The load that gets inside the loop

An amplifier’s specification has a page about gain and bandwidth, a page about noise, and a page about stability with a phase margin on it. What connects them is a number that appears on none of them prominently: the amplifier’s output resistance, which is around fifty ohms for an ordinary part and is the reason a capacitor on the output is a stability problem rather than a load.

The capacitor does not appear in the closed-loop gain. It is not part of the feedback network. Every expression a designer writes down for what the circuit does is untouched by it, to three parts in a million at the far end of the slider.

And it takes ninety degrees of phase margin to thirty.

The pole, and where it is #

Fifty ohms and 2.2 nanofarads make a pole at

$$f = \frac{1}{2\pi \cdot 50\ \Omega \cdot 2.2\ \mathrm{nF}} = 1.45\ \mathrm{MHz}$$

and the loop’s crossover is at 2.50 MHz. So a second pole has appeared below the frequency at which the loop gain reaches one, which is the classic recipe for losing a margin: the first pole has already spent ninety degrees, the second one is spending its own, and what is left at crossover is what is left.

The essential word is inside. The output resistance is between the amplifier’s ideal output and the node the feedback network is connected to, so the pole it forms with the load is in the forward path — the feedback has to live with it rather than correct it. A capacitance on the input side of the feedback network would be outside the loop and would cost bandwidth, not margin.

Measured across the slider:

Forty-five degrees is reached at 905 picofarads, bisected on the netlist rather than estimated from the table. That is about a metre of ordinary coaxial cable, or a modest ceramic capacitor put across an output “to keep it quiet”, or the input capacitance of a long ribbon cable going somewhere else on the board.

The same arithmetic at the other end #

This site already has an essay about a capacitance that costs a margin and changes no gain: nine picofarads at the summing junction takes the same amplifier from ninety degrees to forty-five.

The two are worth holding together, because they are the same conclusion by opposite mechanisms.

At the summing junction, the capacitance is part of the feedback network. It raises the noise gain — the gain the loop closes against — a decade per decade, so the loop closes at forty decibels per decade instead of twenty, and the extra slope is the extra phase. The gain the loop is fighting has changed.

At the output, the capacitance is not part of the feedback network at all. The noise gain is unmoved — 2.005 with and without, to parts in ten million — and what has changed is the forward path, which has grown a pole. The gain the loop has to fight with has changed.

Same symptom, same repair in the end, and completely different diagnoses. It is worth being able to tell them apart, because the fix that works at one end does nothing at the other: a small capacitor across the feedback resistor cures the summing-junction case exactly and leaves the output case alone.

There is a subtlety in that middle curve worth one paragraph, because it is the kind of thing a figure can hide. What is plotted is the amplifier’s own gain divided by the measured loop gain, and that ratio is the gain the loop closes against only while the forward path is the amplifier. A load capacitance stops that being true near crossover — it has put a pole in the forward path, which is the whole point — so the plotted ratio lifts there by 23% even though nothing about the feedback network has changed.

Measured on the feedback network alone, which is the unambiguous quantity: at 2.5 MHz the 905 picofarads on the output leaves the fraction fed back identical to the last bit, and nine picofarads at the junction takes 18% of it away. That is the distinction stated in a way that no choice of plotting can blur, and it is what the gate asserts.

The margins, and what they look like in time #

A margin is a frequency-domain number, and this field’s habit is to measure it twice.

At thirty degrees of margin, a second-order approximation predicts about thirty-five per cent of overshoot on a step, and the field’s own figure finds the two measurements agreeing to a tenth of a degree over most of its range. So the 30.1° above is not a number about a Bode plot: it is a prediction that this circuit, with a load somebody put on it, will overshoot by a third and ring for several cycles.

At 14.4° the ringing is severe enough to be mistaken for oscillation, and on a bench it often is: a circuit with ten degrees of margin driven by anything with edges in it produces a burst of ringing on every edge, which looks exactly like an oscillation that starts and stops.

