Where the two exponents come from

What a resistor in the emitter buys ended with three measurements and no explanation of any of them.

The amplitude at which distortion reaches one per cent moves as the square of the degeneration factor — 1.984 fitted, close enough to two to be suspicious. The amplitude at which the gain is one per cent short moves as the 0.950 power, which is close to nothing in particular. And at a degeneration factor of exactly 1.500 the gain error vanished to second order, which was found because the slider produced a gain edge that was not monotone.

Three numbers, all measured, none derived. This essay derives them, and the route is short because the curve has a property nobody uses.

The curve has no closed form forwards and an exact one backwards #

A bare exponential device gives $i = e^{v/V_T}$. Put a resistor $R_E$ in the emitter and the current depends on itself, because the resistor’s own drop subtracts from what reaches the junction:

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

which has no solution in elementary functions and is why this collection solves it by Newton at every point, the same way it solves an operating point.

Turn it round, though, and it is trivial. Writing $i$ in units of the bias current and $u = v/V_T$, and with $D = 1 + g_m R_E$ the factor by which the degeneration divides the gain:

$$u = \ln i + (D-1)(i-1).$$

That is exact, elementary, and one line. Everything below comes from it.

Reverting the series #

Expand about the bias point, $i = 1$. Differentiating the inverse:

$$\frac{du}{di} = \frac1i + (D-1), \qquad \frac{d^2u}{di^2} = -\frac1{i^2}, \qquad \frac{d^3u}{di^3} = \frac2{i^3}$$

which at $i = 1$ are $D$, $-1$ and $2$. Reverting a series with those three derivatives gives

$$i = 1 + \frac{u}{D} + \frac{u^2}{2D^3} + \frac{(3-2D)\,u^3}{6D^5} + \cdots$$

and the three coefficients are the whole content of the essay. The first is the small-signal gain divided by $D$, which is the definition of $D$ and is a check rather than a result. The second and third are not.

For a cosine drive of amplitude $\hat u$, collecting harmonics gives

$$\mathrm{HD}_2 = \frac{\hat u}{4D^2}, \qquad \mathrm{HD}_3 = \frac{|3-2D|\,\hat u^2}{24D^4}, \qquad \frac{\Delta A}{A} = \frac{|3-2D|\,\hat u^2}{8D^4}.$$

Measured against the sampled transfer curve at $\hat u = 0.02$ and $0.05$, across six degeneration factors, every one of those agrees to four or five significant figures. At $D = 6$ and $\hat u = 0.05$: $3.4723\times10^{-4}$ against $3.4722\times10^{-4}$ for the second harmonic, $7.2340\times10^{-7}$ against $7.2338\times10^{-7}$ for the third, and $2.1702\times10^{-6}$ against $2.1701\times10^{-6}$ for the gain error.

The first exponent: exactly two #

Setting $\mathrm{HD}_2$ to a target $t$ and solving for the amplitude:

$$\hat v_{\text{thd}} = 4 V_T\, t\, D^2$$

so the distortion edge is proportional to $D^2$ exactly, with no correction term at any $D$. The 1.984 the previous rung fitted was two, and the four thousandths were the fourth-order remainder showing at the amplitudes the bisection was working at.

That is a better bargain than the trade is usually described as. Degeneration is spoken of as one-for-one — give up a factor of gain, get a factor of linearity — and it is one-for-two. Dividing the gain by six moves the amplitude at which one per cent of distortion arrives by thirty-six times.

The reason is visible in the expression rather than in the algebra. The resistor does two things: it divides the drive that reaches the junction by $D$, and it linearises what the junction does with what it gets, also by $D$. Only the first of those is what “dividing the gain” means, and the second comes free.

The second exponent, which is not an exponent #

Setting the gain error to a target $g$:

$$\hat v_{\text{gain}} = 2\sqrt2\,\sqrt{g}\; V_T \; \frac{D^2}{\sqrt{|3-2D|}}$$

which is not a power of $D$ at all. It is $D^2$ divided by the root of a linear function of $D$, and the 0.950 the previous rung fitted was a local slope of that expression over the range it happened to be fitted on.

Checking that reading: between $D = 2$ and $D = 6$ the expression gives $4/\sqrt1 = 4$ and $36/\sqrt9 = 12$, a ratio of three over a $D$ ratio of three — a local exponent of exactly 1.000, and the measured edges 29.93 mV and 84.95 mV give 0.949. The fit was measuring a crossover.

Two things follow that a power law would have hidden.

Asymptotically the exponent is $3/2$. For large $D$, $|3-2D| \to 2D$ and the expression tends to $D^{2}/\sqrt{2D} \propto D^{3/2}$. So the gain edge does eventually settle into a power law, and it is a three-halves rather than a first power — evaluated between $D = 100$ and $D = 300$ the expression’s local exponent is 1.494.

