The gain the loop closes against

Everything else in this field measures a loop whose closed-loop gain and whose stability are decided by the same number. Take a follower, or a non-inverting amplifier of gain ten: the fraction of the output fed back is a tenth, the closed-loop gain is ten, and one is the reciprocal of the other. There is nothing to distinguish.

An inverting amplifier is the simplest circuit in the subject where the two come apart. Two equal resistors around an operational amplifier give a gain of −1, and the loop closes against 2. The factor of two is not a subtlety and it is not a correction: it is a different quantity, with a different value and — this is the part that matters — a frequency response of its own that the signal gain does not share.

Two gains, both measured #

Nothing here is written down as an expression. The loop is cut at the amplifier’s inverting input, a signal is injected on one side of the cut, and what comes back to the other side is the return ratio $T$ — the same construction the loop-gain essay uses, and the same one a bench uses. The amplifier’s own gain $A$ is measured on a netlist with no feedback in it at all. The gain the loop divides the amplifier by is then $|A|/|T|$, read off the two solves.

At a kilohertz, with two ten-kilohm resistors and no stray capacitance, that comes out at 2.005.

The 2 is the familiar one plus the resistor ratio. The 0.005 is the amplifier’s own fifty-ohm output resistance, which sits in series with the feedback resistor and therefore appears in the divider the loop sees. Asserting the textbook 2 to a part in a thousand fails here, and the failure is the circuit being right rather than the model being coarse — so the figure asserts $1 + (R_f + r_{out})/R_{in}$ instead, and gets it to a part in $10^4$.

The consequence is immediate and is the first thing this essay is for. A ten-megahertz part in a follower crosses over at ten megahertz. The same part as a unity-gain inverter crosses over at $10\ \mathrm{MHz}/2.005$, and the measured closed-loop corner is 4.988 MHz. Two circuits, both called unity gain, one with half the bandwidth of the other, and the reason is a quantity that appears in no expression for either one’s gain.

The last two rows are the ones worth pinning up. A gain of −10 and a gain of +11 are built from the same two resistors, have the same loop, the same margin and the same bandwidth — 0.9088 MHz to four figures, both — and differ only in which end of the divider the signal is injected at. The bandwidth belongs to the loop, and the loop cannot see which terminal the signal came in at.

The capacitance that only one of them can see #

Now put a capacitance from the summing junction to ground. A metre of coaxial cable from a sensor is a hundred picofarads; a photodiode is a few tens; the amplifier’s own input capacitance is a handful before anything is connected. Nothing in the signal path has changed — the capacitance is at a node the loop holds at ground, so no signal current flows into it and the transfer function’s low-frequency value is untouched.

What it changes is the divider the loop sees. Above $1/2\pi (R_f \parallel R_{in})C$ the feedback network’s own impedance falls, less of the output gets back, and the gain the loop is closing against rises a decade per decade. The amplifier’s gain is falling a decade per decade at the same time, so the two curves close at forty decibels per decade instead of twenty — and the phase margin goes with it.

Forty-five degrees is reached at 9.00 pF, bisected on the netlist. That is less than the input capacitance of the part, less than a scope probe, less than the stray of a socket — and a circuit designed on paper as a unity-gain inverter is already marginal before anybody has connected anything to it.

What the signal gain does meanwhile #

Nothing. That is the whole claim, and the figure asserts it as a number rather than as a remark: the closed-loop gain at a kilohertz is 0.999 980 51 with a hundred picofarads at the junction and 0.999 980 51 without, agreeing to three parts in a million at the far end of the slider and to six parts in $10^8$ at ten picofarads.

So a measurement made at any frequency deep inside the loop — which is to say, any measurement made with a signal generator and a voltmeter — reports a circuit that is exactly as designed. The peaking is at 1.9 MHz and it is twelve decibels: the response of a nominally unity-gain amplifier is four times unity, at a frequency nobody was thinking about. At 470 pF it is 18.5 dB.

That is the shape of fault this collection keeps finding and it is worth naming. A quantity that does not appear in the design equations has moved a quantity that does not appear on the bench, and the two together produce a circuit that rings on an edge and measures perfectly on a sine wave.

A third value fills in the range, and it is the one at which a designer would first notice something wrong on an oscilloscope rather than in a specification.

Why it carries the name it does #

The name comes from the third thing this gain multiplies, and it is the reason the quantity was given a name at all rather than being left as a step in a derivation.

