The rail the load moves

What the load sees looking back measured the impedance a capacitively loaded stage presents to its load, and the step too large to have an impedance and the current above which there is no impedance found the amplitudes at which that impedance stops being one.

All three, and the four rungs below them, put an ideal voltage source where the power supply is. So an ampere of load step is drawn from a node that cannot be disturbed, and the current the output stage takes never gets back into the circuit that took it. Each of those rungs ended by saying so, and the omission has been carried forward five times.

It was attempted once and abandoned, deliberately, and the reason is worth restating because it is the whole difficulty: with an ideal current-source output stage the rail’s impedance does not reach the output at all. Something has to couple the supply to the signal path, and what that something is is a modelling choice rather than a measurement.

This rung makes the choice explicit, offers both, and measures both.

Two objects, and only the second is a choice #

A rail with an impedance is not controversial. An ideal source, fifty milliohms and thirty nanohenries of wiring, and ten microfarads of decoupling with twenty milliohms of equivalent series resistance at the pin. That is the same object the power field’s the pair that is worse than either measures, used here as the thing it is for rather than as the thing being measured.

An output stage that draws its current from that rail is not controversial either. The macro-model is the one this field already uses — a transconductor, a compensation capacitor, a buffer with an output resistance — with the buffer’s output current supplied by the rail rather than appearing from nowhere. That is what an output stage is, and a step into the load pulls that current through the wiring and moves the supply.

What is a choice is how the moved rail gets back into the signal path, and there are two answers in wide use because there are two places a compensation capacitor’s far end can go.

Returned to ground, the gain node holds still while the rail moves. With an ideal buffer this circuit rejects the supply perfectly — which is not a modelling triumph, it is the same omission arriving one level down. What really couples is the input stage’s own supply dependence: a tail current set against the rail, a mirror whose output conductance sees it. That is quoted on every data sheet as an input-referred rejection, and it enters here as one number. At 3 × 10⁻⁵ it is the ninety decibels a decent part specifies.

Returned to the rail, which is what a Miller capacitor across a common-emitter stage physically does when that stage’s collector load returns to the supply, the gain node follows the rail above the compensation corner. That is a feedforward path in parallel with the loop’s own, and the loop never sees it.

Neither is more correct. They are two circuits, both built, and every figure here draws both. The one thing that would be wrong is to draw one of them and call it the rejection.

The rejections separate at a kilohertz and are fifty-four decibels apart by a megahertz: −84.4 decibels either way at ten hertz, and −84.4 against −30.3 at a megahertz. The ground-returned choice never gets much worse than its direct-current value, because its mechanism is an input offset and the loop corrects an input offset wherever the loop has gain. The rail-returned choice gets very much worse, because a feedforward path is not something a loop can correct.

The second and third decoupling values bracket it, and the reason for drawing three is that the impedance a load sees is not monotone in the capacitance: more decoupling moves the resonance rather than removing it.

The channel that caused it is fine #

Here is the result that explains why five rungs left this open and one abandoned it.

Give the rail an impedance and measure the disturbing channel’s own output impedance. It changes by three parts in ten million — 10.0010 ohms either way, across eight decades of frequency.

The loop is why. A load step moves the rail; the moved rail appears at the amplifier’s input through whichever mechanism; and the amplifier’s own feedback corrects it along with every other error at its input. The correction is the same correction that was already reducing the output impedance from fifty ohms to ten, so the supply’s contribution arrives already divided by the loop gain and lands several decades below a quantity that is itself the residue of a division.

So an essay that asked what does a moving rail do to the amplifier that moved it would have found almost nothing, correctly, and would have been the wrong question. That is what the abandoned attempt found, and recording it as an abandonment rather than as a null result is what made it findable.

The channel that did not is not #

The right question has two amplifiers in it.

The third value is the one that separates the two failure modes, and it is worth a line of its own: below it the rail’s inductance decides the impedance, and above it the decoupling capacitor’s own series resistance does.

With the wiring left out the coupling is exactly zero — the floating-point zero, because there is no path — which is what all seven rungs below computed and is the reason none of them found this.

