How much of the amplifier gets through

The argument for negative feedback is usually made in one line. The closed-loop gain depends on the open-loop gain through $1/(1+T)$, so with a loop gain of a thousand a doubling of the device moves the answer by a tenth of a per cent, and the resistors — which are the accurate part — decide everything else.

Every word of that is true at low frequency. This essay measures the quantity rather than quoting it, across seven decades, and three things come out of the measurement that the sentence does not contain.

How it is measured #

Not by differentiating anything. The forward path’s gain is multiplied by $1 + 10^{-5}$ and the closed-loop magnitude is solved again; the fractional change in the answer divided by the fractional change in the device is the sensitivity. That is what the word means and it is what an experiment would do.

The second route is the return ratio. The loop is cut at the amplifier’s input, a test signal is injected, and what comes back round is measured — the same construction the loop-gain essay uses, and no expression for $\beta A$ is written anywhere. The identity being checked is that the sensitivity is the real part of $1/(1+T)$.

The real part, not the magnitude, and the distinction is not pedantic. What is being measured is a magnitude of a complex closed-loop gain, and a magnitude only responds to the component of a change that lies along it. The two routes agree to four parts in $10^5$ of the peak over two hundred and forty frequencies, which is as well as a difference quotient with a $10^{-5}$ nudge can be expected to do.

At low frequency, the textbook answer #

The loop gain at direct current is 995.04, which is the amplifier’s $10^5$ divided by the closed-loop gain of 100, less a little for the loading the divider puts on the output.

The measured sensitivity there is 0.0999%. Double the amplifier’s gain and the closed-loop gain moves by one part in a thousand: from 99.900 to 99.950. That is the whole argument for feedback, it is exactly right, and it is right over the two and a half decades where the loop gain is large.

It is worth noticing what has been achieved. The device’s gain is specified by its manufacturer to a factor of three, part to part and over temperature. The circuit’s gain is set to a tenth of a per cent by that specification, and to whatever the resistors are worth by everything else.

Put the arithmetic in the other direction and the size of the win is clearer. A part whose gain is guaranteed only to be above $10^5$ — which is how an operational amplifier is actually specified — might have $3\times10^5$ in it, three times what was assumed. Three times is a fractional change of two, so the closed-loop gain moves by two-tenths of a per cent: from 99.900 to 99.967. The part-to-part spread of a device has been turned into the third decimal place of a circuit, and nothing was measured, selected or trimmed to do it.

That is worth holding onto, because everything after this section is about where it stops, and the stopping is not a criticism of the technique. It is a statement of the band over which the technique delivers what it is famous for.

It changes sign #

Somewhere below crossover the sensitivity goes negative. At the default setting that is at 258 Hz, and by a kilohertz the sensitivity is −1.4%: making the amplifier better makes the closed-loop gain smaller.

There is no reading of “feedback divides the error by $1+T$” that predicts a sign. The reason is that $T$ is complex, and $1/(1+T)$ is a complex number whose real part changes sign when $1+T$ crosses the imaginary axis — which happens once the loop’s phase has gone past ninety degrees while its magnitude is still above one. Both of those are ordinary conditions inside the bandwidth of an ordinary amplifier.

The effect is not tiny. At a kilohertz it is a per cent and a half of the closed-loop gain per unit fractional change in the device, and at the lowest margin on the slider the negative excursion reaches −1.7 before turning round.

And it does not always happen. Bisected on the second pole, the boundary is at 50.0° of phase margin: below that the sensitivity goes negative somewhere and above it never does. That is the kind of statement this collection exists to produce — a claim that is true of most of the range, false at the end of it, and stated with the boundary measured rather than with the tolerance widened.

The curve, at seven frequencies #

The middle row is the surprising one and the last three are the important ones. Between a hundred hertz and ten kilohertz — two decades, entirely inside the closed-loop bandwidth of a circuit with a crossover at 5.73 kHz — the sensitivity goes from eight parts in ten thousand to one and a half. Three and a half orders of magnitude, in a quantity that a specification records as a single number measured at a kilohertz or at direct current.

Why the sign changes, geometrically #

The sensitivity is $1/(1+T)$, and $1+T$ is a vector from the critical point $-1$ to the loop gain’s locus. So the sensitivity is the reciprocal of that distance, rotated — and its real part is what survives being projected onto a magnitude.

