The drop the lower device did not need

The source that holds to the supply put a second transistor on each branch of a current mirror and measured both halves of what that is always said to do. The output resistance went from 82 kΩ to 7.39 MΩ, and the floor below which the mirror stops working rose by 0.71 V. It measured a third thing the usual account leaves out: the range over which the current actually holds to one per cent went from 1.70 V to 9.09 V, because a plain mirror’s current never stops climbing with its output.

It ended by naming the part of the floor that was not needed. The stacked mirror biases the upper device’s base with a second diode-connected transistor sitting on top of the first, so that base is two base-emitter drops above ground and the lower device’s collector is one drop up — about 0.7 V. The lower device only needs enough collector-emitter voltage to stay out of saturation, which is a few hundred millivolts. The rest of that drop is headroom spent on nothing. This page takes it back and measures what that costs.

The floor and where it comes from #

In the stacked cascode the output can fall until the upper device itself is about to saturate. Its emitter sits at the lower device’s collector, one drop up, so the floor is that drop plus the upper device’s own minimum collector-emitter voltage: 0.91 V on the model the earlier essay used, which treats a transistor as forward-active above 0.2 V and refuses below it.

That model is not good enough here. Every change this page makes is about running a transistor at a few hundred millivolts of collector-emitter voltage, which is exactly where a forward-active model returns a current for a device that is saturating. So every transistor below is modelled with both of its junctions — the base-emitter and the base-collector — in the transport form of Ebers and Moll, with a reverse gain of two. With that model a floor is not assumed but read: the lowest output voltage at which the current is still within one per cent of its value at 5 V.

Measured that way, the stacked mirror’s floor is 0.832 V. Its output resistance at 5 V is 7.39 MΩ and its copy ratio 0.9747, the same values the forward-active model gave at 5 V, where the two models ought to agree and do.

One drop and a margin #

The wide-swing mirror keeps everything about the stacked mirror except where the upper device’s base is held. Instead of a second diode on top of the first, the base is driven from a separate bias at one drop plus a chosen margin: here, the reference transistor’s own base-emitter voltage plus 300 mV. The upper device’s emitter then sits one drop below its base, which puts the lower device’s collector at about 300 mV — out of saturation, and no higher.

The floor falls from 0.832 V to 0.403 V, a little over 430 millivolts — very nearly the drop that was being spent on nothing, less the 300 mV margin put back. On a 1.8 V supply that is the difference between a current source that can swing about a volt and one that can swing nearly a volt and a half.

That much is what the arrangement is known for. What is not usually said is what happens to the output resistance, and the measurement says the opposite of the trade anyone would expect. It does not fall. It rises, from 7.39 to 13.2 MΩ, and the copy ratio improves slightly as well.

Where the resistance came back from #

The earlier essay found the reason and named it. In the stacked mirror the upper device’s base current is drawn from the reference branch, through the second diode. When the output moves, the upper device’s base current moves with it, the reference moves with the base current, and the reference moving is the mirror changing its own bias in the direction that lowers its output resistance. Replacing that diode with a held voltage — a controlled experiment rather than a circuit, in that essay — gave back 1.76 times the resistance.

The wide-swing mirror is that controlled experiment built. Its upper base is driven from a bias that is not the reference, so the base current’s movement goes somewhere that does not feed back into the mirror. The resistance it recovers is not a side effect of the lower floor. It is the second consequence of the same decision: to stop taking the upper device’s bias from the stack.

Across current gains from 25 to 1200 the wide-swing mirror sits on $\beta r_o$ and the stacked one at a little over half of it. The ceiling the device that never sees the swing found for a single cascode stage — $\beta r_o$, set by the upper device’s base shunting the node the feedback works through — is the ceiling here too, and the wide-swing mirror reaches it where the stacked one gave away 45 per cent of it to its own reference.

