Circuit Response
every model has an edge, and the figure carries it
Devices, and the amplitude they stop being linear at

The length that is a voltage

The square law's band is closed from above by a parameter that is not a bias, a current or a temperature: it is the channel length, wearing a voltage's units. Swept, the band goes from a factor of 4.74 on a thirty-micron device to 1.021 at fifty nanometres — and the overdrive a stage actually biases itself to barely moves at all, so the two cross at 7.93 microns and every shorter device is biased outside the band. The band was also measured in the wrong quantity: the square law is exact in the current somewhere at every length, and its error in the transconductance is never below 42.5 per cent at fifty nanometres and reaches one per cent only above 6.502 microns.
20 min read 8 figures Every model has an edgeThe constants decideMeasured, not tabulated

Assumes: The exponent that is a square · A bias point is a solution, not a choice

The exponent that is a square measured a band. A field-effect device is an exponential below threshold and approximately a square above it, the two errors that bound the square law point in opposite directions, and the consequence is that the square law is not wrong on one side and right on the other — it is exact at one overdrive and inside one per cent over a band around that point. On a half-micron channel that band is 152.0 to 161.7 millivolts, a factor of 1.06.

That essay named the parameter that closes the band from above and then left it alone. Velocity saturation enters the device law as a division by 1+Vov/Vc1 + V_{ov}/V_c, and Vc=EcLV_c = E_c L is a critical field times a channel length: a distance, in volts. The critical field is a property of silicon and is about four megavolts a metre. The length is a property of a fabrication process.

So the band’s upper edge is not a bias, a current, a temperature or a supply. It is a dimension of the part, and the whole of what a circuit designer can do about it is choose a different part. Four sampled lengths were quoted; the axis itself was never swept, and until it is there is no way to say where the band goes, whether it closes, or whether the bias a real stage takes is inside it.

What the axis is

The device solved here is the one the rung below solved: a drain current that is the square of an interpolated overdrive, divided by the velocity-saturation term, with a softened threshold underneath that becomes the weak-inversion exponential. Nothing about it is new. What is new is that the velocity-saturation voltage is now the horizontal axis, read as the length it is.

Four megavolts a metre and a two-volt VcV_c is a half-micron channel. Twenty-four volts is six microns; a quarter of a volt is sixty nanometres. Those are three decades of manufacturing history on one axis, and the question this rung asks is what happens to the square law’s range along it.

The band closes over a stage's own bias at 7.93 micronscomputed by solving, not by drawing. The overdrive over which the square law is within 1.0 per cent, against the channel length that sets it — the velocity-saturation voltage is Ec·L, so the axis is a size and not a bias. The shaded region is the band; the curve through it is the overdrive at which the square law is exact, which exists at every length because the subthreshold and velocity-saturation errors have opposite signs. Both edges move: the lower one from 79.6 mV to 252.9 mV and the upper from 81.2 mV to 1200.0 mV, so the band is a factor of 1.021 at 0.050 µm and 4.7 at 30 µm. The fourth curve is the overdrive a common-source stage with a fixed gate voltage and a fixed source resistor solves to, which barely moves at all; it leaves the band at 7.929 microns and is outside it for every shorter device. What the square law would have said about that stage is the last two rows: 220 per cent too much current on the 0.050 µm device and 53 per cent too much efficiency, against 0.15 and 0.66 per cent at 30 µm.100m1100m110channel length (microns), which is Vc / Ecoverdrive VGS − Vth (volts)the bias leaves the band: 7.93 µmthe band, the overdrive it is exact at, and the bias a stage pickstolerance1.0%at 0.050 µm, from79.6 mV…to81.2 mV…a factor of1.0209×at 30 µm, from252.9 mV…to1200.0 mV…a factor of4.74×a stage biases itself at366.1 mV – 439.6 mV…which leaves the band at7.93 µmat that bias, 0.050 µm220% of current…and 30 µm0.15%solved, then checked — one band swept over a sizeoutside its own band below 7.93 µm
Fig. 1 The band against the size of the device it is a model of, from fifty nanometres to thirty microns. The shaded region is where the square law is within the tolerance on the slider; the curve inside it is the overdrive where it is exact. Both edges move — the lower one from 79.6 to 252.9 millivolts and the upper from 81.2 to 1200.0 — so the band is a factor of 1.021 at fifty nanometres and 4.74 at thirty microns. The fourth curve is the overdrive a common-source stage actually solves to, and it crosses out of the band at 7.93 microns.

