The pulse that ends before the heat

The resistance that depends on the reading solved a current shunt’s self-heating as a fixed point. The shunt dissipates, warms through its thermal resistance, and its resistance rises with its temperature, so the resistance the reading is divided by depends on the reading. Along the burden-voltage optimum, where the shunt is chosen so that the voltage across it is 7.75 mV whatever the current, the dissipation is $I u^*$ rather than $I^2R$, and the error came out linear in the current: at 20 K/W and 50 ppm/K, 775 ppm at a hundred amperes. It was the only one of the shunt’s four errors with the current in it.

Everything in that essay was a steady state. The current had been flowing long enough for the shunt, its terminations and the board under it to reach the temperature the thermal resistance sets, which for a board is tens of seconds. The currents a shunt is most often asked to measure are not like that. A motor’s inrush lasts milliseconds, a fault current a few cycles, a switching converter’s peak current microseconds. The question that essay left is what a pulse shorter than the thermal time constants does to the error — and the answer turns out to be that it depends on which time constant, because a shunt’s heat leaves through a ladder network rather than through a single resistance.

The network the heat goes through #

The thermal path is modelled as three sections. The element — a strip of resistive alloy — has a heat capacity of 5 mJ/K and 0.2 K/W to its terminations; the terminations and pads have 0.1 J/K and 4 K/W to the board; the board has 2 J/K and 15.8 K/W to ambient. These are stated assumptions for a metal-element surface-mount shunt, chosen so that the total is the 20 K/W the steady-state essay used, and they give the three sections time constants of about a millisecond, 0.4 s and half a minute. The ladder network’s exact response to a step of power is expanded into its own modes, and a pulse is a step on followed by a step off.

A millisecond pulse of a hundred amperes through the 77.5 µΩ shunt dissipates the same 0.775 W the steady state did, and warms the element by 98.7 mK. At 50 ppm/K that is 4.94 ppm of error at the pulse’s end — against 775 ppm for the same current held. The element has had one of its own time constants to warm, and the terminations and board beneath it have barely moved. The slider on the figure at the head of the page takes the pulse from ten microseconds, where the rise is 1.54 mK and the error 0.077 ppm, to a second, where it is 3.09 K and 154 ppm.

Three regimes in the pulse’s length #

The error at the end of a pulse, drawn against the pulse’s length, has a shape that reads off the ladder network directly.

Below about ten microseconds the heat has nowhere to go. The element warms adiabatically: its rise is the energy delivered over its own heat capacity, $P\,t/C_{element}$, and the solve follows that straight line to a per cent. The error is proportional to the pulse’s length, and the element’s thermal resistance, the terminations and the board play no part at all. Around the element’s own millisecond the curve bends and levels towards the element’s 0.2 K/W: heat now leaves the element as fast as it arrives, and for pulses of a few milliseconds the error is roughly the element’s resistance times the power. Around 0.4 s the terminations fill and the curve climbs again, and past the board’s half-minute it reaches the 20 K/W of the steady state.

So a hundred-microsecond pulse at a hundred amperes carries 0.737 ppm of self-heating error, 1,051 times less than the same current held. The steady-state figure the earlier essay computed is the right answer for a current that flows for a minute and wrong by three orders of magnitude for one that flows for a tenth of a millisecond, and the thermal resistance on a shunt’s data sheet — the number that fixed point was built from — has nothing to do with the short pulse’s error. What decides it is a number data sheets rarely quote: the element’s heat capacity.

Why the heat capacity and not the resistance #

The adiabatic regime is worth understanding rather than just reading off, because it overturns the usual instinct about which part of a thermal design matters. A thermal resistance says how much temperature it takes to push a watt through a path. For that to matter, the heat has to be pushing through the path, which takes time: the element must first warm enough for a gradient to exist between it and its terminations, and that warming is paid for out of its heat capacity. For times short against the element’s own resistance-times-capacitance, almost none of the heat has left it, and the resistance to its surroundings is irrelevant — the element is a closed box being filled.

The pulse the heatsink does not feel found the same fact about a power device at the other end of the scale: above a frequency set by the die’s own heat capacity the junction integrates the pulses, and the heatsink sees only their average. A shunt reading a single short pulse is in the first half of that statement. Its element integrates the pulse’s energy, and nothing beyond the element takes part.

