The copy, and its two errors

A current mirror is two matched transistors with their bases tied and one of them diode-connected. The reference branch sets a current; the other transistor, seeing the same base-emitter voltage, carries the same current. It is the most used circuit in analogue integrated design and it has one error everybody names and a second error that is larger.

The one everybody names is the base current. Both transistors take theirs from the reference branch, so the copy is $\beta/(\beta+2)$ of the reference — 1.32% low at $\beta = 150$ — and the standard repair is a third transistor supplying those base currents from somewhere else.

The one that is usually a footnote is the Early effect. A transistor’s collector current rises with its collector voltage, so the copy is a function of what is connected to it, and over any swing worth having it moves by more than ten per cent.

The solve, and the loop inside it #

The base-current error comes straight out of Newton’s method on the netlist: two exponential devices, a reference resistor, a supply, and a source at the output holding it at a stated voltage so that the current through it can be read as a branch quantity.

The Early effect cannot come out of the same solve, and the reason is structural. This site’s nonlinear element is controlled by exactly one pair of nodes, and a transistor’s collector current is controlled by its base-emitter voltage — there is no second controlling pair for the collector-emitter voltage to use.

So it enters as what it is: a conductance from collector to emitter of value $I_C/V_A$. That value is not known until the circuit is solved, and the solution depends on it, so the pair is iterated. Solve; read the collector currents; restamp the two conductances; solve again. It converges in eight passes to the point where the last one moves the output current by $2.7\times10^{-14}$, because the correction is a part in a few hundred and each pass reduces the residue by that factor.

The figure asserts the convergence rather than assuming it — the iteration count and the last movement are both checked — and it asserts the operating point’s own residual, which is Kirchhoff’s current law rebuilt from the exponentials at the converged voltages and comes out at $2.1\times10^{-15}$.

Two errors, and only one of them is a design choice #

With $\beta = 150$, $V_A = 80$ V, a 9.3 kΩ reference resistor and ten volts of supply, the reference is 0.99853 mA.

The middle column is the mirror. Two of the numbers in the right-hand column are worth separating.

The base-current error is 1.32% and it is constant. It is $2/(\beta+2)$, it does not depend on the resistors, the supply or the output voltage, and it is what the third transistor buys back.

The Early error over a one-volt-to-ten-volt output swing is 11.2%. That is nine times the first one, it is what nobody spends a transistor on, and it is the error a circuit actually sees, because a current source that never moves its output voltage is not doing anything.

The one output voltage at which the copy is exact #

The two transistors have the same base-emitter voltage by construction. They do not have the same collector-emitter voltage: the reference one is diode-connected, so its collector sits at $V_{BE}$, while the output one sits at whatever is connected to it. The Early correction is $1 + V_{CE}/V_A$ for each, so the two cancel exactly when the output is at the reference’s own $V_{BE}$.

Bisecting the solve for where the ratio equals $\beta/(\beta+2)$ gives 0.7043 V, against a measured $V_{BE}$ of 0.7137 V. Nine millivolts under, not at.

The residue is physical rather than sloppy. The reference current is read through the reference resistor, which carries both base currents, so it moves slightly as the output moves — and the crossing shifts by the amount that costs. Which suggests the displacement should go as one over β, and it does:

The last column is constant to four parts in a thousand across a fifty-fold range of β. That was not designed in and the figure asserts it, because a constant that falls out of a solve across a slider is exactly the kind of claim this site exists to make and exactly the kind that is invisible if only one frame is ever drawn.

The compliance, which contains neither the current nor a resistor #

The useful question is not how wrong the mirror is but over what range it is right, and that has a clean answer.

$I(V) = I_0(1 + V/V_A)$, so holding the current to a fraction $\epsilon$ of its value at some reference voltage needs the output to stay within $\epsilon(V_A + V_{ref})$ of that voltage. Bisecting the solve from one volt gives 0.810 V for one per cent, against $V_A/100 = 0.80$ V.

So a mirror is a current source to one per cent over about eight hundred millivolts, and that number contains neither the current it is copying nor any resistor in the circuit. Change the reference from a microamp to ten milliamps and it is the same eight hundred millivolts. It is a property of the device, through one parameter, and it is the honest statement of what a current mirror is worth.

