Circuits that do a job, and the range they do it over

The delay that is two delays

Modelling one comparator delay applied to both transitions makes the two half cycles equal by construction, and the fix is a change of one line. Made, the duty cycle moves by 0.28 per cent for twenty nanoseconds of skew rather than the 0.40 the obvious expression gives, more hysteresis improves the duty cycle and worsens the volt-seconds at the same time, and ten nanoseconds of skew saturates a hundred-turn core in 231 cycles.

14 min read 8 figures Two routes to a number The constants decide Measured, not tabulated

Assumes: Two thresholds because there is a floor · A boundary in volt-seconds · The band a turns ratio holds over

The period a delay lengthens found that a comparator’s propagation delay does not add twice itself to a relaxation oscillator’s period. It adds 4/(1+β) times itself — 2.65 nanoseconds per nanosecond at a divider ratio of a half — because during the delay the capacitor keeps charging past the threshold it has already crossed, so the next half cycle starts further out and takes longer than the delay alone.

That essay’s own list of what it did not do carries this:

The duty cycle of the relaxation oscillator. One delay is modelled, applied to both transitions, so the two half cycles are equal by construction. A real comparator’s rise and fall delays differ, which is a volt-second imbalance of the kind the magnetics field measured — and the machinery for it is a change of one line.

This is the line. It is one line, it produces three results, and the third of them is in a different field.

Two delays, because a comparator has two

A comparator’s output stage is not symmetric. The transistor that pulls the output up and the one that pulls it down are different devices driven from different places, and a data sheet says so: the low-to-high and high-to-low propagation delays are two separate rows, and on ordinary parts they differ by tens of per cent. Nothing in the oscillator’s arithmetic knows that they are the same number, and nothing in the oscillator makes them one.

Call the difference the skew. A hundred and fifty nanosecond comparator with a fifty nanosecond skew delays one transition by 175 ns and the other by 125.

A comparator's delay, at a divider ratio of 0.50. Each nanosecond of comparator delay adds 2.650 nanoseconds to the period — measured as a slope between two delays, both exact multiples of the marching step — against the 2.667 that 4/(1+β) gives and the 2 that counting the delay twice gives. A 50 ns comparator therefore holds the frequency to one per cent only below 75.0 kHz. — Fig. 1 The rung below’s measurement: the period against one delay applied to both transitions, and the 4/(1+β) slope that says the two half cycles are coupled.

The obvious expression, and how wrong it is

Each half cycle is lengthened by the delay that ends it, so the high time is T/2 + d_up, the low time is T/2 + d_down, and the duty cycle is their ratio. That is the expression anybody would write, and for a skew of twenty nanoseconds on a 2.5 microsecond period it gives 50.40 per cent.

Twenty nanoseconds of skew is 0.28% of duty, not 0.40. computed by solving, not by drawing. The duty cycle of a relaxation oscillator whose comparator takes longer to go one way than the other, marched, against the expression that lengthens each half cycle by its own delay. The two part company immediately and by a constant factor of about 1.42: during a delay the capacitor keeps charging past the threshold it already crossed, so the next half cycle starts further out and takes longer, and the two halves partly cancel. At 200 ns of skew on a 2585 ns period the duty is 52.805 per cent where the expression says 54.004. The open circles are the same quantity in closed form — the overshoot is V(1 − (1 − β)e^(−d/τ)) and the next half starts from it — which the march reproduces to parts in ten thousand. — Fig. 2 The marched duty cycle against the skew, with the obvious expression above it and a closed form on top of the measurement.

The measurement is 50.28 per cent. The expression over-states the imbalance by a factor of 1.43, and it does so by a factor that barely moves across a factor of ten in skew — 0.700, 0.700, 0.700 — so it is not a large-signal correction to a small-signal result. It is a term that was left out.

The term is the same one the rung below found and is the other half of it. During the delay that ends the high half, the capacitor overshoots its threshold; the low half then starts further out and takes longer. So the longer delay lengthens the other half cycle as well as its own, and the two partly cancel.

Writing it out exactly is one line of algebra rather than a model. The capacitor at the instant the drive finally reverses is at

V\left(1 - (1-\beta)\,e^{-d/\tau}\right)

and the next half cycle starts from there. Both half cycles in closed form, marched beside them, and they agree to parts in ten thousand across the whole sweep.