It also explains why the steady-state fixed point is so conservative here. The loss that depends on what it causes and the steady-state shunt essay treated heat and resistance as settling together; that is the right description of a current held for minutes. For a pulse the resistance never catches up with the heat, and the feedback loop between them has had no time to close.

The current a pulse may carry #

Along the burden optimum the error is linear in the current, since the shunt is chosen so that its voltage is fixed and its dissipation is the current times that voltage. That makes the inverse question exact: for a stated error, what is the largest current a pulse of a given length may carry?

For 100 ppm, held indefinitely, the answer is 12.9 A. A one-second pulse may carry 64.8 A; ten milliseconds, 936 A; a hundred microseconds, 13.6 kA; a microsecond, 1.29 MA, where the allowance falls as one over the pulse’s length because the heating is adiabatic. None of those currents is one this shunt could survive for long — a hundred amperes held would dissipate its 0.775 W indefinitely, and ten thousand would melt it — but that is a different limit. The point is that for self-heating as a source of measurement error, a pulse’s length matters as much as its size, and the pulse currents a shunt is chosen for are almost never limited by this error.

That changes how the burden optimum should be read for pulsed measurements. The optimum balanced the amplifier’s offset against the shunt’s own voltage drop and found the self-heating term small at modest currents. For a pulse the self-heating term is smaller still, by the ratio of the thermal impedance at the pulse’s length to the steady thermal resistance, and the optimum’s balance of the other two errors is untouched.

A fault current, worked #

Because the error along the optimum is linear in the current, the hundred-ampere figures scale directly. A fault of a kiloampere lasting ten milliseconds, measured by the shunt the optimum picks for a kiloampere — 7.75 µΩ, dissipating 7.75 W while the fault lasts — reads 107 ppm high at the end of the fault, ten times the 10.7 ppm of a hundred amperes. The same kiloampere held would be 7,750 ppm, three-quarters of a per cent, and would also be well past what the part is rated to dissipate.

Set against the shunt’s other errors, a hundred parts per million is small. The burden optimum balances the amplifier’s offset against the shunt’s own drop and leaves 0.129 per cent between them at its best — thirteen hundred parts per million — and the rail that is an input error found a high-side amplifier’s finite common-mode rejection adding to the offset it balances. So for fault measurement the self-heating error that dominated the steady-state picture at high current falls to the bottom of the budget, and what decides the accuracy of a fault reading is the amplifier, not the shunt’s temperature.

Read at the end, or across the pulse #

A converter can read a pulsed current two ways. A sampling converter takes it at an instant, usually near the pulse’s end where the current has settled; an integrating one averages it over the pulse. They do not see the same self-heating error.

While the element heats adiabatically its temperature is a ramp, and the average of a ramp over its length is half its final value: the ratio is 0.500 at every pulse below a hundred microseconds. As the element’s section fills the rise flattens and the ratio climbs, to a first peak of 0.769 near eight milliseconds. Then the terminations take over the ramp and it falls back towards a half, and it climbs again as each slower section fills, reaching nearly one only for pulses long against the board’s time constant. An integrating reading of a short pulse therefore has half the self-heating error of a sampled one at the pulse’s end, and for pulses of a few milliseconds the two differ by a quarter to a third.

That is worth knowing in both directions. A designer comparing a sampled and an integrated reading of the same pulse, and finding them disagree by a few parts per million, has a thermal explanation before looking for an electrical one. And a designer choosing where in a pulse to sample can move the self-heating error by a factor of two by sampling earlier, where the element has had less time to warm, if the current has settled by then.

The thermal model that is usually quoted #

The ladder network’s step response was expanded exactly, into its own modes, and it is worth comparing with the form a thermal model is often quoted in.

A thermal model is often written as a sum of each stage’s own resistance times $1 - e^{-t/RC}$, with each stage’s time constant built from its own resistance and capacitance. That is not the ladder network’s step response, because the stages load each other: the element charges into terminations that are themselves warming. The exact expansion — the ladder network’s own modes, whose resistances sum to its 20 K/W to a part in a billion — agrees with the per-section sum at the shortest times, where only the element matters, and at the longest, where only the total does, and parts from it by 10 per cent at 7.08 ms. A pulse error read off the per-section sum in that range is wrong by that much. Two ladder networks the terminals cannot tell apart priced the same approximation on a power device’s heat path and found it under one per cent there; on the shunt’s path, whose element and terminations are closer together in time constant, it is ten times worse, and it is worst in the middle, which is exactly where a millisecond-pulse measurement sits.

