Where the Q comes from

The Q the amplifier decides found that a quantity a schematic attributes to two capacitors belongs instead to the amplifier: a section built with a part whose gain-bandwidth is a hundred times the corner comes out with a quality factor 1.97 per cent above the designed one and a pole frequency 1.97 per cent below it, and the number is the designed Q divided by the ratio. The ripple that is a temperature then put a temperature into that ratio and found a half-decibel filter crossing a one-decibel specification at 89 °C with no temperature coefficient anywhere in its passives.

Both measurements were taken on one arrangement of the section, and neither said so. There are two, they have been in this collection’s own machinery since the first of those essays was written, and nothing had ever swept between them.

In the unity-gain arrangement the two resistors are equal, the amplifier is a follower, and the quality factor is half the square root of the capacitance ratio. In the equal-R equal-C arrangement every resistor and every capacitor is the same value, and the Q is supplied by the amplifier’s own closed-loop gain: K = 3 − 1/Q. The transfer functions are the same to the last coefficient. Where the Q comes from is not, and that difference has to show up somewhere, because a circuit is not its transfer function.

Two circuits, one filter #

At a designed Q of 2 and a corner of a kilohertz the unity-gain section is a pair of 3.98 kΩ resistors with 160 nF and 10 nF across them. The equal-component section is a pair of 15.9 kΩ resistors, two 10 nF capacitors, and a gain-setting pair that asks the amplifier for a closed-loop gain of exactly 2.5. Not one value is shared between them.

That figure is the instrument’s calibration and not a result, in exactly the sense that the assumption that is a geometry means by the phrase: a comparison of two circuits is worth nothing until there is a case where they must agree exactly, and the case here is the ideal one. Both pole pairs come back out of the determinant at the designed quality factor and the designed frequency to a part in 10¹⁰. Every subsequent disagreement is therefore attributable to whatever was added, and to nothing else.

One asymmetry is visible already and it is not a defect of the measurement. The equal-component arrangement has a passband gain of K, and K is decided by the Q. A section of quality factor 5 has a gain of 2.8; a section of quality factor 12 has 2.9167. It cannot be built at unity gain, and a cascade of three of them has a gain of about twenty-two before anything has been designed. That is a headroom question and a noise-gain question — the same quantity the gain the loop closes against is about — and it is settled by the topology rather than negotiated.

The first departure: an amplifier that is not infinitely fast #

The rung below measured what a finite gain-bandwidth does to the follower arrangement. The gain-stage arrangement has to answer the same question differently, because its amplifier is not a follower: it is a stage of closed-loop gain K, and a stage of closed-loop gain K has K times less bandwidth than the same part used as a follower.

The second law is derived rather than fitted, and the derivation is worth stating because it is what makes the essay’s headline more than a table of measurements. Writing the amplifier’s closed loop as $K/(1 + sK/\omega_t)$ and clearing the denominator turns the section into a cubic in $x = s/\omega_0$:

$$\varepsilon x^3 + (1+3\varepsilon)x^2 + (3 + \varepsilon - K)x + 1 = 0, \qquad \varepsilon = K\,\omega_0/\omega_t$$

For small $\varepsilon$ that cubic factors into a fast real root near $-1/\varepsilon$ and a quadratic $x^2 + (3-K)(1-\varepsilon K)x + (1 - \varepsilon K)$. So the pole frequency falls by $\varepsilon K/2$ and the quality factor rises by $\varepsilon K/2$ — equal and opposite, as the rung below found for the other arrangement — and the coefficient is K²/2 divided by the gain-bandwidth ratio.

The two coefficients behave completely differently in the quantity a designer is asking for. Q is unbounded. K²/2 is not: K = 3 − 1/Q rises from 1.5858 at Q = 0.7071 to 2.9167 at Q = 12 and approaches 3, so K²/2 rises from 1.2573 to 4.2535 and stops at four and a half. A section built the first way is punished in proportion to how hard it was asked to work. A section built the second way is punished by an amount that has an upper bound.

The failure at the low end deserves its own sentence, because it goes the wrong way. Both laws overpredict there, which is not the direction anybody guesses: at ten times the corner the follower arrangement at Q = 2 comes out 12.79 per cent high against its law’s 20, and the gain stage 7.62 against its law’s 31.25. That much is the expansion running out of small parameter. At Q = 5 the gain stage goes further and comes out 17.13 per cent low — below the quality factor it was designed for, with the law still predicting 39.2 per cent above it. The realised quality factor has gone below the designed one. There is no regime in which an amplifier that slow is a design choice, but the sign change is the clearest possible evidence that the coefficient is an asymptote and not an identity, and the assertion in the figure is written to reject an asymptote quietly promoted to a law.

