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

The logarithm is in the collector

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.
14 min read 7 figures Every model has an edgeMeasured, not tabulatedThe constants decide

Assumes: The one current a constant is right at · A bias point is a solution, not a choice

A junction’s current is exponential in its voltage, so its voltage is logarithmic in its current, and that inversion is one of the most useful things a semiconductor does. It is how a circuit compresses six decades of light into a volt, multiplies two signals by adding their logarithms, and measures a ratio of currents as a difference of voltages.

The constant that is a window found that the diode measured in these essays is a logarithm only locally: two conduction mechanisms and a series resistance share its terminals, so its ideality factor is a function of the current and any single pair of constants describes a window. That essay first put the range where the law holds in the wrong place, and its account has been corrected by the measurement this page makes. The range is shorter than it looked and is in a different place — and the obvious way to lengthen it turns out not to be a better diode but a different terminal.

Two currents at one voltage

One junction voltage, two currents: an ideality of 1.95 at the terminals and 1.00 at the collector. computed by solving, not by drawing. The currents the model junction carries against its own voltage at 300 K, on a logarithmic axis. The diode current is both mechanisms together. The collector current of the same junction in a transistor is the diffusion current alone; the base current is the recombination current plus the collector current over a β of 150. Below 0.631 V, where the two mechanisms carry 400 µA each, the diode current is mostly the base's. At 10 µA the diode current's local ideality is 1.953 and the collector current's is 1.0000.
Fig. 1 The currents the model junction carries against its own voltage at 300 K, on a logarithmic axis. The diode current is both mechanisms together. The collector current of the same junction in a transistor is the diffusion current alone; the base current is the recombination current plus the collector current over a β of 150. Below 0.631 V, where the two mechanisms carry 400 µA each, the diode current is mostly the base’s. At 10 µA the diode current’s local ideality is 1.953 and the collector current’s is 1.0000.

A diode’s current is two currents. Carriers that diffuse across the neutral region give the current with an ideality of one; carriers that recombine inside the depletion region give the current with an ideality of two. At the terminals they add, and the sum has whichever ideality the larger of them has, blended where they are comparable. On this junction they are equal only at 0.80 milliamps, and below that recombination carries most of the current: 97.6 per cent at ten microamps, 82.8 per cent at a hundred.

A transistor is the same junction with a third terminal placed where the diffusion current goes. The carriers that cross the base are collected; the carriers that recombine in the emitter’s depletion region are supplied through the base lead, together with the small fraction of the diffusing carriers that the base loses. So the collector current is the diffusion current and nothing else, and the base current is the recombination current plus a βth of the collector’s. The figure draws exactly that, from the same two scales and the same voltage.

At ten microamps the diode current’s local ideality is 1.953 — it is nearly all recombination — and the collector current’s is 1.0000. Nothing about the junction has changed. The two currents were separated by where the wire is.

A logarithm to one per cent

A logarithm to a tolerance has a range that can be measured directly: fit a straight line of voltage against the logarithm of current over the widest span that keeps every point within the tolerance, and read the span.

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.
Fig. 2 Voltage against the logarithm of current for one model junction, as a percentage error of the current from a straight line fitted over the widest range that stays within ±1%. 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, at an ideality of 1.982. At the collector, with recombination supplied from the base and 1.604 Ω remaining, it is 8.50 decades, from the axis’s end at 1 pA to 316 µA, at an ideality of 1.0001.

The diode holds a logarithm over 2.40 decades, and where it holds it is the recombination region: from fifty nanoamps to 12.6 microamps, with a slope that corresponds to an ideality of 1.982. Below fifty nanoamps the recombination current is too close to zero bias to be exponential, and above 12.6 microamps the diffusion current has begun to take over and the slope is changing. There is no stretch of this diode’s terminal current where the ideality-one law describes it to a per cent over even two and a half decades.

