The amplitude nothing linear predicts

The previous essay left a circuit whose poles are in the right half plane, whose envelope is multiplied by 1.88 every cycle at the gain drawn here, and whose amplitude at any time in the future is therefore a millivolt times 1.88 raised to the number of cycles that have passed. That expression has no upper bound in it. Linear circuits do not have amplitudes; they have gains, and a gain applied to a growing thing grows.

So the amplitude of a real oscillator is set by something that is not in the linear description at all, and this essay measures it.

The mechanism, in one sentence #

The feedback resistor Rf sets the gain, and across it sit two diodes in opposite directions, each in series with a resistor of its own. While the output is small the diodes are open circuits, the gain is 1 + Rf/Rg, and the poles are where the previous essay put them. As the output grows the diodes begin to conduct on the peaks, which puts their series resistors in parallel with Rf and reduces the gain — but only near the peaks of the waveform, and only in proportion to how far past conduction the peaks reach.

The amplitude settles where the gain averaged over one cycle has been pulled back to exactly three. Not the instantaneous gain, which is above three near the zero crossings and below it near the peaks; the average around the loop over a whole period.

That is a property of a waveform. It is not a property of a bias point, and no direct-current analysis of this circuit returns it — the direct-current solution is that everything is at zero, which is a perfectly good equilibrium and an unstable one.

The machinery this needed #

The site had two kinds of solver and this circuit needs both at once.

response.integrate marches a netlist forward in time by the trapezoidal rule: each capacitor becomes a conductance and a current source that remember the previous step, and the resulting resistive network is solved once per step. It is what a circuit simulator does, and it is linear.

A bias point is a solution, not a choice is what that half is for: the phrase “the diode drops 0.7 volts” standing in for the root of a transcendental equation, which solved properly is 0.692544 V from a five-volt supply through a kilohm and 0.754459 V from forty-eight volts through the same kilohm.

semi.operatingPoint runs Newton’s method on a netlist: each nonlinear element is replaced by a conductance and a current source that make it agree with its own law at the present guess, and the linearised network is solved; repeat until the guesses stop moving. It is what a bias solver does, and it is one instant.

march is the two joined. Every capacitor and inductor becomes its trapezoidal companion, every nonlinear device becomes its Newton companion, and the resulting purely resistive network goes to the same solveDC as everything else on this site. One Newton loop per step, warm-started from the previous step’s device voltages — which is what takes it from about nine iterations a step to two.

How it is checked #

A new solver is worth exactly as much as the checks on it, and there are two here of quite different kinds.

With the nonlinearity removed it must be the old integrator. Marched with an empty device list, this routine is the same trapezoidal rule applied to the same netlist by different code, so the requirement is not a tolerance but agreement to the last digit. On a four-hundred-step RC step response the worst disagreement is $1.2\times10^{-15}$ volts.

It was not, at first. The first version was 6.2 millivolts out — a relative error of six parts in a thousand, small enough to look like a discretisation difference and large enough to be a bug. It started every capacitor’s current at zero along with its voltage, and an uncharged capacitor across a resistor fed from a volt carries a milliamp at the first instant. The fix is to substitute a voltage source for each capacitor at step zero and read the current off its branch, which is exact.

Kirchhoff’s current law at every step, rebuilt from the element laws. Not from the companion models the step used: each resistor contributes v/R from the solved node voltages, each capacitor the current its own discrete relation gives from the two stored states, each diode its own exponential at its own solved voltage. The sums must vanish at every node, and the worst residual over the runs in this field is $6\times10^{-14}$ of the largest branch current.

It is worth being exact about what that check cannot do, because a check whose limits are not written down gets trusted past them. It cannot catch a wrong rule. The trapezoidal relation is the definition of the discrete solution, so a first-order companion model would satisfy it perfectly while marching a slightly different circuit. What catches that is a third check: halve the step and watch the answer change by a quarter as much.

Four is what a second-order rule gives and two is what a first-order one gives. The ratios are 3.94 and 3.98, converging on four, so the rule is the one it is supposed to be.

What it settles at #

At the gain drawn in the figure — k = 3.2, so a loop gain of 1.067 while the diodes are off — the output settles at 858 millivolts of amplitude, reaching within one per cent of it by the ninth cycle.

