The table that has to be refitted

How many straight lines a diode is built the piecewise-linear diode a circuit simulator would want: a chain of drop-plus-resistance segments, each fitted minimax over its own span of current, meeting at their joins without being asked to. On a real junction it found the segments narrowing where the local ideality is two and widening again at the top, where the series resistance straightens the curve. Eighteen equal-ratio segments from 100 nanoamps to 100 milliamps hold the real part to within 1.886 millivolts.

All of that was at 300 kelvin, and a diode in a circuit is not. It closed by asking the question that decides whether the table is a model or a snapshot: whether a table built at one temperature can be carried to another by moving all of its drops by one amount, or whether its breakpoints must move as well. The answer comes out the other way round from both alternatives. One amount is far from enough, the breakpoints need not move at all, and what does have to change is every segment’s line — its drop and its slope — which is to say the table has to be refitted.

The junction, and the four ways to carry its table #

The junction is the two-mechanism one these essays have used since the constant that is a window: a diffusion current of ideality one and a recombination current of ideality two in parallel, behind 0.6 ohms of series resistance. Two currents with one name gave it its temperature law. The diffusion current’s scale goes as the cube of the absolute temperature times a Boltzmann factor in the full band gap and doubles every 4.49 kelvin at room temperature; the recombination current’s goes as the power one and a half and half the band gap, and doubles every 8.9. The thermal voltage moves with the absolute temperature, and the series resistance is given no coefficient of its own, so that what changes is the junction alone.

The table is eighteen segments of equal current ratio, a third of a decade each, fitted minimax at 300 kelvin. It is carried to each temperature from 250 to 400 kelvin in four ways. Unchanged. With every segment’s drop moved by the one shift that best fits the new curve over all six decades. With each segment’s drop moved by its own best shift, which keeps the breakpoints and the slopes. And refitted at the new temperature on the same breakpoints, which keeps only the breakpoints. The last is not a correction at all but the best any table with these breakpoints can do, and it is the yardstick the other three are read against.

Two millivolts a kelvin is not one number #

Unchanged, the table is wrong by about what anyone would guess: 79.8 millivolts at 325 kelvin, 163 at 250 and 266 at 400. That is the familiar two millivolts a kelvin, and two millivolts a kelvin, and the wrong sign has already measured why the number is a coefficient of one particular current. The obvious repair is to move every drop down by that coefficient times the temperature change. It leaves 22.2 millivolts at 325 kelvin and 38.3 at 350: an eighth of the error remains, and it is ten times the table’s own.

The reason is in the coefficients. Each segment’s drop moves at its own rate, from −3.31 millivolts a kelvin at the bottom of the table to −1.54 at the top, and the junction’s own forward voltage does the same across the six decades, from −3.24 at 100 nanoamps to −1.42 at 100 milliamps. A low current sits further below the junction’s saturation scale, so its voltage is a larger multiple of the thermal voltage, and that multiple’s change with temperature is larger. The coefficient runs over a factor of two across the table, and any one shift splits the difference: −2.30 millivolts a kelvin, too slow for the bottom and too fast for the top.

The error after one shift is therefore a tilt: positive at one end of the table, negative at the other, crossing zero in the middle, 38.3 millivolts at its ends at 350 kelvin. It is not noise and it is not a bow that more segments would reduce; it is a straight-line error in log current, which no correction of the form “move every drop by the same amount” can touch.

The coefficient in closed form #

The spread of coefficients is not an accident of this junction’s numbers; it has a closed form, and the form says why the error after one shift is a straight tilt. For a junction carrying its current by diffusion, $v_j = V_T\ln(i/I_s)$ with $I_s \propto T^3 e^{-E_g/kT}$, and differentiating at a fixed current gives

$$\frac{dv_j}{dT} = \frac{v_j - E_g/q - 3kT/q}{T}.$$

For recombination the ideality is two and the scale goes as $T^{1.5}e^{-E_g/2kT}$, and the two factors of two cancel: the coefficient comes out in exactly the same form. So above a microamp, where the junction is well above both of its scales, the coefficient depends on the junction voltage alone. The figure holds that to half a per cent at every decade from a microamp to a hundred milliamps: at a microamp the junction sits at 321 millivolts and moves −2.91 millivolts a kelvin; at a hundred milliamps it sits at 772 and moves −1.42.

