The resistance that depends on the reading

The burden-voltage optimum is a statement about three errors and its most quoted property is what it leaves out. A shunt’s insertion error is the burden over the supply; the amplifier’s offset error is the offset over the burden; the shunt’s own Johnson noise is four orders below both. None of the three contains the current being measured. The ammeter that is a resistor makes a good deal of that, and it is right to: ten milliamps and a hundred amps want the same 7.75 millivolts of burden, and only the resistance moves.

There is a fourth error and it has the current in it, twice.

A shunt dissipates $I^2R$. Its temperature rises by that times a thermal resistance to whatever it is mounted on. Its resistance rises with its temperature, by whatever its coefficient is. And the current is computed by dividing the measured voltage by the resistance — so the divisor is a function of the dividend, and the reading is the solution of an equation rather than the result of a division.

A fixed point, and the closed form that agrees with it #

The three statements are

$$P = I^2R, \qquad \Delta T = \theta P, \qquad R = R_0\left(1 + \alpha \Delta T\right)$$

and they are circular. Substituting the first two into the third gives $R = R_0(1 + \alpha\theta I^2 R)$, an equation in $R$ whose solution is

$$R = \frac{R_0}{1 - \alpha\theta I^2 R_0}$$

The figure does not use that. It iterates the three statements to convergence — resistance to dissipation to temperature to resistance, four hundred times or until the change is below a part in $10^{15}$ — and then compares the result against the closed form. They agree to $2\times10^{-16}$ across five decades of current, which is the arithmetic’s floor.

That pairing is worth the cost. A fixed point solved by iteration and a fixed point solved in closed form are two computations of one number that share no steps, and the closed form is the kind of expression whose sign is easy to get wrong — the denominator is $1 - \alpha\theta I^2 R_0$ and a first version of almost anybody’s derivation writes $1 +$ there, which gives an error of the right size and the wrong direction.

The iteration also makes the failure visible, which the closed form does not. When $\alpha\theta I^2R_0$ reaches one the denominator is zero and the closed form returns infinity; the iteration simply fails to converge, walking upward for ever. Both are describing thermal runaway, and only one of them looks like it.

Along the optimum the heat is linear in the current #

Here is the part that is specific to a shunt and is not obvious from the expression.

The dissipation is $I^2R$, which is quadratic. But a shunt at the burden optimum has $R = u^*/I$, so

$$P = I^2\cdot\frac{u^*}{I} = I\,u^*$$

The dissipation is linear in the current and the resistance has cancelled out of it. Every shunt sized at the optimum dissipates 7.75 millivolts times its current, whatever that current is: 77.5 milliwatts at ten amps, 775 at a hundred, 7.75 watts at a thousand.

The temperature rise follows: $\Delta T = \theta I u^*$, which is 154.9 millikelvin per ampere at a thermal resistance of twenty kelvin per watt, and contains no resistance either. The fractional error is $\alpha\theta I u^*$ and is therefore the first power of the current rather than the second.

Fitted over five decades the exponent comes out at 1.0007 rather than exactly one, and the departure is the fixed point feeding back on itself: a hotter shunt is a larger shunt is more dissipation. It grows with the coefficient — 1.0000 at ten parts per million a kelvin and 1.0146 at a thousand — which is a measurement of the feedback’s strength and is checked as such.

A shunt of fixed resistance, by contrast, has an exponent of exactly two, and the figure draws both. The difference between the two curves is the whole of what the burden optimum does to the thermal problem: it converts a quadratic error into a linear one, which is a much better position to be in at high current and a slightly worse one at low.

The curvature no calibration removes #

The reading is $V = I\,R_0(1 + \alpha\theta I u^*)$, which expands to

$$V = I R_0 + \alpha\theta u^* R_0\,I^2$$

A term in $I^2$ in a measurement that is supposed to be linear. That is a different kind of error from a gain error or an offset, and the difference matters: a gain error is removed by a single-point calibration, an offset by a zero, and a quadratic term by neither.

Calibrating the gain at full scale makes the reading exact there and leaves it low everywhere below, by $c(I_{\text{fs}} - I)$ as a fraction of the reading. Expressed the way an instrument’s integral non-linearity is specified — as a fraction of full scale — that is $c\,I(I_{\text{fs}}-I)/I_{\text{fs}}$, which is zero at both ends and peaks at mid-scale at exactly a quarter of the full-scale term. At fifty parts per million a kelvin and a hundred amps full scale the full-scale term is 775 ppm, so the worst departure is 194 ppm of full scale — which for a four-and-a-half-digit measurement is two counts and for a six-digit one is two hundred.