What the amplifier’s output impedance actually is #

Treating the output resistance as a constant fifty ohms is the model this figure uses, and it is worth saying where that model stops.

Inside the loop, the closed-loop output impedance is the open-loop one divided by one plus the loop gain — which at low frequency is a tenth of an ohm, rising a decade per decade as the loop gain falls, and reaching the open-loop value somewhere above crossover. This site measures that curve directly in the virtual-earth essay and finds it settling at the two feedback resistors in parallel with the amplifier contributing nothing.

So the pole in the table above is formed against a resistance that is itself frequency-dependent, and the fifty ohms is the value it has reached by the time the load capacitance matters. At frequencies well below crossover the effective resistance is far lower and the pole is far higher, which is why the closed-loop response is undisturbed at a kilohertz.

The model is therefore right where it is used — near crossover — and would be wrong if the same fifty ohms were used to compute, say, the low-frequency output impedance, where the answer is five hundred times smaller.

Why a data sheet cannot settle this #

The number that makes the whole effect is the amplifier’s open-loop output resistance, and it is the one specification of the four involved that is usually absent.

Gain–bandwidth product is on the front page. Phase margin is quoted, usually for unity gain into a stated small load. Output impedance is sometimes given as a graph against frequency — closed loop, which is the useful curve for a designer and the wrong one for this calculation, because it already has the feedback in it and therefore already depends on the gain the part was measured at.

What is needed is the open-loop figure, and where it is given at all it is given as a typical value in a table of characteristics with no tolerance. So the pole in the table above is computed from a number that is a factor of two uncertain in either direction, which moves the 905 picofarads to somewhere between 450 pF and 1.8 nF.

The practical reading of that is not that the calculation is useless but that its answer should be treated as an order of magnitude. A hundred picofarads is safe on almost anything; a nanofarad is a question; ten nanofarads without a series resistor is a mistake on every part that does not say otherwise. Some parts do say otherwise, and the ones that do — the “unity-gain stable with any capacitive load” family — achieve it by having a much lower output resistance or an internal compensation that moves with the load, both of which cost quiescent current.

The stable region is not a range of capacitances #

One more property of this failure is worth naming because it defeats the obvious test.

Increasing a load capacitance does not simply make the margin worse without limit. Beyond a certain size the pole formed with the output resistance drops so far that the loop’s crossover falls with it, and for some amplifiers the margin comes back up. A designer who tries 100 pF, 1 nF and 10 µF may therefore find the first and last acceptable and the middle one ringing, which reads as a measurement error and is not.

The figure’s slider stops at ten nanofarads, well inside the monotone region for this part, and the assertion it makes is that every step of the slider makes the margin worse. That is a claim about the range drawn, and it is stated that way rather than as a general one — a range that does not include the recovery is a range in which the simple story is true.

The three repairs, and what each costs #

Every one of them works by keeping the pole out of the loop or by putting something back in that cancels it.

A series resistor between the output and the load. Twenty or fifty ohms, and the pole now forms against that resistor plus the amplifier’s, but the feedback is taken from the amplifier side of it — so the pole is outside the loop. It costs the divider between that resistor and the load, which for a capacitive load is a rolloff of the signal reaching the load, and it costs any direct-current accuracy at the load if it draws current. This is the standard repair and it is why an output is often drawn with an unexplained resistor in series. The resistor that buys the margin back measures both halves of it: ten ohms restores forty-five degrees and twenty-three restores sixty, and taking the feedback from the load instead — which is what anyone controlling the load would do — makes every value worse, which is the assertion that separates “the resistor damps something” from “the resistor moves a pole out of the loop”. The cost is one per cent of error into a kilohm, at direct current, uncorrected.

