The heat a recovery leaves behind

A diode carrying an ampere forward, with a hundred nanoseconds of carrier lifetime, driven down at a hundred amperes a microsecond, conducts 3.83 amperes backwards for 48 nanoseconds. That was the result of the essay one rung down, computed from a transcendental root and checked against a march of the same charge equation.

It is an amplitude and a duration and nothing else. Both are multiplied by a voltage somewhere before they cost anybody anything, and the first thing worth knowing about this event is where.

The energy is not in the diode #

While the junction is recovering it is still conducting. Its stored charge has not gone, so it cannot block, and a conducting junction holds a few hundred millivolts at most. The current through it is large and negative and the voltage across it is nearly nothing, so the diode dissipates almost nothing during the whole of the recovery interval.

The current has to come from somewhere. What is pulling it down is a transistor turning on, and at that instant the transistor is standing across nearly the whole supply — it has not finished switching, and the diode it is commutating against is still holding the rail. So the recovered charge is delivered through the switch at full voltage:

$$E_\mathrm{switch} = V\,Q_\mathrm{rr}$$

For the numbers above, at a hundred volts, that is 138.9 nanocoulombs times a hundred volts — 13.9 microjoules, in the transistor.

The diode’s own share arrives afterwards, in the interval in which the junction finally blocks: the current returns to zero while the voltage swings to the supply, and the overlap integral of one ramp against another is $\tfrac{1}{6} I_\mathrm{RM} V t_b$, which here is 1.5 microjoules. Ninety per cent of the energy of a diode’s recovery is spent in the transistor, and a design that budgets for it in the diode has put the heatsink on the wrong part.

That is a shape of result this collection meets often enough to be worth naming: the component that exhibits the phenomenon is not the component that pays for it. The essay in this field about the half that never arrives is the same statement about a capacitor and its charging resistor.

One event is nothing; a hundred thousand of them is watts #

Fifteen microjoules is a number nobody would think about. It becomes a design constraint by being repeated.

At ten kilohertz the recovery costs 0.154 watts, against the diode’s own conduction loss of 0.45 watts for a half-duty ampere at 0.9 volts forward. At a hundred kilohertz it is 1.54 watts, three and a half times the conduction loss. The two are equal at 29.2 kilohertz.

That crossing is the number a topology decision is made on, and it moves the way the algebra says it should: it is the conduction loss divided by the energy per event, so halving the supply voltage doubles it, and a diode with a fifth of the carrier lifetime moves it out by about five.

It also explains a piece of practice that looks like superstition from outside. Silicon rectifiers with long lifetimes are cheap and have low forward drops, so they win every comparison made at mains frequency; at fifty kilohertz they lose to a part with twice the forward drop and a tenth of the stored charge, because the quantity that decides is the one nobody measured at fifty hertz. The crossing frequency is where the comparison changes hands, and it is computable from two numbers on each data sheet.

The lifetime rises with temperature, which closes a loop #

Everything above is at one temperature. The stored charge is $\tau I_F$ and the carrier lifetime is not a constant of the device: it rises with temperature, roughly as the 1.8 power of it for silicon, because recombination gets slower as the lattice gets hotter.

So a hotter junction recovers worse. More stored charge, a larger reverse peak, a longer recovery, more energy per event, more heat, a hotter junction. That is a loop with gain in it, and a loop with gain in it either converges or does not.

The fixed point is $T = T_a + R_\mathrm{th}P(T)$ and it is iterated from the ambient upward, because the map is increasing and the lowest solution is the one a device switched on cold actually reaches. At ten kilohertz it converges at 43.7 °C. At a hundred it converges at 50.2. At a megahertz it converges at 174 °C, with the carrier lifetime 2.08 times its cold value — the diode is now storing twice the charge it was specified with, and the recovery it is being asked about is not the one in the data sheet.

Above 1.28 megahertz the iteration does not converge at all. There is no temperature that satisfies the loop: the power rises with temperature faster than forty kelvin per watt of thermal resistance can carry it away, and the device’s temperature is not a large number, it is undefined. The solver reports that as a refusal rather than as the last iterate, which is the only honest thing to return.

