The error that is a distribution

The rung below this one solved a two-transistor mirror with the Early conductance iterated to self-consistency and separated the error that falls with beta from the error that does not. Both come out of a solve, both are properties of the design, and both are what a data sheet’s equations are about.

Neither is what limits a real mirror.

The whole arrangement assumes the two transistors are the same transistor, and they are not. A fractional difference in saturation current is a fractional difference in collector current with nothing to divide it, because the two devices share a base-emitter voltage and the exponential does the rest. Nobody chose that difference and nobody knows it. It is a distribution, and no amount of solving produces one.

Why the exponential does not divide it #

The reason the mismatch passes through undivided is worth spelling out, because the same circuit divides other things very effectively and the difference is not obvious.

Two transistors with their bases and emitters tied together have the same base-emitter voltage by construction. Each one’s collector current is $I_S e^{V_{BE}/V_T}$, so the ratio of the two currents is the ratio of the two saturation currents, full stop: the exponential is the same function evaluated at the same argument, and it cancels exactly. There is no feedback anywhere in the arrangement to notice that one device is carrying more current than the other, because nothing in the circuit measures the difference.

Compare that with what the same pair does to a base-emitter voltage error. An offset of ΔV between the two junctions produces a current ratio of $e^{\Delta V/V_T}$, which for small offsets is $1 + \Delta V/V_T$ — so a millivolt of offset is 3.9 per cent of current, and the thermal voltage is the exchange rate. The two statements are the same statement: a two per cent mismatch in saturation current is half a millivolt of built-in offset, and the mirror cannot tell the two apart because there is no measurement that separates them.

Degeneration works precisely by introducing a measurement. Once a resistor carries the current, the emitter voltage moves with it, and the difference in emitter voltages subtracts from the difference the mismatch was trying to impose. The $1 + g_m R$ is the loop gain of that correction, and the thermal voltage appears in the crossing because $g_m R$ is the degeneration drop divided by it — the same exchange rate, arriving in the answer to a different question.

Three hundred mirrors built to one design #

Every pair in the population is solved rather than sampled from an expression, which matters because the mismatch and the two systematic errors interact through the same nonlinear circuit and there is no guarantee they simply add.

They very nearly do. The mean of the population is 3.836 per cent and the systematic error the rung below computes with identical devices is 3.937: the mismatch does not move the mean, to within the sampling error of three hundred draws. That is worth having as a measurement rather than an assumption, because the base current is stolen from the reference by both devices and there was no obvious reason for a difference between them to leave the average alone.

The spread about that mean is 1.955 per cent for a two per cent device mismatch — the mismatch arriving with nothing dividing it, exactly as the shared exponential predicts.

And the worst pair of the three hundred is 8.43 per cent out. That is the number a specification is written against, and it is not three standard deviations from the mean because the mean is not where the design is; it is four and a half per cent from the design’s intent in one direction and worse in the other.

Which error is the limit #

Put the two together and the rung below’s finding changes status.

The base-current error is 2/β and falls with every improvement in the device. The mismatch does not depend on β at all. They cross at about β = 150, which is where an ordinary small-signal transistor sits, and every modern device is to the right of it.

So the error the rung below separated so carefully is, for any device anybody would use, not the one that decides the answer. Below the crossing the arithmetic is the right thing to look at; above it, the mirror’s accuracy is a statistical statement whose width appears nowhere in that arithmetic.

The Early half of the systematic error is deliberately held out of that comparison, because it behaves differently: it has a floor rather than falling with β, and it is the half a cascode removes by holding the two collectors at the same voltage. That is the design move rung one’s finding actually motivates. It is also the design move that does nothing whatever about the spread measured here — which is the point. Every cure for a systematic error leaves this one untouched.

The one lever, and where it stops working #

There is exactly one thing a designer can do about device mismatch inside the circuit, and it is a resistor in each emitter.

