Circuit Response
every model has an edge, and the figure carries it
Series

Diode model — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. What a 0.7 V constant costs, in the quantity it is used to predict. computed by solving, not by drawing by Newton's method on the exponential at 94 supplies through four resistors. The model is exact at 5.748 mA — the current at which the true drop is 0.7 V — and every curve crosses zero there, at four different supplies. Below it the model is low and above it high, and how much depends on the headroom rather than on the diode. Through the 87 Ω curve the drop is 49 mV out at 0.725 V and 147 mV out at 150.7 V — a factor of 3.0 — while the error in the current falls from -66% to 0.10%, a factor of 674, because the headroom underneath it has grown by 2017. On that curve the model is inside one per cent only above 1.12 V.

    The one current a constant is right at

    Seven-tenths of a volt is the true forward drop at 5.748 milliamperes and at no other current, and every circuit built on it crosses zero error there — four different resistors at four different supplies, all exact at the same current. What decides whether the model is any good is not the diode at all; it is how much of the supply the diode is taking.

    part 1 · limits
  2. 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.

    part 2 · limits
  3. 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.

    part 3 · limits
  4. 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.

    part 4 · limits
  5. 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

    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.

    part 5 · limits
  6. 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

    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.

    part 6 · limits
  7. 6 straight lines over six decades of a diode: ±8.002 mV everywhere, and they meet at every join. computed by solving, not by drawing. A pure exponential junction from a nanoamp to a milliamp, modelled by 6 drop-plus-resistance segments each spanning a current ratio of 10.00, each fitted minimax on its own window. The model's error swings between +8.002 mV and −8.002 mV in every segment, which is the closed form for that ratio, because a span of a logarithm is the same shape wherever it is taken. The dots are the joins: at each one the two neighbouring segments are off by the same amount in the same direction, so they meet, and the chain is continuous although no segment was asked to be.

    How many straight lines a diode is

    One drop-plus-resistance line fitted minimax over a decade of a diode's current is out by ±8.00 millivolts, and it is the same over every decade. Chain such segments over six decades, a nanoamp to a milliamp, and the error of each is a closed form in its current ratio alone. Minimax segments meet at their joins without being asked to, so the chain is a continuous piecewise-linear model for free. The error falls as the inverse square of the segment count: one segment is out by 132 mV, six by 8.00, eighteen by 0.94. A millivolt takes 18 segments and a tenth of one takes 56. A real junction needs more, 23 and 72, because recombination gives it an ideality near two over most of the range and each segment narrows by the square root of the local ideality.

    part 7 · limits
  8. An eighteen-segment diode table built at 300 K, carried to other temperatures four ways: the worst error over six decades of current. computed by solving, not by drawing: the two-mechanism junction's law moved with temperature, eighteen equal-ratio minimax segments from 100 nA to 100 mA fitted at 300 K (worst 1.886 mV). Unchanged, the table is out by 163.0 mV at 250 K and 265.5 mV at 400 K. One shift of every drop leaves 47.14 mV and 38.32 mV at 250 and 350 K. A shift per segment leaves 3.768 mV and 3.839 mV — and 14.67 mV at 400 K. Refitting every segment on the same breakpoints holds 1.573 mV to 2.379 mV at every temperature: the breakpoints never need to move.

    The table that has to be refitted

    Eighteen straight segments model a real diode from 100 nA to 100 mA to within 1.886 mV at 300 K. Carried to 350 K unchanged they are 149 mV out; moved by one common shift, 38.3 mV, because the drops at the bottom of the table fall 3.31 mV a kelvin and those at the top 1.54; moved by a shift each, 3.84 mV. That last correction holds within twice the table's own error over fifty kelvin either way and fails at 400 K, where the bottom of the table has dropped below the junction's own recombination current and is no longer an exponential at all. Refitted on the same breakpoints the table holds 1.57 to 2.38 mV at every temperature: the breakpoints never need to move, and the lines always do.

    part 8 · limits

All series