What the second path costs at the floor

Three essays in this field have measured the same circuit: an amplifier with two nanofarads hung on its output, which is a phase margin problem, and the two repairs for it. An isolation resistor restores the margin and costs the direct-current accuracy of the node the load is on. A second feedback path — the resistor to the load for slow signals, a capacitor from the amplifier’s own output for fast ones — restores both, and costs a band: the capacitor has to be large enough to carry the loop at crossover and small enough not to become the dominant pole, and outside that band the settling time rises by an order while the margin says nothing at all.

That last essay ended by naming the quantity it had not measured. A second feedback path puts a zero in the feedback factor, and a zero in the feedback factor is the classical way a compensated amplifier pays for its stability: the noise gain rises with frequency, and the amplifier’s own voltage noise is multiplied by it across the whole of the bandwidth. The suspicion was that this arrangement’s real cost was there, at the floor, where none of the three earlier measurements looks.

It is not, and the reason is worth the essay.

Three sources, each of them in the netlist #

A noise source is an element and goes in the netlist, which is the instruments field’s rule applied to a quantity nobody can see directly. There are three that matter here and they are in three different places.

The amplifier’s own input-referred voltage noise — four nanovolts per root hertz for an ordinary part — sits in series with its non-inverting input. Each of the two feedback resistors carries its own Johnson noise, $\sqrt{4kTR}$ per root hertz, in series with itself. The signal source is turned off in all three cases, because a noise transfer is measured with the signal off and a voltage source off is a short.

The first thing that falls out is a matter of proportion rather than of principle. The feedback network here is a pair of ten-kilohm resistors, and $\sqrt{4kTR}$ at ten kilohms is 12.66 nanovolts per root hertz — three times the amplifier’s own. So two thirds of the noise power at the load belongs to the resistors, and any argument about what the compensation does to the amplifier’s contribution is an argument about a fifth of the total.

That proportion moves as the compensation grows, and it moves the way it should. At twelve picofarads the amplifier is 22 per cent of the noise power; at two hundred and twenty it is 75 per cent, because the capacitor is progressively short-circuiting the feedback resistors at the frequencies where they were contributing.

The peak that was expected, and the one that is there #

The uncompensated stage does have a peak in its noise gain, and it is large: 3.85 times the low-frequency value, at 2.50 megahertz. That is not a coincidence of numbers — it is the same peaking the step response overshoots by, seen in a different variable, because both are the closed loop’s magnitude near a crossover with too little margin.

The isolation resistor reduces it to 2.64. Taking the feedback to the load instead makes it worse — 4.19 — which is consistent with everything the second essay in this ladder found about that arrangement.

The second path does something else entirely. Its peak falls with every larger capacitor and is gone by twelve picofarads: from there upward the noise gain is flat at its direct-current value of two, across the whole spectrum, to a part in a thousand.

The reason is a distinction that the phrase “a zero in the noise gain” hides. A capacitor from the summing junction to ground — the input capacitance that the noise-gain essay in this field is about — raises the noise gain with frequency, because it lowers the impedance of the leg that is not in the feedback path. A capacitor from the summing junction to the output lowers it, because it lowers the impedance of the leg that is. This is the second kind. It is a feedback capacitor, and a feedback capacitor takes the noise gain down.

What falls is bandwidth, not gain #

If the noise gain never rises above two and starts at two, the compensation cannot be reducing the noise by reducing a gain. What it reduces is the band over which that gain applies.

The equivalent noise bandwidth the amplifier’s own noise sees — the width of the brick wall that would pass the same power — falls from 7.83 megahertz uncompensated to 2.86 at twelve picofarads and 1.76 at two hundred and twenty. The total at the load falls with it: 54.9 microvolts, 28.9, 12.1.

That is the noise-bandwidth field’s own quantity turning up in a circuit designed for something else entirely, and it is the honest description of what is happening. The second path is a low-pass in the feedback network. It slows the loop deliberately, which is what buys the stability, and everything the loop passes — signal, amplifier noise, resistor noise — is slowed with it.

