What a resistor in the emitter buys

Put a resistor in a transistor’s emitter and two things happen. The stage’s gain falls by a factor of $1 + g_m R_E$, and the amplitude it can handle before it distorts goes up.

That is described everywhere as a trade, usually with the implication that the two are the same size: give up a factor of six in gain, get a factor of six in headroom. The exchange rate is worth measuring, because it is not one to one and it is not even the same for the two things “headroom” means.

Two edges, not one #

This field’s foundation essay established that “the linear range” is two numbers rather than one, and that they are far apart.

For a bare exponential — a diode, or a transistor with its emitter at signal ground — the drive at which total harmonic distortion reaches one per cent is 1.03 mV, and the drive at which the gain has fallen one per cent short of its small-signal value is 7.30 mV. A factor of 7.06 between them, and the reason is an order: a second harmonic is first order in the drive, while a gain error is second order, so the harmonic arrives long before the compression does.

So “how much can this stage take” has two answers, and which one matters depends on what the stage is for. An amplifier in a measurement chain cares about the second; an amplifier in an audio chain cares about the first.

Degeneration moves both of them, and this essay is about the fact that it does not move them together.

The curve, and how it is computed #

An emitter resistor makes the transfer curve implicit. The collector current depends on the base-emitter voltage, and the base-emitter voltage depends on the collector current through the resistor:

$$i = \exp\!\left(\frac{v - (i - 1)(k)V_T}{V_T}\right)$$

in units of the bias current, with $k = g_m R_E$. There is no closed form for $i$, so it is solved by Newton at every point of the curve — the same way an operating point is solved everywhere else on this site, rather than assumed or linearised.

The curve that comes out is then sampled, transformed, and its harmonic amplitudes read off, exactly as the bare exponential’s are. Nothing in the measurement knows that a resistor has been added; the degeneration appears only as a different shape of curve.

That matters for the comparison, because it means the two devices are measured by one routine and any systematic error in it cancels out of the ratios below.

The measurement #

Read the third column against the first. Four, nine, thirty-five, a hundred and twelve — against degeneration factors of two, three, six and eleven, whose squares are four, nine, thirty-six and a hundred and twenty-one.

The distortion edge moves as the square of the degeneration factor. That is not the trade as it is usually described, and it is a considerably better bargain than one for one.

The reason is that the resistor does two things at once. It divides the drive that reaches the junction, by the factor $1 + g_m R_E$ — which alone would move the edge in proportion. And it linearises what the junction does with the drive, because the local feedback makes the collector current a function of the resistor rather than of the exponential. Those two are the same factor, applied in series, so the amplitude at which a given distortion appears goes as the square.

At eleven the agreement with the square law has started to loosen — 112 against 121, seven per cent short — because 116 mV of drive on a junction is no longer a small perturbation of anything and the expansion the law comes from is not in its asymptotic regime. That is the same lesson this collection has now recorded three times in different fields, and it is stated here rather than fitted away.

And the edge that does not follow the law #

The fifth column is the interesting one, because there is no clean law in it at all: 4.10, 5.21, 11.64, 25.68 against squares of 4, 9, 36, 121.

At a factor of two the two edges move together — 4.00 and 4.10 — and the distance between them is unchanged at about seven. From three upwards they part company, and by eleven the distortion edge has moved four times further than the gain edge has.

The consequence is in the last column. The two boundaries close up, from 7.06 times apart to 1.61 times apart, and what that means in practice is a reversal:

• A bare stage runs out of linearity first. It is distorting at a millivolt and its gain is still right at seven, so if the specification is on distortion, that is the binding one.
• A well-degenerated stage runs out of gain accuracy first, or nearly so. At eleven times the two edges are within a factor of 1.6, and the amplitude at which its gain is one per cent wrong is not much above the amplitude at which it distorts by one per cent.

Which changes what the stage should be specified on. A designer who degenerates heavily to get linearity, and then relies on the resulting gain being $-R_C/R_E$ to the same accuracy over the same amplitude range, has bought one of the two things they were expecting.

Three halves, where the gain error vanishes #

The gain column has one more thing in it, and it was found by the figure’s slider refusing to work rather than by looking for it.

