What the load sees looking back

This is the fifth rung of one argument, and the first four all asked the same kind of question. An amplifier drives a capacitive load; the load eats its margin; a resistor in series buys the margin back and costs the regulation; a second feedback path buys the regulation back and puts a doublet in the settling; and the doublet turns out to cost nothing at the noise floor. Margin, settling, error, noise.

Every one of those is a measurement of what the amplifier does to the signal. None of them is a measurement of what the load has.

A load is not a capacitor sitting quietly. It is something that draws current — a logic input that switches, a converter that samples, a stage whose bias steps — and what decides how far the node moves when it does is not the loop gain but the impedance it is looking back into. That impedance is what this essay measures, and it separates the three arrangements in a way none of the four rungs below could.

The measurement, which is the site’s usual one pointed the other way #

Ground the input. Put a one-ampere current source into the load node. Read the voltage there. That is what “the impedance the arrangement presents to the load” means when it is measured rather than argued about, and it is the same operation this collection uses for every impedance it draws.

One decision inside it is worth stating because the opposite decision would be defensible and wrong. The load capacitance stays in place. It is part of what the load sees; removing it would measure a circuit nobody has, and the entire subject of this anchor is that the capacitor is there.

Ten ohms, with no loop gain in it at all #

The first result is the one that makes the essay necessary.

With the feedback taken from the amplifier’s own output — the second rung’s arrangement, the one that buys the margin back — the impedance at the load node at direct current is 10.00 ohms. That is the isolation resistor, to four figures, and there is no loop gain in it anywhere.

The reason is structural and is the whole of the second rung’s compromise stated in a new quantity. The loop regulates the node it is fed back from, and it is fed back from the amplifier’s output. The isolation resistor is outside the loop, so what the load is connected to is a regulated node with an unregulated resistor in series, and a hundred decibels of loop gain does not reach past it.

With the feedback taken from the load instead, the same measurement gives 1.2 milliohms — four orders lower, because now the loop is regulating the node the load is on. And the two-path arrangement, which takes its direct current from the load and its fast path from the amplifier, gives 1.2 milliohms as well. It has the good number, exactly.

What a load step actually does #

The impedance curve says everything about this in principle and nothing in practice, because what a designer has is not a swept current source but a load that switches on. Marching a one-milliamp step forward in time turns the curve into the two numbers a designer has:

The first row is the finding restated in the currency of a data sheet. A milliamp of load current leaves ten millivolts of error at the load for ever — not while it settles, not until the loop catches up, permanently — because the isolation resistor is a resistor and the loop cannot see it. On a one-volt signal that is one per cent, and it is the same one per cent the third rung of this argument measured as a direct-current regulation error, arriving here through a completely different door.

The march and the impedance curve share no arithmetic — one is a trapezoidal integration of the netlist in time and the other a solve at $s = 0$ — and they agree on where each trace ends to better than three per cent of the excursion. That agreement is the check that neither is measuring itself.

Where the cost is, and it is not where the last two rungs found theirs #

The fourth rung asked whether the second feedback path costs anything at the noise floor and found that it does not. This rung asks whether it costs anything at the load and finds that it does — in a quantity neither of the two rungs below had a reason to look at.

It is not the excursion. The two-path arrangement’s is the smallest of the three, 19.6 mV against 23.6 and 21.1.

It is not the resonant peak. All three arrangements ring against the load capacitance somewhere above a megahertz, and at 2.2 nF the two-path arrangement’s peak is 28 Ω against the amplifier path’s 37 and the load path’s 59 — again the best of the three.

One number in this section is at a different compensation from the ones above it, and it is worth saying which. The impedance curves are drawn at 100 pF, because the band in which the two-path arrangement has let go of the load node and has not yet resonated only exists there: at the settling optimum of 12 pF the handover is at 1.33 MHz and the resonance is at 2.6, so the two run into each other and there is no middle. The load-step measurements are at the settling optimum, which is the capacitor a designer would actually fit.

