The base current the mirror counts twice

The source that holds to the supply cascoded a current mirror and measured what that buys: an output resistance ninety times higher, and a current that stays within one per cent of its set value over nine volts instead of under two. It also found that forty-three per cent of the resistance that should have been there was missing, and traced the loss to the reference branch, where the upper output transistor’s base current is drawn. The drop the lower device did not need built the wide-swing arrangement, which holds the upper base from a separate bias, and the missing resistance came back.

Neither essay measured noise, and a current mirror is a noise source before it is anything else: its output current is a bias for something, and whatever noise rides on it is injected into whatever it biases. The noise the cascode device does add and the resistor the upper base is held through solved a single cascode stage and found that its upper transistor adds one part in β from its base, and that a bias network at that base is harmless at low frequency and not above it. A cascode mirror has an upper transistor, a reference branch that copies its own noise to the output, and, in the wide-swing form, a separate bias for the upper base. Which of those dominates is the question, and the answer has the same base current in it that the resistance measurement found.

Three mirrors, every source alone #

The three mirrors are NPN sinks at 1 mA, with a current gain of 150, an Early voltage of 80 V and the same capacitances as the cascode stage: 20 pF from base to emitter and 2 pF from collector to base in each transistor. The plain mirror is two transistors, the reference one diode-connected. The stacked cascode adds a diode-connected transistor on top of the reference one and an upper transistor on top of the output one, with the upper output base tied to the top of the reference stack. The wide-swing cascode has a reference branch of one diode, and its upper output base is held by a separate diode-connected transistor at 100 µA, fed by a resistor from the supply. The reference resistor sets 1 mA from a 10 V supply in each.

Every transistor carries collector shot noise 2qI and base shot noise 2qI/β, uncorrelated; every resistor carries 4kT/R. Each source is injected alone and the current it delivers into a short at the output is read, and the powers add. Nothing about which of them the mirror copies, and which it cancels, is assumed.

The headline is that the three are almost the same. At 1 kHz the plain mirror delivers 1.986 times one transistor’s shot noise, the stacked cascode 1.960 and the wide-swing cascode 1.965. A mirror’s noise is two transistors’ worth: the output transistor’s own collector noise, and the reference transistor’s, which the mirror copies faithfully because copying the reference branch’s current is its whole job. Cascoding changes the output resistance by a factor of ninety and the noise by a per cent, and in the direction of less.

That is worth knowing on its own, since a cascode is sometimes avoided in a low-noise bias on the assumption that two more transistors bring two more sets of noise. They bring a great deal less than that, for the reason the single stage found: an upper transistor’s collector noise circulates inside it and is cancelled at the output.

What each source delivers #

The totals hide a difference in composition, and the difference is where the argument is.

In both mirrors the two lower transistors carry nearly everything: the reference one’s collector noise is copied at 0.942 in the stacked mirror and 0.954 in the wide-swing one, and the output one contributes 0.973 and 0.986 of its own. The reference resistor and the lower transistors’ base currents come next, at about half a per cent each.

The upper transistors are where the arrangements part. The stacked mirror’s upper reference transistor contributes 6.65 × 10⁻⁶ — it is a diode, and its collector noise circulates inside the junction it forms without reaching anything. The upper output transistor’s collector noise is cancelled in both, as it was in the single stage. But its base noise is 2.55 per cent of one transistor’s in the stacked mirror and 0.662 per cent in the wide-swing one. One part in β is 0.667 per cent, so the wide-swing mirror counts it once, as the single stage did, and the stacked mirror counts it 3.82 times.

Where the base current goes #

The factor of nearly four is two in amplitude, and it comes from where the upper output transistor’s base current has to be supplied from.

In the wide-swing mirror the base current comes from the separate bias, which is a low impedance to ground. A fluctuation in it is drawn from the node between the output transistors, and the node supplies it once, through the upper transistor’s emitter: one part in β at the output, as in the single stage. The ratio against β is 0.992 at a current gain of 150 and 1.005 at 2000.