With the wiring in, the second channel’s output moves. By 3.84 microvolts per ampere at 271 kilohertz with the compensation capacitor returned to ground, and by 17.3 millivolts per ampere at 2.33 megahertz with it returned to the rail. A factor of four and a half thousand, decided by which node a capacitor’s far end is drawn to.

Seventeen millivolts per ampere is not a subtlety. A power amplifier delivering a two-ampere transient into a loudspeaker puts thirty-four millivolts into the other channel of the same package, which at a hundred millivolts of signal is a channel separation of nine decibels.

That the disturbing channel is unaffected and the quiet one is not is worth stating as a rule rather than as a result. A feedback loop protects the circuit it is around and nothing else. The disturbance is inside channel A’s loop and outside channel B’s, and no amount of gain in A’s loop does anything for B.

The crosstalk readings follow the same three values, and they are the ones a designer actually feels — a rail impedance is invisible until a second load is hung on the same rail.

Why the two worst frequencies are not the same #

The two curves peak in different places, and the arithmetic behind that is the whole design story in miniature. The crosstalk is a product: the rail’s impedance, the second channel’s rejection, and the second channel’s own closed-loop roll-off. The maximum sits wherever the rising factor and the falling one cross, and the two choices have different rising factors.

For the ground-returned amplifier the rejection is flat, so the only thing rising is the rail’s own impedance, and the maximum sits near where that peaks. For the rail-returned one the rejection is itself falling at twenty decibels a decade — that is, the coupling is rising at twenty decibels a decade — and it goes on rising past the rail’s peak until the second channel’s bandwidth stops it. That is why one peaks at 271 kilohertz and the other at 2.33 megahertz on the same rail.

The same shape as a ground return, with one difference that decides everything #

The millivolts in the wire measures the other version of this: two circuits sharing a ground return, with one’s current developing a voltage in the shared impedance that the other reads as signal. The mechanism here is the same — a shared impedance, a current, a voltage — and one difference changes the size of it by decades.

A ground return couples directly. Whatever appears across the shared impedance is in series with the second circuit’s input, full strength, with nothing between.

A supply couples through the rejection. The same voltage appears on the rail and then has to get into the signal path, and how much of it does is the amplifier’s own supply rejection — which the figures above show is a modelling choice worth seventy-three decibels.

So the two mechanisms are not comparable in size and the design responses are different. A shared ground is fixed by not sharing it: separate returns to one point, which is layout. A shared supply cannot be unshared as easily, and the leverage is in the rejection at the frequency where the disturbance lives.

What a decoupling capacitor can and cannot do #

The slider on the crosstalk figure is the decoupling capacitance, and it does something instructive between its ends and nothing at all past one of them.

At a tenth of a microfarad the rail-returned crosstalk is 1.60 volts per ampere, which is a catastrophe and is the wiring inductance resonating with too small a capacitor. At one microfarad it is 101 millivolts. At ten, 17.3. At forty-seven, 16.2. At two hundred and twenty, 16.1.

The improvement stops. The last factor of twenty-two in capacitance buys one part in a hundred and fifty, because what is left is the wiring’s fifty milliohms and a capacitor cannot decouple a resistance. It is a short across the rail at high frequency, so it removes the inductive part; the resistive part is in series with it and stays.

That is the same reading the pair that is worse than either arrives at from the other direction — that a decoupling network has a floor set by what is in series with every branch of it — and it means the useful question is never how much decoupling but what is the impedance at the frequency that matters.

The instrument this suggests, and its limit #

There is a measurement implied and it is a good one, because it isolates the mechanism that the two choices differ over.

Drive a load step into one channel and read the other, with the rail’s impedance known. The crosstalk is the product of the rail’s impedance and the second channel’s rejection, and the first is measurable independently, so dividing gives the rejection at the frequency the disturbance actually lives at — rather than at the frequencies a data sheet’s rejection curve is plotted at, which are almost always sinusoidal and low.

The limit on it is the one this whole rung is built on. The measurement returns a rejection; it does not say which mechanism produced it, and the two mechanisms behave differently under everything a designer might change. The ground-returned one improves with loop gain and is flat with frequency until the loop runs out. The rail-returned one gets worse with frequency at twenty decibels a decade from the compensation corner and does not care about loop gain at all.