The crossing, which is a number rather than an opinion #

Two coefficients, one growing without limit and one saturating, cross exactly once.

Below Q = 3.082 the follower arrangement is the one a slow amplifier treats more kindly; above it, the gain-stage arrangement is. That is the opposite of the received advice, which is that the equal-component form should be kept away from high quality factors — and the received advice is right, for a reason that is not this one and is measured two sections further down.

The crossing moves with the amplifier, and it moves toward a limit rather than wandering. In the asymptotic laws the condition is Q = K²/2, which is 2Q³ − 9Q² + 6Q − 1 = 0 and has its relevant root at Q = 3.73. Measured on the built sections it sits at 2.38 with an amplifier thirty times the corner, 3.08 at a hundred times, 3.43 at five hundred and 3.47 at a thousand — approaching the closed form’s root from below, because both measured errors sit a little under their asymptotes and the gain stage’s sits further under. The two routes agree in the limit and disagree everywhere a real amplifier is, which is the useful half of knowing both.

Why the sensitivity that is enormous does not produce an enormous error #

There is a trap in the arrangement above and stepping into it produces a number wrong by a factor of eighteen and in the wrong direction, so it is worth walking through deliberately.

The quality factor’s sensitivity to the closed-loop gain is 3Q − 1. That is the largest sensitivity anywhere in either circuit: at Q = 2 it is 5, at Q = 12 it is 35. The obvious way to predict what a finite gain-bandwidth does is to ask how much gain the amplifier has lost at the section’s own frequency and multiply. At a hundred times the corner and a closed-loop gain of 2.5, the magnitude of that gain is short by 1/√(1 + (K/r)²) − 1, which is 3.124×10⁻⁴ — three parts in ten thousand. Times five is 0.156 per cent, and it is a loss of gain, so the quality factor should come out low.

The measurement says the quality factor comes out 2.83 per cent high. The prediction is wrong in sign and out by a factor of eighteen, and both failures have one cause: at a hundred times the corner the amplifier has not lost any gain worth naming. It has lost 1.4321 degrees of phase. A single-pole amplifier well below its unity-gain frequency is very nearly an ideal gain with a delay in it, and the magnitude error is second order in K/r where the phase error is first order.

So the sensitivity 3Q − 1 is correct and applies to a component tolerance, which moves the gain’s magnitude, and does not apply to a gain-bandwidth, which moves its phase. Two mechanisms that look like the same perturbation of the same quantity produce results of different order and opposite sign. This is the reason the cubic above is worth deriving rather than approximating away: a first-order argument in the wrong variable is not a rough answer, it is a different answer.

The second departure: parts that are not exactly what they say #

An error budget has a second term and it points the other way.

The quantity is the logarithmic sensitivity — how far the realised Q moves for a given fractional error in one component — and this collection has a field’s worth of machinery for it: every derivative and the one that is zero is where the adjoint method arrived, and the tolerance that can only take away is what happens when the first derivative of a response is zero and the second is not. Here the derivative is taken of a root rather than of a response — the same object the derivative of a root is about — and it is taken numerically: perturb one value by a part in a hundred thousand, re-root the determinant, divide.

Three things in that measurement are exact and none of them was put in by hand.

The follower arrangement’s resistor sensitivities are zero, not small. With the two resistors equal, the Q is half the square root of a capacitance ratio and contains no resistance at all; perturbing either resistor moves the pole frequency and leaves the quality factor where it was. That is a member of the family the three tolerances that do nothing collects, and it is a property of the equal-resistor choice rather than of Sallen–Key sections generally.

The gain-stage arrangement’s numbers are 2Q − ½ and 2Q − 1, which is the same statement twice. The closed-loop gain’s own sensitivity is 3Q − 1: Q = 1/(3 − K), so dQ/dK = Q², and the logarithmic sensitivity is KQ = 3Q − 1. The gain-setting resistor sets only the fraction (K − 1)/K of that gain, so it carries (3Q − 1)(K − 1)/K = 2Q − 1 into the quality factor. At Q = 5 that is 9, at Q = 12 it is 23, and it does not stop.

And the pole frequency’s sensitivities are identical in the two arrangements — every resistor and every capacitor at −½, the gain-setting pair at exactly zero, root-sum-square 1.000 in both. So the two circuits are equally exposed in frequency and unequally exposed in shape, which is precisely the statement that they differ in where the Q comes from and in nothing else.

Which arrangement is better, and what decides it #

The two departures point in opposite directions, so the answer is an arithmetic question rather than a preference.