Setting the degeneration factor to 1.5 puts the gain edge at 71.8 mV — further out than at a factor of two, three, or four and a half, all of which are larger degenerations. The edge is not monotone in the factor, which for a quantity that is supposed to be bought by degeneration is a strange thing.

The cause is a cancellation. A bare exponential’s incremental gain grows with the drive amplitude: average an exponential over a sinusoid and the mean slope is larger than the small-signal one, which is why the gain edge is an edge at all. Local feedback does the opposite — it compresses. The two have the same order in the drive, so at one particular degeneration they cancel, and the second-order term in the gain error is zero.

That factor is exactly three halves. Bisected on the solved curve it comes out at 1.500231 at five millivolts of drive, 1.500919 at ten, and 1.503619 at twenty — converging on 1.5 from above as the drive is reduced, which is the signature of a second-order cancellation with a fourth-order term left over.

Either side of it the gain error has opposite signs. Below three halves a degenerated stage’s gain rises with amplitude; above it, the gain compresses in the ordinary way. And at three halves the one per cent edge is decided by the fourth-order term, which is why it runs out to seventy millivolts.

It is not a design point — the distortion at three halves is still only 2.25 times better than the bare device’s, which is what the square law says — but it is a real feature of the transfer curve, it is the reason this figure’s slider starts where it does, and it is the kind of thing a bisection that assumes one crossing will report as a wrong number rather than as a discovery.

The gain, which is the price #

The price is stated exactly and there is nothing hidden in it: the stage’s transconductance becomes

$$G_m = \frac{g_m}{1 + g_m R_E}$$

so a factor of eleven of linearity-by-the-square costs a factor of eleven of gain, once. Two stages degenerated by three give a factor of nine of gain lost and a distortion edge nine times out on each of them, which is not the same as one stage degenerated by nine — and working out which arrangement is better for a given total gain is the design problem this essay sits underneath.

There is a second price and it is the reason degeneration is not free even when gain is cheap: the resistor’s own Johnson noise appears in series with the input, and the stage’s noise figure rises. The noise field’s optimum-source-resistance argument applies directly, and the two considerations point in opposite directions in exactly the way the impedance-scaling essay’s three do.

Why the two edges behave differently #

The square law for the distortion edge and the absence of one for the gain edge come from the same expansion, and separating them says what each edge is actually measuring.

Expanding the exponential about the operating point gives a series in the drive: a linear term, a quadratic one whose coefficient sets the second harmonic, a cubic one, and so on. Two per cent of second harmonic is a statement about the ratio of the second coefficient to the first.

Degeneration acts on that series in two ways. It divides the drive by $1 + g_m R_E$ before the junction sees it, which scales every term by the appropriate power of the factor. And it feeds the output back, which suppresses the distortion products by the loop gain — the standard result that feedback reduces distortion by one plus the loop gain, with the loop gain here being $g_m R_E$.

Two factors, one from each mechanism, and the harmonic ratio therefore improves as the square.

The gain error is a different quantity. It is the departure of the slope from its small-signal value, and the second of the two mechanisms does not help it: reducing a stage’s sensitivity to its device makes the gain more accurate at every amplitude, but the amplitude at which the slope has fallen one per cent is set by how far the curve has bent, and the bend is a property of the drive reaching the junction. Only the first factor applies.

That is the qualitative statement, and it predicts a linear rather than a quadratic movement of the gain edge. What is measured is neither: 4.10, 5.21, 11.64, 25.68 against factors of 2, 3, 6, 11. It is faster than linear and slower than quadratic, and the reason is that a degenerated stage’s gain error is a mixture of the junction’s own bend and the resistor’s growing share of the total — which have different amplitude dependences and no common expansion. Reporting the column and declining to fit a law to it is the honest treatment.

The other way to make a stage linear #

A differential pair achieves something similar by a completely different route, and putting the two beside each other is instructive.

The pair has no second harmonic at all — its transfer curve is odd, so the even harmonics cancel by symmetry rather than by being made small. Its leading distortion term is the third, which is second order in the drive, which is the order the gain error already had.