It is the time. With the compensation capacitor at 100 pF the two-path arrangement takes 3.38 µs to get back inside two per cent of its excursion, against 0.62 for the amplifier path and 1.00 for the load path. At 220 pF it takes 7.31 µs.

And the mechanism is one this argument has met before, from the other side. The handover between the two feedback paths puts a pole and a zero close together — a doublet — and a doublet is invisible in a margin, nearly invisible in a magnitude, and decisive in a recovery. The third rung of this anchor found exactly that in the signal path. Here it is in the load path, with the same cause and a different victim.

The third quantity the same capacitor decides #

That last sentence is what makes this rung worth the fifth place, because the compensation capacitor now decides three things and the rung below already found that two of them disagree.

Swept from 1 pF to 220, the load-step recovery has a minimum, and it is at 12 pF — which is exactly the capacitor the third rung found settles the signal fastest. The two agree, and they agree to the value on the slider rather than approximately.

The fourth rung found that the capacitor which is quietest is eighteen times that one. At 220 pF the load-step recovery is 7.31 µs against 0.30 — twenty-four times slower.

So the trade the fourth rung named has a third party and the third party takes a side:

• 12 pF: the fastest signal settling, the fastest load recovery, 61.8° of margin;
• 220 pF: the lowest total noise at the load, 46.3° of margin, and a load-step recovery twenty-four times longer.

Two of the three quantities prefer the small capacitor decisively and the third prefers the large one. A designer who read only the fourth rung would take the quiet capacitor and would be right unless the load draws current — and the loads that need this compensation, converters and switching stages, are exactly the loads that draw current.

The peak, and the inductance nobody put in the circuit #

One feature of the impedance curves deserves its own paragraph because it is the same object this field has met twice under other names.

Between direct current and about a megahertz, the impedance of the load-feedback arrangement rises in proportion to frequency. A magnitude proportional to frequency is an inductance, and there is no inductor anywhere in the netlist. What produces it is the loop gain falling: the output impedance is roughly the amplifier’s own output resistance divided by the loop gain, the loop gain falls as one over frequency above the dominant pole, so the quotient rises as frequency.

That synthetic inductance then resonates with the load capacitance, which is where every one of these curves peaks, and the peak is what an ordinary load sees as ringing.

The load capacitance moves the peak and does not remove it. At 0.22 nF all three peak near 53–60 Ω; at 22 nF the peaks are 12–55 Ω and the ordering of two of them has swapped, because the synthetic inductance is a property of the amplifier and the capacitance is not.

Why this is a rung and not an aside #

This collection counts depth in rungs against an idea, and five is further than any argument here has gone. It is worth saying what makes this a fifth rung rather than a fifth figure, because the test is not “another measurement of the same circuit”.

The four rungs below all treat the amplifier as the thing being studied and the load as a condition imposed on it — a capacitance to be tolerated. Each of them asks what the capacitor does to a quantity belonging to the amplifier. That is a coherent way to think about the problem and it is the way every data sheet and every application note thinks about it, and it has a blind spot with a shape: any quantity belonging to the load is invisible from inside it.

The ten ohms is in that blind spot. It is not a degradation of anything the four rungs measured. The margin is fine, the settling is fine, the noise is fine, the direct-current error of the signal was measured and repaired two rungs ago — and a load that draws a milliamp still sits ten millivolts away from where the amplifier thinks it is, permanently, and nothing in the four rungs below reports it.

That is the shape of finding this collection exists for, and it is the same shape as the recovery loss that turned out to be in the switch rather than in the diode, or the noise that folds as many times as the settling needs time constants: the quantity is not wrong, it is being attributed to the wrong component.

Where a load resistance sits in all of this #

Every number above is for a load that draws its current in steps and has no steady resistance. A real load usually has both, and the arithmetic for the steady part is worth one paragraph because it is where the ten ohms turns from a curiosity into a specification.

