The diode that conducts backwards

The diode model this collection uses is one line long. Its current is $I_s(e^{v/nV_T} - 1)$, it is solved by Newton at every operating point and at every step of every march, and the current law is rebuilt from it afterwards to check that the answer closes. It is an excellent model. It has produced the bias point, the load line, the rectifier’s conduction angle, the reservoir’s ripple and the limiter that sets an oscillator’s amplitude.

It contains no time. Not “an approximation to the time dependence” — none at all: the current is a function of the present voltage and of nothing else, so the model says a diode blocks the instant its voltage goes negative.

A real one does not, and the reason is a quantity the model has no variable for. A forward-conducting junction holds a charge of minority carriers on both sides of it, and current cannot stop flowing until that charge is gone. Reverse the applied voltage and the diode goes on conducting — backwards, at whatever current the external circuit is willing to supply — until the charge has been swept out or has recombined.

This essay is about how much current that is and for how long.

One equation, and the ramp the circuit supplies #

The stored charge obeys a linear equation with two terms — supply from the terminal current, loss to recombination at the carrier lifetime:

$$\frac{dQ}{dt} = i(t) - \frac{Q}{\tau}$$

In the steady forward state nothing is changing, so $Q = \tau I_F$: a diode carrying an ampere with a lifetime of a hundred nanoseconds is holding a hundred nanocoulombs.

What the circuit supplies is a ramp. Turning a diode off in any real converter means driving its current down through an inductance — the loop’s own, or the transformer’s leakage — so the current falls at $a = V/L$ rather than jumping. With $i(t) = I_F - at$ the equation integrates in closed form, and the instant the charge runs out is the root of

$$x = r + 1 - e^{-x},\qquad x = t_a/\tau,\quad r = \frac{I_F}{a\tau}$$

from which the peak reverse current is exactly $a\tau\left(1 - e^{-t_a/\tau}\right)$.

The dimensionless $r$ is the whole of the behaviour. It compares the forward current with the current the ramp covers in one carrier lifetime, and the two limits of it are the two expressions everybody uses.

The two limits, and where each is true #

A fast ramp ($r \ll 1$) gives $x \approx \sqrt{2r}$ and therefore

$$I_{RM} \to \sqrt{2 I_F a \tau}$$

which is the expression every reference gives and the one most engineers know.

A slow ramp ($r \gg 1$) gives $I_{RM} \to a\tau$, which contains no forward current at all. A diode turned off gently has a reverse peak set by the slope and the lifetime, and carrying ten times more current forward does not make it worse.

Both are limits, and this collection’s habit is to ask where each is true rather than which one to quote. With an ampere forward and a hundred nanoseconds of lifetime:

The square-root expression is high everywhere, because it is a limit approached from below. It is within ten per cent only above 256 amperes a microsecond, within five above 956, and within one per cent only above sixteen thousand — which is faster than almost anything outside a gate-drive circuit. At the hundred amperes a microsecond a switching converter actually produces it is 17 per cent high.

The slow-ramp expression is within one per cent below 2.76 amperes a microsecond and within ten per cent below 6.7, so between about three and two hundred and fifty amperes a microsecond neither expression is any good and the transcendental root is the only answer.

That range is exactly where power electronics operates.

The boundary this puts on an ideal diode #

The number worth carrying away is the ramp rate at which the reverse peak equals the forward current the diode was carrying. Bisected on the transcendental root rather than inverted from either asymptote — because the interesting region is precisely where neither asymptote holds — it is

12.55 amperes a microsecond, for an ampere forward and a hundred nanoseconds of lifetime.

Above that a diode conducts more backwards than it ever did forwards, and every current rating, conduction-loss estimate and thermal calculation in the design has been made about the smaller of the two numbers. That is this essay’s edge, and it is a boundary in $dI/dt$ — a quantity this collection has not bounded a model with before.

It also scales in a way worth knowing. The edge is where $r$ is about two, so it moves inversely with the carrier lifetime: a fast-recovery diode with a twenty-nanosecond lifetime moves it to 63 amperes a microsecond, and a Schottky diode, which conducts by majority carriers and stores no minority charge at all, does not have this failure mode. The three device choices are the same choice — how much stored charge — seen from the specification sheet.

Two routes to the same number #

The closed form above is a solution of an equation that is easy to get wrong, so it is not trusted.

