Two ladders the terminals cannot tell apart

The pulse the heatsink does not feel drew a junction’s single-pulse thermal impedance two ways. One was a netlist of three resistances and three heat capacities, marched. The other was the sum every account of a thermal path writes down — each stage’s own resistance times $1 - e^{-t/RC}$, with a time constant built from that stage’s own two components — and the essay called it an approximation, on the grounds that a ladder’s stages load each other: the die charges into a case that is itself moving.

That is a true sentence about the arithmetic and it was not a measurement. Two curves that part company tell nobody which of them left. The marched one was called the answer because marching is the more elaborate computation, which is a reason to trust it and not a reason to believe it, and the difference between them was described rather than priced.

The three rungs below all rest on this. The heat a recovery leaves behind computes the energy one switching event costs and turns it into watts; two loops and one heatsink puts two of those on a shared case and iterates to a fixed point. Every one of those numbers is a temperature read off a thermal model, and the model has now been drawn two ways with the difference between them unmeasured.

There is a third route and it is exact.

What is being solved #

The impedance looking into an $RC$ ladder is a continued fraction. The heatsink’s resistance sits in parallel with its own heat capacity; the case’s resistance is in series with that and in parallel with the case’s heat capacity; the die’s resistance is in series with the result and in parallel with the die’s. Three stages give a ratio of two polynomials in $s$, the denominator cubic, and building it needs nothing but the six element values and the arithmetic of polynomials.

An $RC$ ladder’s poles are real, negative and simple — a network of positive resistances and positive capacitances cannot oscillate and cannot have a repeated natural frequency without being built to — so the partial-fraction expansion of $Z(s)/s$ is a sum of ordinary exponentials:

$$z(t) = \sum_i R_i\left(1 - e^{-t/\tau_i}\right), \quad \tau_i = -1/p_i, \quad R_i = \frac{-N(p_i)}{p_i D'(p_i)}$$

No march, no step size, no error that depends on how finely time was cut. And the $R_i$ and $\tau_i$ that come out are not a formula: they are a second network, a chain of parallel resistor–capacitor sections in series, with the same impedance at its terminals as the ladder it came from. The two objects are the standard pair — a ladder whose nodes are the die, the case and the sink, and a chain whose sections are the three natural frequencies.

The calibration, which comes before anything is claimed #

An expansion that finds roots and divides by derivatives is exactly the machinery that returns plausible numbers for a wrong problem, so it is checked three ways before it is asked anything.

The residues must sum to the ladder’s total thermal resistance, because the exponentials have all died by then and $z(\infty)$ is that resistance. They sum to 9.7 kelvin per watt, exactly, at the last digit the arithmetic carries — an identity the root-finding could easily have broken and does not.

The poles must be the network’s own natural frequencies, and this collection already computes those another way: it assembles the netlist’s matrices and samples the determinant of $G + sC$, a computation that shares no line of arithmetic with a continued fraction. The two agree to 4.5 parts in $10^{14}$. That is the pairing one step computed twice insists on for a transient, applied to the poles rather than to the waveform.

And the third: the ladder’s own state equations, three of them, integrated by taking the matrix exponential rather than by stepping. The junction temperature that comes out of that agrees with the expansion to $1.1\times10^{-9}$ kelvin per watt over nine decades of time. Two routes that share the six element values and nothing else after them — the same standard where the behaviour is written down sets for a netlist and a differential equation describing one circuit.

The last of those three is worth a sentence about why it was done that way rather than by marching. The fastest stage is 2.4 milliseconds and the slowest is 480 seconds, a stiffness of two hundred thousand, and any fixed step over that range must either resolve the first or step over it. Taking the exponential of the system matrix has no step in it at all: it costs about twenty multiplications of a three-by-three matrix and is limited by the arithmetic rather than by a discretisation, which is the same reason a sum that is exact prefers a closed form to a fine grid wherever one exists.

The departure, and it is smaller than it was said to be #

With that in place the approximation can be priced instead of characterised.

The worst it does is 0.950 per cent, at 12.9 milliseconds, and the sign of it never changes. Over nine decades of pulse length the sum is high or equal and never low, which is the more useful half of the answer: a closed form that can only over-state a junction temperature is safe to design against, and that is a stronger statement than any bound on its size.

The shape says what the mechanism is. The error is not a property of the ladder; it is a property of each pair of neighbours, and a three-stage path has two pairs and therefore two humps. Where only one stage is charging there is nothing to load it and the sum is exact; where two are moving together the sum has ignored that one of them is charging into a node that is itself rising.

Fifty per cent, divided by the separation #

That mechanism has a size and it depends on one number: how far apart in time the two neighbours are.