A load resistance and the output impedance form a divider. With the feedback from the amplifier, the gain to the load node is divided by $1 + R_\mathrm{iso}/R_L$ — one per cent into a kilohm, which is the number the resistor that buys the margin back measured directly and called a regulation error. Here it is the same ten ohms wearing its other hat: as a series resistance it divides a steady signal, and as an output impedance it converts a load current into an error. One component, two symptoms, and a designer who fixes the first by calibration still has the second.

This is also where the arrangement stops resembling a regulator, which is the circuit a reader is most likely to have in mind. A source below a frequency measures a regulator’s output impedance and finds 0.43 milliohms at direct current, because there the loop is closed around the output node itself and nothing sits between the two. The isolation resistor here is what a regulator does not have, and its cost is the four orders of magnitude between those two numbers.

The two-path arrangement removes both at once, and that is the cleanest statement of what its direct-current path is for. The path that buys the error back is where its price is established, and the price is a range rather than a number: a settling time running from 0.745 microseconds to 9.18 across the load capacitances the design has to cover, with the phase margin at the slow end being the better of the two. It does not reduce the isolation resistor’s value, improve its tolerance or move it inside the loop in any physical sense; it gives the loop a second look at the node the resistor is on, at frequencies where looking is safe.

The three quantities one capacitor decides #

The compensation capacitor on this page decides three things and they do not share an optimum. This one is the impedance the load sees. The path that buys the error back is the settling of the signal path, whose fastest value and best margin are different components. What the second path costs at the floor is the noise, whose optimum is eighteen times away from the fastest settling. The load that gets inside the loop is the problem all three are answers to, and The buffer that is not a buffer is the same inductive output impedance one field over, without a loop around it.

What is checked #

The direct-current impedance of the amplifier-feedback arrangement is asserted to be the isolation resistor to two per cent, which is the claim that the loop does not reach the load at all. The other two are asserted to be a thousandth of it, and the two-path arrangement’s is asserted to equal the load-feedback one’s to five per cent — the three assertions that between them say which arrangement has which number and why.

The two routes are required to agree: every marched trace must arrive within three per cent of its excursion at the value a solve at $s = 0$ predicts. They share no arithmetic, and the two-path arrangement’s tail is long enough that reading the end of the march instead would have reported a settling time against a moving target — which is what a first version did.

The two-path arrangement’s resonant peak is asserted to be below the load-feedback arrangement’s, which holds at every load capacitance on the slider. That it is below the amplifier-feedback one’s as well holds at small capacitance and not at large, so it is a claim about the sweep and lives in the site’s gate.

The recovery time is asserted to be of the order of the handover’s own time constant rather than against a number, because the whole point is that it moves with the compensation capacitor. That its minimum over the capacitor coincides with the settling optimum, and that the quiet capacitor is twenty-four times slower to recover, are claims across settings and are in the gate too.

What is not measured: a load that draws current and has capacitance of its own, which is most real loads and would move the resonance rather than only exciting it; the amplifier’s own supply rejection, so every number here assumes the rail does not move when the load steps; and a load step large enough to slew the amplifier, at which point none of this is linear and the impedance stops being a quantity.

The second and third of those are the two rungs above this one, and both turn out to be larger than the sentence above suggests. The step too large to have an impedance locates where the third begins and finds it low: give the amplifier the differential pair’s own tanh in place of a linear transconductor and the ratio of voltage to current is a number up to 10.6 milliamps and not above, where the input error is 3.63 thermal voltages and the slew rate deciding it is twice the thermal voltage times the gain–bandwidth in radians — an expression with no design choice in it at all. The rail the load moves removes the second assumption and finds something the sentence above could not have predicted: giving the rail an impedance changes this channel’s own output impedance by three parts in ten million, because the loop corrects the supply along with everything else, and instead opens a path to a second amplifier that shares nothing with this one but a wire. How large that path is is a modelling choice rather than a measurement — 3.84 microvolts per ampere with the compensation capacitor returned to ground, 17.3 millivolts with it returned to the rail, a factor of four and a half thousand.

So of the three omissions named here, the one that mattered was not the one that looked largest. The supply was expected to be a small correction to this measurement and is; what it was not expected to be is a coupling to a circuit that is not in the netlist.