One constant, two results, opposite signs

Differentiate that closed form at zero delay and the coupling comes out as a single number:

\frac{\partial t_{\text{low}}}{\partial d_{\text{up}}} = \frac{1-\beta}{1+\beta}

So the period gains 2·(1 + (1−β)/(1+β)) = 4/(1+β) per unit of symmetric delay, which is the rung below’s result, and the difference between the half cycles gains

1 - \frac{1-\beta}{1+\beta} = \frac{2\beta}{1+\beta}

per unit of skew, which is this one. At β = 0.5 those are 2.667 and 0.667, measured 2.650 and 0.700 — the residual in the second being the finite base delay, since the coupling factor is derived at zero delay and measured at a hundred and fifty nanoseconds of it.

The same (1 + β) in both denominators, and the two go in opposite directions. More hysteresis makes the period-stretch smaller and the duty-imbalance factor larger.

Which means the design trade has two answers

More hysteresis fixes the duty cycle and makes the volt-seconds worse. computed by solving, not by drawing. Two consequences of the same 50 ns of comparator skew, against how much hysteresis the comparator has, each normalised to its value at β = 0.1 (the second is divided by three so both fit one axis). The duty imbalance falls from 1.002 per cent to 0.435, because the period a large β buys grows faster than the coupling factor 2β/(1+β) rises. The volt-second imbalance per cycle rises from 18.4 to 46.5 nanoseconds per volt, for the same reason read the other way: a smaller fraction of a longer cycle is a larger area. Which of the two matters depends on what the square wave drives. — Fig. 3 Two consequences of the same fifty nanoseconds of skew, against the hysteresis fraction. One falls and the other rises.

Raising β from 0.1 to 0.85 takes the duty imbalance from 1.002 per cent to 0.435 — better by a factor of 2.3. It takes the volt-second imbalance from 18.4 to 46.5 nanoseconds per volt per cycle — worse by a factor of 2.5.

Both come from the same fact and it is worth stating plainly: a larger β makes the period longer. A fixed skew is a smaller fraction of a longer period, so the duty cycle is better; and it is a larger area, because the area is the fraction times the period and the period grew faster.

Which of the two is the quantity depends entirely on what the square wave is for. Two thresholds because there is a floor sets β against noise: enough hysteresis to stop the comparator chattering on its own floor. If the output is a clock, the duty cycle is what matters and more hysteresis is free improvement. If the output drives a transformer, it is not.

A threshold crossed once, in 2.0e+2k samples of noise. With no hysteresis the comparator changes its mind 223 times on average and as many as 237, on a signal that crosses the threshold once. The vertical bars are the range over twelve seeds. The expected number of extra transitions falls below a tenth at 5.31 standard deviations — so the hysteresis a threshold needs is set by the noise under it and not by the signal over it. — Fig. 4 What the hysteresis is there for in the first place. The floor sets a lower bound on β, and this essay’s trade sits above it.

The volt-seconds, which no transformer gives back

A transformer integrates the voltage across its primary. A square wave whose two halves have equal area produces a flux that returns to where it started every cycle; one whose halves differ by an area leaves that area behind, every cycle, and nothing removes it. The flux that walks is the essay about what happens next, and its input is a fractional imbalance.

Here the fractional imbalance is not assumed. It is twice the departure of the duty cycle from a half, and it comes from a comparator’s data sheet.

Ten nanoseconds of comparator skew saturates a hundred-turn core in 231 cycles. computed by solving, not by drawing. The square wave from the oscillator above, driving a transformer. Each cycle leaves 7.14 nanovolt-seconds per volt more flux than it takes away at the smallest skew here, and nothing in a transformer removes it — so the magnetising flux walks, and the only question is how many cycles the core's headroom lasts. At 10 ns of skew it is 231 cycles, which at 386 kHz is 598 microseconds; at 200 ns it is 12. The open circles are the closed form the magnetics field derived — the headroom divided by the flux each cycle adds — and the number it is given comes from a comparator's data sheet. — Fig. 5 The number of cycles before a hundred-turn core saturates, against the comparator’s skew. The count is the magnetics field’s own closed form given an electrical number.

Ten nanoseconds of skew is a 0.276 per cent imbalance, which at 386 kHz walks a hundred-turn core into saturation in 231 cycles — 598 microseconds. Two hundred nanoseconds does it in twelve cycles. The closed form the magnetics field derived — the flux headroom divided by the area each cycle adds — predicts every one of them to better than six per cent, and the residual is the last partial cycle.

A 0.276% imbalance, and the 231 cycles it survives. computed by solving, not by drawing. The upper panel is the peak flux density, marched cycle by cycle, under a square drive whose positive half is 0.276% larger in area than its negative half. It does not settle. It walks, by the same area every cycle, and reaches 0.35 T after 231 cycles — 4620 ms at 50 Hz — against a closed form of 230.7. The lower panel is the count against the imbalance, and it rises without bound and never becomes infinite. Halving the drive gives 462 cycles, which is exactly twice: reducing the amplitude buys time and not safety, and there is no amplitude at which this design is inside a limit. — Fig. 6 The walk itself: flux against cycle, for a stated imbalance. The staircase does not come back.

