Circuit Response
every model has an edge, and the figure carries it
Where the models stop

The resistance a slow curve cannot see

A diode's series resistance is read off the top of its forward curve, and a bench curve is a slow one: each point is held until the junction has warmed to it. Through 350 kelvin per watt the held curve sits 39.7 millivolts below the pulsed one at 100 milliamps, and the three-parameter fit that reads 0.580 ohms from the pulsed curve reads 0.184 from the held one — with a smaller residual. The fitted resistance reaches zero at 531.7 kelvin per watt, and the resistance it hid is what keeps the junction from folding back: with 0.05 ohms instead of 0.6 the held curve turns over at 87.6 milliamps.
15 min read 8 figures Every model has an edgeTwo routes to a numberThe parasitic is the component

Assumes: The one current a constant is right at · The loss that depends on what it causes

The junction the constant that is a window built to have a known answer contains three things: a recombination current, a diffusion current and a series resistance of 0.6 ohms between the junction and its terminals. At the top of its forward curve the resistance takes over, and the local ideality factor climbs to 3.35 at a hundred milliamps as the reciprocal slope reads a resistive drop as though it were a junction mechanism.

That essay declined to fit all three parameters at once, on the ground that a fit with the generating model in it would simply return the generating numbers — a demonstration of arithmetic and not of anything about diodes. For the curve the junction was built from, at one temperature, that is correct. It is not true of the curve an instrument takes.

A curve tracer or a source-measure unit stepping through a forward characteristic holds each current for long enough to read it. At a hundred milliamps the junction is dissipating about seventy-five milliwatts, and two currents with one name measured what warming does to its drop: about two millivolts less for every kelvin. So each point on a slow curve belongs to a different junction temperature, the curve is bent by that, and the bend is at exactly the end of the curve where the series resistance is read.

One diode curve at two speeds: 39.7 mV apart at 100 mA through 350 K/W. computed by solving, not by drawing. The forward curve of the two-mechanism junction with a 0.6 Ω series resistance, from 1 mA to 300 mA, twice. The dashed curve is pulsed: the die stays at 25 °C. The solid curve is held: each point is the temperature the junction settles at when it dissipates its own current times its own drop through 350 K/W to ambient, found as a fixed point. At 100 mA the junction is 27.8 K warm and the held drop is 39.7 mV lower; the slopes there are 0.865 Ω pulsed and 0.466 Ω held, and at 300 mA the curves are 122.5 mV apart. The series resistance is held constant in temperature.
Fig. 1 The forward curve of the two-mechanism junction with 0.6 Ω in series, from 1 mA to 300 mA, twice. Dashed, pulsed: the die stays at 25 °C. Solid, held: each point at the temperature the junction settles at when it dissipates its own current times its own drop through 350 K/W. At 100 mA the junction is 27.8 K warm and the held drop is 39.7 mV lower; the slopes there are 0.865 Ω pulsed and 0.466 Ω held, and at 300 mA the curves are 122.5 mV apart.

Every point a fixed point

A held point is not a measurement at a known temperature. The junction’s temperature is ambient plus the thermal resistance times the power, and the power is the current times the drop, which depends on the temperature. So each point is a fixed point of that loop, solved rather than assumed — the same structure the loss that depends on what it causes solved for a ferrite, here with the junction’s own drop in place of the core’s loss.

At a milliamp the loop is irrelevant. The junction dissipates about six hundred microwatts, warms by a fifth of a kelvin through 350 kelvin per watt, and the two curves lie on top of each other. At a hundred milliamps it is 27.8 kelvin warm and the curves are 39.7 millivolts apart; at three hundred, 122.5. The gap grows faster than the current because the power is the current times a drop that is itself nearly constant, so the temperature rise is nearly proportional to the current, and the drop’s loss is proportional to that.

For a current-driven junction the loop is stable. The sensor inside its own answer found why: a junction’s drop falls as it warms, so at a fixed current its dissipation falls too, and the loop gain is negative. Every point on the held curve exists and is unique, and the curve tracer is measuring something well defined. It is just not the junction the pulsed curve describes.

