The fifth a resistor adds

The product that is not the third noticed that every distortion result in this field had been odd-order, and that the second-order product had been computed and thrown away. It then measured what degeneration does to that second-order product, and found it removed faster than the third. That left a natural next question, which the earlier measurements of a resistor in the emitter did not have reason to ask. What the fourth order says about the third and the order that stops helping took a truncated series and asked how far it could be trusted. In each case the concern was a low order’s description of a low order. Nobody has asked what the high orders are doing in the meantime.

The usual assumption is that all of them are falling. Local feedback linearises, a linear stage has no harmonics, and the stage is going smoothly from one state towards the other. That picture is true of the total, and it is true of the second order. It is not true of the third, which has a null in the wrong place, and from the fifth upward it is wrong in sign over a range of loop gains that includes ordinary designs. The size of the effect in ratio is very large, and that is rather less alarming than it sounds, since what is rising is starting from nearly nothing. What matters is that a resistor has changed the character of the spectrum, in a way the usual figure of merit cannot show.

Held at a fixed output #

The stage is the field’s own: an exponential junction with a resistance in series. With $x$ the collector current over its quiescent value, $u$ the drive in thermal voltages and $T$ the loop gain $g_mR_e$, its curve is implicit,

$$\ln x + T(x - 1) = u\cos\theta,$$

and it is solved by Newton at 512 points of a cycle and transformed. Every harmonic can be read down to about $10^{-12}$ of the fundamental, far below anything physical but useful for finding the shape of a law.

What is held fixed matters here. Raising $T$ at a fixed drive lowers the output and lowers every harmonic with it, and that says nothing about linearity. So the drive is bisected at each loop gain until the output fundamental is a stated fraction of the quiescent current, here a fifth. A designer comparing two stages does so at a stated output, and it is this fixed output that opens the possibility of a harmonic rising at all.

The implicit function also has a Taylor series that can be written down, and that gives a second route. Its coefficients in $u$ are $1/(1+T)$, $1/2(1+T)^3$ and $(1-2T)/6(1+T)^5$. At a small enough drive the solved harmonics match them to a few parts in a hundred at every loop gain tested, which ties the numerics to the algebra.

Three different behaviours #

Those coefficients already show the first departure from the simple picture.

The second order simply falls. From 4.97 per cent it reaches 2.5 at $T = 1$ and 0.462 at $T = 10$. At a fixed output its coefficient, relative to the fundamental, goes as $1/(1+T)$ and that is what is drawn. It has no structure.

The third has a null. The coefficient has a factor of $1 - 2T$, so at $T = \frac{1}{2}$ the third vanishes at small drive and its sign then reverses. What a resistor in the emitter buys drew the general trade, and at this amplitude the null is sharp enough to matter: 0.165 per cent open-loop, and well under a thousandth of that near $T = \frac{1}{2}$. The third then recovers to 0.0415 at $T = 1$ and falls more slowly from there. So a small loop gain is better for the third than any modestly larger one, up to well past $T = 10$. A designer who chooses a small emitter resistor for a nominal gain, at about this bias, may have stumbled on the best third-order figure the stage has at all. It is the one result in this group that a designer might use deliberately, and it is fragile in the expected way, since its depth depends on a thermal voltage and a resistor.

From the fifth upward, a third behaviour appears, and that is the subject. The fifth starts at 0.814 ppm with no feedback. It falls to 0.503 at $T = 0.2$, is 0.68 at $T = 1$, and reaches 1.16 ppm near $T = 6.7$, higher than where it began, before falling again at high loop gains. The sixth rises from 0.0135 to 0.0706 ppm. The seventh rises from 0.000192 to 0.00472, a factor of 24.6.

Where the new harmonics come from #

The mechanism is old and it is simple to state. A resistor in the emitter feeds back a voltage proportional to the current, and that current already contains a second harmonic. So the voltage across the junction contains one too, and the junction’s own square term mixes it with the fundamental. That second-order product of a second-order signal adds to the third, which is the origin of the $-2T$ in the coefficient: it has the opposite sign to the device’s own third, and so it cancels at a particular $T$. Then the new third mixes with the second to give a fifth, and so on upward. Each order’s added product comes from lower orders that the same resistor has carried back.