This is what makes the result worth having rather than a curiosity about clock symmetry. A duty cycle of 50.28 per cent is a specification nobody would fail. It is also, on a core with no gap and no balancing capacitor, a fault that takes six hundred microseconds to arrive.

A comparator's delay, at a divider ratio of 0.10. Each nanosecond of comparator delay adds 3.633 nanoseconds to the period — measured as a slope between two delays, both exact multiples of the marching step — against the 3.636 that 4/(1+β) gives and the 2 that counting the delay twice gives. A 50 ns comparator therefore holds the frequency to one per cent only below 55.0 kHz. — Fig. 7 A divider ratio of a tenth. The measured period is 3.633 time constants against the closed form’s 4/(1+β) = 3.636, and the oscillator is within one per cent of that expression below 55.0 kHz. The hysteresis is narrow here, so the comparator spends most of the cycle waiting and very little of it deciding.

A boundary in volt-seconds states the limit for a drive that balances. The point of this essay is that the drive does not, for a reason that is on the comparator’s data sheet and not on the transformer’s.

The even harmonics that were not there before

A square wave with equal halves has odd symmetry — the second half is the first half inverted — and odd symmetry forbids even harmonics exactly. A duty cycle that is not a half breaks the symmetry, and the even harmonics arrive in proportion to the departure.

For a duty D the n-th harmonic goes as |sin(nπD)|/n, so the second harmonic is |sin(2πD)|/2, which is zero at D = ½ and rises linearly away from it. Measured on the marched waveform at the 50.28 per cent above it is 40.7 decibels below the fundamental, against the 41.1 the expression gives — from twenty nanoseconds of difference between two rows on a data sheet. At zero skew the same measurement returns −330 dB, which is the check that what is being measured is the asymmetry.

A comparator's delay, at a divider ratio of 0.90. Each nanosecond of comparator delay adds 2.100 nanoseconds to the period — measured as a slope between two delays, both exact multiples of the marching step — against the 2.105 that 4/(1+β) gives and the 2 that counting the delay twice gives. A 50 ns comparator therefore holds the frequency to one per cent only below 95.0 kHz. — Fig. 8 Nine tenths, the other end of the slider: 2.100 measured against 2.105 predicted, and one per cent held to 95.0 kHz. A wider hysteresis shortens the charging leg and lengthens the useful band, because the propagation delay is a fixed fraction of a shorter period rather than of a longer one. The two delays this page is about scale differently, which is the whole of the result.

The same relationship appears from the other side in what a pair cancels, and what it only halves: a differential pair kills even harmonics because it is symmetric, and a mismatch between its halves lets them back in first order. Symmetry is what suppresses even orders, in a device or in a waveform, and a mismatch of any kind is what restores them.

What one line bought, and what it did not

The change was delay becoming delayUp and delayDown, with both defaulting to the old single number so that every figure below this rung is untouched. Ten lines including the closed form, and the duty cycle measured on the output’s transitions rather than on the capacitor’s crossings — which matters, because those two differ by exactly the delay, so measuring the wrong one returns fifty per cent whatever the comparator does.

The measurement of the duty cycle deserves the same care as the model. It is taken between the output’s own transitions and not between the capacitor’s threshold crossings, because those two sequences differ by exactly the delay — which is the quantity in question — so a duty cycle measured on the capacitor comes out at fifty per cent whatever the comparator does. That is a whole essay’s finding measured away by choosing the wrong node, and it is the first thing this rung had to get right.

One fault was found in the writing and is worth recording because it is the kind that passes. The first version decided which interval was the high one from the parity of the array of transitions, which depends on how many transitions the run happened to contain: the duty cycle came out above a half for some values of β and below it for others, with the closed form saying above for all of them. A quantity that alternates in sign with a parameter it cannot depend on is a bug and not a finding, and the only reason it was caught is that the closed form existed to disagree with it.

How much of it there actually is, and how it is specified

The skews used above are not invented. A general-purpose comparator’s data sheet gives its two propagation delays as separate rows with separate limits, and the difference between the two typical figures is commonly ten to twenty per cent of either. A part specified at 150 ns typical with a 20 ns difference is exactly the case drawn here.