With one per cent parts the follower arrangement wins at every quality factor, and it is not close. At Q = 12 it stands at 11.40 per cent — 10.69 of amplifier and 0.71 of parts — against the gain stage’s 50.75 per cent, of which only 1.49 is the amplifier. The gain stage’s advantage against a slow amplifier is real, bounded and swamped: its own tolerance term at that quality factor is 49.26 per cent, seventy times its amplifier term.

So the received advice is right and the reason usually given for it is wrong. The equal-component arrangement is not the one a finite gain-bandwidth treats worse at high Q; it is the one a real component treats worse, by a factor that grows as 4Q − 1. The amplifier is the cheaper of the two things to fix. Buying a part ten times faster costs a few pence and moves the first term by a decade; buying components ten times better costs a great deal more and is the only thing that moves the second.

The crossing at 4.93 in that figure is where the received advice stops applying, and tenth-per-cent resistors and capacitors are not exotic. This is the same shape of answer as a ladder is not a cascade reaches from the other end of the field: two realisations of one response, indistinguishable on paper, separated by two orders of magnitude in what a one per cent part does to them.

The third place, where nothing happens at all #

There is an obvious third quantity in which two arrangements of a resonant section built round an amplifier ought to differ, and it is the one every treatment of Sallen–Key sections warns about: a section with too slow an amplifier is supposed to oscillate. The collection’s own machinery for finding that boundary — bisect on the sign of the pole pair’s real part — has been in place since the first rung of this ladder and had never been run.

Run, it returns no boundary. Not a large one, not one outside the range swept: none.

The enhancement is bounded, and below the maximum the amplifier destroys the Q rather than raising it. At a tenth of the corner the follower arrangement realises a quality factor of 1.456 against a designed 10 and the gain stage 0.500. The curve rises, peaks a little above the design, and comes back down. There is no pole excursion at the end of it because there is no end.

That is not an artefact of the sweep’s range, and it is provable rather than merely unobserved. The cubic above has all-positive coefficients, and the Routh condition for a cubic is that the product of the middle two exceed the product of the outer two. For the gain-stage arrangement that condition is (1 + 3ε)(1/Q + ε) > ε, which holds for every positive ε and every positive Q because every term on the left is positive and one of them is ε itself. The follower arrangement’s cubic gives 4γ + 2γ² + 8Q² + 12Q²γ + 4Q²γ² > 4Q²γ, which holds for the same reason. A single-pole amplifier cannot make a Sallen–Key section unstable, in either arrangement, at any gain-bandwidth and any quality factor.

The instability that everybody has seen is therefore not this. It requires a second pole in the amplifier, or an output impedance that is inductive, or a capacitive load turning the follower into the load that gets inside the loop — mechanisms this collection has, and which are absent from the model used here. What the measurement says is that the one-pole amplifier, which is the model that produced the two per cent and the 89 °C and everything else on this ladder, is refusing to answer a question it was never able to answer. That refusal is worth more than a number would have been: it says where the next rung’s model has to come from.

What it does not say #

It does not say the gain-stage arrangement is a bad circuit. Below Q = 3 the two are within a per cent of each other against the amplifier and within a factor of three against the parts, and the equal-component form buys something neither figure above prices: one resistor value and one capacitor value for the whole section, which on a board is a real saving and in an integrated process is close to the only thing that matches.

It does not say the follower arrangement is insensitive. Its Q sensitivities are ½ and −½ and its frequency sensitivities are −½ four times over, and one per cent parts still leave 0.71 per cent of spread in the quality factor and one per cent in the corner. What it says is that those numbers do not grow with Q, and the other arrangement’s do.

And none of it is about a filter yet. Everything above is one section. A cascade is sections whose quality factors differ by an order of magnitude — the band-edge pair of a fifth-order Chebyshev is near four and a half and the lowest pair near one — so the two arrangements would move the members of a cascade by different amounts, and what changes is the shape rather than the scale. That is the argument the Q the amplifier decides made about ripple, and running it again with the arrangement as a variable is the obvious next measurement rather than one made here.

The number worth carrying #

Two circuits with identical transfer functions and no component in common differ in three quantities, and the three answers are: Q against K²/2, ½ against 2Q − ½, and nothing at all.

The habit that goes with it is the one this collection keeps arriving at from new directions. A transfer function is a description of what a circuit does at a frequency, and it is complete; it is not a description of the circuit. Two realisations of one response are the same object to every measurement made with a signal generator and different objects to a component tolerance, an amplifier’s bandwidth and a temperature — and the way to find out which of those differences matters is to compute all of them and add them up, because the one a textbook warns about turned out here to be the smaller term, and the one it does not mention turned out to be seventy times larger.