Two slopes at one current

The same curve's slope at 100 mA is 0.87 Ω pulsed and 0.47 Ω held. computed by solving, not by drawing. The incremental resistance of the two forward curves of the figure before — pulsed, with the die at 25 °C, and held, with the die at the temperature 350 K/W takes it to — against current, with the V_T/i a small-signal model uses drawn faintly beside them and the 0.6 Ω the junction was built with marked. At 10 mA the slopes are 3.425 and 3.024 Ω; at 100 mA 0.865 and 0.466 Ω. The held slope is taken along the solved curve and checked against the chain rule through the fixed point.
Fig. 2 The incremental resistance of the two curves against current, with the Vₜ/i a small-signal model uses drawn faintly and the 0.6 Ω the junction was built with marked. At 10 mA the slopes are 3.425 Ω pulsed and 3.024 Ω held; at 100 mA, 0.865 and 0.466. The held slope is taken along the solved curve and checked against the chain rule through the fixed point.

At a hundred milliamps the pulsed curve’s slope is 0.865 ohms, which is the junction’s own thermal-voltage-over-current term plus the 0.6 ohms the model was built with. The held slope is 0.466. The difference, four tenths of an ohm, is a negative resistance contributed by the loop: a small increase in current warms the junction a little, the warmer junction drops a little less, and the two effects subtract.

Which slope is right depends on the speed of the question. A small-signal model of the junction — how small is small signal builds one — is asking about signals fast compared with the die’s thermal time constant and slow compared with nothing else, so it wants the pulsed slope. A regulator’s reference string at direct current wants the held one. The pulse the heatsink does not feel put a frequency on the difference for a switching diode, 308 hertz on that part, below which the junction follows the power and above which it integrates it. A junction’s incremental resistance has that frequency in it, and a single quoted value does not say which side of it the value belongs to.

The loss of slope has a property that makes it easy to miss. At ten milliamps the held curve’s slope is 0.401 ohms below the pulsed one, and at a hundred milliamps 0.399 — the same, to a few milliohms, across a decade of current. The loop’s negative resistance is set by the drop, its temperature coefficient and the mounting, and none of the three moves much with the current. So at ten milliamps it is an eighth of a slope dominated by the thermal voltage over the current, and invisible; at a hundred it is nearly half the slope and most of the series resistance. The error is constant, and it hides at exactly the end of the curve where the quantity it corrupts is read.

A better fit to a worse number

Now fit the three parameters a simulator’s diode model carries — a constant, an ideality factor and a series resistance — to the curve between ten and a hundred milliamps, where the resistance is visible, by least squares in the drop.

A fit to the held curve reads the series resistance falling to nothing at 532 K/W. computed by solving, not by drawing. Each point is a three-parameter fit — a constant, an ideality factor and a series resistance — to the held forward curve between 10 and 100 mA, for a junction built with 0.6 Ω and no temperature coefficient on it, mounted at the thermal resistance on the axis. With no thermal resistance the fit returns 0.580 Ω with a residual of 43.9 µV. At 350 K/W it returns 0.184 Ω, an ideality of 1.086 and a residual of 34.0 µV. The resistance it reports reaches zero at 531.7 K/W and is negative beyond.
Fig. 3 Each point is a three-parameter fit — a constant, an ideality and a series resistance — to the held forward curve between 10 and 100 mA, for a junction built with 0.6 Ω, against the thermal resistance it is mounted on. With no thermal resistance the fit returns 0.580 Ω with a residual of 43.9 µV. At 350 K/W it returns 0.184 Ω, an ideality of 1.086 and a residual of 34.0 µV. The resistance it reports reaches zero at 531.7 K/W and is negative beyond.

On the pulsed curve the fit returns 0.580 ohms. It is not exactly 0.600, because the window still contains some of the recombination current and a single ideality cannot absorb it, but it is close and it is honest. On the held curve through 350 kelvin per watt it returns 0.184 ohms, less than a third of the resistance in the part.