In the stacked mirror the upper output base is tied to the top of the reference branch, so its current is taken from the reference branch. A fluctuation drawn from the node between the output transistors enters the reference stack, passes down through its diode-connected transistors, and changes the reference lower transistor’s current by the same amount — and the mirror copies that change into the output lower transistor, which draws it from the same node. The node now has to supply the fluctuation twice: once because it left, and once because the mirror asked for it back. Two in amplitude is four in power, and the solve reads 3.82 at β = 150, 3.96 at 2000, and 3.02 at 20, where the copy itself has lost a noticeable share to the lower transistors’ own base currents.

That is the same current, taking the same path, as the one the source that holds to the supply found removing 43 per cent of the stacked mirror’s output resistance. There, a change in the output voltage changed the upper transistor’s base current, the reference branch copied the change, and the output current moved with the voltage. Here, a fluctuation in that base current is copied by the same branch and adds to itself. The resistance defect and the noise defect are one mechanism, and the wide-swing mirror’s separate bias removes both at once.

The reference resistor takes its place rather than adding #

The other source that changes between the arrangements is the reference resistor, and it behaves in a way worth measuring against the supply, because a low-voltage supply is where cascode mirrors are hardest to fit and where the resistor is smallest.

A resistor carrying a current I with a voltage V across it has thermal noise 4kT/R, and against the current’s shot noise 2qI that is 2kT/(qV) — twice the thermal voltage divided by the drop across it. So as the supply falls towards the junction drops the reference branch stacks, the resistor’s voltage falls and its noise rises. At 15 V it is 0.36 per cent of a transistor’s noise in the stacked mirror and 0.35 in the wide-swing one. At 1.8 V it is 9.65 per cent in the stacked mirror, which stacks two drops under its resistor, and 4.31 per cent in the wide-swing one, which stacks one. That is a real difference between the arrangements, and it favours the wide swing again.

But the reference branch’s total barely moves: 0.950 at 1.8 V and 0.954 at 15 V in the stacked mirror. The resistor shunts the diode node, and the part of the diode’s own noise current that now flows in the resistor instead of being mirrored is almost exactly what the resistor adds. To first order in the ratio of the diode’s resistance to the resistor’s the two cancel, and what is left is second order. The resistor in the same loop found the general form of this for a junction in series with the resistor that carries its current: each element’s noise current has to cross the other, and the shares trade rather than add. The practical consequence is that a low supply costs the mirror’s noise far less than the resistor’s own share suggests.

Above ten megahertz #

At high frequency the composition changes again, and the stacked mirror ends up the quietest of the three for a reason that is not a virtue.

The reference branch’s contribution falls from 0.954 to 0.396 at 100 MHz, and the output lower transistor’s from 0.980 to 0.566. Both fall because the upper output transistor’s base-emitter capacitance sits at the node between the output transistors and takes a growing part of any current arriving there — the output transistor’s own noise and the mirrored copy alike. The reference branch falls further, because the mirror’s own pole, set by the diode’s transconductance against two transistors’ base-emitter capacitance, stops copying the reference at all. Meanwhile the upper output transistor’s collector noise, cancelled at low frequency, comes through the same node admittance that ended its cancellation in the single stage, and the upper transistor’s share rises from 2.57 × 10⁻² to 0.234.

The stacked mirror’s total of 1.196 at 100 MHz is lower than the plain mirror’s 1.681, but not because it is better built: it is lower because its output is losing its grip on the current it is supposed to deliver, noise and signal together. A bias current whose noise falls because the bias itself is shunted is not a quiet bias.

The wide-swing mirror, on the slider, has the extra source the question asked about: its separate bias. At 1 kHz it contributes 1.01 × 10⁻⁶ of a transistor’s noise — the diode at 100 µA is about 260 Ω of dynamic resistance making half a resistor’s noise, and the upper output transistor divides it by $g_m r_o$, exactly as it divided a bias resistance in the single stage. At 100 MHz it has risen to 7.76 × 10⁻³, less than a per cent, because 260 Ω is well down the low side of the curve that essay drew. The third source exists and is the smallest of them.

What two transistors’ noise is, in amperes #

The results are in units of one transistor’s shot noise, which makes the comparisons clean and hides the size. At 1 mA one transistor’s collector shot noise is 2qI = 3.2 × 10⁻²² square amperes per hertz, or 17.9 pA per root hertz, and a mirror delivering 1.96 of it carries 25.1 pA per root hertz on its output current. Across a 20 kHz band that is 3.5 nA rms on a milliampere — three and a half parts per million, which is small for a bias and not small for a current that sets a precision reference’s operating point.