The same equation is marched trapezoidally, from the steady forward state, with the charge integrated step by step and the crossing interpolated rather than taken to the nearest step. The two agree: 48.32 nanoseconds against 48.32 for the conduction time, 3.8318 amperes against 3.8318 for the peak, and at the fastest ramp drawn 44.057 against 44.066 — a part in five thousand, which is the march’s own discretisation and falls as the step is shortened.

That is the site’s standing arrangement and it earns its keep here for a specific reason. The transcendental root is solved by a fixed-point iteration, $x \leftarrow r + 1 - e^{-x}$, which converges for every positive $r$ because the derivative of its right-hand side is $e^{-x}$. A fixed point that converges to the wrong thing looks exactly like one that converges to the right thing from inside the loop, and the march has no fixed point in it at all.

What the loop inductance does at the end of it #

The device question above is answered. The circuit’s question is a different failure and it is the one that actually destroys parts.

At the instant the stored charge runs out, the junction stops conducting — and the inductance that was driving the reverse current still has that current in it. It has nowhere to go but the diode’s own junction capacitance. So the reverse voltage rings up, at a frequency set by the loop inductance and the junction capacitance, to a peak with a surge impedance in it and only incidentally the supply.

With a microhenry of loop and a hundred picofarads of junction, the surge impedance is a hundred ohms and the ring is at 15.9 megahertz. The 3.83 amperes of recovery current from a hundred amperes a microsecond arrives into that and produces 494 volts across a diode in a hundred-volt circuit, marched on the off-state loop with the inductor holding the recovery current as its initial condition.

The expression usually written for that peak is $V + I_{RM}\sqrt{L/C}$, as though the supply and the surge term added. They do not. At the instant the junction blocks, the capacitor holds nothing while the loop is being driven to $-V$, so the ring starts a distance $V$ from where it is heading and with a current in the inductor, and those two are ninety degrees apart in the state plane. The peak is

$$V + \sqrt{V^2 + \left(I_{RM}\sqrt{L/C}\right)^2}$$

which is 496 volts here against the usual expression’s 483 — two per cent, because the current term dominates. At a tenth the recovery current the usual expression gives 130 volts and the correct one 204: thirty-six per cent low, at exactly the recovery currents a well-behaved circuit produces.

Where the charge goes, and what it costs #

Two quantities besides the peak are worth naming because they are what a converter’s efficiency actually depends on.

The recovered charge $Q_{rr}$ is the area under the reverse current, and it is bounded above by the stored charge $\tau I_F$ = 100 nanocoulombs: 116 nanocoulombs of total swept area at ten amperes a microsecond and 147 at a thousand, the excess over the stored charge being the charge the ramp itself delivers during the recovery. Every switching event moves that charge through the full supply voltage, so the loss is $V\,Q_{rr}$ per event and it is proportional to the switching frequency — which is why a converter’s efficiency falls with frequency in a way conduction losses cannot explain.

The duration falls as the ramp steepens: 276 nanoseconds at ten amperes a microsecond, 72 at a hundred, 22 at a thousand. So a faster switch has a shorter recovery and a much larger peak, and it pays for the shorter time with a larger current and a larger overvoltage.

That trade is the whole reason snubbers exist, and it is the same shape as the charging-energy result this collection already has: making the event faster does not make it cheaper, it moves the cost from duration into amplitude.

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

A rectifier’s data sheet quotes a reverse recovery time, usually with a test condition beside it in small print, and the arithmetic above says what that number is and is not.

It is one point on the curve in this essay’s figure. The test condition names a forward current, a ramp rate and sometimes a temperature, and the quoted time is what those produce. Read as a property of the part it is misleading in a specific direction: a part measured at a gentle ramp has a long quoted recovery time and a small peak, and used at a steep one it has a short recovery time and a peak many times larger. The number that moves least between conditions is the stored charge, which is why the better data sheets quote $Q_{rr}$ and the worse ones quote $t_{rr}$.

Two further things it does not contain. The temperature dependence is large — carrier lifetime rises with temperature, so a diode’s stored charge and its recovery peak are worst at the top of its thermal range, which is also where the rest of the circuit is least happy. And the softness, which is how abruptly the reverse current returns to zero after the peak, decides how much of the ring in the figure above actually happens: a soft-recovery part trades a longer tail for a smaller $dI/dt$ at the end of it and therefore a smaller overvoltage. This model takes the softness as a stated fraction and computes neither.

Two things it changes about a circuit that has nothing to do with the diode #

Both are consequences a reader is likelier to meet than the diode’s own destruction.