One thing in this section is the diode’s own and one is not, and the first version of it confused them. The power in the loop above has to be the power dissipated in this junction — its conduction plus the sixth of $I_{RM}Vt_b$ it takes while blocking — and the first version passed the recovery’s whole energy, which is to say it heated the diode with the transistor’s nine tenths on the very rung whose finding is that the transistor has them. Every frequency here was a factor of ten too low as a result. The rung above repairs it and then does the thing that actually replaces it: gives the switch a thermal model of its own and a case in common with the diode, where the pair gives out at 135 kilohertz and the component that goes is the switch.

What the iteration is, and why it is not a formula #

The fixed point above could be written as an equation and solved once, and it is worth saying why it is iterated instead.

The power is a function of the temperature through the lifetime, and the lifetime enters the recovery through a transcendental root: the conduction interval solves $x = r + 1 - e^{-x}$ with $r = I_F/(a\tau)$, and the peak reverse current is $a\tau(1 - e^{-x})$. Substituting a temperature-dependent $\tau$ into that and asking for a closed form in $T$ gives an equation with the temperature inside an exponential inside a transcendental root, which has no useful solution.

So it is iterated, from the ambient upward, and the direction is the load-bearing part. The map $T \mapsto T_a + R_\mathrm{th}P(T)$ is increasing, so starting below the lowest fixed point and iterating upward converges to that lowest fixed point if one exists — and a device switched on cold at ambient temperature does precisely that, physically. Starting from above would find a different solution or none, and would be answering a question about a device that was already hot.

The refusal is what happens when the iterate walks past every plausible temperature without the increments shrinking. That is not a failure of the iteration to converge on an answer that exists; it is the absence of a crossing between the power curve and the straight line the thermal resistance draws, and it is the same shape as the collection’s other refusals — the network with no answer, the amplifier asked to work above its gain–bandwidth, the mirror asked to work saturated. The right response is a name, not a number.

Two edges, and the smaller one is the specification #

There are two frequencies in the figure and it matters which is which.

The junction passes its rated 150 °C at 904 kilohertz. That is a specification: the number exists because a manufacturer wrote it down, it can be argued with, and a part rated to 175 °C moves it.

The loop stops converging at 1.28 megahertz. That is not a specification. It is a property of the positive feedback between temperature and stored charge, it does not care what anybody wrote down, and nothing operates above it in any sense.

The specification binds first, which is the ordinary and comfortable case — a device is out of specification long before it is unstable, and a designer working to the data sheet never meets the second edge. What makes the second edge worth computing anyway is that the margin between them is small: the two are a factor of 1.4 apart, so a design that runs at the rated temperature is running at seven tenths of the frequency at which its own thermal behaviour becomes undefined, and the usual comfort that a rating is conservative does not apply.

The gap narrows as the thermal resistance rises. At a hundred kelvin per watt the two edges are within twenty per cent of each other; the rating and the runaway are almost the same number, and the part is being asked to sit just below a cliff.

The march that the energy is read from #

The energy per event is an integral of a product, and in this essay one of its two factors is modelled rather than solved, which is unusual enough here to be worth flagging.

The current is solved: the charge equation is integrated in closed form and marched trapezoidally, and the two agree to a part in a thousand, so the reverse peak and the conduction interval are measurements. The voltage across the diode during the final interval is not solved — it is taken as a linear ramp from zero to the supply while the current ramps back to zero, which is where the factor of one sixth comes from.

That is a model, and its own edge is known and is in this collection already: the essay below this one measured what really happens at that instant, which is that the loop inductance rings into the junction capacitance and takes the voltage past the supply, by a factor that is a quadrature rather than a sum. The real overlap integral is therefore larger than one sixth of the rectangle, and the diode’s ten per cent is an under-estimate.

It is left as a stated model for one reason: the ring is a property of the layout — a loop inductance and a junction capacitance — and folding it into the energy would make this essay’s headline number depend on a quantity that belongs to somebody’s circuit board rather than to the device. The split is quoted as ninety per cent to ten with the note that the ten is a floor, which is the honest form.

What a data sheet gives, and what it does not #

A rectifier’s data sheet gives $t_\mathrm{rr}$ and $Q_\mathrm{rr}$ at one forward current, one rate of fall and one temperature — usually 25 °C, one ampere, and fifty amperes a microsecond.

Every dependence this essay is about is in the derivatives of those numbers, and the derivatives are not given. The energy per event scales with the supply the part is used at, which the sheet does not know. The stored charge scales with the forward current at the instant of commutation, which is a design variable. And the lifetime’s temperature exponent — the coefficient that decides whether the loop above converges at all — is not on any sheet, is between about 1.5 and 2.5 for silicon, and moves the runaway frequency by nearly a factor of two across that range.