The lower edge moves too, and it was not supposed to

The obvious expectation is that one edge is fixed and the other slides. The lower edge is subthreshold conduction, which is diffusion, whose slope is nVTln⁡10nV_T\ln 10 — a thermal voltage times an ideality factor near unity, and nothing in that expression knows the size of anything. The rung below said as much.

The sweep says otherwise. The lower edge goes from 79.6 millivolts on a fifty-nanometre channel to 252.9 on a thirty-micron one, a factor of 3.18, and it moves monotonically the whole way. The mechanism is that the tolerance is on a ratio rather than on a difference. The square law’s error below threshold is 1−Vov2/u2⋅(1+u/Vc)1 - V_{ov}^2/u^2 \cdot (1 + u/V_c) with uu the interpolated overdrive, and the velocity-saturation factor is in that expression at every overdrive, not only at large ones. It raises the square law’s estimate relative to the device, which partly cancels the subthreshold lift, which pushes the point where the residue reaches one per cent downward. A shorter channel therefore buys a slightly lower bottom edge while losing a great deal of top edge.

That is worth stating because it is the kind of thing four samples hide and a sweep does not. A quantity that was expected to be a constant of the temperature turns out to depend on a fabrication length, weakly, through a term nobody would look for it in.

The band never closes

The second reading is a negative one and it matters more than it looks.

The band does not vanish at some length. It cannot, because the two errors have opposite signs and therefore cross, and at the crossing the square law is exact — so there is always an interval around that point inside any tolerance. What collapses is the interval’s width, and it collapses smoothly, in proportion to the tolerance, with no threshold anywhere in it.

The square law is within 1% over a factor of 1.02 in overdrive. computed by solving, not by drawing. One field-effect device drawn against the two models it is between: the square law, which is zero below threshold and rises as the square of the overdrive, and the weak-inversion exponential, which rises at 77.4 mV per decade. The device is neither and approaches both. The two errors point opposite ways — the subthreshold current lifts it above the square law below, and velocity saturation holds it below above — so the square law is exact at 80.4 mV and the shaded band is where it is inside 1%: 79.7 mV to 81.2 mV, a factor of 1.02. The slider is the velocity-saturation voltage, which is the channel length times a critical field, so what it moves is the band's width and not its position.
Fig. 2 A fifty-nanometre channel, which is the short end of the sweep. The square law is exact at 80.4 millivolts of overdrive, where the drain carries 6.46 microamperes, and it is within one per cent over a factor of 1.02 — under a millivolt of gate voltage on either side of that single point. At ten per cent the same device gives a factor of 1.23.

A factor of 1.02 is not a range. It is a point with a rounding error attached, and a designer who reads “the square law is good to one per cent” and biases anywhere at all is outside it. Loosening the tolerance widens the interval and does not change the shape: at three per cent the same device gives 1.065, and at half a per cent, 1.010.

This is the same construction a band rather than an edge uses for a switch and the edge that is a region uses for an amplifier — a model bounded on two sides by different mechanisms, with the ratio between the sides as the durable number. The difference here is the axis the ratio is a function of, which is a length.

The bias does not move

Now the half of the sweep that could not have been guessed.

A stage’s overdrive is not chosen. It is solved: the gate voltage and the source resistor fix the current, the current fixes the overdrive through a law with no closed inverse, and the answer comes out of Newton’s method on the whole netlist, in the way a bias point is a solution, not a choice establishes for every operating point in this collection. Swept over the same three decades of channel length, with the gate voltage and both resistors held fixed, that solved overdrive moves from 439.6 millivolts to 366.1 — a factor of 1.20 against the band’s 4.6.

