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

Distortion — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. An exponential driven 10.0 mV either side of its bias. computed by solving, not by drawing. A sinusoid in, and out comes a waveform whose peaks are taller than its troughs are deep. The second harmonic is 9.61% of the fundamental, measured by transforming 512 samples and predicted independently as I₂(0.387)/I₁(0.387) = 9.61%. The two routes agree to 5e-10 over the 5 harmonics that stand above the arithmetic's own floor, and share nothing but the amplitude.

    The distortion a linear model cannot have

    A small-signal model's output is a scaled copy of its input by construction, so it has no second harmonic and asking it for one is not a hard question but a meaningless one. Measured on the curve itself, an exponential produces one per cent of harmonic distortion at 1.03 mV of drive — seven times sooner than the 7.30 mV at which its gain is one per cent wrong.

    part 1 · semiconductors
  2. A pair's third-order intercept is 7.8 dB above anything it can produce. computed by solving, not by drawing. Two equal tones through a differential pair, transformed coherently so every product lands in a bin of its own. The fundamental rises with slope 1.000 and the third-order product with slope 3.000, both fitted over the decade marked, and the dashed extensions are the extrapolation a specification quotes. They meet at a drive of 4.00 thermal voltages and an output of 2.00 — against a largest output of 0.8108, which is 8/π² and is what two equal tones give through a limiter. The intercept is 7.84 dB above it, which is π²/4 exactly.

    The point the device is never at

    A device's linearity is specified by one number, and that number is a place on no curve. The third-order intercept is where two straight lines would cross if both went on being straight, and neither does. For a differential pair the crossing sits π²/4 — 7.84 decibels — above the largest output the device can produce at any drive whatever, and the arithmetic that says so contains no tail current and no temperature.

    part 2 · semiconductors
  3. Degeneration removes the second-order product 9 times less well than the third. computed by solving, not by drawing. Both intermodulation products of a degenerated stage at 5 mV a tone, against the degeneration factor. The second-order product falls as D⁻² — the straight reference is exactly that law, anchored at D = 1 — and the fitted exponent is -2.000. The third-order product falls faster, as D⁴/|3 − 2D|, which is one more power of D at large factors, and it collapses altogether at D = 1.5 where its coefficient changes sign. So the ratio between the two goes from 20.7 at D = 1 to 183 at D = 16: the more linear the stage is made, the more completely its distortion is the product a third-order account never measures.

    The product that is not the third

    Every distortion result in this field is odd-order, and every two-tone calculation behind those results computed the second-order product and threw it away. On a bare exponential it is the drive over twice the thermal voltage — 1.934 × 10⁻² of the fundamental at a millivolt, against 1.870 × 10⁻⁴ for the third-order product, a ratio of 4Vₜ/a and a hundred and three to one. A differential pair puts it at 6.2 × 10⁻¹⁶. And degeneration removes it as D⁻² where it removes the third order as D⁴/|3 − 2D|, so a stage linearised until its third-order product is negligible is a stage whose distortion is almost entirely the one nobody measured.

    part 3 · semiconductors
  4. An exponential stage at 20% output, with a loop gain of 1: second 2.5% from 4.97%, third 0.0415% from 0.165%, and fifth 0.68 ppm from 0.814 ppm. computed by solving, not by drawing: the harmonics of an exponential stage with emitter resistance, gm·Re = T, solved point by point on the implicit curve and transformed, at a fixed output fundamental of a fifth of the quiescent current, for T = 1 (dots a) against no feedback (dots b). 2: 2.5%, against 4.97%; 3: 0.0415%, against 0.165%; 4: 5.36 ppm, against 40.9 ppm; 5: 0.68 ppm, against 0.814 ppm; 6: 0.0202 ppm, against 0.0135 ppm; 7: 0.000245 ppm, against 0.000192 ppm; 8: 0.0000478 ppm, against 0.00000239 ppm; 9: 0.00000178 ppm, against 2.62e-8 ppm. The series gives the second and third at small drive: the implicit function's Taylor coefficients are 1/(1 + T), 1/2(1 + T)³ and (1 − 2T)/6(1 + T)⁵, and they agree. The third therefore has a null at T = ½, and past that it has the opposite sign, and feedback that reduces the low orders puts some of them back into the high ones.

    The fifth a resistor adds

    Local feedback's reputation is that it reduces distortion, and on the second order it does: 4.97 per cent falls to 2.5 at a loop gain of one and 0.46 at ten, with the output held fixed. The third has a null at a loop gain of exactly a half and comes back with its sign changed. And from the fifth upward, a harmonic rises above its open-loop value before it falls: the fifth 1.42 times, the sixth 5.23, the seventh 24.6, the ninth 960, peaking at a loop gain between six and twenty. The second order, carried back through the resistor, mixes with the signal and makes them. A total-harmonic reading shows only improvement throughout, because the second dominates it. A differential pair has no second order to carry back, and its fifth never passes its open-loop value.

    part 4 · semiconductors

All series