Which means the arrangement is not free of a noise cost so much as it is paid in advance: the same mechanism that repairs the margin is the mechanism that lowers the noise, and there is no separate price to find.

Two optima, eighteen times apart #

So where is the cost. It is where the previous rung found it — in the settling time — and the new part is that the two quantities want different capacitors.

The settling time to a hundredth of a per cent has a sharp minimum at twelve picofarads, at 0.74 microseconds. The noise falls monotonically with the capacitor and is lowest at the largest value drawn, 12.1 microvolts at two hundred and twenty picofarads, where the settling time is 20.3 microseconds — twenty-seven times the best.

The two optima are eighteen times apart in capacitance. There is no setting that is best at both, and the choice is not the one a designer expects to be making. It looks like a stability choice and it is not: the phase margin is above forty degrees at every capacitor from twelve picofarads upward, and it rises as the capacitor grows, so the measurement that would normally be consulted prefers the slow, quiet end and says nothing about the twenty-seven-fold cost in settling.

There is one pleasing coincidence in the numbers, and it has a reason. The smallest capacitor that flattens the noise gain completely is also the capacitor that settles fastest — both are twelve picofarads, at every load capacitance on the slider. One handover decides both: below it the fast path has not taken over and the loop still sees the isolation resistor’s pole, which is what peaks the noise gain and what rings the step.

The arrangement that is worse, and why it is worth drawing #

Three feedback arrangements are measured here and only two of them are repairs. The third — taking the feedback resistor to the load and stopping there, without the fast capacitor — is the one a designer reaches for first, because it is the arrangement that obviously fixes the direct-current error.

It is the worst of the three in every quantity this essay measures. Its noise gain peaks at 4.19, above the uncompensated stage’s 3.85; its total is 54.9 microvolts, the same as having done nothing at all. The reason is in the loop rather than in the noise: taking the feedback from the load puts the isolation resistor’s pole inside the loop, so the resistor that was bought to move a pole out of the loop is now contributing one to it.

It is drawn because a measurement that shows only the two good options is not a comparison. The site’s habit is to put the arrangement that fails beside the ones that do not, at the same axes, so that the distance between them is the argument rather than an assertion about it.

Where the integral starts and stops, and why it matters #

Every total in this essay is an integral of a density over frequency, and an integral needs limits. They are ten hertz to a hundred megahertz, on a logarithmic grid of three thousand points with a trapezoid in frequency rather than in its logarithm.

The upper limit is the one that could hide something and does not. The amplifier’s own gain–bandwidth is ten megahertz and the closed loop’s noise bandwidth is under eight, so by a hundred megahertz every density in the picture has been falling at twenty decibels a decade for more than a decade: the last decade of the integral contributes under a per cent of the total in every arrangement. Doubling the upper limit moves no number in this essay in its third digit.

The lower limit is the one that is a real omission, and it is stated rather than hidden. Below about ten hertz a real amplifier’s density stops being flat and rises as $1/f$, and none of that is in this model — the density here is white all the way down. What that leaves out is a term that depends on the part and not on the compensation, so it shifts all three arrangements by the same amount and changes none of the comparisons; but a designer who reads 28.9 microvolts as a total in a measurement bandwidth starting at direct current will find more than that.

The site’s rule is that a number is quoted with the band it was integrated over. This one is 10 Hz–100 MHz.

What a designer actually does with this #

The useful form of the result is a rule about which quantity is binding, and it depends on what the stage is driving.

If the load is a converter that must settle to a stated accuracy between samples, the settling time is a hard constraint and the noise is a soft one: choose the capacitor at the settling minimum, accept 28.9 microvolts, and if that is too much, lower the feedback resistors — halving both takes their noise down by $\sqrt{2}$ and does not move the settling at all, since the handover frequency depends on the product of the feedback resistor and the compensation capacitor and both can be traded.

If the load is a slow sensor or a reference, nothing needs settling in a microsecond and the capacitor should be as large as the response time permits. The noise falls, the margin improves, the direct-current accuracy is untouched, and the only thing lost is speed nobody wanted.