At low frequency $T$ is large and real and positive, so $1+T$ points to the right and is long: the sensitivity is small and positive. As frequency rises the loop’s phase falls, $T$ swings clockwise, and $1+T$ swings with it. Once the phase has passed ninety degrees while $|T|$ is still above one, $1+T$ has crossed into the left half plane — its real part is negative, and so is the real part of its reciprocal.

That is why the boundary is a phase margin rather than a frequency. A loop with plenty of margin reaches $|T| = 1$ before its phase has gone far enough for $1+T$ to cross over, and its sensitivity stays positive the whole way. A loop with little margin has the phase gone long before the magnitude arrives.

It also explains the peak’s position. The sensitivity is largest where $|1+T|$ is smallest — the closest approach of the locus to $-1$ — which is a different frequency from either the crossover or the point of minimum phase, and is the quantity the vector margin measures and the two textbook margins do not.

It goes above one #

Near crossover the sensitivity peaks, and the peak is above one.

A sensitivity above one means the closed-loop gain responds to a change in the device more than it would with no feedback at all. At twelve degrees of margin, a one per cent change in the amplifier’s gain moves the closed-loop gain by 3.2% at 2.2 kHz — a frequency inside the closed-loop bandwidth, in a circuit whose whole purpose is to be insensitive to that device.

This is the sensitivity peak a control engineer draws, arrived at here by perturbing one element of a netlist and re-solving. What is worth taking from the measurement rather than the theory is the ordinariness of the numbers: 47° of margin is a comfortable design and it still has a band where feedback is making things 39% worse.

And above crossover it is exactly one #

Far above the crossover the loop gain has gone, $1/(1+T)$ is one, and the measured sensitivity is 1 to a part in a thousand.

That is not a limit being approached from a useful direction. It says the circuit up there is a bare amplifier with two resistors hanging off it, and the whole of its gain is the device’s gain — so whatever the device does with frequency, temperature and part-to-part spread arrives at the output undivided.

The frequency at which that starts is worth naming, and this essay names it as the point where the magnitude of the sensitivity first reaches one: 6.33 kHz at the default margin. That is a little above the crossover, which is where a reader would put it, and it is worth measuring rather than assuming because at low margins it is below the crossover — the peak has already lifted the sensitivity past one before the loop gain has fallen to it.

What this changes about a specification #

A gain accuracy is a statement about a band. “0.1% gain accuracy” from a circuit with a loop gain of a thousand at direct current is a statement about direct current. At a fifth of the crossover the same circuit is passing through most of the device’s variation, and at the crossover it is amplifying it.

Phase margin buys accuracy, not just stability. The table above is a table of gain sensitivity and it is monotone in the margin. The usual argument for margin is overshoot and ringing — two measurements of one margin is where those two are shown to be one quantity, a phase margin computed from the loop gain in the frequency domain returning 34.9° against an overshoot inverted in the time domain returning 34.9° — and this is a third, which applies to circuits that are not being asked to respond to steps at all.

That the margin is read at one frequency is the assumption doing the work, and it is not always available. Stable, and unstable with less gain is the loop where it fails: a three-pole loop with two lead sections is stable between $1.81\times10^6$ and $3.36\times10^7$ of gain and unstable on both sides, its phase crosses −180° at three frequencies rather than one, and neither margin can see any of it because a phase margin describes the loop at one frequency and a gain margin at one other. The sensitivity curve on this page would be a different shape entirely for such a loop, and the summary “sensitivity is monotone in the margin” would have no argument to attach to.

And the sign matters for anything that trims. A circuit trimmed at direct current, where the sensitivity is positive, and operated at a frequency where it is negative, has a trim that moves the answer the wrong way as the device ages. That is a small effect and it is the kind that shows up as an unexplained drift.

The same running-out, on four quantities #

The two figures above are not illustrations of this essay’s result; they are the same result measured on other things a loop is holding, and putting the four side by side is the strongest form of the argument.

The node that is at ground for a while measures the summing junction of an inverting stage and finds it a tenth of an ohm at direct current, ten ohms at a kilohertz and 909 ohms above a megahertz — the two feedback resistors in parallel, with the amplifier contributing nothing at all. Three regions, a decade a decade in the middle one, and a ceiling that is a passive network.