The collector holds 8.50 decades, from the bottom of the axis at a picoamp to 316 microamps, at an ideality of 1.0001. The top of that range is set by 1.604 ohms: the emitter’s 0.6 plus the base spreading resistance divided by the current gain, which together put a resistive drop in series with the junction that grows with the current. The bottom is the axis. A single exponential with nothing in series at all holds a logarithm to one per cent over the whole eleven decades drawn, so nothing in the collector current itself ends the range before a picoamp.

That is 3.5 times the range, from the same junction, by taking the current from a different terminal. It is the reason a logarithmic converter is built round a transistor whose collector current is forced, with its base tied to its collector’s reference, rather than round a diode — the arrangement usually called a transdiode — and the reason is not that the transistor is a better junction. It is the same junction with its second mechanism routed out of the measurement.

A transconductance with nothing else in it

The same separation matters to circuits that have nothing to do with logarithms. A transistor’s small-signal transconductance is the derivative of its collector current with respect to its junction voltage, and at the collector, with an ideality of 1.0000, that derivative is the collector current over the thermal voltage to the precision the figure prints — the relation how small is small signal linearises, exact across the range drawn rather than approximately true near one operating point. The diode connection gives no such relation. At ten microamps its local ideality is 1.953, so the incremental resistance a small-signal model would take as the thermal voltage over the current is out by nearly a factor of two, and out by a different factor at every current.

That is why bias circuits are computed from a collector current and not from a diode’s forward current. A bias point is a solution found a diode’s operating point as the crossing of two curves, one of them the device’s. A transistor biased by its collector current has that curve known in closed form across eight decades, and the terminal it is not measured at carries everything the law leaves out.

The two ends of the range, a decade at a time

At 300 K one junction holds a logarithm to ±0.1% over 1.3 decades as a diode and 6.8 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 ±0.1%. Taken as a diode — both mechanisms and 0.6 Ω of series resistance — the range is 1.25 decades, from 200 nA to 3.55 µA, and its slope is an ideality of 1.989. Taken at the collector, where the recombination current is supplied from the base and 1.604 Ω remains, it is 6.80 decades, from 5.01 pA to 31.6 µA, at an ideality of 1.0000. Nothing arrives beside the collector current, so its lower end on this axis is the axis.
Fig. 3 The same test at ±0.1%. As a diode the range is 1.25 decades, from 200 nA to 3.55 µA, at an ideality of 1.989. At the collector it is 6.80 decades, from 5.01 pA to 31.6 µA, at an ideality of 1.0000.

Ask for a tenth of a per cent and the collector’s range loses a decade at each end, and the two ends lose it for different reasons.

The top falls from 316 microamps to 31.6, exactly a decade for a decade of tolerance. The series resistance adds a drop of iR, which in units of the thermal voltage is an error in the current of iR/Vₜ, and that error is proportional to the current: 316 microamps through 1.604 ohms is half a millivolt, about two per cent of a thermal voltage, and 31.6 microamps is a tenth of that. A tolerance on the top of a logarithm’s range is a current times a resistance, and it moves as the first power.

The bottom rises from a picoamp to 5.01 picoamps, and there the collector current’s own law is responsible. The exponential in the law is really an exponential minus one, and the minus one is negligible only while the current is large against the saturation current, ten femtoamps here. At five picoamps the current is five hundred times the saturation current, and the minus one is a few tenths of a per cent of it. So the bottom of the range is the forward saturation current divided by the tolerance, give or take a small factor, and it is the saturation current two currents with one name found doubling every four and a half kelvin.

The diode loses the most, falling to 1.25 decades. A tighter tolerance leaves it less of the recombination region’s straight part on either side, and no stretch of its terminal current gets longer by asking for less error.