The result is that a pair’s two edges are close together from the start: 18.2 mV for distortion and 10.4 mV for gain error, with the gain giving way first, and only 1.75 times between them. Which is the same place heavy degeneration arrives at, by a different mechanism and without the gain being divided down.

So there are two ways to buy amplitude on this site’s own measurements: cancel the even harmonics by symmetry, or push both edges out by local feedback. The first is free in gain and costs a matched device; the second costs gain and needs no matching.

Where the same factor appears elsewhere #

$1 + g_m R_E$ is one of the most reused quantities in the subject, and it is worth collecting the places it turns up, because they are all the same feedback seen from different sides.

An emitter follower is the extreme case: the emitter resistor is the whole load, the factor is large, and the gain is $g_m R_E/(1 + g_m R_E)$ — which is 0.98 for the stage this field measures.

A degenerated stage’s output impedance is multiplied by nearly the same factor, which is why degeneration is used in current sources as often as in amplifiers.

A current mirror with emitter resistors has its matching improved by the same factor, because a mismatch in the two devices’ saturation currents is divided down exactly as the drive is.

In each case the resistor is trading gain for a property that is a function of a resistor rather than of a device, which is the general reason for using local feedback at all — and it is why the noise penalty is the honest objection to it rather than the gain.

What the degeneration is spent on #

A resistor in the emitter buys amplitude and sells gain, and the field spends the amplitude four ways. The order that stops helping is where the series in the degeneration factor stops converging usefully. Where the two exponents come from is where the two boundaries this page moves are derived rather than measured. The point the device is never at is the intercept the remaining odd harmonic sets. What a pair cancels, and what it only halves is the other way to move the same boundary, by symmetry rather than by feedback, and the two reverse the order of the amplitude edges. How small is small signal is the boundary before any of it, and A bias point is a solution, not a choice is where the operating point comes from.

What is checked #

The figure asserts a different pair of claims for each of its three devices, which is what makes them one figure rather than three.

For the bare exponential, that the distortion edge is less than a third of the gain edge, and that the gain edge is the one this site’s limits module already reports — so the field’s own machinery agrees with itself.

For the pair, that the gain gives way first and that the two edges are within a factor of three, which is the reversal stated as an inequality.

For the degenerated device, that the distortion edge moves as the square of the factor to within nine per cent; that from a factor of three upwards the two edges are closer together than the bare device’s; and that at a factor of two they are not — the last of which is an assertion that the clean story is false at one end of the range, which is why it is written down separately rather than averaged into the others.

Two exponents, and where they came from #

The 2.00 and the 0.950 are fitted here and explained above. Where the two exponents come from reverts the degenerated transfer curve — which has no closed form forwards and an exact one backwards — and gets all three results out of one expression: the distortion exponent is exactly two; the gain edge is proportional to $D^2$ over the root of the absolute value of three minus twice $D$, which is infinite at three halves and tends to a three-halves power; and the two edges are 7.07 apart on the bare device and 1.62 at a factor of eleven, against 7.06 and 1.61 measured here.

That singularity is what the awkward row of this essay’s gate is about. The gain error vanishing at a degeneration factor of exactly three halves reads, from inside this measurement, as a coincidence at one point of a slider; from the reversion it is a pole in a closed form, and the reason the two edges are not closer together at a factor of two is that the design is on the other side of it.

Which is the general form of what a closed form buys after a measurement has been made. It does not improve any number on this page — the fits are what they are — and it turns three separate observations into one object, tells which of them are exact and which are local, and names the value of the design parameter at which the story changes.

The design consequence of the two edges closing up is worth one more line, because it changes what a stage has to be specified on. An undegenerated stage is limited by its distortion and a well-degenerated one is limited by how accurately its gain is known — which moves the difficult quantity from the device to the two resistors setting the degeneration, and those are components with tolerances rather than curves with edges. That is a better place for a difficulty to be, which is the whole reason the technique is used — and it is worth saying that the improvement is a transfer of the problem rather than a removal of it, since a gain known to a per cent is a gain known to whatever two resistors are known to.