A source below a frequency

The networks field’s first essay makes the point that a source is a source only over a range of currents: a nine-volt battery with half an ohm inside it is one per cent low at 180 milliamps. This essay makes the same point about the circuit built specifically to remove that objection, and the range turns out to be in the other variable.

The circuit #

A voltage reference, an error amplifier, a pass device from the rail to the output, a divider from the output back to the amplifier, a capacitor across the output, and a load. Every element is in the netlist and the whole thing is solved by the same solveAt as everything else here.

The pass element is a transconductance from the rail to the output, controlled by the difference between the rail and its own gate — a common-source device, which is what a low-dropout regulator uses. Its own output resistance is a kilohm in the netlist rather than being assumed infinite, because that resistance and the load together set the dominant pole.

The error amplifier is one pole: a transconductance into a resistor and capacitor, so its direct-current gain of $10^4$ and its corner at 15.9 Hz are both properties of elements rather than of a formula. A second pole sits between the amplifier and the pass device’s control terminal — the driver resistance against the gate capacitance, at about 300 kHz — and it is in the model because a real one is there, and because the next essay’s upper edge cannot exist without it.

Solved for its bias, the output comes out at 5.00440 V against the 5.000 the divider ratio and the 1.25 V reference ask for. The 4.4 millivolts is the error the amplifier needs at its input to hold the gate where it must be, and it is the direct-current loop gain divided into the output — the first and mildest of the departures this essay is about.

Two routes to one curve #

The site’s habit is to measure a quantity twice by routes that share as little as possible, and here there are two available that share only the netlist.

Drive it. Put a one-amp test current source into the output node with the reference and the rail at zero, solve, and read the voltage. That is a transimpedance and it is what an instrument would measure.

Divide it. Break the loop at the amplifier’s own input — the one place in this circuit where breaking it is exact, since a transconductance’s control terminals draw no current — drive the break with a test voltage, and read what comes back at the divider. That return ratio is the loop gain T. Measure the impedance again with the loop broken and its drive at zero, which is the same circuit with the feedback disabled, and divide by 1 + T.

Over six decades the two agree to $1.5\times10^{-13}$.

They did not, at first, and the reason is worth recording because it is the kind of error that looks like physics. The first version disagreed by thirteen per cent, uniformly, at every frequency. The rail’s twelve volts had been left in the netlist for the return-ratio measurement — and a direct-current bias in a netlist solved at $s = j\omega$ is not a bias, it is a phasor of twelve volts at whatever frequency is being asked about. Zeroing it is what makes the analysis a small-signal one; leaving it in solves a different circuit, correctly, and answers a question nobody asked.

What the curve says #

The first row is the number a datasheet prints. Everything after it is the loop gain running out.

The impedance doubles by 4.81 Hz, is ten times its floor by 27.0 Hz, a hundred times by 271 Hz, and peaks at 1.95 Ω at 10 kHz — a factor of 4520 above the specification, and about a third above the open-loop impedance at the same frequency, because near crossover the loop is doing slightly worse than nothing.

That last point deserves a sentence of its own. At 10 kHz the loop gain magnitude is 0.95 and its phase is such that $1 + T$ is smaller than one, so dividing by it makes the impedance larger. A feedback loop with 46° of phase margin does not merely stop helping near its crossover frequency; it makes the quantity it is regulating worse there than it would be with the loop cut. The peak in the figure is that, and it is the reason the next essay’s window has a lower edge.

What it means for a load #

A number in ohms is abstract; a load current makes it concrete.

A circuit drawing 100 mA of ripple at 10 kHz from this regulator — an audio power stage, a microcontroller clocking, a switching converter’s input — develops $0.1 \times 1.95 = 195$ mV of ripple on a five-volt rail. That is four per cent, on a supply whose specification says 0.43 mΩ.

The same 100 mA at 100 Hz develops 1.6 mV, and at direct current 43 µV.