A source below a frequency measures a regulator’s output impedance and finds the same three regions on a quantity a data sheet quotes as one number: 0.43 milliohms at direct current, doubled by 4.81 hertz, peaking at 1.95 ohms at ten kilohertz — where it is a third above the open-loop value, because $1+T$ is smaller than one there — and settling above a hundred kilohertz at the passive network the loop has stopped being part of.

The ideal amplifier, and where it stops being one is the same statement about the closed-loop gain itself: 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 which point the ideal answer is not an approximation to anything.

Four quantities, one divisor. What is worth extracting is that they do not fail at the same frequency — the thresholds are on different quantities and each is chosen by a different tolerance — but they fail at the same rate, and the peak near crossover is in all four, because the peak is a property of $1+T$ rather than of anything being divided by it. A designer who has measured one of these on a circuit has measured all four, up to a constant.

What is not measured here #

Only one parameter is perturbed. The forward path’s gain is multiplied by a constant, which scales the amplifier’s response at every frequency together. A real device varies in its gain, its bandwidth and its pole positions somewhat independently, and the sensitivity to each is a different curve. What is measured here is the sensitivity to the one perturbation that has a clean interpretation.

Nothing about the resistors. The closed-loop gain is $1/\beta$ to a part in a thousand, so its sensitivity to the divider is essentially one and there is nothing to draw. That is the point of the arrangement rather than an omission: the accuracy has been moved onto the components that can carry it.

Nothing about the resistors, and there is a way to get it anyway. The paragraph above is right that the sensitivity to the divider is essentially one, and the networks field has since built the instrument that would say so without a second solve. Every derivative, and the one that is zero transposes the matrix and solves once more, which returns the derivative of a response with respect to every element at once, exactly — against a difference quotient’s best possible accuracy of four parts in a hundred million, one component at a time, in two solves each. The measurement on this page is a difference quotient by construction: make the device better, solve again, compare. It is the right instrument for one parameter with a clean interpretation, and it is the expensive instrument for the general question.

What the adjoint route buys is not speed. It is that a sensitivity to everything can be looked at as one object, and objects of that kind have structure that a scan does not reveal. The three tolerances that do nothing is the worked example: the whole second-derivative matrix of a ladder’s magnitude at a ripple peak has two eigenvalues of order one and three nine decades down, which is a three-dimensional subspace of component variations that do nothing at all — and no sample of a five-dimensional box ever lands on it. The sensitivity curve measured here is one number at each frequency; the corresponding matrix for this circuit would say which combinations of device parameters the closed loop is blind to, and the answer is not obviously “all of them but the gain”.

There is a second reason to keep the two instruments apart, and it is the one the derivative of a root draws out. A response’s derivative and a pole’s derivative are different objects and do not agree about which realisation is better: pointed at two realisations of one filter, the response sensitivity puts the ladder ahead by eight orders of magnitude and the pole sensitivity puts it ahead by a factor of 2.17. What is stationary is the magnitude at one frequency, and that says nothing about where the poles are. The same distinction applies to everything on this page, because a desensitivity measured on the gain is silent about the margin — and the margin is what the last column of the table is a function of.

And nothing with a plant in it. Everything above is a netlist with named components, and the sensitivity is of one solved quantity to one element in it. Sensitivity as a design object — weighted functions, loop shaping, anything about a plant that is not a circuit — is the control-systems line this site does not cross.

The gate #

The two routes are asserted to agree, to two parts in $10^4$ of the peak, over two hundred and forty frequencies. The normalisation is to the peak rather than to the local value, and deliberately: the quantity passes through zero, so a ratio taken at each frequency divides by something a millionth of the curve’s own scale and reports six parts in a thousand of disagreement on a curve the two routes track to eleven figures either side.

The low-frequency value is asserted against one over one plus the loop gain, to two per cent, which is the textbook claim being checked rather than assumed.

The peak is asserted to exceed one at every setting of the slider, which is the essay’s main finding.

The sign change is asserted with its regime. Below the bisected boundary of 50.0° it must happen; above it, it must not. The first version asserted it at every setting and failed at 65°, which is a true claim stated as if it were universal.

And the far end is asserted to be exactly one, to a part in a thousand — the statement that the loop has stopped doing anything at all.