The resistance is also flat in the margin, which is the other half of why the trade is not a trade. A cascode’s output resistance comes from the lower device looking like a current source to the upper one’s emitter — its own output resistance, $r_o$, set by the Early voltage over its current — and from the upper device’s base current shunting that node. Neither depends on how many millivolts the lower device has, so long as it has enough to stay out of saturation. Moving its collector from 0.7 V down to 0.3 V changes nothing about either mechanism. The resistance a cascode delivers is a property of whether the lower device is forward-active, not of how far into forward-active it is.

What the margin costs #

So far the margin has been 300 mV and everything has improved. The margin is the new quantity in the design, and the obvious question is how small it can be made, since every millivolt of it is a millivolt of floor.

At 100 mV of margin the lower device is saturating. Its collector-emitter voltage is 0.102 V, its base-collector junction is conducting forwards, and part of the base drive that should reach its collector is going there instead. The copy ratio drops from 0.9758 to 0.9369 — the mirror now delivers four per cent less than it did — and the output resistance falls to 2.07 MΩ, below the stacked mirror’s and a sixth of what the 300 mV margin gave.

The floor reads 0.212 V, lower than before, and it is worth being careful about what that means. The floor is measured against the current at 5 V, and that current is now the wrong current. A mirror that holds a degraded value down to 0.2 V has not got better at holding its current; it has got worse at being a mirror, and more uniform in how bad it is.

The collapse has a size that can be worked out before it is solved. A transistor saturates when its base-collector junction starts to conduct forwards, and that junction’s current is its saturation current over the reverse gain times $e^{V_{BC}/V_T}$. With the lower device’s base at one drop — about 0.65 V at this current — and its collector at the margin, the base-collector voltage is the drop less the margin. Against the ordinary base current, $I_S/\beta \cdot e^{V_{BE}/V_T}$, the base-collector junction’s current is in the ratio $(\beta/\beta_R)\,e^{-m/V_T}$ for a margin $m$. At 300 mV that is 75 times $e^{-11.6}$, seven parts in ten thousand of the base current: nothing. At 100 mV it is 75 times $e^{-3.87}$, about 1.6 times the base current. The junction that should be off is carrying more than the base does, and every milliampere it carries is taken from the collector the mirror is copying into. The factor $e^{-m/V_T}$ is why the collapse is abrupt: every 26 millivolts of margin changes it by a factor of e.

The sweep shows the whole shape. Below about 150 mV the lower device saturates and the mirror collapses in two ways at once: at 50 mV the resistance is 0.51 MΩ and the copy ratio 0.7665, a quarter of the current gone into the lower device’s base-collector junction. Above about 200 mV the resistance is flat at 13.2 MΩ, and the floor rises exactly as fast as the margin, one millivolt for one. The 50 mV point’s floor reads about a volt, which is the other symptom of the same collapse: with a saturated lower device the current never settles, and a one-per-cent band around its value at 5 V is left only far up the curve.

So the margin has an optimum in the ordinary sense. Every millivolt above about 200 mV is spent on floor and buys nothing; every millivolt below about 150 mV costs resistance and accuracy much faster than it saves floor. The stacked mirror had no such decision in it, because two drops is far more margin than the lower device needs, and the stack produces it automatically.

The cost is in the bias #

That is where the real price of the wide-swing mirror lives, and nothing on this page’s figures charges for it. The figures hold the upper base at exactly one drop plus the margin — the reference transistor’s own base-emitter voltage, read from the solve, plus 300 mV — which is what a well-designed bias generator is trying to do. A real one has to produce that voltage from somewhere: typically a separate transistor run at a lower current density, or with a resistor in its emitter, so that its drop is the mirror’s drop plus the margin.

Everything that moves a base-emitter drop moves that margin with it. The drop falls by about two millivolts a kelvin, and if the generator’s drop and the mirror’s drop do not track, the margin changes with temperature. Two currents with one name measures a junction’s drop against temperature, and the difference between two junctions at different current densities is itself proportional to absolute temperature — so a margin built from a current-density ratio grows as the part warms and shrinks as it cools. A margin designed at 300 mV on the bench that falls towards 150 mV in the cold is a mirror that collapses in the cold.

