The peak that only has a bound

Two solves that add, and the one that does not drew the line this sequence of essays is about: every node voltage and every branch current in a linear network is the sum of the per-source solves, to the last bit of a double, and every power is not. What made that essay a measurement rather than a warning is that the failure has a closed form. The gap between the true power and the sum of the separate powers is $2\,\mathrm{Re}(I_1 I_2^{*})R$, exactly, at every amplitude ratio and every angle, so the discrepancy is a quantity rather than an error bar.

There is a second class of quantity that does not superpose, and it is worse in a specific way: it has no cross term at all. Only an inequality.

The one that is exactly additive #

Start with the quantity that does work, because the essay before it already established why and this is the case it applies to most cleanly.

A fundamental and a harmonic are orthogonal over a period: the integral of their product is exactly zero, because one completes a whole number of cycles while the other completes a different whole number. So their mean squares add, and the root-mean-square value of the sum is

$$V_\mathrm{rms} = \sqrt{V_1^2 + V_2^2}$$

which for ten volts and 3.333 is 7.454 volts. Measured across a hundred and eighty-one relative phases, it is 7.454 volts at every one of them, to better than a part in a million.

That is that essay’s quadrature case, and it is worth noting how much stronger it is here than there. There, two sources at the same frequency had to be ninety degrees apart for their powers to add, and the slider had one setting at which the shortcut was exact. Here the two components are at different frequencies, so the cross term averages to nothing over every whole number of both periods and the addition is exact at every setting of the slider. There is no phase at which a harmonic’s power fails to add to a fundamental’s.

Which is the whole reason a distorted current’s heating can be computed from its harmonic amplitudes — what the power field calls the total harmonic distortion arithmetic, and what the current that does no work is careful about in a different direction. Every harmonic’s contribution to the heating in a resistance is independent of every other’s, including of their phases, so a spectrum is a complete description for that purpose.

The one that only has a bound #

Now ask the same signal for its peak.

The peak of a sum is bounded above by the sum of the peaks — the triangle inequality, and it is the only general statement there is — and where it sits inside that bound is a function of the relative phase:

One root-mean-square value, and a peak that moves by forty-one per cent. The crest factor moves by the same forty-one per cent, because it is the ratio of one thing that moves to one thing that does not.

The two ends of that table are the two shapes anybody working with distorted waveforms would recognise. At 180 degrees the harmonic’s crest coincides with the fundamental’s and the peak is 13.333 volts, which is exactly the sum of the two amplitudes. At 0 degrees the harmonic subtracts at the fundamental’s crest, the top of the waveform is flattened, and the peak has moved off the fundamental’s own crest to somewhere near fifty degrees — where it is 9.428 volts, and where there is no closed form for it at all, because it is the maximum of a transcendental function of the phase.

Which end reaches the bound, and why it is not the obvious one #

That the bound is reached exactly at one end of the sweep is worth being precise about, because the natural way to state it is wrong.

At the fundamental’s crest, $\theta = 90°$, the $n$-th harmonic contributes $\sin(90n + \varphi)$. For a third harmonic that is $\sin(270 + \varphi)$, which is $+1$ at $\varphi = 180°$ — so the two crests coincide in antiphase. For a fifth it is $\sin(450 + \varphi) = \sin(90 + \varphi)$, which is $+1$ at $\varphi = 0$ — so they coincide in phase.

So the end of the sweep at which the peak reaches the bound is decided by the harmonic number modulo four and by nothing else. “The antiphase case is the worst” is a true statement about a third harmonic and a false one about a fifth, and the difference is a property of the arithmetic rather than of the amplitudes. Written as a check on the figure it is the kind of claim that would have survived review and failed a slider, and this one did: the requirement was first written for the third harmonic and refused the fifth.

For a fifth harmonic at the same amplitude ratio the peak runs from 11.61 volts to 13.333 — a spread of fifteen per cent rather than forty-one, with the same 7.454 volts of root-mean-square throughout. So the size of the phase dependence is a property of the harmonic too, and only its existence is general.

What a design pays for each of them #

The two quantities buy different things and it is worth separating them, because a specification usually gives one and needs both.

The root-mean-square value decides the heating, and the heating is what a conductor, a resistor and a transformer are sized for. It also decides what a true-rms meter reads, and what a meter multiplies by is where the distinction between that and an averaging meter’s reading is measured — an averaging meter scaled for a sinusoid reads a distorted waveform wrong by its form factor, which is a third quantity again.

The peak decides the headroom, and the headroom is what an amplifier’s rails, a converter’s full scale and an insulation rating are sized for. Nothing about the root-mean-square value bounds it usefully: a crest factor can be anything from one upwards, and this figure shows it moving by 41 per cent with the amplitudes held fixed.