What a data sheet does not give is the skew’s own tolerance, and that is the number this argument needs. The two delays are each specified with a maximum, and a reader who subtracts one maximum from the other gets a bound that is far too wide, because the two are correlated: they move together with supply voltage, with overdrive and with temperature. The honest statement is that the skew is a difference of two quantities specified separately and correlated unknown-ly, which is precisely the situation the tolerance that is not on any part is about — a quantity whose specification exists for its two halves and not for itself.

The overdrive dependence is worth one sentence because it couples back into this oscillator. A comparator’s delay falls as the overdrive rises, and in a relaxation oscillator the overdrive at the threshold is the capacitor’s own slope, which is not the same on the two transitions unless the charging and discharging resistances are equal. So even a comparator with no intrinsic skew acquires one from the circuit it is in, and the mechanism is the same one the gain that is exactly one uses to set an oscillator’s amplitude: the loop’s own state decides the parameter that decides the loop.

What is not in this model

No asymmetric slew rate. The comparator’s output is a step here, delayed. A real one has a rise time and a fall time that differ as well, which adds an area of its own — of order the transition time times the amplitude — that is a second imbalance with a different dependence on frequency.

No temperature. Both delays move with temperature and they do not move together, so the skew is itself a function of temperature and the walk that takes six hundred microseconds at 25 °C is a different number when the part is warm. That is the same gap the inductance the current decides names for the core’s own saturation flux density, on the other side of the same interface.

And no balancing. The standard cures are a series capacitor, which forbids a direct component outright, and current-mode control, which measures the magnetising current and corrects it. Neither is modelled; what is modelled is what happens with neither. The band a turns ratio holds over is where the transformer’s own limits are, and the walk above reaches the lowest of them from a direction the transformer has no way to see.

What a measurement of this looks like

The duty cycle is easy to measure and the volt-second imbalance is not, and the gap between those two statements is where this failure lives.

A counter with a duty-cycle function reads 50.28 per cent in a second and is accurate to a few parts in ten thousand, so the electrical half of the argument is a bench measurement. What it cannot see is whether the number matters, because that depends on the transformer and on the core’s remaining headroom, neither of which is on the oscillator’s schematic.

The magnetic half is measured by putting a current probe in the primary and looking at the magnetising current rather than at the voltage. A balanced drive gives a triangle centred on zero; an imbalanced one gives the same triangle walking upward, and the walk is over in a few hundred microseconds, so a slow acquisition sees only the end state — a current spike once per cycle, which looks like a shorted turn or a fast diode and is neither.

That timescale is the reason the fault is diagnosed wrongly. Six hundred microseconds is long against a switching period and short against anything a person watches, so the transformer appears to saturate immediately on start-up, which points at the design rather than at a twenty-nanosecond asymmetry. The first cycle, which no steady state contains is the essay about the other transient of the same kind, and its lesson applies here in reverse: that one is a genuine start-up effect that a steady-state analysis cannot see, and this one is a steady-state effect that looks like a start-up one.

The habit this belongs to

A model with one delay in it where the object has two is not an approximation with a small error. It is a model with a symmetry the object does not have, and a symmetry that is imposed rather than measured suppresses a whole class of result: the duty cycle was exactly a half, the even harmonics were exactly zero, and the flux came back exactly to where it started, all by construction.

The distinction matters because the collection has the opposite case as well, and the two look identical in a table. What the limiter charges for reports every even harmonic of an oscillator’s output at the arithmetic’s floor, and there the zero is real: two diodes facing opposite ways make a genuinely odd characteristic, and breaking the symmetry — one diode instead of two — fills the lines in immediately, which is what makes the claim a test. Here the zeroes were the model’s rather than the circuit’s, and no experiment performed on the model could have distinguished them. The only thing that could was asking what the object has that the model does not.

The period a delay lengthens knew this and said so, in one paragraph, with an estimate of what the fix would cost — two delays instead of one, marched separately, with the difference between the half cycles reported rather than their sum, which it described as a change of a line because the delay is applied by a closure that already sees which transition is pending. The estimate was right. What it could not know was that the fix would produce a number in the magnetics field, which is the argument for writing the paragraph down rather than remembering it.

It is worth being precise about which of that rung’s results survive the change and which do not, because the answer is nearly all of them. The period is the sum of the two half cycles, and the sum is unmoved by a skew to first order — so $4/(1+\beta)$ times the mean delay remains the right expression for the period, at every divider ratio, and the frequency edges computed from it stand. What was wrong was not a number but an absence: a quantity that had been reported as exactly zero because the model could not represent it. Those are the ones worth going back for, and they are hard to notice precisely because a zero looks like a result.