The mechanism is the one the degeneration essay measures for a single stage: the shared base-emitter voltage now sets a current through a resistance rather than through an exponential, so a fractional error in the saturation current is divided by 1 + gₘR. That expression is the feedback field’s desensitivity arriving in a circuit with no amplifier in it, which is worth noticing: it is the same algebra because it is the same idea, a loop gain dividing an error, with the loop closed through a resistor instead of through a differential pair.

The spread falls, flattens and stops. It stops at 0.48 per cent, which is the half per cent the resistors were given, because the same degeneration that divides the device mismatch multiplies the resistors’ own by gₘR/(1 + gₘR). Adding more never crosses that floor and the improvement per millivolt collapses long before it.

The crossing — where the two contributions are equal — is at gₘR equal to the ratio of the two tolerances, which makes it a voltage:

$$V_{\text{degen}} = V_T \cdot \frac{\sigma_{\text{device}}}{\sigma_{\text{resistor}}}$$

For two per cent devices and half per cent resistors that is 103 millivolts, and it contains nothing about the circuit except that ratio. Not the current, not the resistance, not the transistor’s β, not the supply. Past it, more degeneration costs headroom and buys nothing.

The measured spread follows the closed form to within about five per cent across the sweep, and the measurement is consistently below the prediction — the expression uses the nominal transconductance while the solved circuit’s is slightly lower at the degenerated operating point.

What the trade actually is #

The design instruction that falls out is more specific than “add degeneration”.

Below a hundred millivolts of drop, degeneration is nearly free: each millivolt divides the dominant error and the resistors contribute almost nothing. Above it, each millivolt costs a millivolt of compliance and buys a diminishing fraction of a floor that is already reached. A mirror with a volt of degeneration is a design decision that has spent a volt of output swing for nothing measurable — and compliance is exactly what a mirror is short of, which is why the cascode’s cost is quoted in the same units.

The second half of the instruction is about the resistors. The floor is their tolerance, so the degeneration is worth having only if they are better matched than the devices. In an integrated circuit that is nearly always true — thin-film resistors match to a fraction of a per cent while bipolar saturation currents do not — and on a circuit board with ordinary one per cent parts the ratio is nearer unity and the crossing is at 25 millivolts, so there is very little to gain.

That is the whole trade, and both of its numbers are ratios of tolerances rather than component values.

The population is drawn with a seed, and that is not a detail #

Every number above comes from a sample, so it is worth saying what makes a sampled number quotable.

The draws come from a counter-based generator with an explicit seed, and the seed is part of the figure rather than an implementation detail: a sampled figure that changed every time it was drawn would make every assertion about it a one-off, and there would be no way to say that a number quoted in the prose is the number in the drawing. Three hundred pairs at one seed is one experiment, and the summaries — mean, spread, extreme — are statements about it.

That is why the assertions in the site’s gate are written as bounds with slack in them rather than as equalities. The mean is required to be within a quarter of the systematic error and the spread within a fifth of the device mismatch, and both of those tolerances are wider than the effect being claimed and narrower than the effect being excluded. A tighter bound would fail on a different seed and a looser one would pass a circuit whose exponential divided the mismatch, which is exactly the error the assertion exists to catch.

The extreme is the summary that behaves worst under sampling. The worst of three hundred draws from a normal distribution is around 2.9 standard deviations and it moves by a good fraction of one from seed to seed, which is why it is asserted only to exceed two — and why the sentence in the figure’s own description says worst of the three hundred rather than worst case. They are different quantities and only the first one has been measured.

What a designer should take from the width #

Three consequences, in the order they bite.

A mirror ratio is not a design parameter below a few per cent. An arrangement that needs two currents to match to a per cent needs either degeneration, or a matched pair on one die with its mismatch specified, or a trim. Building it out of two discrete transistors from the same tape and hoping is a design whose accuracy is 8 per cent worst case, and that is what the population above says.

The specification has to be written on the population. A mean of 3.84 per cent and a spread of 1.96 is not “about four per cent”; it is a distribution whose tail decides the yield. A design that tolerates five per cent passes most of the time and fails about a fifth of the time, and a design that tolerates ten passes essentially always. Which of those two a specification chooses is a commercial decision and it needs the width to make it.