There is a more useful way to hold the number. Two transistors’ shot noise, 4qI, is the thermal noise current of a resistor of kT/(qI) — the reciprocal of one transistor’s transconductance, 26 Ω at 1 mA. A cascode mirror presents several megohms of output resistance and carries the noise current of a 26 Ω resistor. That is the whole case for a mirror as a quiet bias, and its limit: it holds the current far better than a resistor could, and its noise is fixed by the current it carries rather than by the resistance it presents. The floor a current sets found the same equality from the resistor’s side, with a current’s noise and a resistance’s meeting where the drop is twice the thermal voltage.

What a designer should take #

A cascode current mirror is not noisier than the plain mirror it replaces. At low frequency all three deliver two transistors’ shot noise within one and a half per cent of each other, and the two cascodes deliver slightly less. The noise of a mirror is set by its lower transistors, and reducing it is a matter of those — degeneration, or a lower current — not of the cascode.

Between the two cascodes the wide-swing mirror is the better noise source at low frequency, by a margin that is small in total and exact in cause: its upper output base counts once, where the stacked mirror’s counts nearly four times, and its reference resistor has more voltage across it at a given supply. The separate bias it needs contributes a millionth at low frequency and under a per cent at 100 MHz.

And the stacked mirror’s defects come as a pair. The base current that its reference branch supplies costs it output resistance and noise together, by the same path, and the arrangement that removes one removes the other.

Two routes to one number #

The factor of four is predicted by an argument — the node supplies the fluctuation twice — and measured by a solve that knows nothing of the argument. The approach to four as β rises, 3.02 → 3.82 → 3.96, is the check: the argument ignores the base currents of the mirror’s own lower transistors, which take a share of any copied current of the order of 1/β, and the solve’s shortfall from four shrinks steadily as β grows: 0.98 at 20, 0.18 at 150, 0.04 at 2000. The wide-swing mirror’s 0.992 and 1.005 on either side of one make the same check on the other arrangement.

The reference resistor’s first-order cancellation is checked the same way: the resistor’s own share at 1.8 V is 9.65 per cent, and the branch total moves by less than half a per cent between 1.8 V and 15 V. An argument that the resistor adds its share would predict a nine per cent rise, and the solve shows none.

What was left out #

Mismatch. Every transistor pair here is matched, so the mirror copies at exactly its intended ratio. The copy and its two errors measured what a real pair does, and a mismatched mirror copies its reference noise at its mismatched ratio — a two per cent error is a four per cent change in the copied noise power, which is as large as the difference between the two cascodes found here. On real transistors the choice between them is settled by output resistance and headroom, not by noise.

Flicker noise. Only shot and thermal noise are modelled. Below some corner frequency a transistor’s base current carries flicker noise as well, and in the stacked mirror it would be counted four times like the shot noise. The factor is a property of the path, not of the spectrum.

The bias source’s own design. The wide-swing mirror’s bias is a single diode at 100 µA fed by a resistor. A practical bias is often a scaled copy of the reference branch, which adds its own mirrored noise and correlates it with the reference’s.

Still open: degeneration, a scaled bias, and the output impedance’s own noise #

Emitter degeneration in the lower transistors. A resistor in each lower emitter reduces the copied noise, because the mirror then copies a voltage across a resistance rather than a junction’s current, and the resistor in the same loop has the arithmetic. How much it removes from the two-transistor floor, and what it costs the cascode’s headroom, is the next measurement on this mirror.

A bias derived from the reference. Holding the wide-swing upper base from a scaled copy of the reference branch shares noise between the two paths. Whether the correlation makes the bias’s contribution larger than the independent diode’s millionth, or cancels part of the reference’s, needs the correlated solve.

The noise a load sees. A mirror drives a load, not a short, and its high output resistance means its noise current flows into that load almost whatever it is. Driven into a finite load, the cascode’s output resistance decides how much of the lower transistors’ noise becomes a voltage there, and that is where its hundredfold resistance might finally show up in the noise.