The mismatch that cancels itself

The error that is a distribution solved three hundred current mirrors built to one design, each with its own devices drawn from a normal distribution, and reported what a population does that a single solve cannot: the mean is the systematic error, and the spread about it is something the systematic calculation contains nowhere. At two per cent device mismatch the mean came out at 3.836 per cent against a systematic 3.937, and the spread at 1.955.

The sentence attached to that spread was that it is the saturation-current mismatch, arriving with nothing dividing it. It is, and it had never been shown to be. Three quantities were drawn on every one of those three hundred solves — the two saturation currents, the two current gains and the two emitter resistors — and the check that ran was that the total spread equals one of the three inputs. A total agreeing with one contribution is consistent with the other two being negligible and equally consistent with two of them cancelling, and neither had been looked at.

The three ways two transistors differ #

A mirror’s two devices are nominally identical and are three separate things in the model. Each has a saturation current, which is proportional to emitter area and carries the process’s own scatter; each has a current gain, which is set by base width and by recombination and is the loosest parameter a bipolar data sheet prints, quoted as a range rather than as a value; and each has whatever emitter resistor is under it, with a tolerance of its own.

Their nominal spreads are not comparable either, which is part of why the question is worth asking. The sizes this collection’s populations have been built with are two per cent on the saturation currents, ten on the current gains and a half on the resistors — area matching on one die being the tightest of the three and the current gain the loosest by a wide margin. A budget that added those three in quadrature at face value would put the current gain first.

Turning them on one at a time is a controlled experiment and costs nothing but the runs. Each point below is a full population — every pair a Newton solve of the netlist, with the Early conductance iterated to self-consistency against the current that sets it, exactly as a bias point is a solution requires. The draws are scaled from one set of standard normal numbers, so the populations at different mismatch sizes are the same population rescaled: the slope of each line is a measurement of an order rather than a fit through sampling noise, which is what makes a slope of 2.003 a statement about the circuit rather than about how many pairs were solved.

The answer is three orders of magnitude, and it is not a small coefficient. It is a different exponent. Doubling the spread of saturation currents doubles the copy error’s spread; doubling the spread of current gains quadruples it, from a starting point four thousand times lower.

Read the other way round, that is a statement about how much gain mismatch would be needed to matter. For the current gains to contribute as much spread as two per cent of area does, they would have to differ by rather more than a factor of two between the two devices — which is not a mismatch, it is two different transistors. Within any range a real pair can occupy, the term is absent.

There is no third line in that figure, and the reason is worth saying plainly rather than treating as an omission: with no emitter resistors in the circuit there is nothing for a resistor tolerance to be a tolerance of. The population that produced the 1.955 above was drawn with a half per cent resistor spread and no resistors, so that input never reached the output at all.

Why the exponent is two #

A population is a blunt instrument for a question about order, so the same statement is made on one pair.

The mechanism is in the expression rather than in the size of anything. A mirror’s reference branch carries the collector current plus both base currents, so the copy is

$$\frac{I_\mathrm{out}}{I_\mathrm{ref}} = \frac{1}{1 + 1/\beta_1 + 1/\beta_2}$$

and the two gains appear only through the sum of their reciprocals. Write them as $\beta(1 \pm d/2)$ and that sum is $(2/\beta)/(1 - d^2/4)$: the two first-order terms are equal and opposite and there is no linear term at all. What is left is $d^2/2\beta$, which the solve reproduces to within a twelfth at every mismatch drawn, and to within three per cent across most of the range.

Two things about that derivation are worth separating, because only one of them is the finding. That the base currents subtract is elementary and is the first thing any account of a mirror says. That the difference between them does not subtract is the part nothing says, and it is not visible in the usual form of the expression, which is written with one $\beta$ because both devices are assumed identical. Assuming them identical is exactly what removes the term the question is about, so the answer had to come from a model that keeps them apart — the schematic naming one device where the netlist has two is the general form of that.

That is a stationary point, not a small number. It is the same shape of argument as what a pair cancels, and what it only halves: a symmetry in the circuit removes a term rather than shrinking it, and what survives is the next one. The distinction matters because a small coefficient can be made large by a different operating point and a missing term cannot.