So a measured rejection at one frequency extrapolates in two quite different ways depending on a fact about the inside of a part that is not on its data sheet. That is the same trap as the exponent nobody put in’s fitted coefficient — a number that is right where it was measured and whose extrapolation depends on which mechanism is behind it — and it is why this rung draws two curves where a data sheet prints one.

Where the fix goes #

The design instinct is to attack the rail’s impedance peak, and whether that works depends on which choice is inside the part — which is exactly the thing a designer cannot see.

For the ground-returned amplifier it works: its worst frequency is near the rail’s own, so lowering the peak lowers the crosstalk. For the rail-returned one the worst frequency is a decade above anything the decoupling controls, and moving the peak does nothing at all.

What the choice costs a designer who cannot see it #

The two compensation returns are inside a part. A data sheet gives one power-supply rejection curve, usually measured with the output unloaded and a small sinusoid on the supply, and that measurement does not separate the two mechanisms because at the frequencies it is plotted over they can look alike.

What separates them is the shape: flat until the loop runs out, against falling at twenty decibels a decade from the compensation corner. A curve plotted from ten hertz to a hundred kilohertz on a part whose compensation corner is at ten hertz shows the second as a straight fall and the first as a plateau, so the shape is visible when the plot goes low enough — and many do not.

The practical consequence is that two parts with the same quoted rejection at a kilohertz can differ by fifty decibels at a megahertz, which is where a load step’s energy is. A specification taken at one frequency is a measurement of one point on a curve whose slope is a fact about a topology.

What this rung still does not do #

The rail is small-signal. A real load step drives the output stage into a region where its transconductance moves, and the current above which there is no impedance established that above a current this stage has no impedance at all. Both channels here are linear, so the crosstalk is a transfer function and not a waveform.

That matters more here than on the rungs below, because a crosstalk measurement is about a second circuit and the two need not be operating in the same regime. The disturbing channel is the one driven hard enough to move the rail, so it is the one most likely to be outside the linear range; the step too large to have an impedance puts that boundary at 10.6 milliamps for this arrangement, which is a modest load current. The disturbed channel is by construction quiet and is comfortably linear. So the honest statement of the mechanism is that a nonlinear source is coupling into a linear victim, and the coupled waveform is therefore not a scaled copy of anything — which is the same asymmetry the split-supply paragraph below describes, arriving from the current axis rather than from the topology.

And there is one rail. A split supply has two, the output stage takes its current from whichever one is sourcing, and the disturbance is therefore a rectified version of the signal rather than a copy of it — which puts even harmonics into the other channel where a linear model puts none. That is a distortion mechanism rather than a crosstalk mechanism and it needs the nonlinear output stage this rung does without.

And the rail’s impedance is one number. It is a resistance here, and the applied field has measured what the thing supplying it actually presents. A source below a frequency finds a regulator at 0.43 milliohms at direct current, doubled by 4.81 hertz, and 1.95 ohms at ten kilohertz — four and a half thousand times its own specification, and a third above the same circuit’s open-loop value there, because near crossover $1+T$ is smaller than one. So the crosstalk measured on this page is not a flat number times a frequency response; it is a frequency response multiplied by another one that peaks, and the peak sits where a load’s own switching current usually is.

The capacitors fitted to fix that do not simply lower the curve either. The pair that is worse than either puts a bulk capacitor beside a ceramic and finds a frequency at which the pair presents six times the impedance either does alone, with the two exchanging 5.87 amps for every amp the load draws, and the capacitor that is not where the load is adds three nanohenries of ordinary copper and finds the anti-resonance moving to 5.63 megahertz and 1.29 ohms, with the twentieth capacitor worse than the second. The shared impedance this essay’s crosstalk is proportional to therefore has a peak in it that no component on the board is responsible for, and the design decision this essay identifies — where the compensation capacitor returns — sets how much of that peak reaches the other channel.

That is the honest place to leave the number. The factor of four and a half thousand between the two modelling choices is measured against a rail whose impedance is a constant; against a real rail the same choice decides which of two frequency-dependent curves the crosstalk follows, and the ratio between them at any one frequency is not the ratio measured here.