The reason the bias hardly moves is that the source resistor is doing the deciding. A gate held at a fixed voltage above a resistor to ground fixes the sum of the overdrive and the drop across that resistor, and the device is steep enough at these currents that almost all of the adjustment lands on the current rather than on the overdrive. Shortening the channel takes current away — 133.9 microamperes at thirty microns against 60.4 at fifty nanometres, at a residual of 1.2×10−151.2\times10^{-15} and 3.4×10−163.4\times10^{-16} amperes respectively — and the overdrive that corresponds moves by a fifth while the band it has to sit in moves by a factor of nearly five.

So the two curves cross, once, and the crossing is the number this rung exists to produce. At 7.93 microns the stage’s own bias leaves the one-per-cent band, and every shorter device is biased outside it. At two per cent the crossing is 4.26 microns and at half a per cent it is 13.9, which says the ordering is not an artefact of where the tolerance was set: on any device anyone has fabricated in forty years, the bias a stage picks is outside the band the square law is quoted with.

What that costs is in the same figure. At the shortest length drawn the square law would have said the stage carries 220 per cent more current than it does, and that its transconductance efficiency is 53 per cent higher than it is. At thirty microns the same two errors are 0.15 and 0.66 per cent. The expression did not become approximate; it became a different answer.

The band was measured in the wrong quantity

There is a sharper problem underneath all of that, and it is about which number the band is a band in.

The band above measures the square law against the device’s current. That is the natural thing to measure, because the current is what the expression computes. But almost nothing a designer does with the square law uses the current. What gets used is the derivative — the transconductance, and specifically the transconductance divided by the current, which is what a stage gives back for the current spent on it. And the error in a derivative is not the error in the value.

The square law is exact in the current at every length and never within 42% in the gain at 0.050 µm. computed by solving, not by drawing. Two errors of one expression, each minimised over every overdrive, against the channel length. The lower curve is the square law against the device's current: it passes through zero at one overdrive whatever the length, because the subthreshold and velocity-saturation errors have opposite signs, so it is below a thousandth everywhere. The upper curve is the same expression differentiated — 2/Vov against the device's own gₘ/ID — and it has no zero: the best it does on a 0.050 µm channel is 42.5 per cent, at 222.7 mV of overdrive, and ten per cent needs a channel longer than 0.406 microns while one per cent needs 6.502. A band quoted in the current says nothing about the gain.
Fig. 3 Two errors of one expression, each minimised over every overdrive, against the channel length. The lower curve is the square law against the current: it passes through zero somewhere at every length, so its best is below 5.4 × 10⁻⁴ over the whole sweep. The upper curve is 2/Vov against the device’s own transconductance efficiency, and it has no zero at all — 42.5 per cent at fifty nanometres, and one per cent only above 6.502 microns.

The two do not merely differ in size. They differ in kind. The current’s error crosses zero, so a band exists around the crossing at every length whatever. The efficiency’s error has an interior minimum that never reaches zero, because the two mechanisms that bound it do not cancel in the derivative — subthreshold conduction lowers the efficiency below 2/Vov2/V_{ov} from one side and velocity saturation lowers it from the other, so both errors have the same sign and there is nothing to cross.

That the two errors share a sign in the derivative is not obvious from the fact that they oppose each other in the value, and it is the whole of why the second band does not exist. Subthreshold conduction adds current without adding the transconductance a square law would need to go with it, and velocity saturation removes transconductance without removing the corresponding current. Both therefore push the measured efficiency below 2/Vov2/V_{ov}, from opposite ends of the bias range, and a sum of two same-signed terms has no zero to put a band around.

The consequence is a single sentence: the square law is within one per cent of the device’s transconductance at no bias whatever unless the channel is longer than 6.502 microns. Ten per cent needs a channel longer than 0.406 microns. On the half-micron device the rung below took as its worked example, the closest 2/Vov2/V_{ov} ever comes is 8.5 per cent, at 305.1 millivolts of overdrive — which is not inside the current’s band either, so the two quantities are not even wrong in the same place.

A model quoted with a range in one quantity and used in another is the failure every model has an edge is about, arriving from an unusual direction. Nothing here is a bad model or a bad measurement. The measurement is of the thing the model computes, and the thing the model is used for is something else.

The axis a designer actually holds

The other consequence of the same point is that the transconductance efficiency belongs on a current axis. A stage is allotted a current, not an overdrive, and the question is what that current buys.

