The optimum that hands back a bandwidth

The ammeter that is a resistor found a result whose whole value is what it does not contain. A shunt has two errors pointing opposite ways — a larger shunt changes the current being measured, a smaller one leaves less voltage for the amplifier’s offset to be compared against — and written in terms of the burden voltage they are the burden over the supply and the offset over the burden. The best of them is at the geometric mean: 7.75 millivolts on a twelve-volt rail, with a worst-case error of 0.129 per cent.

Neither number contains a resistance and neither contains the current. Ten milliamps and a hundred amps want the same 7.75 millivolts, and only the resistance moves — three orders of magnitude of it, from 775 milliohms to 77.5 microohms.

That is the whole of the argument and it is about direct current. Nothing in it mentions frequency, and a shunt is not a resistor above one.

Two nanohenries, whatever it is made of #

A shunt is a piece of metal with two connections. Whatever its resistance, it has a loop area between its terminals, and a loop area is an inductance — a few nanohenries for a small surface-mount part, a few tens for a four-terminal bar with leads.

The resistance can be made almost anything by choosing the alloy and the geometry. The inductance cannot, because it is set by the size of the part, and the size is set by the power it has to dissipate rather than by anything electrical. A shunt for a hundred amps is physically large and has more inductance than one for a hundred milliamps, not less.

Two nanohenries is a reasonable figure for a four-terminal chip shunt and it is what is used here. The impedance is $R + sL$ and it has a zero at $R/2\pi L$, above which the part is an inductor with a resistor in parallel rather than the other way round.

The corner is a function of the current, and the optimum is why #

Here is the consequence, and the shape of it is the point rather than the size.

The burden optimum says $u^* = IR$ is a constant, so $R = u^*/I$. Substituting into the corner frequency,

$$f_z = \frac{R}{2\pi L} = \frac{u^*}{2\pi L I}$$

and the current is in the denominator. The corner falls in exact proportion to the current being measured, for the same piece of metal, because the optimum that contains no current has fixed the resistance in terms of one.

A hundred-amp shunt sized for the best direct-current accuracy has a corner at six kilohertz. That is not a radio-frequency problem; it is inside the control bandwidth of every switching converter ever built, and a current-mode controller reading that shunt is reading an inductor for most of its band.

The direction is the awkward part. High currents are exactly where a shunt is most attractive relative to a current transformer or a Hall sensor — the burden is low, the dissipation is manageable, the part is cheap — and high currents are where its bandwidth is worst. The optimum from the essay before it did not cause that, but it did fix the exchange rate, and stating it in burden voltage is what makes the trade invisible: a quantity with no current in it handed the bandwidth a current dependence through the back door.

What that does to a step #

A current step of rise time $t_r$ through a shunt of time constant $\tau = L/R$ produces a reading that overshoots. During the ramp the inductive term is $L\,\mathrm{d}i/\mathrm{d}t$ on top of the resistive $iR$, and the fractional overshoot at the top of the ramp is $\tau/t_r$ — the shunt’s own time constant divided by the edge’s.

At one ampere and 258.2 ns of time constant, a ten-nanosecond edge is read 2580 per cent high. That is not an error, it is a different waveform: the instrument reports a spike twenty-six times the current and then decays to the right answer with a time constant of a quarter of a microsecond. Anybody who has put an oscilloscope on a shunt in a switching converter has seen it and has usually been told it is ringing in the probe.

It is not the probe. The probe is part of the circuit measures what a probe does to a node and the effect here is upstream of it: the voltage across the shunt genuinely has that spike in it, because the shunt’s impedance genuinely rises with frequency, and a perfect probe would show the same thing.

The expression has a useful reading. $\tau/t_r$ says that what matters is not the shunt’s corner against the signal’s frequency but the shunt’s time constant against the edge’s, and a switching converter has edges two orders faster than its switching period. A shunt whose corner is ten times the switching frequency is still wrong by a factor of ten on the edges — which is where a current limit looks.

The edge slow enough to be read correctly #

Turn the overshoot expression round and it becomes a specification rather than a complaint. To read a current to within one per cent during its edge, the edge has to be slower than a hundred time constants:

$$t_r > 100\,\frac{L}{R} = 100\,\frac{LI}{u^*}$$

A shunt sized at the burden optimum for one ampere reads a one-ampere edge correctly only if that edge takes more than twenty-six microseconds. Nothing in power electronics has edges that slow, and very little in instrumentation does either. So the uncompensated shunt at its accuracy optimum is not a slightly-too-slow instrument; for transient work it is the wrong instrument, by two or three orders of magnitude, and the direct-current accuracy it was optimised for is irrelevant to what it is being asked to do.

