The frequency that is not the formula

An oscillator is usually specified by its frequency, so it is worth asking whether the frequency is the one the design equation gives. It is not, and this essay measures the two separate reasons and finds that only one of them can be bought off.

The formula is exact about the network #

The zero-phase frequency of the Wien network is $1/2\pi RC$, and there is nothing approximate about it. The network’s transfer is

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

whose imaginary part vanishes at $x = 1$ exactly, where the magnitude is exactly 1/3. Both are algebraic identities, both are reproduced by the solver to the last digit, and neither contains an approximation to be violated.

What the formula is not exact about is the circuit, which contains two other things.

Which frequency is being measured #

Before either mechanism, it is worth saying what the measured number is, because on a waveform that is not a sinusoid there is more than one candidate and they are not obviously the same.

The measurement here is the reciprocal of the period: the interval between the first and last upward zero crossing of the last four cycles, divided by the number of intervals, with each crossing instant interpolated between the two samples either side of it rather than taken as the later of them.

The alternative would be the position of the fundamental in the spectrum. For a genuinely periodic waveform those two are the same number by definition — the fundamental of a waveform of period T is at 1/T, and the harmonics are at multiples of it — so there is no ambiguity to resolve, only a choice of which is cheaper to measure. The period is, and it does not need a coherent record.

What it does need is a settled limit cycle, and the previous essay’s flag covers that: a run still growing has no period to measure, and its zero crossings are those of an expanding waveform whose successive intervals are not equal. The interpolation matters as well — at two hundred steps a cycle, taking the later sample instead of interpolating would put a quarter of a per cent of error into every crossing, which is a third of the total effect this essay is about.

The first mechanism: the amplifier has phase of its own #

The Barkhausen condition is that the total phase round the loop is zero, and the loop contains an amplifier as well as a network. An amplifier with a finite gain–bandwidth product contributes a lag, so the network has to contribute a lead of the same size, and it does that at a frequency below its own zero-phase point.

The size is estimable. A one-pole amplifier closed for a gain k has a closed-loop lag of about $k f_0/\mathrm{GBW}$ radians at the oscillation frequency; write $\rho = \mathrm{GBW}/f_0$ and that is $k/\rho$. The network’s phase slope near $x = 1$ is $-2/3$ radians per unit of x, so restoring the balance needs

$$\Delta x \approx -\frac{3}{2}\cdot\frac{k}{\rho} = -\frac{4.8}{\rho}$$

for the k = 3.2 used here. The measurement gives 4.45/ρ, seven per cent below the estimate, and the product of shift and ratio is constant to 6.7% over two decades of ρ — which is what makes it a 1/ρ law rather than a coincidence at one point.

The right-hand column is the left-hand one with the ideal amplifier’s answer subtracted, and the subtraction is what makes the two mechanisms separable at all. It is only available because the collection can solve the same circuit with a nullor — an amplifier with infinite gain at every frequency, which is an element and not a limit.

Two other essays make that subtraction legitimate rather than convenient. The ideal amplifier, and where it stops being one measures the part being subtracted from: a closed-loop gain already a tenth of a per cent low at direct current, one per cent low by 1.35 kHz, and with no loop gain left above 10 kHz — so the lag this essay attributes to the amplifier is the same quantity that essay reports as a magnitude error, read at ninety degrees to it. And the phase the magnitude already knows is the reason the two readings are not independent: for a minimum-phase network the phase is fixed everywhere by the magnitude through Bode’s integral, returning 39.289 degrees against a solved 39.289 and tracking the whole curve to two hundredths of a degree. An amplifier’s gain–bandwidth roll-off is minimum phase, so its lag at the oscillation frequency is not a second specification. It is the first one, restated.

That is worth having because it says which part of the 1/ρ law is a property of the part and which is a property of this circuit. The lag is forced by the roll-off and could not have been made smaller at a given gain–bandwidth; what the circuit contributes is the phase slope of the Wien network, −2/3 radians per unit of x, which decides how much frequency has to move to supply a given lead. A network with a steeper phase slope would convert the same amplifier lag into a smaller frequency error, which is the whole of what a high quality factor buys an oscillator.

The second mechanism: the waveform is not a sinusoid #

Set the amplifier to a nullor, so that it contributes no phase at any frequency, and the frequency is still not $1/2\pi RC$. At a gain of 3.2 it is 0.771 per cent low, which is 12.3 Hz out of 1591.5.