On that axis the square law is a single straight line. Vov=2ID/βV_{ov} = \sqrt{2I_D/\beta}, so gm/ID=2β/IDg_m/I_D = \sqrt{2\beta/I_D}: slope exactly −12-\tfrac12 on logarithmic axes, from a nanoampere to an ampere, with neither end in it.

On a current axis the square law is one line of slope −½ and the device leaves it at both ends. computed by solving, not by drawing. Transconductance divided by drain current, against drain current, for one 0.500 µm device — the axis a designer works on, because a stage is given a current rather than an overdrive. The square law is √(2β/ID) here, a single straight line of slope −½, and it is above the device at every current on the plot. Below, the subthreshold exponential flattens the efficiency at 1/nVₜ = 29.76 per volt against a bipolar's 38.68. Above, velocity saturation steepens the slope from -0.54 to -0.98, because the current becomes proportional to the overdrive while the transconductance stops rising. The two upper regimes cross where the overdrive equals Vc, which is 2 mA and is a current named by a length. The closest 2/Vov ever comes to the device is 8.5 per cent, at 305.1 mV of overdrive.
Fig. 4 A half-micron device on the axis a design is done on. The straight line is √(2β/ID), above the measurement at every current on the plot; the measured slope is −0.54 where the square law comes nearest and −0.98 at eight times Vc. The efficiency stops at 29.76 per volt against a bipolar transistor’s 38.68, and the two upper regimes cross at 2.00 mA — a current named by a length.

The device leaves that line at both ends and for unrelated reasons. Below, the subthreshold exponential takes over and the efficiency flattens at 1/nVT1/nV_T — 29.76 per volt, against the 38.68 a bipolar transistor has at every current whatever. Above, the carriers stop accelerating: the current becomes proportional to the overdrive rather than to its square while the transconductance stops rising at all, so the efficiency falls as 1/ID1/I_D and the slope steepens from −12-\tfrac12 to −1-1. Measured on the curve rather than named, it is −0.98 at eight times VcV_c on every length swept.

Between those two limits there is an exact statement, and it is the one that makes the length a current. In strong inversion the efficiency is (2+Vov/Vc)/(Vov(1+Vov/Vc))(2 + V_{ov}/V_c)/(V_{ov}(1 + V_{ov}/V_c)), so at an overdrive equal to VcV_c it is exactly 3/(2Vov)3/(2V_{ov}) — three quarters of what the square law promises, at any width, any temperature and any process. The figure asserts that number to a part in a thousand, and the current it happens at is βVc2/4\beta V_c^2/4 exactly: 2.00 milliamperes for the half-micron device, 29.1 microamperes for a sixty-nanometre one, and five amperes for a twenty-five-micron one, which is a current no such device is ever at. A channel length names a current density with no bias anywhere in the expression.

On a current axis the square law is one line of slope −½ and the device leaves it at both ends. computed by solving, not by drawing. Transconductance divided by drain current, against drain current, for one 0.0600 µm device — the axis a designer works on, because a stage is given a current rather than an overdrive. The square law is √(2β/ID) here, a single straight line of slope −½, and it is above the device at every current on the plot. Below, the subthreshold exponential flattens the efficiency at 1/nVₜ = 29.76 per volt against a bipolar's 38.68. Above, velocity saturation steepens the slope from -0.69 to -0.98, because the current becomes proportional to the overdrive while the transconductance stops rising. The two upper regimes cross where the overdrive equals Vc, which is 29.1 µA and is a current named by a length. The closest 2/Vov ever comes to the device is 38.2 per cent, at 224.9 mV of overdrive.
Fig. 5 Sixty nanometres, the same axes. The measured slope where the square law comes nearest is already −0.69 rather than −½, and the closest 2/Vov ever comes to this device at any current at all is 38.2 per cent, at 224.9 millivolts of overdrive. There is no stretch of this curve on which the square law is a description of the gain.

Between those two lengths the curve does not jump; it slides. A quarter of a micron is the length at which a designer stops thinking of a device as long and starts thinking of it as short, and it is worth seeing that nothing happens there — the slope, the knee and the best agreement all move smoothly through it, which is why no threshold appears in any of the numbers above.

