What the limiter charges for

Everything in the previous essay’s circuit is linear except two diodes, and those two diodes are the only reason it has an amplitude. It follows that they are also the only reason it has harmonics, and that the purity of the output and the stability of its amplitude are bought from the same account.

This essay measures the exchange rate.

Where the numbers come from #

The march returns a trace and the trace has to become a spectrum, and how that is done matters enough to state.

The settled period is measured first, from the interval between the first and last upward zero crossing of the last four cycles divided by the number of them. A record of exactly 512 points spanning exactly one measured period is then resampled out of the trace — cubically, not linearly, because linear interpolation of a sinusoid at two hundred points a cycle puts $1.2\times10^{-4}$ of second harmonic into a perfectly clean wave, which is the same size as some of the distortions being measured here.

That record goes through the same harmonics the semiconductors and power fields use, and Parseval’s identity is checked on it: the mean square of the samples against the sum of the squares of the lines. Over the whole gain sweep the worst gap is $6.5\times10^{-11}$, so no energy has been lost or invented between the waveform and its spectrum.

The even harmonics are absent #

At a gain of 3.2 the lines come out:

The even entries are not small. They are absent: $6.8\times10^{-7}$ of the fundamental is the floor of double-precision arithmetic in a transform of this length, and the gate asserts they are below $10^{-5}$ while the odd ones are above $10^{-2}$ — three orders apart, so the claim cannot be satisfied by a merely quiet even harmonic.

The reason is symmetry and nothing else. Two diodes facing opposite ways make a characteristic that is an odd function of the voltage across it: whatever it does to a positive peak it does identically to a negative one. An odd characteristic driven by a symmetric waveform produces only odd harmonics, exactly, and a single diode instead of a pair would fill the even lines in immediately.

What a pair cancels, and what it only halves is the first time, and it is worth reading beside this one: a differential pair’s transfer characteristic is an odd function, so its second harmonic comes out at $10^{-16}$ of the fundamental — the arithmetic’s own floor — while its third comes out at exactly half the single stage’s, which is a reduction and not a cancellation. The same distinction applies here, and it is why the odd lines in the table above are quoted as measurements and the even ones as floors.

This is the second time the collection has met that argument, and it is worth putting them side by side because they are the same theorem about two different circuits.

The trade, in two exponents #

Fitting the sweep across the gain slider gives the exchange rate as two power laws in the excess gain k/3 − 1, which runs from 0.67% to 26.7% — a factor of forty:

$$\text{amplitude} \propto (\text{excess})^{0.110}, \qquad \text{distortion} \propto (\text{excess})^{0.745}$$

Over that fortyfold range the amplitude moves by a factor of 1.51 and the distortion by 15.4.

Neither exponent is a round number and neither is claimed to be one; they are fits over 1.6 decades, quoted because the ratio of the two is the design fact. Distortion is nearly seven times more sensitive to the gain than the amplitude is, so a design that wants a clean output should run as close to the critical gain as it dares, and one that wants to start quickly should not.

And why “as close as it dares” is a real constraint #

The previous essay’s start-up arithmetic sets the other end of the trade.

At k = 3.02 the envelope grows by 1.021 a cycle. Starting from the Johnson noise of the resistors — a few hundred nanovolts — and heading for 700 millivolts, that is about 350 cycles, or a fifth of a second at 1.59 kHz. At k = 3.4 it is 25 cycles, or 16 milliseconds. So the distortion and the start-up time are the same trade read in the other direction:

And there is a floor under how close to three a design can sit that has nothing to do with either column: the gain has to stay above three across temperature, ageing and the whole tolerance band of two resistors, or the oscillator does not start at all. The gain that is exactly one is where that arithmetic is made — the poles reach the imaginary axis at 3.000000000000, an equality rather than a range, and three per cent either side of it multiplies or divides the envelope by 1.0987 every cycle. A one per cent resistor pair gives ±1.4%, so a design built from one per cent parts must aim at about k = 3.05 to be sure of exceeding three — and 3.05 is 1.9% of distortion, whether or not the particular unit on the bench needed it.