This is the limiter, and the reason is that the whole apparatus of transfer functions and zero-phase frequencies is about sinusoids. The waveform going round this loop is not one — it carries 5.8% of third harmonic at this gain — and the period of a periodic waveform that is not a sinusoid is not decided by the phase of the network at the fundamental alone. The harmonics go round the loop too, they are shifted in phase by the network by their own amounts, and they come back to add to the waveform at instants that are not where a sinusoid would have put them. The period that satisfies the whole loop is a little longer than the one that satisfies the fundamental.

The size follows the distortion. Fitting across the gain slider:

$$\left|\frac{\Delta f}{f_0}\right| \propto \mathrm{THD}^{1.84}$$

over 1.2 decades of distortion — near enough to a square that the mechanism is recognisable as a second-order effect, and far enough from it that the exponent is quoted as measured rather than rounded.

Where the two cross #

The useful number is the ratio at which the two mechanisms are equal, because it says when a faster amplifier stops being the answer.

At k = 3.2 the limiter’s floor is 0.771% and the amplifier’s share is 4.45/ρ, so they are equal at

$$\rho = \frac{4.45}{0.00771} = 578$$

which for this circuit is a gain–bandwidth product of 920 kHz. Below that, doubling the amplifier’s speed halves the error. Above it, doubling the amplifier’s speed changes the frequency by less than a tenth of what the diodes are already doing, and the design has to attack the distortion instead.

The crossing moves with the gain, because the floor does. At k = 3.05 the floor is 0.0968% and the crossing is at ρ = 4600; at k = 3.4 the floor is 1.84% and it is at ρ = 240. So the same slider that trades purity against start-up time in the previous essay also decides how fast an amplifier is worth buying.

Where the first mechanism stops being a shift #

The 1/ρ law is a small-signal statement about a phase lag, and it has a range like everything else here. Walking the slider down past the bottom of the table finds the end of it.

Between ρ = 8.5 and ρ = 9 the circuit stops oscillating at all. Not gradually: at ρ = 9 it settles to two thirds of a volt in a couple of hundred cycles, and at ρ = 8.5 the envelope is heading for zero from the millivolt it was started with.

The reason is that the shift is not free. Moving the frequency down the Wien network’s skirt costs loop gain — the network passes less than 1/3 anywhere except at $x = 1$ — and the amplifier’s own magnitude is falling as well. Below about nine times the oscillation frequency the two together take the loop gain under one at every frequency, and there is nowhere left for the condition to be satisfied.

So the 1/ρ law describes an amplifier that is at least an order faster than the oscillation, and below that the correct statement is not “a larger shift” but “no oscillation”. That is the ordinary shape of a boundary on this site: a quantity that is well described by a first-order term over a range, and then a range where the model has nothing to say because the thing it describes has stopped happening.

What this does to a calibration #

A Wien-bridge oscillator is the classical laboratory audio source precisely because it tunes with one dual potentiometer over a wide range, and the dial is calibrated from $1/2\pi RC$.

The measurements above say what that dial is worth. With a diode limiter at a sensible margin above the critical gain — k = 3.05, so that a one per cent resistor pair still starts — the limiter contributes 0.097% and an amplifier of ρ = 1000 contributes 0.44%, for a total of about half a per cent. That is five hertz at a kilohertz, which is a good deal worse than the resistors and capacitors themselves are.

And it is not a constant offset that a calibration could absorb, because ρ is a ratio: an amplifier of fixed gain–bandwidth gives ρ = 1000 at 1 kHz and ρ = 100 at 10 kHz, so the error grows by a factor of ten across a decade of the dial. A one-megahertz part on a dial marked 100 Hz to 10 kHz is at ρ = 10 000 at the bottom and ρ = 100 at the top, which is 0.14% and 4.9%.

The frequency error of a tuned oscillator is a function of where the dial is set, and the shape of that function is the amplifier’s, not the network’s.

The same lag, in a circuit that is not an oscillator #

The first of the two mechanisms is not about oscillators at all. It is what a finite gain–bandwidth does to any frequency a passive network was supposed to set, and the filters field measures it on a circuit with no limiter in it and therefore with only one mechanism to separate.