On a current axis the square law is one line of slope −½ and the device leaves it at both ends. computed by solving, not by drawing. Transconductance divided by drain current, against drain current, for one 0.250 µm device — the axis a designer works on, because a stage is given a current rather than an overdrive. The square law is √(2β/ID) here, a single straight line of slope −½, and it is above the device at every current on the plot. Below, the subthreshold exponential flattens the efficiency at 1/nVₜ = 29.76 per volt against a bipolar's 38.68. Above, velocity saturation steepens the slope from -0.57 to -0.98, because the current becomes proportional to the overdrive while the transconductance stops rising. The two upper regimes cross where the overdrive equals Vc, which is 500 µA and is a current named by a length. The closest 2/Vov ever comes to the device is 14.5 per cent, at 270.9 mV of overdrive.
Fig. 6 A quarter of a micron, between the two. The slope is −0.57, the knee has moved out to 500 microamperes, and the best the square law does is 14.5 per cent. The progression across the three lengths is 38.2, 14.5 and 8.5 per cent, and the axis it is a function of is a dimension.

What it does not say

It does not say the square law is useless, and the reason is in the same three figures.

The ordering survives everywhere: more overdrive is more current and more transconductance, at every length, and no amount of velocity saturation reverses it. The efficiency ceiling survives and is exact — 1/nVT1/nV_T, the bipolar’s own number divided by the ideality factor, with no dimension in it. And the shape of the trade survives: transconductance bought with overdrive, paid for in drive headroom, which is what what a resistor in the emitter buys is about for the other device family and is why designers still think in overdrive.

What does not survive is any number obtained by squaring, and any number obtained by differentiating the square. The first is what the band measures; the second is what the band was being read as.

The square law is within 1% over a factor of 1.41 in overdrive. computed by solving, not by drawing. One field-effect device drawn against the two models it is between: the square law, which is zero below threshold and rises as the square of the overdrive, and the weak-inversion exponential, which rises at 77.4 mV per decade. The device is neither and approaches both. The two errors point opposite ways — the subthreshold current lifts it above the square law below, and velocity saturation holds it below above — so the square law is exact at 259.3 mV and the shaded band is where it is inside 1%: 227.6 mV to 322.1 mV, a factor of 1.41. The slider is the velocity-saturation voltage, which is the channel length times a critical field, so what it moves is the band's width and not its position.
Fig. 7 Six microns, which is a device of about 1980 and is the length at which the transconductance error first falls under about one per cent. The current band is a factor of 1.41 and the square law is exact at 259.3 millivolts. This is roughly where the expression was a description rather than a mnemonic — and the crossing that matters is not this figure’s band but the other one’s.

There is also a limit on the device model itself. It carries one velocity-saturation term and one subthreshold interpolation, and a real short-channel part has several more effects that are also functions of the length — a threshold that falls as the drain voltage rises, an output resistance that stops being a constant, a mobility that degrades with the vertical field. Each of those would move the numbers here and none of them would move the argument, which is about what happens to a closed form when the quantity that bounds it is a dimension. The same caution how small is small signal states for the exponential applies: the boundary is computed from the model’s own parameters, and the model is a model.

An edge that is a length

This collection keeps a list of boundaries that are distances rather than frequencies or amplitudes, and the edges that are lengths is where it keeps them. Most of them are about a circuit being physically large — a board long enough that Kirchhoff’s laws lose a degree, a lead long enough to resonate.

Where four of this site's models stop being true. In order: the ideal operational amplifier at 1.42 kHz, a 10 V output at full amplitude at 7.96 kHz, Kirchhoff's laws on 10.0 cm at 3.97 MHz, the ideal 100 nF capacitor at 4.69 MHz. The fifth boundary is an amplitude rather than a frequency and cannot share this axis: a small-signal model is 1% wrong above 7.3 mV, at every frequency there is.
Fig. 8 Four of this collection’s boundaries on one frequency axis, with a fifth that is an amplitude and cannot share it. The one this essay measures cannot share it either, and for a third reason: it is a length inside a device, and it moves the model’s range without moving any frequency at all.