A die at 2.4 milliseconds, a case at 200 and a heatsink at 480 seconds are 83 and 2,400 times apart, and 50 per cent divided by 83 is the 0.95 measured above. A path whose two fastest stages sit a factor of five apart gives 10.7 per cent, and there the closed form genuinely is the wrong object.

So the honest statement about the approximation is neither that it is good nor that it is an approximation. It is that it costs about half divided by the separation, and that a real package — a die of milligrams, a case of grams, a heatsink of kilograms — separates its stages by construction. The sum is accurate for the same reason the thermal model is a ladder in the first place, which is a better sort of reason than an empirical one.

Why the separations are large, and what would make them small #

The coefficient is a fact about arithmetic; the separations are a fact about parts, and they are large for a reason that has nothing to do with the electrical problem.

A thermal time constant is a resistance times a heat capacity, and a heat capacity is a mass. A die weighs milligrams, its package and the grease under it weigh grams, and a heatsink weighs hundreds. Each stage therefore carries about three orders of magnitude more heat than the one above it, while the resistances move the other way by rather less, and what is left is the two factors of 83 and 2,400 that the ladder has. A thermal path is stiff by construction, and the closed form is accurate for the same reason the lumped model is legitimate at all.

The case where it is not is where two stages are built to be similar — a die attached directly to a copper slug of comparable mass, or a two-layer model of a single die’s interior, which is what a hot-spot calculation needs. There the separation falls to a factor of a few, the error goes to the ten per cent measured above, and the closed form has stopped being a convenience. The diode that conducts backwards is the event that would need such a model: a recovery is tens of nanoseconds, and at that scale the object being heated is a few square microns of silicon and not a die at all.

The second network is not the first one rearranged #

The expansion produces element values, and it is worth looking at what they are.

Six element values and not one of them survives. The differences are small — a per cent — and they go both ways, so they are not a scaling that could be undone. The chain is 1.18787 kelvin per watt and 2.38792 milliseconds where the ladder is 1.2 and 2.4; 0.505409 and 199.674 milliseconds where the ladder is 0.5 and 200; 8.00672 and 483.217 seconds where the ladder is 8 and 480.

And yet the two networks are indistinguishable at their terminals. That is not an approximation holding well; it is an identity. The impedance is one rational function and the two networks are two factorisations of it, in the same way that the two numbers a source can be described by are two factorisations of one linear behaviour.

A pole is not a place #

Which raises the question the pair of networks exists to answer, and it has a sharper answer than “the element values differ”.

A data sheet does not publish a die’s heat capacity. It publishes a transient impedance curve, and what can be extracted from a curve is exactly the chain: a handful of resistance–time-constant pairs that reproduce it. Those pairs are then used in a simulator, and the simulator has nodes in it, and the node between the first section and the second looks like the case.

It is not the case.

The exponents are the whole content. A real case is two integrations away from the power that heats it — the power charges the die, the die’s excess drives current through the first resistance, that current charges the case — so its rise starts as $t^2$. The sink is three integrations away and starts as $t^3$; fitted over the same window it comes out at 2.984. The chain has no such structure: every one of its sections hangs directly off the terminal, so every internal node is one integration from the source and starts as $t$.

The consequences are quantitative and they are large where it matters. The chain’s inner node reads 5.59 times the real case rise at a millisecond, 1.33 times at ten, and comes within ten per cent only past 32.3 milliseconds and within one per cent only past 45.2 seconds. A converter switching at any ordinary frequency lives entirely inside the region where the number is wrong by an order.

This is the same defect exact outside and wrong within collects, arriving from an unusual direction: the model is not approximate and then wrong beyond its range — it is exact for the quantity it was fitted to and meaningless for every other quantity in it. Nothing in the curve it came from ever contained a case temperature, so nothing in the network built from that curve can.

The largest disagreement is not about heat #

There is a third source of difference in a picture of this kind and it turns out to be the biggest one, and it is not a model at all.

The march is essentially exact. It steps uniformly inside each decade, hands its state on at the boundary, and its worst sample over 848 of them is out by five parts in a hundred thousand — the trapezoidal rule doing what it is supposed to and no more. What costs three per cent is asking that run for a value it did not compute and being given the next one it did. Near the bottom of a decade the next sample is a whole uniform step later, and on a curve that is climbing steeply that is three per cent of impedance.

The same arithmetic applies to the reported grid itself. Sixty rows over six and a half decades sit 28.8 per cent apart in time, so a value read off such a curve at “one millisecond” is the row at 1.2526 milliseconds. The exact impedance at a millisecond is 0.40897 kelvin per watt and at that row it is 0.488056 — 19.3 per cent higher — before any read-back error is added.