What a designer should take #

For a pulsed current measurement, the self-heating error is the power times the thermal impedance at the pulse’s length, times the resistance’s temperature coefficient — not the thermal resistance. For pulses short against the element’s own time constant, which for a metal-element surface-mount part is around a millisecond, the error is adiabatic: $\alpha\,P\,t/C_{element}$, proportional to the energy delivered and independent of how the part is mounted. Ask for the element’s heat capacity, or measure the thermal impedance at the pulse lengths that matter; the steady thermal resistance answers a different question.

Decide whether the converter samples or integrates, since an integrating reading of an adiabatic pulse carries half the error. And when a thermal model is needed at the millisecond scale where the sections overlap, use the ladder network’s exact response rather than the sum of its sections.

The larger point is the one the optimum that hands back a bandwidth and its successors kept arriving at: a shunt’s errors have different dependences on the current, and their balance depends on the conditions of the measurement. Self-heating was the one error with the current in it; for a pulse it also has the time in it, and it becomes the smallest of the four by a large margin.

How the numbers were obtained #

The thermal path is a Cauer ladder of three stages, each a capacitance to ambient at a node and a resistance to the next node, and its step response at the element’s node is expanded exactly: the ladder network’s impedance is built as a continued fraction in polynomial arithmetic, its real negative poles found and polished on the undeflated polynomial, and the residues checked to sum to the total resistance. A pulse of length $t_p$ is the step response minus the same step delayed by $t_p$; the average over a pulse is the step response’s integral, in closed form from the modes. The electrical side is the burden optimum of the earlier essays, a shunt of $u^*/I$ with $u^* = 7.75$ mV, so that the power is $I u^*$; the temperature coefficient is 50 ppm/K throughout.

What it leaves out #

The fixed point within the pulse. The resistance rises as the element warms, so the dissipation rises with it. At the few kelvin a second-long pulse reaches, that is a correction of a part in ten thousand to the power and so to the error; the steady-state essay solved it as a fixed point, and for a pulse it is smaller still.

The element’s own distribution. The element is one node here. A real strip heats unevenly, hottest in the middle and coolest at the terminations, and its average temperature is what the resistance follows. The adiabatic regime is unaffected, since every part of the strip then warms at the rate its own dissipation sets; the millisecond regime, where heat is flowing along the strip to the terminations, would change in shape. The loop gain one temperature understates measured what a single-node model misses when a part heats unevenly, and found the lumped figure a bound rather than a value; the same caution applies to a strip in its millisecond regime.

The temperature coefficient’s curvature. A shunt alloy’s coefficient is specified at one temperature with a curvature about it. For the millikelvin rises of short pulses the linear coefficient is exact; for the kelvin rises of long ones, the curvature matters, which is the earlier essay’s second open question.

Still open: the curved coefficient, the repeated pulse, and the unheated sense element #

The coefficient that is parabolic. A manganin-like alloy’s resistance is flat at one temperature and curves either side. For a pulse starting at an ambient away from the flat point, the first millikelvin of rise sees the full local slope; a pulse starting at the flat point sees almost none, and its error is second order in the rise. The pulse error with the curvature in it would say whether choosing the operating temperature can make a shunt’s pulse error vanish to first order.

A train of pulses. A pulse repeated at a duty cycle heats the board towards the average power’s steady rise while each pulse adds its own adiabatic ramp on top. The error then has a slowly rising floor and a fast ripple, and the reading at each pulse depends on how many came before it — the same structure a switching device’s junction has under a pulse train, which the pulse the heatsink does not feel solved for a die and its heatsink.

A sense element that carries no current. The earlier essay asked whether a shunt could be built with an unheated reference element in the same thermal environment, differenced against the heated one. For a pulse, the reference would see the terminations’ and board’s rise but not the element’s adiabatic ramp, so it would cancel the slow part of the error and leave the fast part — which is the small one. Whether that is worth building depends on which part dominates at the pulse lengths in use.