The base current is where the other current went

One junction voltage, two currents: an ideality of 1.95 at the terminals and 1.00 at the collector. computed by solving, not by drawing. The currents the model junction carries against its own voltage at 300 K, on a logarithmic axis. The diode current is both mechanisms together. The collector current of the same junction in a transistor is the diffusion current alone; the base current is the recombination current plus the collector current over a β of 50. Below 0.631 V, where the two mechanisms carry 400 µA each, the diode current is mostly the base's. At 10 µA the diode current's local ideality is 1.953 and the collector current's is 1.0000.
Fig. 4 The same junction with a current gain of 50 rather than 150. The base current carries a larger share of the collector’s; the diode current and the collector current are unchanged, with the mechanisms equal at 0.631 V and 400 µA each, and at 10 µA the diode current’s local ideality is 1.953 and the collector current’s 1.0000.

A current gain of fifty rather than a hundred and fifty changes the base current and nothing that matters to the logarithm. The collector current is still the diffusion current, with an ideality of 1.0000 at ten microamps, and the recombination current is still being supplied through the base whatever the gain is.

The gain does reach the logarithm in one place, through the series resistance. The base spreading resistance appears in series with the emitter junction divided by the current gain, so a lower gain puts more of it in the path and lowers the top of the range. Tested the same way, the collector at a gain of fifty has 3.612 ohms in series and holds its logarithm to one per cent from the axis to 141 microamps, 8.15 decades; at a gain of four hundred it has 0.976 ohms and reaches 501 microamps, 8.70 decades. So the gain moves the top of the range by a factor of 3.6 between those two values, close to the ratio of the two resistances, and moves nothing at the bottom. What it does not do is let any of the recombination current into the collector, which is the effect that separated 2.40 decades from 8.50.

A millivolt switches the leakage on

The collector test so far had nothing arriving at the collector except the current being measured. In a transdiode the collector is held at the base’s potential by an amplifier, and an amplifier holds a node at its own offset, not at zero.

At 300 K one junction holds a logarithm to ±1% over 2.4 decades as a diode and 5.1 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 5.10 decades, from 2 nA to 251 µA, at an ideality of 1.0013. The collector junction is held at 1 mV rather than zero and conducts 38.3 pA beside the current being measured, which sets the lower end.
Fig. 5 The same test at ±1% with the collector junction held at 1 mV rather than zero. It conducts 38.3 pA beside the current being measured, and the collector’s range is 5.10 decades, from 2 nA to 251 µA, at an ideality of 1.0013; the diode’s is unchanged at 2.40 decades.

One millivolt across the collector junction makes it conduct 38.3 picoamps beside the current being measured, and the bottom of the range moves from the axis’s picoamp to two nanoamps — fifty-two times the leakage, because a current has to be comfortably larger than an error before the error is within a per cent of it. The range falls from 8.50 decades to 5.10.

The size of that leakage has a short account. A junction at zero bias is a resistance, and two currents with one name found this junction’s to be 25.9 megohms once the generation current is in the model. A millivolt across 25.9 megohms is 38.6 picoamps, within a per cent of the leakage the figure finds. The bottom of a logarithmic converter’s range is set by the amplifier’s offset divided by the collector junction’s zero-bias resistance, and that resistance is set by the generation current, not by the forward saturation current.

The current the instrument draws is the same arithmetic from the amplifier’s side: an input current is nothing until it flows somewhere, and a converter’s input node is exactly where the photocurrent or the sensor current being logged arrives. The offset term and the bias current term both land at the bottom of the range, and both are specified on the amplifier’s data sheet rather than the transistor’s.

The same amplifier holds the other end of the connection as well. A photodiode feeding a logarithmic converter sits at the amplifier’s input at the same offset, and the junction that is a resistor at zero volts found that a photodiode held at zero volts has a dark noise equal to its own shunt resistance’s Johnson noise — 4.00 femtoamps per root hertz at a gigaohm. So the offset acts twice at the bottom of the range, across the collector junction’s zero-bias resistance and across the photodiode’s, and neither is set by the forward saturation current that a conformity test with a perfect amplifier finds at the bottom.