And there is a pole at $D = 3/2$. The factor $|3-2D|$ is zero there, and the predicted gain edge is infinite.

The three halves, explained #

The previous rung found the null empirically: at $D = 1.500231$ at five millivolts of drive, $1.500919$ at ten and $1.503619$ at twenty, converging on 1.5 from above.

The expression says why. The gain error and the third harmonic share the coefficient $(3-2D)$, and it is zero at $D = 3/2$ exactly. Both quantities vanish there, to second order in the drive, and what is left is the fourth-order remainder — which is why the measured null sits slightly above 1.5 and converges on it as the drive falls.

The physical reading is a cancellation between two mechanisms that were always there. A bare exponential’s incremental gain rises with amplitude, because the curve is convex. Local feedback compresses gain with amplitude, because more current means more drop across the resistor. Both are second order in the drive, so at one particular amount of feedback they cancel, and $3/2$ is that amount.

The check is a strong one, because it predicts something the previous rung did not measure: if the mechanism is a shared coefficient then the third harmonic must vanish at $D = 3/2$ too. It does. At $D = 1.5$ and $\hat u = 0.05$ the third harmonic measures $9.5\times10^{-10}$ against $1.04\times10^{-4}$ on the bare device — five orders down, and the residue is the fourth-order term rather than a small non-zero coefficient.

The distance between the two edges #

Dividing one edge by the other, the $D^2$ cancels:

$$\frac{\hat v_{\text{gain}}}{\hat v_{\text{thd}}} = \frac{\sqrt{2g}}{2t\sqrt{|3-2D|}}$$

and with both targets at one per cent that is $7.0711/\sqrt{|3-2D|}$.

At $D = 1$ it is 7.071, and the previous rung measured 7.06 on the bare device.

At $D = 11$ it is $7.0711/\sqrt{19} =$ 1.622, and the previous rung measured 1.61.

Both numbers were measurements with nothing behind them, and both come out of one expression with the degeneration factor in a single place. The figure asserts the closed form against the bisected edges at every setting of the slider except $D = 3/2$, where the expression divides by zero and the assertion is replaced by the claim that the ratio exceeds twenty — the same statement made where it can be checked.

The design reading is what the previous rung stated and could not justify. The two boundaries close up as the degeneration rises, so a well-degenerated stage runs out of gain accuracy before it runs out of linearity — and now the crossing point is computable rather than observed. They are equal when $|3-2D| = 50$, which is $D = 26.5$: a stage whose gain has been divided by twenty-six and a half, with both edges at 726 millivolts — an amplitude twenty-eight thermal voltages out, where the expansion that produced the number has long since stopped applying. So the crossing is a statement about the trend rather than a design point, and the useful reading is the trend: the two edges close as $D$ rises and there is no degeneration factor at which the gain edge is the far one again.

Why reverting is worth the trouble #

The route this essay takes is unusual enough to be worth defending, because there is an obvious alternative that does not work.

The obvious route is to expand the forward relation. But the forward relation does not exist in closed form, so the expansion has to be built by implicit differentiation of $i = \exp((v-iR_E)/V_T)$, which gives each coefficient in terms of the ones before it and produces expressions that grow quickly and carry no structure. Two terms in and the $(3-2D)$ is nowhere visible.

The inverse route is different because the inverse is a sum of two elementary functions — a logarithm and a straight line — whose derivatives at the bias point are $D$, $-1$, $2$ and, if wanted, $-6$. The degeneration appears in exactly one of them, the first, and every later derivative is the bare exponential’s. So reverting the series puts $D$ in the denominators as powers and lets the numerators carry the device’s own shape, which is where $(3-2D)$ comes from: it is $3$ from the logarithm’s own third-order behaviour minus $2D$ from the linear term interacting with the second-order one.

That is the general lesson and it is not specific to transistors. A relation that is implicit one way round and explicit the other should be expanded the explicit way and reverted, and the structure of the answer survives the reversion. It is the same reason a resistance is easier to work with than a conductance in some networks and the other way round in others, applied to a series rather than to a matrix.

Where the closed form itself stops #

The expansion is a small-signal one, which means it is a model with a range like everything else here, and the figure is explicit about where it ends.

At small degeneration the edges are small — a millivolt for distortion on the bare device — and the series is deep inside its own region of validity: the predicted edge is 1.0341 mV against 1.0341 measured, which is agreement to five figures.

At $D = 11$ the distortion edge has moved to 116 millivolts, which is four and a half thermal voltages, and the third-order term the series was truncated after is no longer small. The prediction is 125.1 mV against 116.3 measured — seven per cent out, in the direction of optimism.

So the closed form’s own boundary is an amplitude, in units of $V_T$, exactly like the boundary it is describing. Below about two thermal voltages it is good to a per cent; above four it is out by several. That is the honest scope of it and it is worth stating plainly, because a closed form printed without one is the thing this collection exists to avoid.