Across the gain slider:

The right-hand column is the next essay. The middle one is this one, and the fact worth extracting from it is how flat it is: a fortyfold change in the excess gain moves the amplitude by a factor of 1.51.

That is the whole point of the arrangement. The amplitude is set by where the diodes start to conduct — which is a property of an exponential and of the resistors around it — and the gain resistors decide only how hard the diodes have to work to pull the average back to three. A circuit whose amplitude depended on the gain as sharply as the growth rate does would be no better than the linear one.

The trap this measurement has #

The measurement is “run it and see where it stops”, and there is a way for that to be wrong which is worth recording because the first version of the sweep above fell into it.

At k = 3.02 the envelope grows by 1.021 a cycle. Starting from a millivolt and heading for 700, it needs $\ln(700)/\ln(1.021)$ — about 315 — cycles to get there. A forty-cycle run stops in the middle of the growth and reports the envelope where it stopped: 51 millivolts, against a true settled amplitude of 703. A factor of fourteen, with nothing about the number to say it is wrong. The waveform looks like a clean sinusoid, the residual is at $10^{-14}$, and the answer is the answer to a different question.

So the run length is now computed from the gain rather than fixed — $6.55/\pi(k-3)$ cycles for the growth plus forty to settle — and every run reports whether its envelope was still moving over its last eight cycles. The gate checks both halves: that a deliberately short run at k = 3.02 declares itself unsettled, and that a long one does not.

This is the third time the collection has met a measurement that is a limit rather than a value. A ladder is not a line is the clearest of the three: a transmission line is introduced as the limit of a chain of inductors and capacitors as the number of sections goes to infinity, which is true and gives entirely the wrong impression of how close a finite chain gets — forty sections still ring through every plateau by five per cent, and extrapolating the fitted convergence puts one per cent at about nine hundred and sixty. The truncated reconstruction needed the same treatment across term counts; and here the limit is in time.

What the three have in common is that the wrong answer is well formed. A forty-section ladder produces a clean waveform, a forty-cycle run produces a clean sinusoid, and neither carries any mark of being an approach rather than an arrival. The repair in all three cases is the same and it is not a tighter tolerance: it is a second measurement at a different value of whatever is being taken to a limit, and a rule that reports the trend rather than the endpoint.

What the linear model is still for #

None of this makes the pole calculation useless, and it is worth being precise about which questions each half answers, because the split is clean.

The linear model gives the frequency and the growth rate. The envelope in the figure’s lower panel follows the poles’ $\exp(2\pi\sigma/\omega_d)$ while it is small, and the gate checks the fit over the cycles that are actually small — which at a large excess gain is the first two and no more.

The nonlinear model gives the amplitude. And nothing else gives it: there is no operating point to find, no transfer function to evaluate, and no small-signal expansion, because the quantity being measured is the size of the excursion the small-signal expansion is about.

That is a cleaner division than most of this collection’s boundaries, and it is why the field needed the marcher rather than merely benefiting from it.

Where the mechanism runs out #

The limiter has a range of its own, and past it nothing is setting the amplitude.

The shunt resistors in series with the diodes decide how much the gain falls once they conduct. At 10 kΩ against a 22 kΩ feedback resistor, hard conduction takes the gain from 3.2 to about 1.7, which is well below three and stops the growth firmly. At 47 kΩ it takes it to about 2.7 — still below three, but only just — and the amplitude at k = 3.8 runs to 3.17 volts instead of 1.06, with the distortion no longer rising with gain at all.

The gate holds both of those as claims, because they are the boundary of this mechanism rather than of the oscillator: a limiter more than about twice the feedback resistor stops setting the amplitude, and the tidy trade the next essay is about stops holding there too.

In a real circuit the amplitude would not run to three volts; it would run to the supply rails, and then the limiting would be done by the amplifier’s output stage saturating. That is also a nonlinearity and it also produces a limit cycle, and the reason nobody designs an oscillator that way is entirely in the next essay.