So a signal characterised by its spectrum is characterised for the first purpose and not the second, and the missing information is the phases. That is an unusual thing for a spectrum to be missing, because a magnitude spectrum is normally treated as a complete description of what a signal does to a linear system — and for a linear system it is, which is that essay’s point about voltages superposing. It stops being complete the moment something asks for a maximum, and the things that ask for maxima are the things that break: a rail, a full scale, a breakdown voltage.

Two of the neighbouring measurements are about exactly that gap. The direct voltage that is a sawtooth finds a rectifier drawing its 157 milliamps of average as a 2.098-ampere pulse — a crest factor of 13.4, which gets worse as the reservoir capacitor is made larger, so the component that improves one quantity degrades the other. And the first cycle no steady state contains finds the first conduction after switch-on at 32.4 amperes against a repetitive peak of 1.23 — a factor of twenty-six that no steady-state spectrum contains at all, because it happens once.

Half a bit, from a phase #

The place this costs something measurable is in front of a converter, because a converter has both quantities in its specification and they are specified against different things.

A converter’s full scale is a peak limit: exceed it and the output clips, which is not a graceful degradation. Its signal-to-noise ratio is an rms quantity: the quantisation floor is $q/\sqrt{12}$ and what it is compared against is the signal’s root-mean-square value. So the useful dynamic range of the pair is the rms value a signal can have with its peak at full scale, which is full scale divided by the crest factor — and this figure’s crest factor moves by forty-one per cent with a phase.

Three decibels between the two ends of the sweep, which is half a bit — and the two signals are spectrally identical. A twelve-bit converter fed the favourable phase performs like a twelve-and-a-quarter bit one and fed the unfavourable phase like an eleven-and-three-quarter, with the same two amplitudes at the same two frequencies in both cases.

Three decibels is a small number and the reason to state it is what it is not available from. It is not available from the spectrum, it is not available from the converter’s data sheet, and it is not available from the signal’s rms value — and a designer chasing the last half-bit of a converter’s performance, which is a thing people do, is chasing a quantity of the same size as this. The digital field’s own the floor a converter sets measures where a converter’s quantisation floor crosses its source resistance’s thermal one — 18.8 bits for a kilohm in a hundred kilohertz — and the half-bit here sits comfortably inside the range where that crossing matters.

It is also the one term in this arithmetic a design can choose. Amplitudes are set by the signal and the converter’s bits by the part, and the relative phase of a harmonic is set by whatever filtering sits in front — so an anti-alias filter chosen for its skirt, which what the filter in front costs prices in clock rate, is also choosing this half-bit without anybody noticing. A Bessel filter’s flat delay leaves harmonic phases where they were; a Chebyshev’s 49 per cent delay variation does not.

Where the phase comes from, when nobody chose it #

The slider on the figure is a free parameter, which is a way of avoiding the question of what sets it in a real signal. Three answers, and they are different in kind.

A nonlinearity sets it. The harmonics a distorting element produces have phases fixed by the nonlinearity’s own shape: a symmetric compression produces odd harmonics in the phase that flattens the crest, which is the 0° end of this sweep and is why clipping reduces the peak-to-rms ratio rather than raising it. That is the one case where the phase is not free and the answer is the favourable one.

A filter moves it. Any network with a frequency-dependent phase — which is every network — shifts a harmonic relative to its fundamental by the difference between the phase responses at the two frequencies. So a distorted waveform’s crest factor changes as it passes through a filter that does not change its spectrum at all. That is worth stating as a sentence because it sounds impossible: a filter flat in magnitude over the band containing both components alters the peak by tens of per cent while leaving every amplitude and the root-mean-square value exactly where they were. The filters field’s own group-delay measurements are measurements of precisely that phase difference.

And two independent sources leave it undetermined. Two converters running at unrelated frequencies, two signals from different equipment, a fundamental and interference: the relative phase drifts, so the peak drifts with it and the worst case is the bound. A design that must not clip has to be built for the bound, and the bound is the arithmetic sum of the amplitudes — which for $N$ components is $\sum |a_k|$ where the heating is $\sqrt{\sum a_k^2}$, and those two diverge as $\sqrt N$.

That last ratio is the useful form of the whole essay. A signal of $N$ comparable components has a worst-case crest factor $\sqrt N$ times a single component’s, and its heating has not changed at all. Sixteen interferers of equal size need four times the headroom of one and put the same total power into a load.

Two routes to a number with no closed form #

The site’s habit is to compute a quantity twice by routes sharing nothing but the netlist, and a maximum is the one quantity on this page where that is awkward — which is worth saying, because it is the reason the figure’s numbers are stated to four figures rather than to twelve.