The same relation read the other way is where the Early voltage’s name comes from: extrapolating the output characteristic back to zero current hits the voltage axis at $-V_A$. The figure fits that intercept from two points on the solved curve and gets $-80.0$ V against the 80 V stamped, which is the definition being checked rather than assumed.

What the reference resistor is not doing #

It is worth noticing what does not appear in either error, because it is what a designer spends most of the effort on.

The reference resistor sets the current and nothing else. Change it from 9.3 kΩ to 93 kΩ and the reference falls by ten; the base-current error is still $2/(\beta+2)$, the slope is still $1/(V_A+V)$ per volt, and the one per cent compliance is still 0.81 V. Every quantity in this essay is a fraction, and the fractions are properties of the devices.

That is a stronger statement than it looks. It means the accuracy of a current mirror cannot be bought with better resistors, more supply voltage or a different operating current — the three things that improve almost everything else in analogue design. What buys accuracy is topology: a helper transistor for the first error, a cascode for the second.

The same reading explains why mirrors are ratioed by emitter area rather than by resistors. Two transistors of different area at the same base-emitter voltage carry currents in the ratio of their areas, exactly, with the base-current error becoming $(1 + n)/\beta$ for a ratio of $n$ — so a ten-to-one mirror has ten times the base-current error of a one-to-one, which is the first thing that breaks when a ratio is pushed. Nothing about the Early term changes at all.

Where the model refuses #

Below about two-tenths of a volt at the output, a real transistor saturates: its base-collector junction forward-biases, the collector current collapses, and the output stops being a current source in a way that has nothing to do with the Early effect.

The model here has no base-collector junction in it. It would happily return a forward-active current for an output at fifty millivolts, which would be a confident number about a device that is not in that state. So it refuses instead, by name, and the refusal is run rather than described — the figure calls the solver at 0.05 V and requires it to decline.

That is the site’s standing rule and it earns its place here, because the left-hand end of a compliance range is exactly where somebody is tempted to extrapolate. The useful range is bounded below by a saturation this model cannot compute and above by a supply, and the essay says so rather than drawing a curve into a region it has nothing to say about.

Three consequences #

A cascode is not about gain. Stacking a second transistor on the mirror’s output holds the lower one’s collector voltage still, so the Early term stops moving — the compliance for one per cent goes from 0.8 V to tens of volts. Everything a cascode is used for downstream follows from that, and the base-current error is untouched by it.

A current source is a resistance. $\partial I/\partial V = I_0/V_A$ is a conductance, so this whole essay is the statement that the mirror’s output impedance is $V_A/I$ — 80 kΩ at a milliamp. A “current source” with 80 kΩ across it is what it is, and every gain that depends on it depends on that number rather than on infinity.

Matching does not help with this. Two transistors on the same die have base-emitter voltages matched to a millivolt, which is what makes the mirror work at all. It does nothing for the Early term, because that error is not a mismatch — it is both devices behaving identically at different collector voltages.

The two errors have different signatures #

They are worth telling apart on a bench, and they are easy to.

The base-current error is a fixed fraction. Measure the reference and the copy at one output voltage and the ratio is a number; move the supply, the resistor or the temperature and the ratio stays put, because both currents scale together. It shows up as a gain error in whatever the mirror is biasing.

The Early error is a slope. It shows up as an output impedance — 80 kΩ at a milliamp — so it appears wherever that impedance is in parallel with something: a gain that is lower than the transconductance times the load resistance predicts, a filter corner that has moved, a current source whose current depends on what the stage it feeds is doing. It also shows up as a dependence on supply voltage, which is what makes it look like poor rejection rather than like a mirror problem.

The distinction matters because the repairs are unrelated and each is useless against the other error.

What this essay does not claim #

That $V_A$ is a constant. It is not: a real device’s output characteristic is not straight and the extrapolated intercept moves with the current. Eighty volts is a number for one device at one bias, and the linear model built on it is a model with a range like everything else here.

That the iteration is the only way in. It is not; a simulator with a proper Gummel–Poon model solves the whole thing at once. The iteration exists because this site’s nonlinear element takes one controlling pair, and it is written out rather than hidden because a self-consistent loop that nobody mentions is the kind of thing that quietly stops converging.