An amplifier’s input offset voltage and its own input-referred noise sit in series with the inverting terminal, on the same side of the summing junction as the feedback divider. So they are amplified by whatever the loop is closing against, not by the signal gain. A unity-gain inverter with a one-millivolt offset has two millivolts at its output; a four-input summing amplifier of unity gain per input has five. The same factor multiplies the part’s voltage-noise density, so a 4 nV/√Hz amplifier in that summer contributes 20 nV/√Hz at the output while its signal gain is one.

Two consequences follow that are hard to see any other way. Adding an input to a summing amplifier degrades its noise and its offset without touching any existing channel’s gain — the channels are independent and the noise gain is not. And the transimpedance case is the extreme version of both: the noise gain rises with frequency, so the output-referred noise density of a photodiode amplifier rises with frequency too, peaking near the crossover for exactly the reason the response peaks there. The two peaks are the same curve.

That is also the honest answer to why the quantity is worth separating rather than folding into “the feedback factor”. It is the multiplier on three different things at once — bandwidth, stability, and the amplifier’s own imperfections — and none of the three is the signal gain.

The repair, and the constant in it, follows from reading the two curves rather than either one: the feedback capacitor that restores the margin is sized against the junction capacitance and the feedback resistance, and the signal bandwidth it costs is the price.

The repair, and the constant in it #

The standard fix is a capacitor across the feedback resistor, which puts a pole in the feedback network to cancel the zero the input capacitance put there. The usual expression for it is

$$C_f = \sqrt{\frac{C_{in}}{2\pi R_f\, f_{gbw}}}$$

which is the value that makes the two break frequencies coincide, and it gives 12.6 pF here. Bisecting the netlist for a stated margin instead gives 7.75 pF for forty-five degrees and 14.8 pF for sixty-five. The closed form sits between the two, which is what one would hope: it is derived from a condition on the asymptotes rather than on the phase, and the phase is what stability is.

The cost is bandwidth. With 14.8 pF of feedback capacitance and a comfortable sixty-five degrees, the closed-loop corner is 1.25 MHz — a quarter of the 4.99 MHz the circuit had before the input capacitance arrived, and two thirds of the 1.93 MHz it has while it is peaking. The peaking is not bandwidth; it is the loop failing to control the response, and removing it costs some of what it was pretending to give.

Where else the two gains come apart #

A summing amplifier. Four inputs of gain −1 through four ten-kilohm resistors into one ten-kilohm feedback resistor give a noise gain of five, not one. The bandwidth is a fifth of the single-input case and the offset voltage is amplified five times, both of which surprise people who counted the gain per input.

A transimpedance amplifier. The input is a current source, so the signal gain is a resistance and has no dimensionless value at all. The noise gain is $1 + Z_f/Z_{source}$, and for a photodiode the source impedance is a capacitance — so the noise gain rises from one at direct current with no flat region anywhere. Everything in this essay is that circuit’s ordinary operating condition rather than a fault in it.

A difference amplifier fed from unequal sources. The two halves of the divider are no longer matched once source resistance is included, and the noise gain differs between the two paths — which is why a difference amplifier’s common-mode rejection falls with frequency in a way its resistor tolerance does not predict.

The measurement that would have found it #

Nothing in the low-frequency gain measurement can see any of this, so it is worth saying what can.

A square wave, not a sine. A ten-kilohertz square wave at the input of the hundred-picofarad circuit rings at 1.9 MHz on every edge, with an overshoot the fourteen-degree margin predicts. It is the same information the peaking carries and it arrives on the instrument everybody already has connected.

A gain measurement taken high enough. The peaking is at 1.9 MHz and the corner without the capacitance was 4.99 MHz, so a sweep that stops at a megahertz — a reasonable place to stop for a circuit specified at audio — sees nothing at all. The frequency to sweep to is set by the loop rather than by the application, which is an uncomfortable rule and the correct one.

A step at the summing junction rather than at the input. Injecting into the feedback network and watching the output is the cut-loop measurement done with a pulse generator, and it shows the margin directly. It is also the only one of the three that would have found the problem before the board was built, because it is the one this essay’s figure is.

What this essay does not claim #

That the noise gain is a noise quantity. The name comes from what it does to the amplifier’s own input-referred noise and offset, which it multiplies. It is not a noise density, it has nothing to do with the noise field’s floors, and it would be better called the feedback attenuation’s reciprocal — but the name is universal and inventing another one would help nobody.

That the fifty-ohm output resistance always matters. It contributes 0.005 to a noise gain of 2 with ten-kilohm resistors, and it would contribute 0.5 with hundred-ohm ones. The point of asserting $1 + (R_f + r_{out})/R_{in}$ rather than $1 + R_f/R_{in}$ is that the site’s rule is to assert what the netlist actually does, and the netlist has an output resistance in it.