Feedback taken from the far side of a series resistor at direct current only. A resistor in series with the output, feedback taken from the load through the feedback resistor and from the amplifier through a small capacitor. The loop then sees the amplifier’s output at high frequency, where the margin matters, and the load at low frequency, where the accuracy does. It costs a component and some care. The path that buys the error back prices that care and finds the price is not an error and not a margin but a range: at twelve picofarads the circuit settles to a hundredth of a per cent in 0.745 microseconds, faster than the arrangement it repairs, and at a hundred it takes 9.18 — with the phase margin at the slow value being the better of the two.

A snubber across the load. A resistor in series with a capacitor, in parallel with the load, sized so that above the frequency where the load capacitance is a low impedance the resistor is what the amplifier sees. It costs current and it costs dissipation, and it is what a regulator does.

The regulator essays are all about the third of these, which is where the trade is at its sharpest: an output capacitor is required for the load step and its equivalent series resistance is required for the stability, and the window between too little and too much is narrow enough to be worth a figure of its own.

The other place a load capacitance costs a margin #

A follower has no feedback network a designer drew, and it has exactly the same problem in a form that is harder to see.

An emitter follower’s output impedance is small at low frequency and inductive above it, because the current gain that made it small is falling. An inductive output impedance and a capacitive load are a resonant circuit, and the semiconductors field measures the result: a follower driven from a kilohm presents 19 Ω at low frequency and peaks at 67 Ω at 29 MHz with a hundred picofarads on it.

The same repair works there for the same reason, and the same trap is set: the peaking is worst at an intermediate source resistance, so it cannot be avoided by making the driving impedance smaller or larger.

What is checked #

The figure asserts the same four things in both of its modes, and the pair of them is what makes the comparison worth having.

That the closed-loop gain at a kilohertz is unmoved by the capacitance, which is the claim that makes the whole thing surprising — measured at eight significant figures rather than asserted.

That the noise gain is one plus the resistor ratio, and specifically $1 + (R_f + r_{\mathrm{out}})/R_{\mathrm{in}}$ = 2.005 rather than 2, because the amplifier’s own output resistance is in series with the feedback resistor. Asserting the textbook 2 to a part in a thousand fails, and the failure is the circuit being right.

That in the load mode the noise gain does not move with the capacitance, to parts in ten million, and in the summing-junction mode it does — the two halves of the mechanism, stated as opposite assertions so that a figure drawn in the wrong mode cannot pass.

And that there is a capacitance at which the margin is forty-five degrees, found by bisection on the netlist: 905 pF at the output, 9.00 pF at the junction. A factor of a hundred between the two, and the same amplifier.

What the ladder above this one does with it #

Eight rungs stand on this measurement and it is worth naming what each adds, because the capacitance on the axis of this essay’s figure turns out to be only the first of the things that change with the load.

The three immediately above are the repairs and their prices — the resistor, the second path, and what that path costs at the floor, which is where the expected noise penalty is looked for and found to be absent, the total at the load falling from 54.9 microvolts to 28.9 as the capacitor is added while the settling gets eighteen times worse.

Then the question turns round. What the load sees looking back asks what a load drawing its own current meets, which none of the rungs below it does, and finds the isolation resistor with no loop gain in it at all. The step too large to have an impedance then finds where that answer stops being a number: replace the linear transconductor with a differential pair’s own tanh and the ratio of voltage to current is constant only up to 10.6 milliamps.

And the last two rungs remove idealisations this essay makes without comment. The current above which there is no impedance gives the output stage a rating of its own and finds the two limits binding at different loads, with 2 254 Ω reported for a quantity that was 37 above the rating. The rail the load moves gives the supply an impedance — seven rungs having assumed a node that cannot be disturbed — and finds the disturbing channel’s own output impedance moving by three parts in ten million while a path opens to a second amplifier sharing nothing with it but a wire, at 3.84 microvolts per ampere or 17.3 millivolts depending on where one compensation capacitor returns.

The fifty ohms named on this page is in every one of those. It is the quantity the whole ladder is about, and it is still the quantity no data sheet page puts next to the stability page. Fifty ohms is printed, on the page about output drive; the stability page is elsewhere, and nothing connects them.