The measured edges do not have that limitation, which is why the figure keeps bisecting them rather than evaluating the expression. The expression is there to say why the numbers are what they are; the bisection says what they are.

What the resistor is doing to the feedback, in the loop’s language #

There is a second way to see all of this that connects it to the feedback field, and it makes the $D^2$ obvious rather than algebraic.

Local degeneration is series feedback. The loop gain around the device is $g_m R_E = D - 1$, and the closed-loop transconductance is $g_m/D$, which is the usual desensitivity result. Feedback theory then says that a distortion generated inside the loop is reduced by the return difference, $D$ — one factor.

The second factor is not a feedback effect at all. It comes from the drive at the junction being smaller by $D$ for the same input, and the second harmonic of an exponential being first order in the drive. So the harmonic that would have been produced is itself $D$ times smaller before the feedback reduces it by another $D$.

That decomposition also explains why the gain error does not get the same treatment. Gain compression is second order in the drive, so dividing the drive by $D$ reduces it by $D^2$, and the return difference reduces it by another $D$ — three factors, and an edge that should move as $D^{3/2}$. It does, asymptotically. The $|3-2D|$ in the denominator is the cancellation term that dominates at small $D$ and dies away, leaving the feedback counting to take over.

So the two exponents are the same argument counted twice, once for a first-order quantity and once for a second-order one, and the only thing feedback theory does not supply is the sign that makes them cancel at three halves.

Where the two boundaries are met #

Two amplitude boundaries that swap order depending on the device is the finding this field’s amplitude essays all rest on. What a resistor in the emitter buys is the parameter that moves them and the distance between them. What a pair cancels, and what it only halves is where the order reverses, because the pair removes the even harmonic and leaves the gain error first. The order that stops helping and What the fourth order says about the third are the series behind the exponents, with its own radius. How small is small signal is the undegenerated case, and The distortion a linear model cannot have is the measurement both exponents are fitted to.

What is checked #

Three assertions, and each is the closed form against a route that shares no arithmetic with it.

That the second harmonic is $\hat u/4D^2$, to five parts in a thousand, against the harmonic content of a curve sampled and transformed. The prediction comes from reverting an inverse; the measurement comes from a discrete transform of a Newton-solved curve.

That the gain error is $|3-2D|\hat u^2/8D^4$, to two per cent, at every degeneration factor except the null itself — where it is zero and a relative comparison is meaningless.

And that the ratio of the two edges is $\sqrt{2g}/2t\sqrt{|3-2D|}$, to ten per cent, against the edges bisected on the sampled curve. That is the assertion that connects this essay to what a resistor in the emitter buys: two numbers measured there with nothing behind them — a distortion edge moving as the square of the degeneration factor and a gain edge as its 0.950 power, with the gain error vanishing at a factor of exactly three halves — coming out of one expression that was not available when they were found.

What a reversion buys, and what it costs #

The degenerated transfer curve has no closed form forwards and an exact one backwards, which is the fact this essay is built on, and it is worth saying what that kind of result is generally good for.

It gives exponents rather than values, and exponents are what transfer between designs. That the distortion exponent is exactly two rather than 2.00-as-measured means it will still be two on a different device at a different bias, and that the gain edge goes as $D^2/\sqrt{|3-2D|}$ explains a singularity at $D = 3/2$ that no amount of measuring would have identified as exact.

What it costs is a range, and the range is measured on the rungs above. What the fourth order says about the third carries the reversion one term further and finds the leading expression 6.89 per cent optimistic where it predicts 7.33, taking the residue to 0.362 per cent — so the closed forms on this page are themselves truncations with their own edges. And the order that stops helping finds where carrying more terms stops being worth it: the error of an expression truncated at order $m$ grows as the $m$-th power of the drive, so every added order buys a range ending sooner than the last, and above 8.9 thermal voltages the six-term expression is further from the device than the four-term one.

Which is the honest frame for everything here. A reversion turns a measurement into a structure, the structure is exact in a limit, and the limit’s edge is another measurement.

The singularity at $D = 3/2$ is the part of that structure most worth carrying, because it is the one thing on this page that a measurement could not have produced. A gain edge proportional to $D^2$ over the root of $|3-2D|$ is infinite at one value of the design parameter, which reads on a sweep as a point where the error happens to vanish and reads in the expression as a pole. The difference matters for a design: a value at which an error is small is a value with a tolerance around it, and a value at which an expression diverges is a value the design should not be near, because the next term decides what happens there and it is not in this expression.

That is the same caution what the fourth order says about the third supplies in the other variable, and the two together bound this essay’s results from both sides: in the drive, by the order at which the series was truncated, and in the degeneration, by the point at which the truncated expression stops being finite. Two bounds on one closed form, both measured, and between them the region in which the three exponents on this page are what a design should be sized by.