It applies to the departure above as well, which is why the two pictures on this page disagree about it. 0.950 per cent at 12.91 milliseconds is what the figure at the head of this essay measures: the exact expansion against the sum, scanned at fifteen hundred pulse lengths. The sixty-length picture reports 0.94, and the whole of the difference is grid and route — its lengths straddle the peak, the nearest sitting at 12.19 milliseconds where the departure is 0.9495, and the curve it subtracts is the march rather than the expansion, which sits a part in six thousand towards the sum there and closes the gap a little further. Two grids and a substituted route, for 1.2 per cent of the answer. The figure above is the one this essay means throughout, because it is the only one of the two with an exact curve in it.

None of that is heat flow. It is a logarithmic axis read off a linear grid, and it was three times the closed form’s error and six hundred times the march’s. Ranking the three was the point: 2.919 per cent from the reading, 0.950 from the model, 0.0047 from the integration. A picture in which two curves visibly separate invites the reader to attribute the gap to the two models it labels, and most of that one belonged to neither.

Past tense, because the middle curve is the repair and it is one line. Interpolating between the two samples either side of each reported time takes the worst reported error from 2.919 per cent to 0.0152, at no cost in anything — the samples are dense enough that a step is under two per cent of the elapsed time, so a straight line across one of them is wrong by less than the march is. The single-pulse curve at the head of this essay is the interpolated one; it was the nearest-later one until this measurement was made, and the two-route comparison drawn on it was reporting a defect in the reading as a difference between two models of heat flow.

What it does not say #

It does not say the sum of exponentials should be replaced. Inside a package whose stages are two or three orders apart it is under one per cent and always conservative, it costs three exponentials to evaluate, and it superposes — which is the property a pulse train needs and the reason data sheets publish it in the first place. A duty cycle’s worth of steps added and subtracted is a sum of terms in the same three time constants, where the same calculation on a ladder is another march.

It does not say the march was wrong. It was right to five parts in a hundred thousand, which is far better than anything the model’s own linearity deserves — the sink’s convective resistance falls with temperature and the die’s heat capacity rises with it, corrections the rung below puts at of order ten per cent over the range these numbers cover, which is the edge that is a region rather than a line.

And it does not say the chain is a bad model. It is an exact one, of the junction. What it has no opinion about is the inside, and the failure is in reading a network as a picture of a package rather than as a factorisation of an impedance.

Nor does the ranking of the three errors carry over unchanged to a different question. It is a ranking of the three at their worst, and the three have their worst in different places: the model’s is between two time constants, the reading’s is at the bottom of a decade, and the march’s is wherever the curvature is greatest. At a pulse of one millisecond the closed form is 0.599 per cent high and the reading 0.389; at ten seconds they are 0.413 and 0.242. The order of the two reverses away from the reading’s own worst place, so a single number for “the accuracy of the curve” is the wrong shape of answer, in the way every model has an edge is about.

What it opens #

Two things, and both are about identification rather than about heat.

The first is that the direction of the derivation only runs one way. Given a ladder, the chain follows in closed form. Given a chain — which is all a measured curve can yield — the ladder does not follow, because the three sections carry three resistances and three time constants and the ladder has six free values with a different meaning. Recovering a physical ladder needs information the terminal never had, which is the same shortfall two numbers without solving for the waveform runs into from the other side.

The second is that the pricing above is a template. Any place this collection compares a closed form with a march is a comparison of two inexact things, and the ranking found here — reading, then model, then integration — is not obviously peculiar to a thermal path. It is the ordering that follows whenever the reported grid is coarser than the computed one, which is nearly always, and it is invisible without a third route.

The number worth carrying #

Half a per cent divided by the separation, in a picture whose read-back cost three until it was interpolated.

The habit that goes with it is the harder half. Two computations of one quantity that disagree do not say which is wrong, and the older, cheaper, more approximate-looking one is not reliably the guilty party — here it was right to a per cent while the reading of the elaborate one was wrong by three, and the repair belonged to neither of them. What settles it is an exact case, and for a network of resistances and capacitances an exact case is usually available: the poles are real, the expansion is finite, and the identity that the residues sum to the direct-current answer is a check the arithmetic cannot pass by accident.

The second habit is narrower and is worth stating because it cost three per cent here. A computed curve and a reported curve are different objects, and the grid a result is printed on is part of the result. Sixty rows over six and a half decades cannot resolve anything narrower than 29 per cent in time, however exactly each row was computed, and a number lifted off such a grid at a round value carries that spacing with it — the digits the arithmetic did not have, arriving one level up, where what is missing is not precision but resolution.