That the base-current repair is free. The third transistor takes its own base current from the reference, so the error becomes $2/(\beta^2 + 2\beta + 2)$ rather than zero — six parts in a hundred thousand at $\beta = 150$ — and it costs a base-emitter drop of headroom at the reference. It is an excellent trade and it is a trade.

That this generalises to a Wilson or a Widlar. It does not. Those change the topology and change both errors, and each would need its own solve. What is measured here is the two-transistor mirror, which is the one in every textbook and the one the two errors are usually quoted about.

Where the two errors are paid for #

A mirror is the current source every other stage in this field is drawn with, so its two errors turn up everywhere. The device that never sees the swing is biased by one and needs its compliance, which is the second error read as a voltage. The source that is not a source is the ideal this page is a departure from, and A bias point is a solution, not a choice is why the departure is a root rather than a number. Two millivolts a kelvin, and the wrong sign is what happens to both errors when the part gets warm, and The exponent that is a square is the same accounting on a device whose law is not an exponential — where the base current disappears and the second error does not.

The gate #

The Early conductance and the current that sets it are iterated to self-consistency, and both the iteration count and the last movement are asserted — eight passes, $2.7\times10^{-14}$ — rather than the loop being run a fixed number of times and trusted.

The operating point’s residual is asserted, rebuilt from the exponentials at the converged voltages and required below $10^{-9}$. A Newton loop that has converged to the wrong thing looks exactly like success from inside the loop, and this is the check that is not inside it.

The Early voltage is checked as an intercept, not as a slope. The fractional slope depends on which current it is normalised to — 1.21% per volt at three volts rather than 1.25%, because $I(V) = I_0(1+V/V_A)$ makes it $1/(V_A + V)$ — and the x-intercept of the same chord has no such choice in it. That distinction cost this figure one wrong assertion before it was noticed.

The displacement of the exact point is asserted to go as $1/(\beta+2)$ across the whole slider, to one per cent, because it is an unplanned result and an unplanned result that is not asserted is an anecdote.

And the refusal is run. The mirror is asked for its output current at fifty millivolts and must decline, so that the assertion is still rejecting something rather than having quietly stopped.

Neither of these two is what limits a real mirror #

The two errors separated here — one that falls with beta and one that does not — are the two every account of a current mirror leads with, and the rungs above this one find that neither is the binding one.

The error that is a distribution is the finding: two transistors on the same die differ, a fractional difference in saturation current is a fractional difference in collector current with nothing dividing it, and the honest object is a spread rather than a number — mean 3.84 per cent, standard deviation 1.96, worst of three hundred 8.43. Against 1.32 per cent from base currents and 11.2 per cent of Early effect over a ten-volt swing, that is comparable with the second and four times the first, and unlike either it is different in every unit.

The repair for it is degeneration, and where it stops is the sharpest number on that page: divided by one plus $g_m R$ until the resistors’ own tolerance takes over, and where it stops is a voltage — a hundred millivolts across the degeneration resistor, containing nothing but the ratio of two tolerances. Not a resistance, not a current, and not a device parameter.

And the refusal, and what it was protecting is where the model’s own boundary — the two hundred millivolts this essay’s gate exercises — turns out to have been placed conservatively: putting the base-collector junction in shows the forward-active model still right to seven parts in ten thousand there, and locates the mirror’s real failure somewhere else entirely, at six and a half per cent off an output sitting at five volts.

Which leaves this essay’s own result as the one about beta and nothing else, and that is the part worth keeping. Moving beta from 20 to 1000 changes the base-current error by a factor of forty-five and the Early effect by nothing at all — so a designer told that a mirror’s accuracy improves with a better transistor is being told something true about one of the two terms and false about the larger one. The repair for the first is a third transistor; the repair for the second is a cascode, which the source that holds to the supply prices at 0.71 volts of headroom for a factor of ninety in output resistance, with forty-three per cent of what should have been there missing in the reference branch.

The general lesson is about which term an account leads with. Every textbook treatment of a mirror opens on the base currents and files the Early effect as a footnote, and the ratio measured here is nine to one the other way over any useful swing. That ordering is not an error about either quantity; it is an ordering inherited from an era of low supply voltages and small swings, kept because the base current is the term with the tidier expression.