The Q the amplifier decides

A unity-gain Sallen–Key section is four passive components and a follower, and its design is two lines. The pole frequency is one over the square root of the product of the two resistances and the two capacitances; with the resistances equal, the quality factor is half the square root of the capacitance ratio. Neither expression mentions the amplifier, and that is not an approximation that has been made — the amplifier does not appear in the derivation at all, because it was taken to be a follower with infinite gain at every frequency.

This collection has an element for that assumption. A nullor draws no input current and holds its inputs at the same potential, and it is what the inductor that is an amplifier is built out of. Built with one, the section has exactly the pole frequency and exactly the quality factor it was designed for, to eight digits. Built with an amplifier that has a gain and a corner, it does not.

Poles rooted, not fitted #

The obvious way to measure a Q is to sweep the response and read the height of the peak. That would beg the question, because the height of the peak is exactly the quantity in dispute — and it would also be measuring a response, when what changed is the network.

So the poles are recovered the way everything on this site recovers poles: the determinant of the nodal matrix is sampled along the imaginary axis, the polynomial is rooted, and the pair nearest the design point is picked out of what comes back. That machinery carried a defect until the derivative of a root found it and recorded the fix, and with that repaired it is exact for a network of this size.

What comes back is worth noting on its own. The section as drawn has two poles. As built it has three, because the amplifier brought one with it, and the pair is not where it was put.

One number, in two directions #

At a gain-bandwidth a hundred times the section’s corner — which is the rule of thumb everybody is given — the quality factor comes out 1.97 per cent high and the pole frequency comes out 1.97 per cent low. Not approximately: the two agree to better than a twentieth of themselves everywhere the amplifier is well clear of the section.

And the number is the designed Q divided by the gain-bandwidth ratio. Q of two at a ratio of a hundred is two per cent; Q of five is five per cent; Q of twelve is eleven. The error is proportional to the very quantity it is an error in.

That proportionality is the whole of why this matters, and it is worth saying why it is not obvious. A section with a high Q is a section whose two poles are close to the imaginary axis, which means the loop that positions them is working against a small quantity: the damping is a difference between larger terms. The amplifier’s phase lag subtracts from that difference directly, so a fixed lag makes a fixed absolute change in the damping and therefore a change in Q proportional to Q itself.

The identity is not an identity #

The equal-and-opposite result is asserted with its own departure, because it is an asymptote rather than a law. At a gain-bandwidth of ten times the corner the Q is 12.8 per cent high and the pole is 16.0 per cent low — a fifth apart, not equal — and there is no small parameter left to expand in.

That is the shape of assertion the rung below’s proximity argument settled on and the reason is the same: an asymptote quietly promoted to an identity is a claim that has stopped being checkable. Requiring the two errors to be equal and requiring them to be visibly unequal at the low end is a pair of statements that a wrong model fails.

What it does to a filter, which is not what it does to a section #

A section moving by two per cent is a statement about a section. A filter is a cascade of sections, and a ladder is not a cascade is where the difference between those two objects was measured. A fifth-order half-decibel Chebyshev has three sections: one pair with a Q of about four and a half at the band edge, one with a Q of about one, and a real pole.

They move by different fractions, because the error is proportional to Q. So the passband does not shift, it tilts.

At a hundred times the corner the ripple is 2.10 dB against the 0.51 the design asks for. It takes a thousand times the corner to bring it back to 0.63. The familiar instruction — use an amplifier at least a hundred times the corner frequency — is out by a decade, and the factor it is out by is the highest section’s Q.

Trimming cannot recover it, and the reason is in the sentence above: what moved is not one frequency. Three sections moved by three different fractions and one of them also changed shape. A single adjustment has one degree of freedom against a three-parameter error.

Where the shift comes from, in one sentence per term #

It is worth saying what the amplifier’s pole actually does to the loop, because the result is more memorable than the number.