Checked against $\beta$ rather than against $d$, the same expression holds from a current gain of thirty to one of fifteen hundred, with the measured departure between 0.92 and 1.04 times $d^2/2\beta$ across that range. So it is not a coincidence of one bias point — the same standard one step computed twice sets for a transient, applied to an algebraic prediction and a Newton solve of the netlist it describes.

What that says about matching a mirror #

The practical statement is short and it contradicts a common instinct.

Matching a current mirror is a statement about emitter area and about nothing else. A pair specified on current gain has bought nothing measurable: at the two per cent that a good layout achieves on area, a matched gain is worth four decimal places of a per cent, and the same two per cent of area is worth two whole points. A device pair on one die, at one temperature, with its transistors interdigitated, is doing something entirely about the first quantity and incidentally about the second.

The layout techniques bear this out, which is the reassuring half. Common-centroid arrangements, dummy devices at the ends of a row, and unit transistors wired in parallel rather than one wide device are all methods of making two areas equal in the presence of a gradient across the wafer. None of them does anything for base width. The practice has been right for a long time; what has been missing is the measurement that says which of the two parameters it is buying, and therefore what a different technique would have to improve to be worth anything.

It also settles a smaller question about what to specify when buying parts. A dual transistor sold as matched quotes a base-emitter voltage difference, in millivolts, and that is a statement about the ratio of the two saturation currents: 26 millivolts of thermal voltage times the logarithm of the ratio, so a one-millivolt part is matched to about four per cent in area. Some also quote a gain match. On this measurement the first number is the whole of the specification and the second is decoration.

That does not make the current gain irrelevant to a mirror; it makes it irrelevant to the spread. The gain still sets the systematic error — the two base currents stolen from the reference are $2/\beta$ of it, which is 1.32 per cent low at a gain of 150 — and that error is the same in every mirror built, so it is a design error that can be calculated and repaired.

Two errors of the same name and opposite character: one is common to every part and can be designed out, the other differs part to part and cannot. What has just been measured is that the current gain belongs entirely to the first.

The cure un-cancels it #

The standard repair for the spread is a resistor in each emitter, and its arithmetic is well settled: the device mismatch is divided by $1 + g_m R$ while the resistors’ own tolerance is multiplied by $g_m R/(1 + g_m R)$, so the total falls, flattens, and stops at whatever the resistors were given.

That account has two terms in it. The measurement has three.

The third line is the finding. With no degeneration, the two current gains reach the output only through the sum of their reciprocals and cancel to first order. With a resistor under each emitter they no longer do, because the emitter current is the collector current times $1 + 1/\beta$, and that current is what develops a voltage across the resistor. Two different gains now drop two different voltages, the difference appears directly in the base-emitter voltage available to each device, and the transconductance turns it back into a current error. The first-order term that had cancelled is put back by the resistor.

Its size follows the same expression the other two do: $\sigma\, g_m R/\big((1 + g_m R)\beta\big)$, which the solve tracks to within an eighth over eighty times the degeneration. All three sensitivities in that figure lie on their own closed forms, which is what makes the reversal a mechanism rather than an anomaly.

The factor $g_m R/(1 + g_m R)$ in it is the same one that multiplies the resistors’ own tolerance, and that is not a coincidence: both terms are voltages developed across the emitter resistor rather than across the junction, so both are handed to the output by the same divider. What differs is the size of the voltage. A resistor tolerance moves the whole emitter drop by its own fraction; a gain mismatch moves it by the fraction $1/\beta$ of that. So the gain term is the resistor term divided by the current gain, which is exactly the ratio the measurement shows — 0.011653 against 1.8591 points at four hundred millivolts, a factor of 160 where $\beta$ is 150.

The same divider is the desensitivity a feedback loop buys, arriving in a circuit that contains no amplifier. It divides what the devices contribute and multiplies what the resistors contribute, and the third thing it does is put back a term that a symmetry had removed. Two of those three are in every account of degeneration and the third is in none.