The step that is too big measures the boundary that limiter would be working against, and the number is smaller than the phrase “the rails” suggests: a real amplifier’s output can only move at a fixed rate, so a linear circuit’s scaling — double the input and the output doubles, exactly — stops holding at about eighty millivolts for an ordinary part. No transfer function contains that number, because no transfer function can. So an oscillator limited by its own amplifier is limited by a mechanism that arrives at a tenth of the amplitude the diodes settle at here, and it arrives as a rate limit rather than as a level — which is why its waveform is a triangle at the peaks rather than a rounded sinusoid, and why the harmonics it produces are of a different order entirely.

What the amplitude depends on instead #

If the gain resistors barely move the amplitude, something else must decide it, and it is worth naming what.

The diodes begin to matter where their exponential rises through the current the feedback resistor is already carrying. That current is the output voltage divided by Rf, so the amplitude at which limiting begins scales with Rf; and the voltage at which the exponential arrives depends on the saturation current and on the thermal voltage. Every one of those is a property of a part rather than of a design.

The temperature dependence is the one that shows. Two millivolts a kelvin, and the wrong sign measures it on the element itself and finds 1.828 millivolts per kelvin, downwards — although the thermal voltage in the exponent is rising, because the saturation current rises by nine orders of magnitude across the same range, and the two dependences of opposite sign come out of one solve. A thirty-kelvin rise therefore moves the conduction knee by about 55 millivolts, roughly six per cent of this oscillator’s amplitude, from nothing but the ambient temperature.

That the whole of this essay is quoted at one temperature and does not say so is the general failure the edges that move with the room collects: every boundary in this collection is stated at one temperature, most of them are functions of it, and the small-signal edge alone runs from 5.67 millivolts at −40 °C to 9.69 at +125 — a factor of 1.71, which is the ratio of the absolute temperatures exactly. An amplitude set by a diode’s knee is a boundary of exactly that kind, and the six per cent above is what it is worth here.

Which is why precision oscillators do not use diodes. The classical answer is a small filament lamp in the Rg position, whose resistance rises as it heats, so the gain falls smoothly with the average power rather than abruptly at a peak — a limiter with a time constant of seconds rather than of microseconds, which touches the waveform hardly at all. What it costs is settling time and an amplitude that wanders during a temperature transient, and what it buys is the subject of the next essay.

Three routes to the same number #

There is a way of predicting the settled amplitude without marching, and it is worth stating because it is the classical one and because it agrees.

Treat the limiter as an element whose gain depends on the amplitude of the sinusoid through it — a describing function — and solve for the amplitude at which that gain equals three. For a symmetric pair of exponential diodes it is a transcendental equation and the solution is close to the marched answer wherever the waveform is close to a sinusoid, which at 0.8% of distortion it is and at 12.8% it is not.

This collection prefers the march for a reason that has come up in every field: the describing function assumes the answer is a sinusoid, and one of the things being measured is how far from a sinusoid it is. The march assumes nothing about the waveform’s shape. Where the two agree, the describing function is a fast route to a number; where they disagree, the disagreement is the fifth harmonic and the march is right.

The claim, in one line #

An oscillator’s frequency is a property of its linear part and its amplitude is a property of its nonlinear part, and this is the first circuit in the collection where both halves are needed to say anything at all about what it does.

Both halves of that sentence are qualified further up and further down the ladder. The gain that is exactly one is what the linear part gives and what it does not: a crossing at 3.000000000000, bisected on the netlist, with a growth rate either side of it that reaches the rails in sixty-one milliseconds and no third case in between. What the limiter charges for prices the nonlinear part, and the price is not in the amplitude column of the table above but in the one beside it: across the same fortyfold change in excess gain the amplitude moves by 1.51 and the distortion by 15.4, which is what makes the flatness measured here a design feature rather than an accident.

And the first half of the sentence turns out to be wrong in a way this essay could not have seen. The frequency that is not the formula measures the frequency of the settled limit cycle and finds it 0.771 per cent below the network’s own zero-phase point even with a perfect amplifier — because the harmonics the limiter makes are part of the waveform whose period is being measured. So the nonlinear part sets the amplitude and contributes to the frequency, and the clean division claimed here holds only to first order.

The gate holds the machinery — the exact reduction to the linear integrator, the current law at every step, the second-order convergence — and then holds the two facts about the result: that the settled amplitude rises with gain at every step of the sweep, and that it rises by a factor of one and a half while the excess gain rises by forty.