And its residual is smaller: 34.0 microvolts against 43.9. The curve that has been bent by heating is better described by the three-parameter law than the curve the law was written for, because the heating straightens the top of the curve and the law’s own curvature was never quite the curvature of a two-mechanism junction. The constant that is a window found an ideality factor whose tightest fit was its worst extrapolation; this is the same inversion, one parameter over and caused by a different physical effect. The diagnostic an extraction produces for free again reports that the extraction is excellent.

The size of the error has a short account. The loop contributes a resistance of about the drop’s temperature coefficient times the thermal resistance times the drop, since the power is nearly the current times a constant drop. At a hundred milliamps the coefficient is about 1.43 millivolts per kelvin, and 350 kelvin per watt times seven-tenths of a volt times that is about 0.35 ohms — the size of what vanished from the fit. Set equal to the 0.58 the pulsed fit reads, the same estimate puts the fitted resistance at zero near 580 kelvin per watt, against the 531.7 the fits find. It is an estimate, and the figure’s points are the measurement; what the estimate shows is that nothing mysterious is involved. A thermal resistance times a voltage times a temperature coefficient is a resistance.

Beyond 531.7 kelvin per watt the fitted resistance is negative. That at least is a number nobody would accept, and it is the first reading in the whole sweep that betrays the extraction. Everything short of it is plausible, and a series resistance of a fifth of an ohm on a small signal diode is a very ordinary thing to read.

Three mountings

One diode curve at two speeds: 17.4 mV apart at 100 mA through 150 K/W. computed by solving, not by drawing. The forward curve of the two-mechanism junction with a 0.6 Ω series resistance, from 1 mA to 300 mA, twice. The dashed curve is pulsed: the die stays at 25 °C. The solid curve is held: each point is the temperature the junction settles at when it dissipates its own current times its own drop through 150 K/W to ambient, found as a fixed point. At 100 mA the junction is 12.3 K warm and the held drop is 17.4 mV lower; the slopes there are 0.865 Ω pulsed and 0.686 Ω held, and at 300 mA the curves are 55.8 mV apart. The series resistance is held constant in temperature.
Fig. 4 The same junction through 150 K/W, which is a part with generous copper. At 100 mA it is 12.3 K warm and 17.4 mV low; the held slope there is 0.686 Ω against 0.865 pulsed, and at 300 mA the curves are 55.8 mV apart.

Through a hundred and fifty kelvin per watt the loss of slope is less than half of what it was — 0.686 ohms held against 0.865 pulsed — and a fit taken from this curve would keep more of the resistance. The error scales with the mounting, which is the useful half of the estimate above: a part on a small pad in still air is at the far end of the range, and the same part soldered to a plane is near the start.

One diode curve at two speeds: 76.3 mV apart at 100 mA through 700 K/W. computed by solving, not by drawing. The forward curve of the two-mechanism junction with a 0.6 Ω series resistance, from 1 mA to 300 mA, twice. The dashed curve is pulsed: the die stays at 25 °C. The solid curve is held: each point is the temperature the junction settles at when it dissipates its own current times its own drop through 700 K/W to ambient, found as a fixed point. At 100 mA the junction is 53.1 K warm and the held drop is 76.3 mV lower; the slopes there are 0.865 Ω pulsed and 0.125 Ω held, and at 300 mA the curves are 220.9 mV apart. The series resistance is held constant in temperature.
Fig. 5 Through 700 K/W, a small part on a thin trace. At 100 mA the junction is 53.1 K warm and the held drop is 76.3 mV lower; the held slope there is 0.125 Ω against 0.865 pulsed, and at 300 mA the curves are 220.9 mV apart.

Through seven hundred the held slope at a hundred milliamps is an eighth of an ohm, and the curve above there is flatter than the junction’s own thermal voltage allows. That is past the thermal resistance at which the three-parameter fit reports a negative resistance, so a curve tracer pointed at this mounting would produce a model with a negative series resistance in it — which is, perversely, the safest outcome, because nobody will use it by mistake.