What it does not say #

It does not say degeneration is a bad idea. The term it revives is still the smallest of the three by two orders: at a hundred millivolts of drop a two per cent gain mismatch costs 0.0102 points where the same two per cent of area costs 0.380 and two per cent of resistor tolerance costs 1.612. The cure has changed what kind of quantity the gain mismatch is, and it has not changed the budget.

It does not say the resistor is worse than the device. At a half per cent tolerance — which is what an ordinary thin-film resistor gives — the resistors’ contribution at a hundred millivolts is 0.403 points against the devices’ 0.380, and the two are comparable rather than one dominating. The crossing is at 25.5 millivolts for equal tolerances, which is one thermal voltage and contains neither the current nor any resistance, in the way the constants that decide collects.

And it does not extend past this circuit unexamined. The cancellation is a property of the reference node being shared — both base currents come out of one branch, so both reciprocals enter one sum. An arrangement that does not share it does not get the cancellation, and a cascoded mirror is exactly such an arrangement: its upper device’s base current is charged twice, once into the inner node and once back into the reference, and 43 per cent of the output resistance the arrangement should have is missing because of it.

What a single solve could not have said #

It is worth being clear about which of the two instruments used here did the work, because they answer different questions and the collection has both.

A population says how wide a distribution is. It is the right object for a specification, since a design is built many times and the pair nobody has is the one that fails, and it is what the error that is a distribution exists to produce. What it cannot do is attribute: three inputs go in and one number comes out, and the number is the same whether a contribution is small or cancelled.

A pair says what the mechanism is. Sweeping one difference on one pair over two decades and fitting an exponent is a question a distribution cannot be asked, and the answer — an exponent of 2.002 rather than a coefficient — is what turns “the current gains do not matter here” into “the current gains cannot matter until something breaks the symmetry”.

The two had better agree, and they do: 2.003 from the population and 2.002 from the pair, computed from the same solver on different inputs and with nothing shared but the netlist. That agreement is the check on both, and it is the reason the reversal under degeneration can be believed, because it shows up in both instruments and neither was written expecting it.

The population’s answer under degeneration is a slope of 1.001 where it was 2.003, and the pair’s is 1.024 where it was 2.002. Two instruments, one reversal, and it is the resistors that caused it. A figure that had drawn only the total spread would have shown a curve falling and flattening, exactly as expected, with the whole of this underneath it.

What it opens #

The first thing is a measurement that is now worth making elsewhere. Every place this collection computes a spread from a population, it draws several parameters at once, and a total that matches one input has not established that the others are absent. The refusal and what it was protecting makes the related point about a guard that never fires; this is the same idea about a check that never separates.

The second is that a stationary point is a thing to look for rather than to notice. Any quantity that enters a network only through a symmetric function of two nominally equal parameters has no first-order sensitivity to their difference, and the list of such places is longer than a mirror: a differential pair’s tail, a bridge’s two arms, two windings of a transformer. Where one exists, the matching requirement on that parameter is loose by a factor of the parameter itself.

The third is the warning that goes with it. A cure applied to one sensitivity is applied to the whole circuit, and it can move a term that was not being looked at — here it moved one from second order to first, and only a sweep that had all three sensitivities in it at once could see that. Every model has an edge is usually about a range in frequency or amplitude; this one is about a range in the design, which is the harder kind to notice, because nothing about the circuit has left its band.

The number worth carrying #

Two per cent of area is two points of copy error and two per cent of current gain is a ten-thousandth of one — until a resistor is added under each emitter, at which point the second becomes 0.0102 and rises with every millivolt of degeneration.

And the third number belongs beside them: 86, which is what the second becomes as a multiple of itself once four hundred millivolts of degeneration is in the circuit. A repair applied for one reason has moved a quantity nobody was watching by nearly two orders, and left it still small.

The habit is to ask, of any parameter a specification is written on, not merely how large its effect is but what order it enters at. The two questions have different answers and only the second is stable: a first-order term can be reduced by a factor and remains first order, while a term that cancels is absent until something in the circuit breaks the symmetry that cancelled it — and what breaks it is usually the repair for something else.