This is the known effect that an earlier generation of audio designers noticed when a little feedback was applied to a stage that had a smooth low-order spectrum. Feedback removes what is there and adds orders that were not, and it reduces the second, third and so forth only at the rate of loop gain while creating the higher ones at a different rate. Eventually a large enough loop gain reduces all of them. Before that happens, in a range that depends on the order, the addition wins.

The order in which things happen is clear from the figure and it is worth stating. The second never rises, since there is nothing below it to make it from. The third gets a contribution, so it has a null, but it never climbs back above the open-loop value. The fourth likewise. Only the fifth and above actually exceed their open-loop values. It takes two steps of mixing, a second mixing into a third mixing into a fifth, before an added product is large enough, relative to the very small original, to overtake it.

A ratio that grows with the order #

The largest value each harmonic reaches, over its open-loop value, is the number that this kind of statement rests on.

The rise grows steeply with the order: 1.42 at the fifth, 5.23 at the sixth, 24.6 at the seventh, 141 at the eighth and 960 at the ninth. The loop gain at which each peaks moves up slowly, from about 6.7 to 18. These are very ordinary amounts of degeneration: a 1 mA stage with 26 Ω of intrinsic emitter resistance has a loop gain of ten with a 260 Ω emitter resistor, which is a common choice for a gain of a few.

The size of these ratios needs to be read together with where they start. An exponential has a very fast-falling spectrum, the Bessel coefficients that the distortion a linear model cannot have measured, and with this modest drive the ninth harmonic starts at $2.6 \times 10^{-14}$ of the fundamental. A factor of 960 takes it to $2.5 \times 10^{-11}$. Nothing is going to read that. So the ninth harmonic’s rise is a mathematical fact about the spectrum and has no practical meaning at this level, and it would be wrong to lean on the headline ratio. What the ratios establish is the direction of the change, in a stage whose high orders were effectively absent: the spectrum’s shape changes from steep to shallow. At a large enough drive that starts to matter.

Whether it is worth noticing #

Drive is the variable that decides this. The fifth is of the fourth power of the drive, relative to the fundamental, so its absolute level rises very quickly.

At a tenth of the bias the fifth reaches 0.0709 ppm and it does not concern anyone. At six-tenths of the bias, a fairly hard-driven stage, it reaches 119 ppm at $T = 8.9$: twice its open-loop value, and at a level that a modest converter will see. The peak moves up with drive, from $T = 6.4$ to 8.9.

At every drive the total harmonic distortion is dominated by the second order. So it falls throughout, smoothly, and it reads 1.8 per cent at the point where the fifth is at its worst. A total-harmonic reading will therefore show this design getting steadily better. That is a correct reading of a number that weights all orders by their amplitude, and since the second is at least a hundred times the fifth in all of this, the fifth cannot possibly appear in it. Weighting the orders by their square over four, as has often been proposed in order to count high orders as worse, changes almost nothing here: at half the bias and a loop gain of one the plain total is 6.32 per cent and the weighted one 6.34, since a second order that large leaves no room for anything above it.

The thing that does read the fifth is anything looking at a narrow band. A converter’s spur-free figure reads a harmonic at its own frequency, and a radio’s adjacent-channel measurement reads products at their own frequencies. The word length that is not a threshold considers how such spurs are bounded. For a measurement of that kind, a stage is not getting steadily better as its loop gain rises from zero to ten. At the fifth it is getting worse.

Symmetry takes the mechanism away #

If the second order is the raw material, then a device with no second order should not show the effect, and there is a good one to test.

A differential pair’s output is a tanh, which is odd. It has no even orders at all, as symmetry and a resistor together measured, and no second-order current to be carried back through its resistors. It is the natural comparison. Its third falls from the start and has no null: 2.23 per cent with no feedback, 1.55 at $T = 0.5$, 1.19 at 1 and 0.417 at 5. Its fifth does have a null, since a third mixing with a third can feed a fifth with a sign that opposes the device’s own, and it shows a dip below 61 ppm near $T = 1$ and a recovery to 124 ppm at $T = 5$. But at no loop gain does it reach its open-loop 0.0602 per cent, and the same is true of its seventh. The single stage at the same output, by comparison, rises to 1.84 and 38 times its open-loop values.