So the regulator is a voltage source over a band, and the band’s upper edge depends on how good a voltage source is required. Two times the floor is 4.8 Hz; ten times is 27 Hz; a hundred times is 271 Hz. There is no single frequency at which a regulator stops being a source, which is the same shape of answer the magnetics field gives for a transformer’s band and the frequency field gives for a capacitor: an edge defined by a tolerance, quoted with the tolerance attached.

The three regions, and what each is made of #

The curve has three parts and each is a different circuit doing the work.

Below a kilohertz it is the loop. The impedance is the open-loop impedance — 4.97 Ω, which is the load in parallel with the pass device’s own output resistance and the divider — divided by 1 + T. Since T is falling at twenty decibels a decade from the amplifier’s 15.9 Hz corner, the impedance is rising at twenty decibels a decade, and it does so over three decades. Every point in that stretch is the open-loop 4.97 Ω divided by whatever loop gain is left.

Near ten kilohertz it is the crossover. The loop gain passes one, $1 + T$ is at its smallest, and the impedance peaks. The peak’s height is set by the phase margin: 46° here, and a design with 30° would peak higher and one with 70° hardly at all. This is the same peaking the transients field’s step responses show as overshoot, seen in the frequency domain and in a different variable.

Above a hundred kilohertz it is the passive components. The loop is gone entirely, and what a load sees is the output capacitor’s series resistance in parallel with the load resistance and the pass device’s output resistance:

$$5\ \Omega \parallel 1\ \Omega \parallel 1\ \mathrm{k}\Omega = 0.833\ \Omega$$

against a measured 0.835 Ω at a megahertz. The regulator is not in the circuit at all up there — what a fast load current sees is a resistor and a capacitor, and which resistor it is depends entirely on what was chosen for the capacitor.

What is left at crossover is where the loop gain in the second column of that table comes from, and how: cut the loop, drive one side of the cut, and measure what comes back to the other, which is the same operation performed on a bench and the same one performed here. The margin it reads at the crossing is the single number the peak’s height is a function of.

That last identity is why the next essay exists. The series resistance that sets this floor is the same series resistance the loop needs for its stability, and the two requirements want it to be different sizes.

Load regulation, which is the direct-current version #

The specification a regulator is usually judged by is load regulation: how much the output moves when the load changes. Marched from the operating point with a 100 mA step, this one settles 40 µV low — 0.0008% of five volts, or 0.4 mΩ of effective direct-current impedance, which is the first row of the table arrived at from the time domain instead of the frequency domain.

That is the specification, and the figure above is the reason the specification is nearly irrelevant to how a supply behaves in a circuit. Loads do not change once; they change at the rate the circuit they power switches at. A regulator with excellent load regulation and a 10 kHz crossover is a regulator with a 1.95 Ω output impedance where its load is actually operating.

The decoupling capacitor that everybody puts next to every integrated circuit exists for exactly this reason, and it is worth seeing what it is doing in these terms: it is a second source, in parallel with the regulator, whose impedance falls with frequency where the regulator’s rises. Below a few hundred hertz the regulator is the lower of the two; above it, the capacitor is.

The obvious conclusion from that sentence — that together they cover the range — is the one the power field has measured and found false. The pair that is worse than either puts a bulk capacitor beside a ceramic, each good where the other is not, and finds a frequency between them at which the pair presents six times the impedance either one does alone. The peak is a parallel resonance between one part’s inductance and the other’s capacitance; its height is one over the series resistance every data sheet asks to be minimised; and at it the two capacitors exchange 5.87 amps for every amp the load draws. Two impedance curves that cross do not add to the lower of the two. They add to something with a peak in it, and the peak sits exactly where the crossing is — which is to say, exactly where this essay’s argument put the handover.

The same is true of the handover measured here, and for the same algebra: the regulator’s rising impedance is inductive in character above its crossover, since it is a resistance divided by a loop gain falling at twenty decibels a decade, and the capacitor beside it is a capacitance. An inductive source and a capacitive shunt in parallel resonate. What keeps the peak finite is the loss in both, which is the series resistance the next rung is entirely about.