That is why one shift fails the way it does. The junction voltage rises by a fixed amount for each decade of current — about a hundred and twenty millivolts a decade where the ideality is two — so the coefficient changes by a fixed amount a decade, about four tenths of a millivolt a kelvin, and the error left by any single shift is a straight line in log current whose slope grows with the temperature change. Eighteen shifts, one a segment, are eighteen samples of that straight line, and they remove it.

The voltage every coefficient points at #

The closed form can be read the other way round. Rearranged, it says that a junction voltage at a fixed current, extended back along its own temperature coefficient, reaches $E_g/q + 3kT/q$ at zero coefficient — 1.198 volts for silicon at 300 kelvin — whatever the current. Every segment’s drop in the table is a line through its own voltage pointing at the same place, steeper the further below it the segment sits, and the segments’ eighteen coefficients are eighteen distances from one voltage divided by one temperature.

The voltage at which the slope is zero found a close relative of this number from the other side: 1.1717 volts, the output of a reference that cancels a junction’s fall with a rise proportional to temperature. That is $E_g/q$ plus two thermal voltages rather than three, because a reference sets the slope of a sum to zero at one temperature, where this extrapolation follows a single junction’s coefficient to the point where it would vanish; the two differ by one thermal voltage, 25.9 millivolts, and they are measuring the same fact. A band-gap reference is the one circuit that makes use of the spread this essay is fighting: it needs a junction voltage whose coefficient is set by how far it sits below 1.2 volts, and it gets one at whatever current it runs. A table that must follow a junction across temperature needs the same fact for eighteen currents at once.

It also says why the coefficient at a given current is not a property of the part in the way the drop is. Two diodes of one type at one current differ in their saturation current, so they sit at different junction voltages, and so they have different coefficients — by a few microvolts a kelvin for each millivolt of difference in their drops. That is the effect two millivolts a kelvin, and the wrong sign measured as a coefficient belonging to a current, and here it is the reason a table’s coefficients cannot be borrowed from a data sheet’s single number.

One shift a segment #

Giving each segment its own shift removes the tilt, because a tilt across eighteen segments is eighteen small offsets. What is left is a bow within each segment: at 350 kelvin, 3.84 millivolts at worst, against the 2.18 a refit achieves. Over the whole range from 250 to 350 kelvin a shift per segment stays within twice and a half of the refit, and a table that carries eighteen drop coefficients, one a segment, is a reasonable engineering model over that range.

The bow it leaves is a change of slope. A shift moves a segment’s line up or down; it cannot rotate it, and the line’s slope — the segment’s resistance — has changed.

Between a microamp and a milliamp every segment’s resistance rises 0.328 per cent a kelvin, within a per cent and a half of one part in three hundred. That is the signature of a thermal voltage: a segment’s resistance there is the local ideality times $kT/q$ over a current, and it moves as the absolute temperature. At the top it rises less, 0.130 per cent, because the series resistance — which carries no coefficient in this model — is most of the slope; at the bottom 0.227, for a reason the next figure gives. So the second correction a table needs is a resistance per segment scaled by $T/T_0$, and with it the shifts and the slopes together are close to a refit.

Where a shift stops being enough #

Past 350 kelvin something else happens, and it shows at the bottom of the table first.

At 400 kelvin a shift per segment leaves 14.67 millivolts in the bottom segment, while a refit there leaves 79 microvolts — far less than the segment’s 1.9 millivolts at 300 kelvin. The bottom of the curve has become nearly straight. The recombination current’s scale, two nanoamps at 300 kelvin, is 0.692 microamps at 400; the table’s lowest current is a hundred nanoamps, so at 400 kelvin the junction there is below its own saturation current, barely forward biased, and its current is close to proportional to its voltage. A diode below its saturation current is a resistor, a hundred nanoamps of it carried by a few millivolts.