That comparison needs care in one direction. The pair’s open-loop fifth is 20 times the single stage’s, so it has further to fall, and the pair’s absolute fifth is larger than the single stage’s at most loop gains on the figure. A pair’s virtue is in its low orders, and it is much less clean than a single stage at the fifth. So symmetry has not given an absolute improvement in the high orders. It has removed a particular mechanism, which leaves the pair’s high orders falling, and the single stage’s rising. Saying “a differential stage has better high-order behaviour under feedback” would be a confusion of those two things.

Why this was not visible before #

There are good reasons this has been out of view. The field’s existing measurements were of low orders by design, and the order that stops helping found that the series truncated at a low order is a good model of what it describes. Nothing in those results is wrong here. A truncation at the fifth order, used to predict the third, is doing well; it was never asked to predict the fifth itself.

A deeper reason is that total harmonic distortion is the customary measure, and it is a sum of squares dominated by the largest term. Any measure of that kind is a statement about the low orders. It is much the same point as the coefficient that is about one reading: a number chosen because it is convenient and well-behaved cannot report on what its definition has already discounted. A two-tone test does better for odd orders, as the third intercept shows, and it is still fundamentally a low-order reading.

What the model leaves out #

The device here is a pure exponential with a pure resistor. A real transistor has a base resistance and a finite current gain, which change the effective loop gain at high current and add a small nonlinearity of their own. Its collector-base capacitance varies with voltage, which adds a frequency-dependent distortion that has nothing to do with the emitter resistor. Thermal effects at low frequency also modulate the bias and give a different kind of third order, on a bias that a bias point is a solution showed is itself an answer to an equation. All of those will sit on top of the structure here, and at the very low levels of the high orders they are likely to be larger than what was computed, at least beyond the seventh. So the figures past the sixth should be read as a statement about the ideal device’s arithmetic.

The output is measured as the collector current, with an ideal load. A real load’s own nonlinearity is absent. So is the second mechanism of the same kind, which is global feedback around several stages: the same reasoning applies with a different amount of delay in the loop, and the delay changes the phases with which added products combine.

The numbers worth carrying #

With the output held fixed, local feedback sends the second order straight down: 4.97 per cent, then 2.5 at a loop gain of one, then 0.462 at ten. The third has a null at $T = \frac{1}{2}$, where the coefficient’s $1 - 2T$ vanishes. From the fifth upward every order rises above its open-loop value before falling: 1.42 times at the fifth, 24.6 at the seventh, 960 at the ninth, peaking at a loop gain of six to twenty. They are made by the second order, carried back through the resistor and mixed. They are very small at modest drive and reach 119 ppm at six-tenths of the bias, and a total-harmonic reading cannot see them. A differential pair, with no second order to carry back, has high orders that fall.

Still open: the second harmonic of the source, global loops, and an emitter resistor that is a capacitor at high frequency #

A source with a second harmonic of its own. A preceding stage feeds this one its own second harmonic, and the input’s phase decides whether it adds to or cancels the one the resistor is carrying back. A chain of stages may have much larger or much smaller high orders than one stage, depending on an interstage phase nobody designs for.

A loop around several stages. Global feedback adds a delay between what the output does and what the input hears. With delay in the loop the added products have phases that depend on frequency, and the rise here could be much worse at frequencies near where the loop is running out. Whether a loop of a stated gain raises or lowers the fifth at its crossover is a direct extension of this calculation.

An emitter resistor bypassed at high frequency. A capacitor across part of the emitter resistor gives a loop gain that falls with frequency. Harmonics of a low fundamental then see a smaller loop gain than the fundamental does, and the mixing changes with their frequency. That is a common way to set a stage’s gain and bandwidth, and it should move every peak here in a frequency-dependent way.