Across the three mountings the lost slope is nearly proportional to the thermal resistance. At a hundred milliamps it is 0.179 ohms through 150 kelvin per watt, 0.399 through 350 and 0.740 through 700 — 1.19, 1.14 and 1.06 milliohms for every kelvin per watt, falling slightly as the junction runs warmer. That near-proportionality is what lets the account of the fitted resistance be written as one product, and it is what makes the mounting, rather than the part, the first thing a fitted series resistance depends on.

When the resistance warms as well

The model so far has held the series resistance constant in temperature. A silicon bulk resistance is not constant: the mobility falls as the lattice warms, so the resistance usually rises with temperature, and how fast depends on the doping.

A fit to the held curve reads 0.347 Ω at 350 K/W, with 0.973 for its ideality. computed by solving, not by drawing. Each point is a three-parameter fit — a constant, an ideality factor and a series resistance — to the held forward curve between 10 and 100 mA, for a junction built with 0.6 Ω rising 0.60 per cent per kelvin, mounted at the thermal resistance on the axis. With no thermal resistance the fit returns 0.573 Ω with a residual of 43.9 µV. At 350 K/W it returns 0.347 Ω, an ideality of 0.973 and a residual of 256.2 µV. It does not reach zero on this axis.
Fig. 6 The same fits for a junction whose 0.6 Ω rises 0.60 per cent per kelvin. With no thermal resistance the fit returns 0.573 Ω and a residual of 43.9 µV. At 350 K/W it returns 0.347 Ω, an ideality of 0.973 and a residual of 256.2 µV. The fitted resistance does not reach zero on this axis.

A rising resistance works against the loop, so less of it disappears: 0.347 ohms read at 350 kelvin per watt rather than 0.184. And this time the extraction gives itself away twice. Its residual is six times the pulsed fit’s, 256.2 microvolts, and its ideality factor is 0.973, below one — a value that neither mechanism in the junction has and that a fit should not be believed at.

So whether the extraction warns depends on a property of the silicon that nobody measured, and the two cases lie on either side of the most dangerous line: a resistance with no temperature coefficient produces a smooth, confident, wrong model, and one with a coefficient of 0.6 per cent produces a model that refuses to look right. A real part’s coefficient is somewhere, and this page does not say where; what it says is that the safe outcome is the one with the larger error.

The resistance that holds the junction still

Driven from a current source the loop is stable at every point. Driven from a stiff voltage it need not be, because at a fixed voltage a warmer junction draws more current, dissipates more and warms further, and the loop gain is positive.

With 0.05 Ω in it the held curve is flat at 87.6 mA, and a supply above 0.743 V runs it away. computed by solving, not by drawing. The forward curve of the two-mechanism junction with a series resistance of 0.05 Ω, pulsed and held at 350 K/W, with the held curve for 0.6 Ω beside it; each held curve is stopped where the junction reaches 150 K above ambient. The held curve's slope reaches zero at 87.6 mA, with the junction 22.8 K warm and dropping 0.7430 V. The same junction held at that voltage has a thermal loop gain of 1.0000 there, solved separately, so above it a stiff supply has no temperature to settle at. With 0.6 Ω it does not turn over below 1 A.
Fig. 7 The forward curve of the junction with a series resistance of 0.05 Ω, pulsed and held at 350 K/W, with the held curve for 0.6 Ω beside it; each held curve is stopped where the junction reaches 150 K above ambient. The held curve’s slope reaches zero at 87.6 mA, with the junction 22.8 K warm and dropping 0.7430 V. Held at that voltage the junction has a thermal loop gain of 1.0000 there, solved separately, so above it a stiff supply has no temperature to settle at. With 0.6 Ω the held curve does not turn over below 1 A.

With a twentieth of an ohm in series the held curve flattens at 87.6 milliamps and then folds back: past there, more current needs less voltage. The point where it is flat is the point where a voltage-driven junction’s loop gain is exactly one, and the figure checks that with a separate solution rather than taking it from the curve. Put 0.7430 volts or more across this junction from a stiff supply and there is no temperature for it to settle at.