The ±1.4% is itself the optimistic reading of the tolerance, and the tolerance that is not on any part is the essay that separates the three numbers hiding inside it. The worst case of a combination of one per cent parts is one per cent and does not improve with the number of them; the measured spread over a population is 0.29 of a per cent and shrinks as one over the root of the count; and the root-sum-square bound quoted everywhere as though it were a standard deviation is 1.70 of one. A designer choosing the margin above three is choosing which of those three to believe, and the choice is visible in the output as distortion rather than as a failure. Aim at the spread and a few units in a hundred never start; aim at the worst case and every unit that ships is dirtier than it needed to be. There is no setting that is right for both, which is what makes the tolerance a design parameter here rather than an error bar.

The distortion of a diode-limited oscillator is therefore set by the resistor tolerance, which is a strange sentence and a true one, and it is the reason precision oscillators use a slow limiter instead.

What the harmonics are made of #

The odd lines fall in a pattern worth reading. At a gain of 3.2 the third is 5.69% of the fundamental, the fifth 0.98%, the seventh 0.46% and the ninth 0.18% — ratios of 5.79, 2.15 and 2.48 between successive pairs.

That is not the geometric fall of a soft cubic nonlinearity, which would take the third to the fifth by roughly the square of the drive. It is the signature of something closer to a clip: a characteristic that is nearly linear over most of the cycle and turns quite sharply near the peaks produces a slowly falling series of odd harmonics, and the sharper the turn the slower they fall.

Which is a statement about the diode. A diode’s exponential moves 59.5 millivolts per decade of current, so the transition from “not conducting” to “conducting hard” is spread over about a decade of amplitude rather than happening at a voltage — and that is precisely why the waveform is still recognisably a sinusoid with 5.8% of distortion on it, rather than a triangle.

The loop does almost nothing to clean it up #

There is a step in this argument that is easy to skip, and skipping it makes the numbers above look worse than they should. The diodes generate harmonics at the amplifier’s output; those harmonics then go round the loop through the Wien network, which is a frequency-dependent thing, and come back. So what appears at the output is not what the diodes generated — it is what the diodes generated, filtered by whatever selectivity the loop has.

The Wien network’s transfer is

$$\beta(j\omega) = \frac{1}{3 + j\left(x - 1/x\right)}, \qquad x = \omega/\omega_0$$

which is 1/3 at the fundamental and falls away either side. At the harmonics:

Two and a half decibels. That is the whole of the loop’s contribution to cleaning up its own third harmonic, and the reason is that the Wien network is not a resonator. Its selectivity comes from a single first-order section against another, and the effective quality factor of the arrangement is a third — the same 1/3 that sets the gain condition. A network whose peak is that broad is a phase reference and not a filter.

Compare the alternative. An LC tank of quality factor 100 attenuates a third harmonic by

$$\frac{1}{\sqrt{1 + Q^2\left(x - 1/x\right)^2}} = 3.75\times10^{-3}$$

which is −48.5 dB — forty-six decibels more filtering than the Wien network provides, for the same job in the same place in the loop. At Q = 10 it is still −28.5 dB.

That is the real reason a Wien-bridge oscillator’s distortion figure is what it is, and why the same limiting mechanism in an LC oscillator produces an output two orders cleaner. The distortion is generated in the same place in both; only one of the two loops removes it afterwards.

What a spectrum does not carry #

There is a limit to what this measurement says, and it is the same one the noise field records about its own.

Total harmonic distortion is a single number summarising a whole spectrum, and two circuits with the same figure can sound and behave quite differently — a fifth harmonic at 5% is a different object from a fifty-first at 5%, and this measurement adds them in quadrature without comment. When a floor stops being a floor is the collection’s measurement of exactly that distinction, made where it can be watched: the share of a converter’s error power sitting in harmonics of the input rises from 0.29 per cent to 76 per cent as the amplitude falls, while the floor itself does not move at all. The total is the same number throughout and the object it describes has changed completely. An oscillator’s single percentage hides the same information, and this essay’s own table of nine lines is the part worth quoting when it matters.

There is a second summary in this field with the same hole in it. The point the device is never at takes the number a linearity specification is usually given as and finds it is a place on no curve: the third-order intercept is where two straight lines would cross if both went on being straight, and for a differential pair the crossing sits π²/4 — 7.84 decibels — above the largest output the device can produce at any drive whatever. A distortion figure quoted at one amplitude and an intercept quoted at no amplitude at all are the two ends of the same problem, and the measurement between them is a sweep, which is what the table above is.