A segment fitted to a logarithm at 300 kelvin cannot be shifted into a straight line at 400; it has to be refitted, and when it is the fit is better than it ever was. The local ideality says the same from the other side: at 100 nanoamps it is 1.96 at 300 kelvin and 0.252 at 400, and at a microamp 1.99 against 1.18. An ideality below one is not a property any junction has; it is what a curve that has become linear in voltage looks like when it is read as an exponential.

The breakpoints, which never move #

The refitted curve in the hero figure holds between 1.57 and 2.38 millivolts at every temperature from 250 to 400 kelvin, rising roughly in proportion to the absolute temperature. That is the answer to the question the earlier essay asked about the breakpoints. A segment’s error depends on how curved the characteristic is across it, and over a third of a decade of current the curvature of a logarithm is set by the thermal voltage times the local ideality. Both scale smoothly with temperature, so the same breakpoints that suited 300 kelvin suit 400, with an error that grows only as the thermal voltage does. Nothing about the shape of the junction’s law moves the places where straight lines should meet.

That is a stronger statement than it looks. The straight line between two models found the best single line depending strongly on which decade it was fitted over; here eighteen lines fitted on fixed breakpoints track a curve that moves by hundreds of millivolts, through two changes of mechanism at the bottom, without any of their breakpoints moving at all. A simulator needs one set of breakpoints and a rule for each line.

What a simulator should store #

The figures say what a table that covers temperature has to carry. For each segment: its breakpoints, which are fixed; its drop at a reference temperature with a coefficient of its own, from −3.31 to −1.54 millivolts a kelvin across this table; and its resistance, scaled by the absolute temperature where the junction dominates and by the series resistance’s own coefficient where that does. Over fifty kelvin either side of the reference that is within twice the table’s own error. Past that the honest answer is to refit, which for a table of eighteen segments is eighteen minimax fits — the same work as building the table — and the junction’s own law is needed to do it.

That last point is the reason the table cannot replace the law. The logarithm is in the collector found the junction’s own logarithm holding over only a few decades; a table can be as accurate as anyone likes at one temperature, but carrying it to another needs the physics the table was supposed to save. The resistance a slow curve cannot see found self-heating moving a held curve’s top decade by its own temperature; a table used inside a circuit that heats its diode has to be refitted as the junction warms, which in a simulator means evaluating the law anyway.

What the model leaves out #

The series resistance is given no temperature coefficient, which separates the junction’s behaviour from the bulk’s and is not physical: a real bulk resistance rises or falls with temperature according to its doping, and the top segments would then move with it. The ideality factors are fixed at one and two; a real junction’s recombination ideality varies a little with temperature and with bias. And the law is a diffusion and a recombination current only; at the bottom of a hot junction a surface leakage path often sets the current long before the recombination current reaches its scale, which would make the bottom of the table linear sooner than the figures show.

The number to keep #

A diode table of eighteen segments built at 300 K is 80 millivolts out at 325 K and 266 at 400 unchanged. One shift leaves a tilt of 22 to 59 millivolts, because the segments’ drops move at rates from −3.31 to −1.54 millivolts a kelvin. A shift per segment is within twice the table’s own error over fifty kelvin either way, and fails at 400 K, where the bottom of the table is below the junction’s recombination current and is a resistor. Refitted on the same breakpoints the table holds 1.57 to 2.38 millivolts at every temperature: the breakpoints stay, the lines move.

Still open: what a segment saves a solver, the reverse segment, and the heated junction #

What a segment saves a solver. A table is searched and evaluated with a multiply and an add, where the exponential is a function call and a Newton step that can overflow; a table that must be refitted with temperature gives some of that saving back. Counting the iterations a solver takes on each model, on a circuit that drives the diode across six decades at two temperatures, would price the table against the law in the only currency a solver uses.

The reverse segment. Below zero bias the junction conducts a generated current the forward segments say nothing about, and at 400 kelvin it is hundreds of times larger than at 300. Whether one reverse segment suffices, and whether it too keeps its breakpoints while its line moves, would complete the table.

A junction that heats itself. In a circuit the junction’s temperature is set by its own dissipation, so the table’s temperature is a function of where on the table the diode sits. A table refitted once, at the temperature its operating current produces, against one refitted at every current, is the measurement that decides whether a self-heated diode needs a table per bias point.