With the 0.6 ohms the model was built with, the held curve at the same mounting does not turn over at all below an ampere. The series resistance is a negative feedback on the loop: a junction trying to draw more current drops more across the resistance, which leaves less for the junction. The ohms that the fit could not see are precisely the ohms keeping the part stable.

With 0.6 Ω in it the held curve is flat at 221 mA, and a supply above 0.764 V runs it away. computed by solving, not by drawing. The forward curve of the two-mechanism junction with a series resistance of 0.6 Ω, pulsed and held at 700 K/W, with the held curve for 0.05 Ω beside it; each held curve is stopped where the junction reaches 150 K above ambient. The held curve's slope reaches zero at 221 mA, with the junction 118.0 K warm and dropping 0.7639 V. The same junction held at that voltage has a thermal loop gain of 1.0000 there, solved separately, so above it a stiff supply has no temperature to settle at. With 0.05 Ω the held curve is flat at 40.3 mA.
Fig. 8 The 0.6 Ω junction through 700 K/W. Its held curve is flat at 221 mA, with the junction 118.0 K warm and dropping 0.7639 V, where the thermal loop gain is again 1.0000; with 0.05 Ω on the same mounting the held curve is flat at 40.3 mA.

Through seven hundred kelvin per watt the real 0.6 ohms is no longer enough: the curve flattens at 221 milliamps, a hundred and eighteen kelvin above ambient. And a twentieth of an ohm on the same mounting folds back at 40.3.

Put the two halves of this essay beside each other and they make one uncomfortable sentence. A slow curve through 350 kelvin per watt reads the series resistance as 0.184 ohms, and the resistance it left out is the ballast that kept the measured part from folding back — a twentieth of an ohm on the same mounting folded back at 87.6 milliamps. A model extracted this way understates the one parameter that decides how far the part is from running away, and whether that makes a simulation optimistic or pessimistic depends on whether the simulation also knows the thermal resistance the fit absorbed.

Taking the curve the model wants

The remedy follows from where the error came from, and there are two. The first is to take the curve the junction’s own law describes: pulsed, with each point held for much less than the die’s thermal time constant, so that every point is at the temperature of the mounting. The pulse the heatsink does not feel put a frequency on that time constant for one switching diode. A dwell long enough for a meter to settle is long on most dies, and how short a pulse has to be for a particular part depends on its own thermal network, which is not solved here.

The second is to keep the heating and give it somewhere to go: fit the held curve with the thermal resistance as an element of the model rather than folding its effect into the other three parameters. A model that carries a junction and a thermal path separately can reproduce both curves from one set of numbers. A model that carries a junction alone can reproduce only the curve it was fitted to, and it quotes a series resistance as though that were a property of the part.

Both remedies ask the extraction for something a three-parameter fit never asks for: a statement of the conditions the curve was taken under. The constant that is a window found that a fitted ideality needs its window of current beside it. A fitted series resistance needs a thermal condition beside it as well, and its residual will never say so.

What is and is not in the model

The junction is the two-mechanism model with its temperature law from two currents with one name: a diffusion constant doubling every four and a half kelvin, a recombination one every nine. The thermal path is a single resistance to an ambient of 25 degrees with no heat capacity, so every held point is a settled one; a curve tracer that holds each point for less than the die’s thermal time constant is taking a curve between the two drawn. The series resistance is constant unless a coefficient is given, and the fold-back curves stop at 150 kelvin above ambient, well before the junction would stop being a junction.

Nothing here says what thermal resistance a particular part and board have, what its bulk resistance’s coefficient is, or how long a particular instrument dwells at each point. Those are the three conditions a fitted series resistance needs beside it, and a model file carries none of them.

Still open: dwell time, a fourth parameter, and the collector

The curve between the two. The pulsed and held curves are limits. An instrument dwelling for a time comparable with the die’s thermal time constant takes a curve between them, and which one depends on the dwell. The pulse the heatsink does not feel has the thermal network that would say, for a given die, how short a pulse has to be for the fitted resistance to be within a per cent of the part’s — a boundary in time rather than in current.