The other thing a spectrum does not carry is phase noise, which is what an oscillator is usually specified by. The harmonics measured here are deterministic — they are at exact multiples of the fundamental, they are the same on every cycle, and a filter removes them. What a real oscillator also has is a fundamental that is not at exactly one frequency, because the noise in the loop modulates the instant at which each cycle crosses zero. That is a different measurement, it needs a stochastic march rather than a deterministic one, and this field does not make it.

The three ways out, and what each costs #

Since the distortion is generated by the amplitude-setting mechanism and barely filtered by the loop, there are only three places to attack it, and the collection can say something about each.

Make the limiter slower. A limiter whose gain responds to the average power over many cycles rather than to the instantaneous peak of each one barely distorts the waveform at all, because within any single cycle its gain is constant. This is the filament lamp of the classical design, and it is why that design’s distortion is quoted in parts per million rather than in per cent. What it costs is a second, much slower loop with its own stability problem: the lamp’s thermal time constant against the amplitude’s own dynamics, which can and does produce a slow amplitude oscillation — an envelope that breathes at a few hertz on an output that is otherwise clean.

Make the loop selective. Replace the Wien network with a resonator and the harmonics are divided by something like Q rather than by 1.34. This is what every radio-frequency oscillator does, and the cost is that a resonator is not tunable over a decade with a potentiometer.

It has a second cost that the frequency field measures and that the arithmetic above quietly assumed away. The Q the components allow finds that the quality factor is not a design parameter but a ceiling: the reciprocals of the component quality factors add, so the total sits below the smallest of them, and an inductor of 79 beside a capacitor of 1 581 gives a resonator of 75. The worst component decides and the best one cannot help. So the −48.5 dB computed above at Q = 100 is not available from ordinary parts at audio frequencies, where an inductor good enough to reach it would be large and expensive, and the honest figure for a wound resonator in this band is nearer the −28.5 dB quoted for Q = 10. Resonance, and the bandwidth it sets exactly supplies the other half of what a quality factor means here: the half-power bandwidth is $f_0/Q$ to every digit the arithmetic has, so the same number that divides the harmonics is the number that decides how far the frequency can be pulled, and a resonator selective enough to clean the output is one that cannot be tuned.

The obvious escape is to build the inductor rather than wind it, and the filters field has measured what that does. The inductor that is an amplifier presents one henry at a node out of four resistors, a capacitor and two amplifiers — an inductance with nothing magnetic in it, good to within one per cent over three and a half decades — and then finds the thing that matters to an oscillator: its series resistance goes negative at 63 hertz, well inside the band where it is still an excellent inductor. A resonator built around it there does not have a high quality factor; it has a loss of the wrong sign, and starts on its own noise. Which is to say that the synthetic resonator does not merely fail to solve this essay’s problem — at the wrong frequency it becomes an oscillator whose amplitude is set by whatever nonlinearity it reaches first, and this essay is about what such a limiter charges.

Take the output from somewhere else. The waveform at the amplifier’s output is the distorted one; the voltage at the junction of the Wien network’s series arm has been through the network once more and is therefore cleaner by the table above — 2.5 dB on the third harmonic and 5.5 on the fifth. It is a small improvement, it costs a buffer, and it is worth mentioning mainly because it shows the distortion is not a single number attached to the circuit but a quantity that differs from node to node.

That last point is the one this collection keeps arriving at. A measurement is of a node in a network, not of a device, which is the instruments field’s whole subject.

The gate #

Four claims, and the third is the one with teeth.

The spectrum holds all of the waveform’s energy. Parseval’s gap below $10^{-9}$ across the whole sweep, so the lines are the waveform and not an artefact of the resampling.

Distortion rises with gain at every step of the sweep, monotonically, over six gains.

Every even harmonic is at the arithmetic’s floor and every odd one is three orders above it. A single diode instead of a pair breaks this immediately, which is what makes it a test rather than an observation.

Distortion moves an order further than amplitude over the same fortyfold change in the gain. The whole essay is that sentence with numbers attached.

The third of those is the one a design can be built on, because it is a symmetry rather than a magnitude. A quantity that is small can be made smaller by care and larger by carelessness; a quantity that is absent by symmetry stays absent under every change that preserves the symmetry, and returns the moment one diode is a different part number from the other. That is the difference between a measurement and a test, and it is why the even harmonics are worth four lines of a table that would otherwise read as seven zeroes.