The Q the amplifier decides builds a second-order section whose quality factor is a capacitance ratio and whose pole frequency is a product of four passive values, neither expression containing the amplifier, and then builds it with an amplifier a hundred times the corner — the usual rule, and the same ρ = 100 row as the table above. The Q comes out 1.97 per cent high and the pole 1.97 per cent low: the same number in both directions, and the number is the designed Q divided by the ratio. Against this essay’s −4.95% at ρ = 100 that is a smaller error for a reason worth naming — a filter section’s frequency is set by where the magnitudes balance and an oscillator’s by where the phases do, and phase is the quantity that moves first.

What that essay’s own rung above it adds is the thing this one lists as unmeasured. The ripple that is a temperature takes the same two per cent and asks what is inside it: with a tail current set by a resistor the transconductance falls as one over absolute temperature, so a design whose passives have no temperature coefficient at all goes from 0.82 decibels of ripple at −40 °C to 1.05 at +125, crossing a one-decibel specification at 89 °C — and with the other bias arrangement it does not move at all. The amplifier’s share of an oscillator’s frequency error has exactly that structure, because it is exactly that quantity, so the dial calibrated above drifts with the room by an amount that is a property of how the amplifier is biased rather than of the oscillator.

Two things this does not measure #

Frequency stability, as opposed to frequency accuracy. Everything here is a steady-state measurement of where the limit cycle sits. What a specification usually wants is how much that moves with temperature and time, and both mechanisms above supply a route for it to move: the amplifier’s gain–bandwidth has a temperature coefficient, as the section above sets out, and the limiter’s contribution depends on the diode drop, which two millivolts a kelvin, and the wrong sign measures at 1.828 mV/K — downwards, although the thermal voltage in the exponent is rising, because the saturation current rises by nine orders of magnitude over the same range. That the whole of this essay is quoted at one temperature and not marked as such is the general failure the edges that move with the room is about: the numbers are right and the condition attached to them was left off, and it is the same condition every time.

Phase noise. The cycle-to-cycle jitter of the zero crossings is a different quantity from the mean period, it comes from the noise in the loop rather than from either mechanism here, and measuring it needs a stochastic march. A picosecond, read as bits is the collection’s nearest approach and it is measuring something else — a sampled value wrong by the slope times the timing error, falling at exactly twenty decibels a decade of input frequency, which is the effect of timing error rather than its origin. The distinction that would matter here is the one a floor, or a line draws: two clocks with identical jitter in picoseconds put the same total error power in two entirely different places, one spread over two thousand bins and one into two lines twenty-eight decibels higher, and nothing in a figure quoted in picoseconds root-mean-square distinguishes them. An oscillator’s specification has the same hole in it.

The ladder, finished #

Four rungs, each a question the one below produced.

The gain that is exactly one said the design condition is an exact equality on the gain — 3.000000000000, bisected on the netlist — met on a set of measure zero, with a growth rate either side of it that multiplies the envelope by 1.0987 every cycle and reaches the rails in sixty-one milliseconds. That forced the second question: what stops it?

The amplitude nothing linear predicts said a nonlinearity stops it, at 858 millivolts — an amplitude no pole, no transfer function and no bias point contains — and that getting the number needs a netlist with a nonlinearity in it marched forward in time. That forced the third: what does the nonlinearity do to the waveform?

What the limiter charges for said it produces harmonics that the loop barely filters, with the amplitude going as the 0.11 power of the excess gain and the distortion as the 0.74 — so the exchange rate is not one to one, and the same slider that buys purity costs start-up time. It also found no even harmonic at all, at seven parts in ten million, because two diodes facing opposite ways make an odd characteristic. That forced the fourth: if the waveform is not a sinusoid, is the frequency still the sinusoidal one?

It is not, and neither of the two reasons is the one a reader would guess. The amplifier’s share is the one everybody expects and it is buyable; the limiter’s is the one nobody mentions and it is the floor.

The gate #

The amplifier’s share is inversely proportional to its gain–bandwidth, with the product of shift and ratio constant to 6.7% over 1.5 decades.

With a perfect amplifier the frequency is still not 1/2πRC, by 0.771% at the gain drawn — which is the claim that the limiter has a share of its own.

Every measured frequency is below the network’s zero-phase frequency, at every amplifier speed and every gain tested. Both mechanisms push the same way, which is not obvious in advance and is what makes the total have a floor rather than a null.