer where the output is small, and with no square root anywhere — and it arrives at the explicit converter’s answer at half the time constant to 3.7 × 10⁻¹³ of the reading.

A reading falls more slowly than it rises #

The averaging time also decides how long a reading takes to arrive after the input changes, and here the square root does something else.

The averaged square relaxes exponentially between the two squares, 0.25 and 1, and it relaxes the same way in both directions. What differs is the target. A reading within one per cent of 1 needs the square within about 2 per cent of 1, which is 2.7 per cent of the gap between the squares; a reading within one per cent of 0.5 needs the square within 2 per cent of 0.25, which is 0.67 per cent of the same gap. The fall has a target four times narrower in the quantity that is actually relaxing, and it takes 1.379 times as long to hit it.

So the settling time of a true-RMS meter is two numbers, and the one a user meets when a load is switched off is the longer. For a converter with a 100 ms averager, a reading that halves is within one per cent after half a second, and one that doubles after 363 ms.

It is a small instance of the complaint the corner that says nothing about an edge makes about a coupled input: one time constant printed on a specification describes a response to sinusoids, and the response to a change is shaped by whatever the instrument does after its filter. It also makes a reading taken just after a load is switched untrustworthy in a known direction — too high for a while after a fall, too low after a rise — during the few time constants that the first cycle, which no steady state contains would call the only interval in which anything interesting happened.

How the numbers were obtained #

Every steady state is computed exactly for the converter’s own equation. The input is sampled 4,096 times a period and its square joined by straight lines; each interval is then solved in closed form, one period becomes an affine map of the averaged square onto itself, and the periodic solution is its fixed point, so nothing is marched in from a start-up and nothing is truncated. The second route integrates both converters’ equations forwards with a fourth-order rule, stepping on the sample boundaries, from the true reading until twelve time constants and at least three periods have passed, and averages the reading over two whole periods. Each one per cent edge is found by bisection on the exact steady state, and each settling time by marching a step and held against the closed form $\tau\ln\big((r_0^2 - r_1^2)/(r_e^2 - r_1^2)\big)$, for a step from a reading $r_0$ to $r_1$ that counts as settled once it passes $r_e$.

What it does not say #

It treats a one-pole averager and nothing else. Converters use multi-pole averagers to trade settling for ripple, and a second pole before the root does reduce the bias, where one after it does not; how much, for a given settling time, is a different measurement. The squarer and the root are ideal, so none of the bandwidth the true-RMS meter ran out of is here, and nor is the offset that limits a converter at small inputs.

It says nothing about a thermal converter, which averages the heating in a resistor and compares it with a second heater rather than taking a root of an averaged square, and whose low-frequency error has a thermal time constant in it instead.

And every input is deterministic. A true-RMS reading of noise has an averaged square that ripples at random rather than at twice a frequency, and the same concavity applies to it.

Still open: noise, whole cycles, and a heater #

A true-RMS reading of noise. The averaged square of a white noise voltage has a random ripple whose power is set by the ratio of the averager’s noise bandwidth to the noise’s own, and the root pulls its mean low by an eighth of that power just as it does a sine’s. If the arithmetic carries over, a converter reading noise of a kilohertz of bandwidth through a 100 ms averager is low by a part in a few thousand, and a narrow-band noise measurement is low by much more. That is a prediction from the series above rather than a measurement, and the seeded measurement is the next distinct question on this subject. For flicker noise the ratio has no floor at all, which is where the corner where averaging stops working found averaging itself running out.

A converter that knows the cycle. Summing squares over whole periods removes the bias exactly and moves the error into the period estimate. How far a sampling converter’s reading moves when its assumed cycle is one per cent long — which is the same one per cent that leaves a mean forty decibels short of its null — would put a number on what the exact method costs when the frequency drifts.

The heater and the thermocouple. A thermal converter averages power in a mass and never takes the root of a rippling quantity, and it is what calibrates every other instrument. Solved as a thermal network driven by a slow sine, it would say whether its lowest frequency is set by ripple in temperature — which falls only as the first power of frequency — or by something it shares with the converters here.