And the geometry between the two is not free either. The capacitor that is not where the load is puts three nanohenries of ordinary copper into that arrangement — one of mounting loop per part and two of plane between the bank and the load — and the anti-resonance moves down to 5.63 megahertz and up to 1.29 ohms, the twentieth capacitor is worse than the second, and a probe touching the ceramic reads a twelfth of what the load sees at 16.7 megahertz and three and a half times too much at 8.35. That last is the sharpest warning available about the measurement this essay makes: an output impedance is a property of a node, and the node the load is on is not the node the instrument is on unless somebody has made sure of it.

Above all of that the capacitor stops being one. The capacitor that is an inductor measures a hundred nanofarads following $1/2\pi fC$ for four decades and then turning round and climbing, with the crossing at 14.5 MHz and its impedance a decade further up ninety-nine times what its capacitance predicts — all of it caused by about a nanohenry of lead and via that nobody chose and nobody drew. So the second source in the parallel pair has an upper edge of its own, and the honest statement is that a supply rail is a voltage source over a band assembled out of several components, each of which is itself a component over a band, with a resonance at every junction between them.

Why this measurement is not the one on the datasheet #

It is worth asking why a quantity this consequential is usually quoted only at direct current, and the answer is that measuring it properly is awkward in a way the collection has already met.

Measuring an output impedance means driving a known current into the output and reading the resulting voltage. The current has to be known at the frequency of interest, which means a current source with a useful output impedance of its own up there; the voltage is a few hundred microvolts on top of five volts, which means an instrument that can reject the five volts; and both connect to the same two terminals, so the leads carry the drive current and the sense signal at once.

That last problem is one this collection has already solved twice, in the instruments field.

None of which makes the number less real. It makes it a measurement that a specification writer can decline to make, and a curve that a circuit meets whether or not anybody has drawn it.

What the model leaves out #

The pass device’s own limits. It is a transconductance with an output resistance, which is a small-signal model of a transistor. It has no maximum current, no saturation and no dropout: asked for more current than a real device could give, this one gives it. The dropout voltage — the minimum rail above the output at which regulation is still possible — is therefore not in this model at all, and it is one of the two numbers a datasheet leads with.

Thermal effects. The pass device dissipates the difference between rail and output times the current, which for a 12 V rail, a 5 V output and an amp is seven watts. Its parameters move with temperature, the reference moves with temperature, and the output moves with both. That is a slow loop with its own dynamics, and this essay’s fastest measurement is at a megahertz.

Everything nonlinear. The whole of this essay is a small-signal analysis about a bias point, which is exactly the model the semiconductors field spends its time bounding. How small is small signal puts a number on the condition: the amplitude at which linearising an exponential is one per cent wrong is 7.3 millivolts at room temperature, which is 28 per cent of the thermal voltage rather than a small fraction of it. A 100 mA step on a 1 A bias is inside the range; a step from zero to full load is not, and the answer would need the marcher.

Two of the three omissions above have their own essays on this rung. Two requirements pulling one capacitor marches a load step rather than solving for an impedance, and finds the same output capacitor’s series resistance to be a stability requirement and a transient requirement at once, with the best droop sitting ten per cent inside the region the loop must not be built in. What gets through from the rail asks the complementary question to this essay’s and gets an answer that barely depends on the loop at all: the same netlist with one node moved, the same loop gain and the same 46.5 degrees of margin, rejects 60.009 decibels better at every frequency in six decades. Both of those are measured on this circuit, and neither is visible in the curve this essay draws.

And the thing this whole rung is a repair for is one field back. The source that is not a source states the general case: an ideal voltage source holds its voltage at any current, which makes it the flattest line in the subject and the most commonly assumed model in it, and its edge is a current set by one resistance nobody draws — past which the model is not approximately right but describing a different object. A regulator moves that edge from the current axis onto the frequency axis. It does not remove it.

The gate #

The two routes agree over six decades, to $1.5\times10^{-13}$, sharing only the netlist.

The impedance rises by three orders inside the audio band, from 0.43 mΩ to 1.95 Ω.

The edges are ordered and spaced — twice at 4.81 Hz, ten times at 27.0 Hz — so what is being described is a slope rather than a step, and quoting one number for “the bandwidth of the regulator” would be quoting a tolerance without saying so.