This one is a different member of that class and the difference is worth naming. The lengths in that essay are the circuit’s — chosen by whoever laid the board out, measurable with a ruler, and changeable by moving a part. This length is inside the transistor. It is chosen by the person who sold the wafer, it appears in no schematic, and the only way a designer meets it is that an expression stops working.

Which makes it the most durable kind of edge in the collection. A frequency limit can be pushed by spending current; an amplitude limit can be pushed by spending supply voltage. A limit set by a fabrication length cannot be pushed at all — it can only be measured, and then designed around by not using the expression. The same is true of the ideality factor in the efficiency ceiling and of the thermal voltage in the subthreshold slope, which is what the one current a constant is right at finds for the constant-drop diode: a model’s accuracy set by something outside the design rather than by the device.

Two routes, and what they share

Every number above comes from one device law, which is one route, and it is worth being explicit about what that means for the claims.

The band, the crossing at 7.93 microns and the two error curves all read the same device law and differ only in what they ask it. That is not two routes and is not offered as two. What is independent is the bias: the overdrive drawn through the band comes from Newton’s method on a four-element netlist, solved by the same machinery that answers every other operating point here, and checked afterwards by rebuilding Kirchhoff’s current law from the device’s own expression rather than from anything the iteration produced. So the curve that crosses the band was not evaluated from the band’s arithmetic, and the crossing is a meeting of two computations rather than a rearrangement of one.

The identities are the stronger check. Three quarters at Vov=VcV_{ov} = V_c, and βVc2/4\beta V_c^2/4 for the current there, are closed forms with no scan and no solver in them, and the figure asserts the measurement against both. A device model that had drifted — a wrong interpolation, a wrong derivative — would still produce a plausible band and would fail those two immediately. That is the same move the straight lines, and where they are not the curve makes for an asymptotic sketch: find the place the approximation is exact by construction, and quote every inexact result against agreement there.

The number worth carrying

A factor of 4.74 at thirty microns and 1.021 at fifty nanometres, with the bias a stage picks crossing out of the band at 7.93 microns — and, underneath that, a transconductance error that never reaches one per cent below 6.502 microns and never reaches zero at any length at all.

The habit that goes with it is about which quantity a range belongs to. A closed form is validated against the thing it computes, because that is the comparison available, and it is then used for something derived from that thing — a slope, a ratio, a difference of two nearly equal values. The range does not transfer. It has to be measured again in the quantity that will actually be used, and when the two mechanisms that bound the value happen to cancel there, they will not cancel in its derivative, because a cancellation is a coincidence between two functions and not between their slopes.

The other half of the habit is about parameters that are dimensions. A model whose validity is bounded by a current, a frequency or an amplitude has a boundary a designer can move. A model bounded by a length inside the part has one that arrives already decided, and the only useful response is to find out which side of it the part is on. On this device law, for every channel anyone now fabricates, the answer is the wrong side — which is why the copy, and its two errors and the distortion a linear model cannot have are both written about a device whose law is an exponential: an exponent that is a constant of nature is a model that does not have this problem.

Part 2 on square law

One argument about Square law, and one of 2 essays on it so far, each part numbered by how much of the idea it assumes. What sits either side of it:

The square law is within 1% over a factor of 1.06 in overdrive. computed by solving, not by drawing. One field-effect device drawn against the two models it is between: the square law, which is zero below threshold and rises as the square of the overdrive, and the weak-inversion exponential, which rises at 77.4 mV per decade. The device is neither and approaches both. The two errors point opposite ways — the subthreshold current lifts it above the square law below, and velocity saturation holds it below above — so the square law is exact at 156.5 mV and the shaded band is where it is inside 1%: 152.0 mV to 161.7 mV, a factor of 1.06. The slider is the velocity-saturation voltage, which is the channel length times a critical field, so what it moves is the band's width and not its position. The exponent that is a square Part 1 — Every device in this collection so far has been an exponential, and the whole of its arithmetic — 59.5 millivolts a decade, a distortion edge at 1.03 millivolts, 3,333 parts per million a kelvin — comes out of that one law. A field-effect device obeys a different one, and the interesting part is that it obeys both: an exponential below threshold and a square above it. The square law is within one per cent over a factor of 1.06 in overdrive at half a micron, and the two errors that bound it point opposite ways.