A warmer room takes the bottom

At 350 K one junction holds a logarithm to ±1% over 1.6 decades as a diode and 6.0 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 1.60 decades, from 794 nA to 31.6 µA, and its slope is an ideality of 1.937. Taken at the collector, where the recombination current is supplied from the base and 1.604 Ω remains, it is 5.95 decades, from 398 pA to 355 µA, at an ideality of 1.0000. Nothing arrives beside the collector current, so its lower end on this axis is the axis.
Fig. 6 The same test at 350 K with nothing arriving at the collector. As a diode the range is 1.60 decades, from 794 nA to 31.6 µA, at an ideality of 1.937. At the collector it is 5.95 decades, from 398 pA to 355 µA, at an ideality of 1.0000.

Fifty kelvin takes 2.55 decades from the collector’s range, and the loss is at the bottom. The forward saturation current has risen to 7.74 picoamps at 350 kelvin, so the minus one in the law now matters below a few hundred picoamps, and the range starts at 398. The top has risen slightly, from 316 microamps to 355, because the error the series resistance makes is its drop in units of the thermal voltage, and a warmer junction’s thermal voltage is larger — at 400 kelvin the held collector’s top has moved from 251 microamps to 562.

The collector's logarithm shrinks from 5.1 to 3.2 decades between 300 and 400 K, from the bottom. computed by solving, not by drawing. The widest range over which each connection of the model junction stays within ±1% of a logarithm, at five temperatures. The diode holds 2.40 decades at 300 K and 1.15 at 400 K. The collector with nothing arriving beside its current holds 8.50 and 3.85. With its collector junction held at 1 mV — the generation current of the same junction model, 38.3 pA at 300 K and 10 nA at 400 K — it holds 5.10 and 3.20, and every decade it loses is lost at the low end.
Fig. 7 The widest range within ±1% of a logarithm for each connection of the junction at five temperatures. The diode holds 2.40 decades at 300 K and 1.15 at 400 K. The collector with nothing beside its current holds 8.50 and 3.85. With the collector junction held at 1 mV — 38.3 pA of leakage at 300 K and 10 nA at 400 K — it holds 5.10 and 3.20, and every decade it loses is lost at the low end.

Across a hundred kelvin the picture is consistent. With a perfect amplifier the collector’s range shrinks from 8.50 decades to 3.85 as the forward saturation current rises into it; with a millivolt of offset it shrinks from 5.10 to 3.20 as the leakage, 38.3 picoamps at room temperature, rises to 10 nanoamps. In both cases the top of the range holds or improves, and every decade is lost at the bottom — which is where a converter is usually asked for its dynamic range, since the top is set by the largest signal and the bottom by the smallest.

The two currents with one name are therefore the two floors of a logarithm. The forward saturation current sets the bottom with an ideal amplifier, and the generation current sets it with a real one, and the second doubles every nine kelvin rather than every four and a half. At room temperature, with a millivolt of offset, the leakage floor is the higher of the two by a wide margin; at 400 kelvin the two are much closer, 3.85 decades of range against 3.20.

What the logarithm’s slope is doing meanwhile

Everything above is about the range of the logarithm — where the straight line holds. The slope of the line is a separate matter and this page does not correct for it. It is the ideality times the thermal voltage, and the thermal voltage is proportional to absolute temperature: 59.5 millivolts a decade at 300 kelvin at an ideality of one, a third more at 400. A converter whose range is intact at 350 kelvin is still reporting every decade as a different voltage than it did at room temperature.

The edges that move with the room found the small-signal boundary moving with the same thermal voltage, and the remedy there was a condition stated beside a number. A logarithmic converter needs the condition built into the circuit instead, usually by dividing by a quantity that is itself proportional to absolute temperature, and whether that division is accurate over the range drawn is a question this model could answer and has not.