The quarter is worth keeping because it is exact and because it is the signature of a quadratic. A term in $I^2$ calibrated out at one point always leaves a parabola whose peak is a quarter of what was removed, and an instrument whose non-linearity is a quarter of its full-scale gain error has a quadratic in it whatever else is said about the mechanism.

So the honest specification of a shunt-based instrument has an integral non-linearity in it, that non-linearity is thermal in origin, and it depends on the mounting rather than on the part. The same shunt on a large copper plane has a third of the thermal resistance and a third of the non-linearity, and nothing about the electrical design has changed.

This is exactly the shape of the sensor inside its own answer, where a sensor’s own excitation changes the quantity it is measuring, and of the bridge that is linear near one point, where the linearity is a property of an operating region rather than of a circuit. A measurement that dissipates has a non-linearity, and the only questions are how large and what shape.

The runaway that exists and cannot be reached #

A fixed point of this kind has a condition for existing, and it has turned up here before: $\alpha\theta I^2R_0 < 1$. Past that there is no solution and the resistance runs away.

Along the burden optimum, substituting $R_0 = u^*/I$ gives the condition $\alpha\theta u^* I < 1$, so the critical current is

$$I_{\text{crit}} = \frac{1}{\alpha\theta u^*} = \frac{1}{50\times10^{-6}\cdot 20\cdot 7.746\times10^{-3}} = 129\ \text{kA}$$

A hundred and twenty-nine kiloamperes. At a thousand parts per million a kelvin it is 6.45 kA and at ten it is 645 kA. None of those is reachable by a shunt, which would have vaporised at a thousandth of any of them — 129 kA at a 7.75 mV burden is a kilowatt of dissipation in a part sized for milliwatts.

That is a genuine and slightly unusual result here, where the usual finding is that a boundary is closer than expected. Here the boundary is real, it is the same mathematics that produces thermal runaway in the ratio that does not walk to one and in a bipolar transistor’s second breakdown, and for a shunt it is six orders of magnitude away from anything that can happen.

The reason is the linearity of the previous section. A device whose dissipation is quadratic in its drive has a runaway condition that scales as $1/\sqrt{\alpha\theta R_0}$ and can be met; the burden optimum makes the dissipation linear, so the condition scales as $1/\alpha\theta u^*$ and is fixed by a burden voltage of a few millivolts. Holding the burden constant has not merely improved the error — it has moved the instability out of reach entirely.

Which of the three thermal resistances matters #

The single $\theta$ above hides three paths and only one of them is a design variable.

The element’s own path to its terminations is set by the part and is typically a few kelvin per watt. The terminations’ path into the board is set by the pad geometry and the copper attached to it, and is where most of the twenty comes from. The board’s path to the ambient is the largest of the three and the slowest, and it is shared with everything else on the board.

Only the middle one is the designer’s, and it is worth a factor of three or four: a shunt on two small pads sees perhaps thirty kelvin per watt into the board, and the same shunt on pads with an acre of copper and a dozen thermal vias sees eight. That is a factor of nearly four on every error in the table, bought with board area and nothing else — no extra part, no calibration, no bandwidth cost, which is unusual enough in this field to be worth stating on its own.

It also means the error is not a property of the shunt. Two identical parts on two boards have different non-linearities, and a specification written from the data sheet alone is not a specification of the instrument. The quantity to measure is the rise per ampere, which is $\theta u^*$ and has no resistance in it, so it can be measured once on the assembly with a thermocouple and a current source and used for every range the instrument has.

The refusal, and what it is protecting #

With the temperature coefficient set to zero the loop is not a fixed point at all. One pass through it returns the cold resistance and so does every subsequent pass, at every current, and the curvature in the reading is absent rather than small.

That is worth claiming because the alternative — a very small coefficient — looks the same in a figure and is not the same statement. A zero-coefficient shunt is exactly linear, so a single-point calibration is exactly right, and an instrument built on one has no thermal non-linearity to specify. A ten-parts-per-million shunt has one of 155 ppm at a hundred amps and needs it in the specification.

The distinction is the difference between a term that is negligible and a term that is not there, and it is kept deliberately. It is the same distinction as a reactance is not warm — where a lossless capacitor’s noise is zero rather than small — and it matters for the same reason: a term that is absent stays absent when the conditions change, and a term that is small does not.