That is the reading worth carrying out of the table. The essay before it produced a number good to 0.129 per cent and this one shows that the number describes a measurement almost nobody is making.

The cure is a divider whose time constants match #

The repair is old, exact and the same one a scope probe uses.

Put a resistor and a capacitor in series across the shunt and take the reading across the capacitor. The shunt’s impedance is $R(1 + s\tau)$ and the divider’s transfer is $1/(1 + sR_cC_c)$, so the product is flat whenever

$$R_cC_c = \tau = L/R$$

and the inductance cancels exactly rather than approximately. It is the compensated divider of the capacitor that was right once, used to cancel a zero instead of a pole, and it is the same balance condition.

The figure finds the capacitor by searching the solved response for flatness rather than substituting the expression, because the expression is what is being tested. With a kilohm compensating resistor at one ampere the search lands on 258.2 pF against $L/(R\cdot R_c) = 258.2$ pF, agreeing to eight parts in a hundred thousand, and the reading is then flat to $7.8\times10^{-5}$ per cent across six decades against 2580 per cent for the bare shunt.

And it is a value rather than a direction: a fifth too much capacitance is ten times worse than the right value. Under-compensated the reading still peaks; over-compensated it sags. There is one answer and both sides of it are wrong, which is exactly the situation a scope probe’s trimmer exists to resolve and exactly why a shunt’s compensation network has to be trimmed too.

What a designer actually has to choose between #

The compensation works and it costs something, and what it costs is the reason the whole trade is worth drawing.

The compensating network divides. Taking the reading across $C_c$ with $R_c$ in series means the amplifier sees the shunt voltage through a network whose direct-current gain is one — so far so good — but the network has $R_c$ in series with the amplifier’s input, which adds $R_c$’s Johnson noise and the amplifier’s bias current times $R_c$ to the measurement. A kilohm contributes 4.00 nV per root hertz, which against a 7.75 mV signal is 0.5 parts per million per root hertz and is nothing; a bias current of a nanoamp across it is a microvolt, which against 7.75 mV is 129 parts per million and is comparable with the whole error budget from the essay before it.

So the compensation is nearly free for a low-bias amplifier and is not free for a bipolar one, and the decision is the same shape as every other one in this field: a network that fixes a frequency response by adding an impedance has put that impedance in series with an instrument’s input, which is what the errors that arrive before the gain is about.

The alternative is to give up the burden optimum. A shunt ten times larger has ten times the corner frequency and ten times the burden — 77.5 mV at a hundred amps, which is 7.75 watts of dissipation and a 0.65 per cent insertion error against the optimum’s 0.129. That is a real choice and it is usually the right one for a current-mode controller, where 0.65 per cent of accuracy is worth nothing and 61.6 kHz of bandwidth is worth everything.

Which is the honest way to state that essay’s result: 7.75 millivolts is the best burden for accuracy alone, and accuracy alone is not what most current measurements are for.

And the result all of this is a consequence of, drawn once more so that the two pictures can be held together.

Where the compensation network has to sit, and what it costs to put it there #

The divider cancels the shunt’s zero when $R_cC_c = L/R$, and both sides of that equation are awkward for a different reason.

$L/R$ is not a specified quantity. A shunt’s data sheet gives a resistance to a tenth of a per cent and a tolerance on it, and gives the inductance either as a typical figure or not at all — so the time constant the network has to match is known to perhaps a factor of two before anything is measured. A trimmer is therefore not a convenience; the network cannot be designed without one, and the trim has to be done on a real edge with a real probe, which is the same procedure a scope probe’s compensation uses and for exactly the same reason.

$R_cC_c$ is not stable. The compensating capacitor is a small ceramic, and if it is a class II dielectric its value falls with applied voltage and with age — the capacitor that was right once measures both. Across the shunt the applied voltage is millivolts, so the voltage coefficient is harmless; the ageing is not, and a compensation trimmed at manufacture drifts out over years by a few per cent. A few per cent of mis-compensation on a 2580 per cent overshoot leaves a few tens of per cent, which is enough to matter.

And the network has to be physically at the shunt. The whole defect is a loop area of a couple of square millimetres between the sense terminals; a compensation network placed at the amplifier, several centimetres away, adds its own loop in series with the one it is cancelling and makes the time constant it is trying to match into something larger and less well known. The sense connection itself is part of the parasitic, which is the point the millivolts in the wire makes about a connection that was assumed to be a node.