And the two cures are for different errors. A cascode removes the Early half of the systematic error and does nothing about the spread. Degeneration divides the spread and does nothing about the base-current half. They are compatible and often used together, and a designer who applies one to the problem the other solves has spent headroom for no accuracy at all — which is exactly the mistake that follows from reading the rung below and stopping there.

Why this is not the tolerance essay #

The tolerance that is not on any part draws the distribution of a divider’s output over its components’ tolerance bands and compares it against the worst case and the root-sum-square bound. This is the same kind of object and a different kind of quantity, and the difference is worth stating.

There, the tolerance is specified. A one per cent resistor is a promise about a distribution, the distribution is roughly known, and the arithmetic connecting it to the answer is linear. Here the mismatch is not specified anywhere on a data sheet for a discrete pair, it depends on geometry and process in ways the buyer cannot see, and the arithmetic connecting it to the answer runs through an exponential.

What the two share is the shape of the conclusion: the honest answer to “how accurate is it” is a distribution, the worst case is far from the mean, and the number a solve returns is a property of a circuit nobody has.

What is not modelled #

The mismatch is normal and it is not. A real population of saturation currents is roughly log-normal, and the tail matters more than the width for a specification written at a yield. Using a normal distribution on the fractional difference is close for small mismatches and diverges in the tail, which is exactly where a worst-case number is read.

And it is one number. In an integrated pair the mismatch scales inversely with the square root of the emitter area — a physical law with a great deal behind it — so a designer buys matching with silicon, and the trade against degeneration is really a three-way one between area, headroom and accuracy. That third axis is a process parameter rather than a circuit quantity and is not here.

The betas are drawn independently. They should not be: β and saturation current are correlated on the same die because both depend on the same doping. What that changes is the width of the distribution rather than any of the boundaries, and getting it right needs a joint model this collection has no source for.

The temperature is 300 K everywhere. A mismatch in saturation current appears as a mismatch in base-emitter voltage, and that mismatch has a temperature coefficient of its own — about 3.3 µV per kelvin per millivolt of offset — so a pair trimmed to match at room temperature drifts apart. That is the mechanism the differential-pair drift essay measures and the same treatment applies here unchanged.

And nothing is trimmed. A production mirror can be laser-trimmed or digitally corrected, which removes the mean and part of the spread and leaves the drift. That is a different subject and it is where the interesting modern answer lies.

What the gate checks #

The mean of the population is asserted against the systematic error a single solve gives with identical devices, to a quarter — which is the claim that the mismatch does not move the mean, and is not a tautology, because both are computed independently.

The spread is asserted against the device mismatch itself, to a fifth, so a version whose exponential divided the mismatch by something would fail rather than be quietly plotted.

The worst pair is asserted to be beyond two standard deviations, which is what says a specification cannot be written at one.

The degeneration sweep carries three claims: that the spread never rises with added degeneration, that it stops at the resistors’ own tolerance rather than at zero, and that the measured spread tracks σ_Is/(1 + gₘR) and σ_R·gₘR/(1 + gₘR) added in quadrature at every drop on the sweep. The third is what makes the crossing a mechanism rather than a coincidence of two curves.

And the crossing with the base-current error is asserted to exist inside the range of β drawn, which is the assertion that carries this rung’s finding: if it did not, the rung below’s arithmetic would still be the limit and this essay would be about a correction.

What a solve is for #

It is worth being clear about what has and has not been given up.

Nothing here says the solve was wrong. Every one of the three hundred pairs is the same Newton iteration on the same netlist that the rung below runs, and each returns an exact answer for the devices it was given. What changed is the question: not what does this mirror do but what does a mirror built to this design do, and the second question has an answer with a width.

A bias point is a solution rather than a choice, which is this field’s founding statement, and it remains true. So does the pair that cancels what a single device cannot, for the same reason and with the same caveat attached. It is a solution to a problem whose coefficients are not known to better than a couple of per cent, and a solution computed to twelve digits from coefficients known to two is a description of one instance rather than of a design.