A fourth parameter. The three-parameter fit fails because heating adds a term it has no place for. A fit given the thermal resistance as a fourth unknown, fed the held curve alone, might recover both — or might find the two indistinguishable over a decade of current, which would be a statement that the extraction is ill-conditioned rather than merely wrong. Which of the two happens is a measurable question.

The logarithm at the collector. The resistance and the heating both live at the top of the curve. The bottom belongs to the recombination current, which the logarithm is in the collector removes by taking the current at a transistor’s collector instead of at a diode’s terminals — and there both ends of the logarithm’s range can be measured.

Part 4 on diode model

One argument about Diode model, and one of 6 essays on it so far, each part numbered by how much of the idea it assumes. What sits either side of it:

Two currents called saturation: one doubles every 4.49 K, the other every 8.98 K. computed by solving, not by drawing. The two current scales of one model junction against temperature, on a logarithmic axis. The saturation current of the exponential law goes as the square of the intrinsic carrier density — a cube of the temperature and the whole band gap in a Boltzmann factor — and doubles every 4.489 K at 300 K. The generation current a reverse-biased junction actually conducts goes as the density itself, with half the band gap, and doubles every 8.978 K: exactly twice as long, at every temperature. The dashed line is "doubles every ten kelvin" drawn through the generation scale, which is the current the rule belongs to. At 300 K this junction's two scales are 10 fA and 2 nA, which are its own parameters and not a property of silicon, and they become equal only at 616.8 K, or 343.6 °C. Two currents with one name Part 3 — A junction's saturation current is two currents with one name. The one in the forward law doubles every 4.49 kelvin; the one a reverse-biased junction actually conducts is generated in its depletion region, doubles every 8.98, and on this model junction is 3.06 × 10⁵ times larger at a volt of reverse bias. Doubles every ten kelvin is the second current's rule, and applied to the first it turns the forward drop's −1.81 millivolts per kelvin into +0.39. At 300 K one junction holds a logarithm to ±1% over 2.4 decades as a diode and 8.5 at its collector. computed by solving, not by drawing. The voltage of one model junction against the logarithm of the current it carries, as a percentage error of that current from a straight line fitted over the widest range that stays within ±1%. Taken as a diode — both mechanisms and 0.6 Ω of series resistance — the range is 2.40 decades, from 50.1 nA to 12.6 µA, and its slope is an ideality of 1.982. Taken at the collector, where the recombination current is supplied from the base and 1.604 Ω remains, it is 8.50 decades, from the axis's own end at 1 pA to 316 µA, at an ideality of 1.0001. Nothing arrives beside the collector current, so its lower end on this axis is the axis. The logarithm is in the collector Part 5 — A diode is the textbook logarithm, and a real junction holds one to within one per cent over only 2.40 decades — from 50 nanoamps to 12.6 microamps, at an ideality of 1.98 — because two mechanisms and a series resistance share its terminals. The same junction read at a transistor's collector, with its recombination current supplied from the base, holds 8.50 decades at an ideality of 1.0001. How far the logarithm reaches is decided by which terminal the current is taken from, and at the bottom of the range by a leakage current a millivolt is enough to switch on.