What one thermal resistance does not describe #

That twenty kelvin per watt is the right thermal resistance. It is a reasonable figure for a surface-mount shunt on an ordinary board and it is the one thing in this analysis that belongs to the assembly rather than to the part. On a heavy copper plane with thermal vias it might be five; hanging in free air it might be a hundred. Every temperature rise and every error above scales in direct proportion, and the runaway current inversely.

That the thermal path is a single resistance. It is not — it is a network with several time constants, which is what two loops, and one heatsink measures — and the steady state computed here is the limit of that network after long enough. A current pulse shorter than the shunt’s own thermal time constant produces a smaller rise and a smaller error, and the figure’s numbers are the worst case.

That the temperature coefficient is a constant. Shunt alloys are chosen because their coefficient is small and because it is roughly parabolic with a maximum near room temperature, so the effective coefficient over a rise is smaller than the specified one at one end of the range and larger at the other. The linear model here is the first term of that and is honest about being one.

That the current is the only heat source. A shunt on a board near a power device is warmed by it, and that rise produces exactly the same resistance error with none of the current dependence — an offset in the gain rather than a curvature. The two are distinguishable only by measuring at two currents.

A fixed point iterated and in closed form, and the exponent fitted #

The fixed point is solved twice. By iteration to a part in $10^{15}$, and by the closed form $R_0/(1 - \alpha\theta I^2R_0)$, agreeing to $2\times10^{-16}$ over five decades of current.

The exponent is fitted rather than claimed, over five decades, and required to be one within two per cent — and separately checked never to be below one, because the fixed point feeds its own output back and any departure must be upward.

The temperature rise per ampere is checked against $\theta u^*$ to a part in a thousand, which is the statement that the resistance has cancelled and is the whole reason the error is linear.

The runaway current is checked to be above a kiloampere at every coefficient on the slider, so that “the fixed point never fails for a shunt” is a measurement rather than an expectation.

And the loop is refused at zero coefficient, where it returns the cold resistance at every current and the curvature is absent.

The one error with the current in it #

Four errors, and the first three do not know what current is flowing.

That is a strange property for a current measurement to have and it is the reason the burden-voltage result is worth carrying: it means one number sizes every shunt, from a milliamp to a kiloamp, and the design does not have to be redone for each range. The three errors it balances are properties of the instrument rather than of the measurement.

The fourth is a property of the measurement, and its arrival changes what kind of specification the instrument has. Three current-free errors give a fixed accuracy — 0.129 per cent, everywhere. A fourth error proportional to the current gives an accuracy that degrades across the range, and one proportional to its square would give a curvature that no calibration removes. The optimum turns the second of those into the first, which is the best available outcome short of removing the term.

What it cannot do is remove it, because the term is a consequence of measuring a current by dissipating some of it. Every instrument that works that way has it. A current transformer does not — it dissipates nothing in the primary, which is the whole of its appeal and which the ammeter that is not in the circuit prices in a different currency entirely.

So the three essays after the first one have found three quantities the optimum handed back: a bandwidth that falls with the current, a burden that scales with the rail, and a non-linearity that grows with the current. The optimum is still the right place to start and it is still resistance-free and current-free. What it is not is the whole design, and the pattern across the three is the same: a constraint that eliminates a variable from one quantity puts it into the others.

Still open: the transient rise, the alloy whose coefficient is not linear, and the four-wire connection that does not help #

The pulse shorter than the thermal time constant. Everything here is a steady state. A shunt has a thermal time constant of some tens of milliseconds to its board and perhaps a millisecond to its own element, so a current pulse of a hundred microseconds produces a rise set by the element’s heat capacity alone and much smaller than the steady figure. Solving the thermal network rather than the single resistance would give the error as a function of pulse length, which is what a fault-current measurement actually needs.

The coefficient that is parabolic. A shunt alloy’s resistance is specified as a coefficient at twenty degrees and a curvature, and the effective coefficient over a fifteen-kelvin rise from an ambient of twenty-five is not the specified one. Substituting the real curve into the fixed point would say how much of the 775 ppm at a hundred amps is a real prediction and how much is an artefact of linearising a parabola about the wrong point.

And the four-wire connection, which repairs a different defect. Two terminals measure the leads as well removes the leads’ resistance from a measurement, and a Kelvin-connected shunt has the sense terminals inside the current terminals for exactly that reason. It does nothing whatever about self-heating: the element between the sense points is the part that is warming, and no connection scheme puts the measurement outside it. Whether a shunt could be built with a sense element that carries no current — a second, unheated element in the same thermal environment, differenced against the first — is a construction question this analysis makes precise.