That the feedback capacitor is always the right repair. It is the right repair when the bandwidth it leaves is enough. When it is not, the alternatives are a faster part, a smaller feedback resistance, or reducing the capacitance — and a sum that is exact, and the estimate that is not is the tool for deciding which. Add each capacitor’s value times the resistance seen at its own terminals with the others removed, and the total is the ratio of the first two coefficients of the denominator polynomial — a theorem, holding to a part in a billion at every spread tested — so the term that dominates the sum is the term worth attacking. What that essay is careful about is the step after: dividing one by two pi times the sum gives a bandwidth estimate that is 14 per cent low with three equal capacitors and never once optimistic, and two settings of its slider have identical time constants and bandwidths two per cent apart. So the sum identifies the offender exactly and predicts the answer only approximately, which is the right division of labour for a repair decision.

That a smaller feedback resistance is free. It is the obvious move once the sum above has named the summing junction’s own time constant as the offender, and what it costs is measured in the noise field. The floor a circuit has puts a device’s two generators at 4 nV/√Hz in series and 0.6 pA/√Hz across the input, with a source resistance of 6.67 kΩ at which their sum is least — so a feedback network scaled down for bandwidth walks along that curve, and past the minimum the current generator into the smaller resistance is what dominates. There is a resistance at which the circuit is quietest and it is not the resistance at which it is fastest.

That a ninety-degree margin means no ringing anywhere. It means none from this loop. A single-pole loop with resistive feedback has ninety degrees by construction, which is why the zero-picofarad row of the table is not a design achievement.

Where the same quantity is decisive elsewhere #

The noise gain is worth separating from the signal gain because four other measurements in this collection are functions of it, and in each of them the signal gain would give the wrong answer.

The load that gets inside the loop is the closest relative: it hangs the capacitance on the output instead of the summing junction, finds forty-five degrees at 905 picofarads against this essay’s 9.00, and — the part that identifies the mechanism — finds the noise gain not moving with the capacitance in that arrangement while it moves here. The same amplifier, the same margin, a factor of a hundred in the capacitance, and the difference is entirely which node the capacitance shares with the feedback network.

Where the trouble is at the input is the arrangement where the capacitance at the summing junction is not a parasitic to be minimised but the thing being bought: a photodiode’s junction capacitance is the price of its area, and area is what a photodiode is for. Everything on this page then reads as a design constraint rather than a warning, and the essay’s noise result follows from the same rising noise gain — the feedback resistor’s own 127 nV/√Hz against the amplifier’s 4, with the amplifier still ninety-five per cent of the total at a large diode, because the noise gain has multiplied its four and not the resistor’s hundred and twenty-seven.

What the second path costs at the floor is the measurement that expected this mechanism and did not find it: a compensation suspected of paying for itself in noise, because that is how compensations usually pay, turning out to have no peak in its noise gain at all and a total that falls from 54.9 microvolts to 28.9 as its capacitor is added.

And how much of the amplifier gets through is the reason the distinction is not merely bookkeeping. What a loop divides a device change by is $1+T$, and $T$ is the amplifier’s gain divided by the noise gain — so a circuit whose noise gain is two has half the desensitivity of a follower built from the same part, at every frequency, along with half the bandwidth measured here.

Where the two gains come apart again #

A signal gain that does not move while the noise gain does is the distinction three later essays are about. The load that gets inside the loop is the same failure at the output rather than the summing junction. Where the trouble is at the input is the transimpedance case, where the noise gain multiplies the amplifier’s own voltage noise by the detector’s capacitance. What is left at crossover is where the crossover the noise gain moves is defined, and The node that is at ground for a while is the impedance at the junction that all of it happens at.

The gate #

Both gains are measured, neither is written down. The return ratio comes from a cut loop and the amplifier’s own gain from a netlist with no feedback in it; the noise gain is their ratio.

The signal gain is asserted to be unmoved, to a part in $10^5$, at every setting of a slider that takes the phase margin from ninety degrees to under seven. That is the essay’s claim and it is asserted as a number rather than described.

The margin is asserted to fall monotonically across the whole slider, not merely to be small at the setting drawn, because a figure whose caption is about a trend has to have the trend tested at every frame.

The forty-five-degree capacitance is bisected on the netlist — 9.00 pF — rather than evaluated from the asymptote condition, and the closed form is quoted in the prose beside the two bisected values so a reader can see that it sits between them.