The recovery current flows through whatever is on the other side of the loop, which in a converter is the switch that was turning on. So the switch sees a current spike of the recovery peak on top of the load current it was expecting, at exactly the instant its voltage is still high — which is a current and a voltage simultaneously, and therefore a power spike rather than a current one. In a hundred-volt circuit with the numbers above that is nearly four hundred watts for a few tens of nanoseconds, once per cycle.

And the ring is a radiator. Sixteen megahertz in a loop of a microhenry is a current of several amperes circulating in a physical loop of some area, which is the same arrangement this collection already measures as an antenna: a small loop’s radiation goes as the fourth power of frequency and the square of the area, so the recovery ring is very often the dominant emission from a switching supply and the reason snubbers are fitted in circuits whose diodes are in no danger at all.

What the recovery costs, and where it is met #

Reverse recovery is the one place in this field where a device model has to carry a memory, and three later results depend on it. The heat a recovery leaves behind turns the charge measured here into a temperature and finds a frequency above which the iteration has no fixed point. The direct voltage that is a sawtooth is where the diode is doing its ordinary job, and the conduction angle it reports is what decides how often this recovery happens. The half that never arrives is the other loss that does not depend on the resistance it happens in. And One step, computed twice is the machinery all of it rests on, because a charge that flows backwards for ninety nanoseconds is not a thing a residue expansion can be asked about.

The recovery is also what makes the switching loss on a converter’s diode a function of how fast it is turned off rather than of how much current it was carrying, which is the reading The first cycle, which no steady state contains needs and cannot get from a steady-state model.

What is checked #

Five assertions, and the first two are the closed form against the march.

That the closed form and a trapezoidal march of the same charge equation agree on the peak reverse current and on how long the junction goes on conducting, to a part in a thousand, at every ramp rate the slider offers.

That the square-root expression is an over-estimate at every slope, because it is a limit approached from below — a direction rather than a tolerance, which a sign error would fail.

That the slow-ramp limit is an over-estimate too, and contains no forward current at all.

That there is a ramp rate at which the reverse peak equals the forward current, solved rather than estimated.

And, on the ring, that the marched peak sits below the undamped quadrature form by the damping and no more — bracketed between the undamped answer and $e^{-\zeta\pi}$ times it — with Kirchhoff’s current law rebuilt from the element laws closing to $10^{-11}$ over the whole march.

A model with no time in it, given one #

Every diode elsewhere in this collection is an instantaneous function of its own voltage, which is what a bias point is a solution, not a choice establishes and what every march since has used: each nonlinear element replaced by a conductance and a current source agreeing with its own law at the present guess. That model is exact for an operating point and has no state at all.

This essay gives it one, and the consequence is the same one the magnetics field met when it gave a core a memory. The core the solver has to remember is the parallel: a saturating inductor’s state is one number because the current is a function of the flux linkage, and a hysteretic core’s is not — half an amp is one flux on the way up and a different flux on the way down — so the march’s state vector gains twenty-four numbers the Newton loop is forbidden to touch. A charge-storing junction is the same construction with one extra number rather than twenty-four.

What both have in common is the thing that makes them worth the machinery. A model with state can produce a behaviour that no static model can approximate at any level of refinement: a diode that conducts backwards is not a small correction to one that does not, and a core that dissipates is not a lossy version of one that cannot. The three point eight three amperes measured here is the size of a behaviour, not the size of an error.

The seventeen per cent by which the standard expression for that peak is high, and its being right to a per cent only above sixteen thousand amperes a microsecond, is the ordinary shape of an asymptotic result used inside its own asymptote. It is the same shape the resistance that grows with frequency finds in the skin-effect rule of thumb — already 2.05 per cent up at the frequency the rule names as where the effect begins, and agreeing to 0.01 per cent two decades higher — and the same practical consequence: the expression is excellent where nobody needs it and wrong where the design is.

Which makes the seventeen per cent the useful half of that result rather than the sixteen thousand amperes a microsecond. A designer at a hundred amperes a microsecond needs to know that the standard expression overstates the peak by about a sixth; being told the rate at which it becomes reliable is being told about a circuit nobody builds. Which is the ordinary use of an asymptotic result and the ordinary way it disappoints: a designer at the rate the expression is right for is a designer whose recovery current is so large that the part has already been chosen for something else.

The measured seventeen per cent is also the right size to be missed. An expression a sixth high looks like a conservative estimate rather than a wrong one, and a design sized on it is oversized rather than unsafe — which is why the error survives: it costs money quietly instead of failing loudly, and nothing in a working circuit reports it.