What a designer should not do is set the capacitor from a stability measurement. Every value here has an adequate margin. The margin cannot distinguish the settling optimum from a capacitor twenty times too large, and it will not warn about either.

What this ladder measured, in the order it measured it #

Four rungs on one circuit is unusual for this collection and it is worth setting out what each one added, because the shape is the argument.

The load that gets inside the loop measured the problem: two nanofarads on the output of an ordinary amplifier takes the phase margin to a few degrees, and the reason is fifty ohms of output resistance that no data sheet page puts next to the stability page, with forty-five degrees arriving at 905 picofarads — a metre of coaxial cable. The resistor that buys the margin back measured the standard repair and its price: forty-five degrees restored at ten ohms and sixty at twenty-three, with the loop no longer regulating the node the load is actually on, which costs about one per cent of accuracy into a kilohm. And the path that buys the error back measured the repair for that, and found its cost was neither a margin nor an error but a band of capacitor values, outside which the settling time rises by an order at a margin that improves.

This one asked the quantity none of the three had looked at, expecting to find the price there, and found the opposite: the arrangement that repairs the margin and the accuracy also lowers the noise, by almost a factor of two at the settling optimum and by four and a half at the quiet end.

Which is a result worth being suspicious of, and the suspicion has a specific form. A repair that improves four quantities at once usually means a fifth is being spent and has not been looked at. The candidates are named in the closing section: the amplifier’s current noise, its flicker corner, and the output impedance the arrangement presents to the load at high frequency, which is the isolation resistor’s — none of which this netlist contains. The claim here is bounded accordingly: of the four quantities this ladder has measured, the second path is better in three and neutral in the fourth.

The suspicion was right, and the fifth quantity was found by the rung above rather than by looking harder at any of these four. What the load sees looking back asks what a load drawing its own current meets, which none of the four rungs below it does, and the answer with the feedback taken from the amplifier is the isolation resistor with no loop gain in it whatever: ten ohms, and a load step leaving an error that never goes away. The two-path arrangement recovers to a thousandth of it — and charges for that in a quantity none of the four had measured. So the honest form of this essay’s closing claim is narrower than it reads: the second path is better in three quantities and neutral in a fourth among the quantities a signal source cares about, and the ladder had to change which terminal it was standing at before the price appeared.

Two more rungs then removed idealisations that were doing quiet work in all five measurements. The step too large to have an impedance found the impedance stopping being a number at 10.6 milliamps, and the current above which there is no impedance found a second limit that is nothing but design choice, binding at different loads from the first — 19.4 per cent against 4.2 at 0.47 nF and 0.9 against 2.3 at 22 nF, with an excursion above the output stage’s rating that does not come back at all. Neither is visible in a noise integration, and both decide whether the arrangement measured here is the one that is running.

The three quantities the same capacitor decides #

One compensation capacitor decides three things with three different optima. The path that buys the error back is the settling time, fastest at twelve picofarads. What the load sees looking back is the impedance the load sees. This page is the noise, quietest eighteen times away from the fastest settling. The load that gets inside the loop is the problem all three answer, and The gain the loop closes against is the distinction between signal gain and noise gain that makes the third quantity possible at all.

What is checked #

The claims of this essay are comparisons across settings, so most of them are asserted here rather than inside any one figure.

The total noise is asserted to fall at every larger capacitor. The peak in the noise gain is asserted to fall with every larger capacitor rather than to move somewhere else, and to be gone entirely above twelve picofarads, which is asserted to be the settling optimum to within a factor of two and a half at every load. The uncompensated stage’s own peak is asserted to be larger than the compensated one’s at every capacitor. The margin is asserted to be above forty degrees over the range in which the arrangement is doing its job, which is what makes it useless as a selector. And the three noise sources are asserted to sum in power rather than in amplitude, by comparing the total against the root of the sum of squares of the three solved separately.

What is not measured here: the amplifier’s current noise, which flows in the feedback network and adds a term that scales with the resistors rather than with the capacitor, and the amplifier’s own 1/f corner, which puts a rising density below a few tens of hertz that this integration starts above. Both are named in the noise field and neither is in this netlist.