The root-mean-square value has two routes and they are both exact. Integrate the square of the waveform over a period, numerically; or take the quadrature sum of the amplitudes, in closed form. They agree to a part in a million, and the residue is the quadrature rather than the physics.

The peak has one route, and it is a search. The waveform’s maximum is where its derivative vanishes, which for a fundamental plus a third harmonic is $\cos\theta + 3\cdot\tfrac13\cos(3\theta + \varphi) = 0$ — a cubic in $\cos\theta$ whose roots exist in closed form and are not worth having, because there are three of them and which is the maximum depends on the phase. So the figure sweeps twenty thousand points of a period and takes the largest, which locates the peak to a part in $10^8$ of the period and therefore to about a part in $10^{16}$ of the value, since the function is stationary there.

That is the one case in this collection where a fine sweep is the right instrument rather than a fallback, and the reason is the stationarity: a quantity being searched for at a maximum is quadratically insensitive to the search’s own resolution. The site’s usual complaint about locating a feature on a grid — the loop gain of a three-pole amplifier being bisected rather than read off a sixty-point sweep, because four points near a crossover is several degrees of margin — does not apply here, and the difference is that a crossover is located where a curve is steep.

One consequence worth having: the two ends of the sweep are not equally trustworthy. At 180° the peak is at the fundamental’s own crest, exactly, so the value is the sum of the amplitudes in closed form and is required to a part in $10^9$. At 0° the peak has moved off the crest to somewhere near fifty degrees, there is no closed form, and the figure reports a number from a search. The essay’s two headline numbers are therefore of different kinds — one a theorem and one a measurement — and the check treats them differently for that reason.

What this does not claim #

That the peak is unbounded. It is bounded, tightly, by the sum of the amplitudes, and the bound is reached — so a worst case is always available and is always computable from a magnitude spectrum alone. What is unavailable is the actual peak of an actual signal, which needs the phases.

That the root-mean-square value is always exactly additive. It is additive for orthogonal components, and two components are orthogonal over an interval containing a whole number of both periods. Over an arbitrary window they are not, which is the whole content of spectral leakage and is what the bandwidth a bin is not measures in the noise field. A reading taken over 3.7 cycles of a fundamental has a cross term in it, and its size is a property of the window rather than of the signal.

And nothing here is about a nonlinear network. Both quantities are computed from one waveform. What happens when that waveform drives something whose response depends on its amplitude is that essay’s diode result — three correct solves added giving a wrong answer, with the added diode drop being a voltage no silicon diode has ever had — and a peak is exactly the quantity such an element responds to.

Still open: the filter that changes a crest factor, and the worst case over N #

The filter with no magnitude in it. The claim above that an all-pass network changes a crest factor by tens of per cent while changing no amplitude is stated and not drawn. It is a one-figure measurement — a distorted waveform through a delay equaliser, with the spectrum before and after and the peak before and after — and it would be the sharpest available demonstration that a magnitude spectrum is not a complete description of a signal. The filters field already has the sections for it.

The bound at N components, and how often it is approached. $\sum|a_k|$ against $\sqrt{\sum a_k^2}$ diverges as $\sqrt N$, which is the worst case; what a design needs is how much headroom to allow for a given probability of exceeding it, which for random phases is a distribution rather than a bound. That is a measurement over seeded phases and it is the arithmetic behind every crest-factor allowance in a multi-tone system.

And the nonlinearity that fixes the phase. The section above requires that a symmetric compression produces its odd harmonics in the crest-flattening phase, which is why clipping lowers a crest factor. It follows from the shape of the nonlinearity and it is not solved here — and the case that matters is the asymmetric one, where the harmonics arrive in a phase nothing obvious predicts.

What is checked #

The root-mean-square value is required to be the quadrature sum at every phase, to a part in a million, across a hundred and eighty-one settings. That is the theorem, and requiring it at one setting would say nothing, because the claim is that the phase is absent.

The peak is required not to exceed the bound, which is the only general statement about it, and required to move by more than ten per cent across the sweep, which is the claim that the bound is not a description.

The bound is required to be reached exactly, to a part in $10^9$, at the end of the sweep that the harmonic number modulo four selects — and required to be missed by at least five per cent at the other. Written for the third harmonic alone the first of those said “in antiphase”, which is true of a third and false of a fifth; the measurement refused the fifth, which is how the modulo-four rule got into the figure.

And the variation threshold is ten per cent rather than thirty. A third harmonic moves the peak by 41 per cent and a fifth by 15, so a bound fitted to the third refuses the fifth — the standing defect in this ledger, where a requirement with a number in it turns out to be a fact about the first placement rather than about the family.