Which is why the slider on the figure is the thermal resistance and not the exponent. The thermal resistance is a design choice made with a heatsink; the exponent is a property of a part that nobody states, and a figure whose headline number depends most strongly on an unstated parameter is a figure that should say so rather than pretend to a precision it has not got.

Three ways out, and what each costs #

The result above is a constraint rather than a verdict, and the ways around it are worth setting beside each other because they are the design space this measurement describes.

Less stored charge. A fast-recovery part with a twenty-nanosecond lifetime holds a fifth of the charge and moves every frequency in this essay out by about five. It pays in forward drop — the lifetime is short because the silicon is doped or irradiated to make it short, and that raises the resistivity — so the conduction loss rises, and the crossing frequency at which recovery overtakes conduction moves out for both reasons at once.

No stored charge at all. A Schottky diode conducts by majority carriers and has no minority-carrier charge to sweep out, so this entire essay does not apply to one: there is no recovery, no switch loss from it, and no thermal loop. It pays in reverse leakage, which rises steeply with temperature and closes a different positive-feedback loop, and in the reverse voltage it can hold — which is why the technology choice is made on the rail voltage first and the frequency second.

Slow the current down. The recovered charge falls as the rate of fall falls, and the essay below measured the whole of that dependence: at three amperes a microsecond the peak is a tenth of an ampere rather than 3.83. It pays in the transistor’s own switching loss, which rises as the transition is lengthened, and the sum of the two has a minimum that is not at either end. That minimum is the real design point of a hard-switched converter, and this essay supplies one of its two terms.

The three quantities this iteration is made of #

The fixed point on this page is built from three measurements that share no arithmetic. The diode that conducts backwards supplies the charge, at a fixed lifetime. Two millivolts a kelvin, and the wrong sign supplies the temperature dependence that turns the charge into a loop. The direct voltage that is a sawtooth supplies the conduction angle that decides how often the loop is traversed. The half that never arrives is the other loss in the same circuit that does not depend on the resistance it happens in, and One step, computed twice is the march all of it is computed on.

What is checked #

The energy split is asserted rather than assumed: the switch’s share is required to be above three quarters at every softness a junction has, and the two contributions are required to sum to the total. The settled junction temperature is asserted to rise monotonically with switching frequency up to the frequency at which it stops settling, and the sweep is asserted to contain at least one frequency at which there is no fixed point — a check that the refusal still refuses.

The two edges are asserted in the right order: the rated temperature must be reached at a lower frequency than the runaway, at every thermal resistance the slider offers. That is the claim a designer would use, and it is the one that would fail first if the lifetime model were changed.

What is not modelled: the reverse leakage current, which also rises with temperature and adds a second positive-feedback path that this loop does not contain; and the transistor’s own junction, which is where ninety per cent of the energy actually goes and which has a thermal loop of its own. This essay computes the diode’s temperature and prices the switch’s dissipation without heating it, which is stated here because the omission is in the direction that makes the answer optimistic.

Where the nine tenths goes, and what the rung above does with it #

Attributing nine tenths of a recovery’s energy to the transistor rather than to the diode is a reassignment rather than a reduction, and the two rungs above this one are what happens when the reassignment is taken seriously.

Two loops, and one heatsink is the immediate consequence: the diode alone survives to 1.28 megahertz instead of 128 kilohertz — a factor of exactly the ninety per cent — and what replaces that number is the arrangement that exists, two devices with two different positive temperature loops on one piece of aluminium, giving out at 135 kilohertz with the switch being the part that goes. So the repair improved the component that was being blamed and left the system limit almost where it was, on a different component.

The pulse the heatsink does not feel then removes the assumption underneath all three rungs: that a fixed point computed from an average power is a temperature. It is one above 308 hertz, where the die’s own heat capacity integrates, and by the hundred kilohertz this rung switches at the fixed point is exact to five parts in ten thousand; at one hertz the same average power on the same heatsink puts the junction three hundred kelvin hotter. A hundred and fifty watts for two per cent of the time is three watts only if the thermal mass is fast enough to believe it.

Which is the shape of this whole ladder in one sentence. Each rung takes a quantity the one below it computed correctly and asks what it was attributed to — an energy to a device, a power to a temperature, a temperature to a fixed point — and each time the attribution is the thing that moves.