Where this goes next

The square law is within 1% over a factor of 1.06 in overdrive. computed by solving, not by drawing. One field-effect device drawn against the two models it is between: the square law, which is zero below threshold and rises as the square of the overdrive, and the weak-inversion exponential, which rises at 77.4 mV per decade. The device is neither and approaches both. The two errors point opposite ways — the subthreshold current lifts it above the square law below, and velocity saturation holds it below above — so the square law is exact at 156.5 mV and the shaded band is where it is inside 1%: 152.0 mV to 161.7 mV, a factor of 1.06. The slider is the velocity-saturation voltage, which is the channel length times a critical field, so what it moves is the band's width and not its position. The exponent that is a square Every device in this collection so far has been an exponential, and the whole of its arithmetic — 59.5 millivolts a decade, a distortion edge at 1.03 millivolts, 3,333 parts per million a kelvin — comes out of that one law. A field-effect device obeys a different one, and the interesting part is that it obeys both: an exponential below threshold and a square above it. The square law is within one per cent over a factor of 1.06 in overdrive at half a micron, and the two errors that bound it point opposite ways. A diode fed from 5 V through 1.0 kΩ. computed by solving, not by drawing. The operating point is where the exponential meets the load line: 0.692544 V and 4.3075 mA, reached in 13 damped Newton steps from a cold start. The one-line Newton on Vs = v + R·i(v), which touches no matrix, gives 0.692544 V. The "drop" is not a constant: it moves 59.53 mV per decade of current, measured between two solved operating points. A bias point is a solution, not a choice The phrase "the diode drops 0.7 volts" is a constant standing in for the root of a transcendental equation. Solved properly, from a five-volt supply through a kilohm, it drops 0.692544 V — and from forty-eight volts through the same kilohm it drops 0.754459 V, because the drop moves about sixty millivolts for every decade of current through it. Where four of this site's models stop being true. In order: the ideal operational amplifier at 1.42 kHz, a 10 V output at full amplitude at 7.96 kHz, Kirchhoff's laws on 10.0 cm at 3.97 MHz, the ideal 100 nF capacitor at 4.69 MHz. The fifth boundary is an amplitude rather than a frequency and cannot share this axis: a small-signal model is 1% wrong above 7.3 mV, at every frequency there is. The edges that are lengths Almost every boundary in this collection is a frequency or an amplitude, and both of those are things a circuit designer chooses. A handful are lengths — the 0.60 millimetres a gap's field reaches into a window, the 200 microns between a track and its plane, the 10 centimetres at which Kirchhoff's laws are a degree out — and they behave differently in one way that matters: nobody chooses them at the schematic, they are set by whoever builds the thing, and they appear in no netlist at all. Where four of this site's models stop being true. In order: the ideal operational amplifier at 1.42 kHz, a 10 V output at full amplitude at 7.96 kHz, Kirchhoff's laws on 10.0 cm at 3.97 MHz, the ideal 100 nF capacitor at 4.69 MHz. The fifth boundary is an amplitude rather than a frequency and cannot share this axis: a small-signal model is 1% wrong above 7.3 mV, at every frequency there is. Every model has an edge Four assumptions this collection runs on, with the frequency at which each stops being true, on one axis. The ordering is not the one most readers would guess — an ordinary amplifier circuit runs out of model at 1.42 kHz, three thousand times sooner than a ten-centimetre circuit board does. Linearising an exponential at 27 °C, and what it costs. The linear model understates the gain by 1% at 7.30 mV and by 10% at 22.8 mV. The thermal voltage at this temperature is 25.9 mV, so "small compared with Vₜ" is not the criterion — 28% of Vₜ is already 1% wrong. How small is small signal Linearising an exponential replaces a curve by its tangent, which is exact at a point and progressively wrong away from it. The amplitude at which it is one per cent wrong is 7.3 millivolts at room temperature — 28 per cent of the thermal voltage, not a small fraction of it, and a good deal smaller than "small signal" suggests.

The objects named here

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

Channel lengthDesign tradeoffModel rangeOverdrive voltageSquare lawSubthreshold conductionTransconductanceTransconductance efficiencyVelocity saturation