The follower’s job in a Sallen–Key section is to hold the output equal to the voltage on the second capacitor while driving the top of the first one. That first capacitor is the positive feedback path, and the section’s Q is entirely built out of it: the higher the Q, the more of the response at the pole frequency is being supplied by that path rather than by the input. An amplifier with finite gain-bandwidth returns that voltage late, and a positive feedback signal returned late is a signal returned with a component in quadrature — which is to say, with the sign that reduces damping.

So the direction of the error is not an accident. Q always rises and the pole frequency always falls; no choice of amplifier makes them go the other way, because the amplifier’s phase can only lag. That is the same one-sidedness the phase the magnitude already knows describes for a minimum-phase network: a physically realisable lag has a sign, and a quantity built out of it inherits that sign.

Which realisation, and whether it matters #

There are two textbook Sallen–Key arrangements and they have the same transfer function. One uses equal resistors and a capacitance ratio, with the amplifier as a follower. The other uses equal resistors and equal capacitors, and gets the Q from the amplifier’s own closed-loop gain, K = 3 − 1/Q.

The transfer functions match to the last coefficient. The sensitivity to the amplifier does not: at a gain-bandwidth of three hundred times the corner the follower form is 0.67 per cent high in Q and the gain-K form is 1.01. Where the Q comes from is a property of the realisation, and the transfer function does not carry it.

Neither form oscillates, which is worth reporting because the folklore says the gain-K form can. In this model it does not, at any gain-bandwidth down to a tenth of the corner and at any Q up to twenty: the poles move a very long way and stay in the left half plane. What actually happens at a gain-bandwidth below the corner is that the section stops being second order in any useful sense — its pole frequency is out by eighty per cent and its Q is unrecognisable — and calling that stable is technically true and practically irrelevant.

The same defect, in the arrangement that has no amplifier in the signal path #

Eight amplifiers, and what they add measured a different active realisation — a gyrator ladder, where the amplifiers synthesise the inductors rather than closing a section’s loop — and found its own departure from the passive design as the impedance scale rises. The two defects are the same defect wearing different clothes: an amplifier with finite gain-bandwidth cannot hold a node exactly, and every active filter is built on the assumption that one node is held exactly.

Where they differ is in what the shortfall damages. A Sallen–Key section’s error appears as a Q, which moves the response’s shape; a gyrator’s appears as a series resistance in a synthesised inductor, which fills the notches and raises the floor. Neither shows up in a direct-current measurement and both are invisible to a component tolerance analysis, because no component has changed.

What a designer does about it #

Three responses are in use and the measurements above say what each is worth.

Predistort the design. Since the shift is systematic and its size is Q over the ratio, a section can be designed at a Q two per cent low and a pole frequency two per cent high, and it comes out right. This works, it costs nothing, and it is exact only at one gain-bandwidth — so it turns an amplifier’s specification into a filter’s component values, and a second-source part with a different gain-bandwidth is a different filter.

Buy the ratio. A thousand times the corner brings the ripple to within a quarter of a decibel of the design. For an audio filter at twenty kilohertz that is a twenty-megahertz amplifier, which is ordinary; for a two-hundred-kilohertz anti-alias filter it is two hundred megahertz, which is not, and the power it costs is real.

Change the realisation. A state-variable or biquad arrangement uses three amplifiers and puts each one inside its own integrator loop, where the finite gain-bandwidth appears as an integrator that is not quite an integrator rather than as a phase error in a positive feedback path. The Q error is then proportional to Q squared over the ratio in some arrangements and to Q over the ratio in others, and which it is depends on the topology rather than on the transfer function — which is the essay’s point restated at the next level up.

That last connection is the one worth carrying away. A steeper specification needs a higher order, a higher order puts its band-edge pole pair closer to the axis, a closer pair is a higher Q, and a higher Q needs a proportionally faster amplifier. The amplifier’s gain-bandwidth is therefore set by the stopband requirement, through three steps none of which mentions it.

What is not measured here #

No slew rate and no swing. Everything above is small-signal: the amplifier has a gain and a pole and no limit anywhere. A high-Q section has internal nodes that swing several times the output, so the amplitude at which a real one departs is lower than its output stage’s rating suggests — the step too large to have an impedance has the machinery and this essay does not use it.