So the compensated shunt is a small assembly rather than a component: a shunt, two parts across it, a trim, and a layout rule. That is not a criticism of the technique — it is the cheapest accurate wideband current measurement there is — but it is the honest inventory of what “just add an RC” costs, and it is why the alternative of accepting a larger burden is chosen as often as it is.

What two nanohenries does not stand for #

That two nanohenries is the right figure for every shunt. It is a reasonable one for a four-terminal chip part. A bar shunt with bolted terminals has tens of nanohenries and a coaxial shunt — built so that the return current flows back through a tube around the element, cancelling the loop — has a fraction of one. Every number in the table scales inversely with whatever is used, and the shape of the dependence on current does not change at all.

That the four-terminal connection helps with this. It does not. Two terminals measure the leads as well is about resistance, and a Kelvin connection removes the leads’ resistance from the measurement. It does not remove the loop area between the sense terminals, which is where this inductance is. The two defects have the same cure in the ohmmeter case and different cures here.

That the compensation cancels the inductance. It cancels the inductance’s effect on the reading. The shunt’s impedance still rises with frequency, so the current being measured is still being disturbed by an impedance that is larger than it should be — the insertion error grows with frequency even when the reading is flat, and nothing in the divider touches that.

That the overshoot is a small-signal statement. It is linear and exact, and it says a ten nanosecond edge reads twenty-six times high at one ampere. What a real amplifier does when handed twenty-six times its full-scale input is a separate question with a separate answer, usually involving several microseconds of recovery.

The corner two ways, and the capacitor found by search #

The corner is bisected on the solved impedance at every current on the slider and checked against $R/2\pi L$ — two routes to one frequency, one from a probe current into the netlist and one from an expression.

Its proportionality to the current is checked as a ratio between adjacent decades, exactly ten, because the claim of the essay is about the dependence rather than about any one value.

The compensating capacitor is found by golden-section search on the solved flatness, not substituted, and checked against $L/(R\cdot R_c)$ to five parts in a thousand.

And the compensation is checked to be a value rather than a direction: a fifth too much capacitance is required to be at least ten times worse than the right amount, which is the property that makes it a trimmer.

An optimum in one variable, priced in another #

The structure worth carrying is that an optimum stated in the right variable can hide a dependence rather than remove it.

The burden-voltage result is genuinely resistance-free and current-free, and that is genuinely useful: it means one number covers every shunt a designer will ever size. But it fixes $R$ in terms of $I$, and every quantity that depends on $R$ therefore acquires a dependence on $I$ that it did not have before. Bandwidth is one. Dissipation is another — $I u^*$, linear in the current — and self-heating follows it.

So the optimum did not make the design current-free; it moved the current out of the error budget and into everything else. Reading it as “the answer does not depend on the current” is the mistake this essay exists to prevent, and the general form is worth stating: a constraint that eliminates a variable from one quantity introduces it into every other quantity the constraint touches.

The same shape appears wherever a design is reduced to a single figure of merit. The three tolerances that do nothing finds combinations of component variations that the response does not respond to, and the flat directions are flat only for that response. A number that does not move is a number that has been held still by something, and the thing holding it is where the cost went.

Still open: the coaxial shunt, the insertion error against frequency, and the sensor with no burden #

The geometry that removes the inductance rather than compensating it. A coaxial shunt returns the current through a tube surrounding the element, so the loop area between the sense terminals is nearly zero and the inductance is a fraction of a nanohenry. Solving one as a distributed structure rather than as a lumped $R + sL$ would say where its own model stops — it becomes a transmission line at some frequency — and whether the residual is an inductance or something else entirely.

The insertion error, which the compensation does not touch. The reading can be made flat while the disturbance to the circuit grows with frequency, so there are two bandwidths here and only one of them has been measured. Solving the loaded circuit rather than the shunt alone would give the second, and the ratio between them would say whether a compensated shunt is honest at frequencies where it is flat.

And the instrument with no burden at all. The ammeter that is not in the circuit measured a current transformer, whose errors are of a completely different kind — no response at direct current, a ratio error that stops falling at one minus the coupling. A shunt with a compensation network and a current transformer with a burden resistor are two instruments whose bandwidths are limited by opposite mechanisms, and drawing their usable bands on one axis against current would say where the crossover between them actually is.