What the model junction is

The transistor here is the two-mechanism diode junction with its recombination current routed to the base, a current gain of 150, a base resistance of 150 ohms and an emitter resistance of 0.6. It has no high-injection effects, so its current gain does not fall at large currents; no dependence of the collector current on the collector voltage beyond the leakage; and no temperature dependence of its resistances. A real transistor’s logarithm ends at the top for more reasons than 1.6 ohms, and a real converter’s range is bounded below by its package and board leakage as well as by its junction. The figures establish the mechanism and its order of magnitude on one set of parameters, and the numbers belong to those parameters.

Still open: the slope, the ratio, and a straight line

The slope that moves with the room. The range measured here survives warming better than the scale factor does. Solving a logarithmic converter with its temperature correction in the netlist, rather than assumed, would measure how well the correction holds across the range this page found — and whether the residual error is at the ends of the range, where the leakage and the resistance are, or spread through its middle.

The ratio of two logarithms. Two matched transistors carrying two currents produce a voltage difference that depends only on the ratio of the currents and the thermal voltage, with the saturation current cancelled. That cancellation is exact only for matched junctions at one temperature, and measuring how a mismatch in their saturation currents and a gradient between their temperatures break it would be the logarithm’s version of the copy and its two errors.

The straight line between a constant and a logarithm. The model a designer reaches for between a constant drop and an exponential is a drop plus a resistance. Its best fit to a pure exponential over any one decade is out by ±8.00 millivolts, whichever decade is chosen, where the best constant is out by ±29.76. On the real two-mechanism curve the straight line should be at its best in the top decade, where the series resistance straightens the curve, and the constant at its worst there — a pair of claims the one current a constant is right at made without measuring either.

Part 5 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:

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. The resistance a slow curve cannot see Part 4 — 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. Over one decade a constant is out by ±29.76 mV and a drop plus a resistance by ±8.002 mV. computed by solving, not by drawing. The forward drop of a pure exponential junction over the decade from 1e-3 to 1e-2 amperes, with the best constant drop and the best drop-plus-resistance drawn across it. Both are fitted minimax — the model whose WORST error over the window is smallest, which is what a design has to tolerate — rather than by least squares. The constant is 684.6 mV and is out by ±29.76 mV; the line is 656.2 mV plus 6.61 Ω and is out by ±8.002 mV, which is 3.72 times better. The best constant needs no search: it is the midpoint of the window's highest and lowest voltage, and its error is half their difference. The straight line between two models Part 6 — Between a constant seven-tenths of a volt and an exponential sits the model a designer actually reaches for: a drop plus a resistance. Fitted so that its worst error over a decade of current is as small as it can be, it is out by ±8.00 millivolts where the best constant is out by ±29.76 — and both numbers are the same over every decade, because a decade of a logarithm is the same shape wherever it is taken. On a real two-mechanism junction the line does best in the top decade, ±8.71 against a constant's ±58.53, because a series resistance is exactly the term the line has and the constant has none. And the resistance the fit returns is the part's plus 701 milliohms that is not there.

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 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. 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. Linearising an exponential at 125 °C, and what it costs. The linear model understates the gain by 1% at 9.69 mV and by 10% at 30.2 mV. The thermal voltage at this temperature is 34.3 mV, so "small compared with Vₜ" is not the criterion — 28% of Vₜ is already 1% wrong. The edges that move with the room Every boundary in this collection is quoted at one temperature and most of them are functions of it. The small-signal edge is proportional to the thermal voltage, so it runs from 5.67 millivolts at −40 degrees to 9.69 at +125 — a factor of 1.71, the ratio of the absolute temperatures exactly. A realised Q is 1.54 per cent high at one end of that range and 2.58 at the other. The numbers are right; the condition attached to them was left off, and it is the same condition every time.

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 resistanceIdeality factorLeakage currentMeasurement conditionModel rangeOperating pointSaturation currentThermal voltageTransfer curve