Where this goes next

One diode curve, eight one-decade fits, and eight different ideality factors. computed by solving, not by drawing. A junction with two conduction mechanisms — recombination near n = 2 at low current, diffusion near n = 1 above it — and a series resistance, which is what a real diode is. Fitting ln(i) against v over each decade in turn returns an ideality factor for each, and they run from 1.227 to 1.984 without being monotonic: the factor rises through the recombination region, falls through the diffusion region, and rises again where the series resistance takes over. Two of the windows are straight to a few parts in a thousand, so the residual gives no warning. The bars are what each fit predicts for the forward voltage at 1 mA: the worst is out by -186 millivolts, which is a current 0.01 times the truth. The constant that is a window A diode's ideality factor is quoted as a number and defined as a derivative, which means it has a value at every current and no value anywhere. Eight one-decade fits to one curve return factors from 1.23 to 1.98, two of them straight to a few parts in a thousand — so the residual gives no warning at all. Asked for the forward voltage at a milliamp, the window containing it is right to a third of a millivolt and the worst is out by 186, which is a current a hundredth of the truth. Two currents called saturation: one doubles every 4.49 K, the other every 8.98 K. computed by solving, not by drawing. The two current scales of one model junction against temperature, on a logarithmic axis. The saturation current of the exponential law goes as the square of the intrinsic carrier density — a cube of the temperature and the whole band gap in a Boltzmann factor — and doubles every 4.489 K at 300 K. The generation current a reverse-biased junction actually conducts goes as the density itself, with half the band gap, and doubles every 8.978 K: exactly twice as long, at every temperature. The dashed line is "doubles every ten kelvin" drawn through the generation scale, which is the current the rule belongs to. At 300 K this junction's two scales are 10 fA and 2 nA, which are its own parameters and not a property of silicon, and they become equal only at 616.8 K, or 343.6 °C. Two currents with one name A junction's saturation current is two currents with one name. The one in the forward law doubles every 4.49 kelvin; the one a reverse-biased junction actually conducts is generated in its depletion region, doubles every 8.98, and on this model junction is 3.06 × 10⁵ times larger at a volt of reverse bias. Doubles every ten kelvin is the second current's rule, and applied to the first it turns the forward drop's −1.81 millivolts per kelvin into +0.39. A diode thermometer measuring its own sense current, at 600 K/W. computed by solving, not by drawing. The same fixed point as the core and the thermistor, on a junction: the dissipation is I·V and V falls with temperature, so the loop gain is negative and the equation has one root at every current. What it costs is two errors. The junction sits above ambient by 0.29 millikelvin at a microamp and 4.26 kelvin at ten milliamperes, which a calibration removes; and a kelvin of ambient produces less than a kelvin of junction, by 1/(1 − R_th·dP/dT), which it does not. After a calibration at 25 degrees the reading at 85 is out by -583 millikelvin at ten milliamperes and -0.09 at a microamp. The coefficient itself moves too — -2.403 against -1.613 millivolts a kelvin — so a quoted tempco carries a sense current as well as a junction. The sensor inside its own answer A junction driven from a current source cannot run away, because its forward voltage falls with temperature and its loop gain is therefore negative. What that costs is a thermometer that is warmer than what it is measuring by 4.26 kelvin at ten milliamperes, and — the part a calibration cannot remove — under-reports every change in ambient by 9,584 parts per million, because the sense current's own dissipation falls as the reading rises. Calibrated at 25 degrees, it is out by 583 millikelvin at 85. The frequency at which a pulse train becomes an average. computed by solving, not by drawing. The same 5 watts of average dissipation at every frequency, delivered 2 per cent at a time. The flat line is the steady-state answer, which does not know about the frequency. The falling curve is the marched peak junction temperature, which does. They meet at 308 Hz, and that frequency is not a property of the converter: it is a fraction of one junction time constant per period — f·τ = 0.738 at this duty, with τ = 2.40 ms, and between 0.78 and 0.56 across the duties on the slider. A hundred-kilohertz converter fits 240 periods inside that time constant, and at the top of the sweep — 10.0 kHz — the steady state is already exact to 0.46 per cent, so the averaged-power fixed point is right and this is the measurement that says why. The march puts 48 steps inside each pulse, which is what the answer is sensitive to: at six it put the boundary 19 per cent too high. The pulse the heatsink does not feel A thermal resistance iterated to a fixed point with a diode or a switch is a statement about a power — so it assumes that a hundred and fifty watts for two per cent of the time is three watts. The die's own heat capacity decides whether that is true, and it decides it at a frequency: above 308 hertz the junction integrates, by a hundred kilohertz the fixed point is exact to five parts in ten thousand, and at one hertz the same average power on the same heatsink puts the junction three hundred kelvin hotter.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down.

The objects named here

The third axis, after the field and the idea: the things themselves, and every essay that touches each one.

Bulk resistanceDynamic resistanceIdeality factorLoop gainMeasurement conditionModel rangeOperating pointThermal resistanceVerification