No component tolerance. Every value here is exact, so the two per cent measured is entirely the amplifier’s. A one per cent capacitor pair contributes about half a per cent of Q error on its own, and the two effects add in a way that is worth knowing: the amplifier’s error is systematic and one directional, so it shifts the whole batch, while the tolerance’s is a spread. Every derivative, and the one that is zero is where that distinction is measured properly.

And no second amplifier pole. The macro-model here has one, which is what a compensated operational amplifier is designed to look like over its useful range. A real part has a second pole near its gain-bandwidth product, and a section built at a tenth of the gain-bandwidth is well inside the region where it matters.

The measurement a bench makes, and what it can and cannot see #

A filter built and swept on a bench shows a passband that is 2 dB rather than 0.5, and the natural first suspicion is the components. It is worth knowing which observations distinguish the two causes, because they are different and the repairs are different.

A component-tolerance error is a spread: build ten and they are different from each other, some better than the design and some worse. The amplifier’s error is systematic: build ten and they are the same, and all ten are wrong in the same direction. Ten boards agreeing with each other and disagreeing with the simulation is the signature.

The second distinguishing observation is the corner. The amplifier’s error moves the pole frequency down by the same fraction it raises the Q, so the whole passband edge is low as well as peaky — a filter whose ripple has grown and whose corner has fallen, together, in proportion. Component tolerance moves those two independently.

And the third is temperature. An amplifier’s gain-bandwidth falls with temperature by something like a quarter over the military range, and by the law above that turns straight into Q. A filter whose ripple grows when the oven is hot and whose components have a low temperature coefficient is being told something specific. Two millivolts a kelvin is the machinery for the device end of that, and it is not applied here.

The number that was not in the design #

Two lines of design equations, four passive values, and a quantity that appears in neither: the amplifier’s gain-bandwidth product. It changes the Q by the designed Q over the ratio, changes the pole frequency by the same fraction in the other direction, and changes a filter’s ripple by a factor of four at the ratio everybody uses.

The measurement that says so is not a response and not a fit. It is the poles of the network as built, taken from the determinant, compared with the poles of the network as drawn — two objects that the design procedure treats as the same object and that differ by one element’s finite gain.

The same gain–bandwidth, three other places #

The number this essay is about is one specification of one part, and three other measurements in this collection are functions of it — which is the argument for treating a gain–bandwidth as a design input rather than as a headroom figure.

The ideal amplifier, and where it stops being one is the direct reading: a closed-loop gain a tenth of a per cent low at direct current, one per cent low by 1.35 kHz, and above 10 kHz no loop gain left at all. The rule of thumb this essay tests — a hundred times the corner — is chosen against that curve, where a hundred times is a comfortable margin, and the two per cent measured here is what the same margin is worth once the quantity being protected is a pole position rather than a gain.

The frequency that is not the formula finds the same lag deciding an oscillator’s frequency, and its arithmetic is the cleanest available comparison: the shift is 4.45 divided by the ratio of gain–bandwidth to oscillation frequency, so at a ratio of a hundred the frequency is 4.5 per cent low against this essay’s 1.97 per cent. Twice as large, on the same ratio, because an oscillator’s frequency is set by where the phases balance and a filter section’s by where the magnitudes do — and phase is the quantity that moves first.

The gain the loop closes against supplies the correction most likely to be forgotten when the rule of thumb is applied. A gain–bandwidth product is quoted for unity gain, and what a section actually has is that product divided by its own noise gain — so a Sallen–Key section built for a gain of two has half the ratio its designer thought it had, and the error measured here doubles before anything has gone wrong.

Taken together those say the ratio in this essay’s denominator is not a number a data sheet gives. It is the data sheet’s number divided by the section’s noise gain, evaluated against the section’s own pole frequency rather than its corner, and the two per cent it produces should be read as a lower bound on what a real build sees.