The size of the risk is easy to bound. If the generator produces a fixed voltage and the mirror’s drop falls at two millivolts a kelvin, the margin grows by two millivolts for every kelvin of warming and shrinks by two for every kelvin of cooling: a 300 mV margin set at room temperature is 180 mV at −35 °C and 150 mV — the edge of saturation on the sweep above — at −50 °C. If the generator instead produces a margin proportional to absolute temperature, as a current-density ratio does, a margin of 300 mV at 300 K is 240 mV at 240 K, which is comfortable. The second arrangement is why wide-swing bias generators are built around a ratio of junctions rather than around a resistor, and the arithmetic says the choice between them is worth about ninety millivolts in the cold.

The stacked mirror’s two drops carry none of that risk: its margin is a whole drop, and a drop does not fall to 150 mV at any temperature a circuit runs at. So the honest statement of the trade is not “headroom for resistance”, which the measurement contradicts, but “headroom for a bias that must be held”: the wide-swing mirror is better on every figure drawn here, provided a margin of a couple of hundred millivolts is held across everything the circuit will see.

What it says about the usual account #

The usual account of cascode mirrors has one axis: more devices stacked, more resistance and less swing. The stacked and wide-swing mirrors have the same number of devices in the output branch, and the wide-swing one has both more resistance and more swing. The extra axis is where the upper base’s bias comes from. Taking it from the reference branch, as the stack does, is convenient and costs both a drop of headroom and nearly half the resistance; taking it from outside costs a bias circuit and a margin that has to be kept.

The copy, and its two errors measured the plain mirror’s copy error as a base-current effect and an Early-effect one. Here the same base current, in the upper device, turns out to decide the output resistance as well, and the wide-swing arrangement is a way of routing it where it does no harm. And the device that never sees the swing found the cascode’s resistance ceiling to be $\beta r_o$ rather than the $g_m r_o^2$ that is usually quoted; the wide-swing mirror is the arrangement that actually reaches that ceiling.

The same idea is older and more common in field-effect circuits, where it is the standard way to cascode a mirror on a low supply. There the lower device needs its overdrive voltage rather than a saturation voltage, the upper gate is held one threshold and two overdrives up instead of two thresholds and two overdrives, and the margin is the overdrive — typically a couple of hundred millivolts, and chosen by the device’s width rather than by a bias voltage. The bipolar version measured here has one difference that matters: a field-effect gate draws no current, so the stacked field-effect mirror never gave half its resistance to its reference in the first place. The resistance recovered on this page is a bipolar gain, and it is a reason the wide-swing arrangement is worth more in a bipolar circuit than its usual description as a headroom trick suggests.

A built mirror can be checked for which side of the margin it is on without any of this apparatus. The lower device’s collector-emitter voltage is a node that can be probed, and a reading below about 150 mV says the mirror is in the collapsed region whatever its current looks like. The output resistance is the other test: a wide-swing mirror measuring far below $\beta r_o$ has a saturated lower device, and one measuring near half of it is behaving like a stack.

How the numbers were obtained #

Every transistor is the transport form of the Ebers–Moll model, with a saturation current of 10⁻¹⁵ A, a forward gain of 150 unless swept, a reverse gain of 2, and an Early voltage of 80 V applied as an output resistance $V_A/I_C$ recomputed from each solve until the output current changes by less than a part in 10¹³. The circuit is solved by Newton’s method with the step limited, and each solution is checked against Kirchhoff’s current law rebuilt from the nonlinear laws. The wide-swing mirror’s upper base is an ideal source at the reference transistor’s solved base-emitter voltage plus the margin, refreshed each iteration. Output resistance is the difference of two solves 50 mV either side of 5 V. The floor is bisected on the solved current against its value at 5 V. The reference resistor is 9.3 kΩ from a 10 V supply.

What it leaves out #

The bias generator, as above. The upper base is held by an ideal source here, which is the best any generator can do, and what a real generator’s output resistance and temperature tracking do to the margin is not solved.

Mismatch. Every transistor is identical; the copy errors here are the systematic ones. A wide-swing mirror’s lower devices run at different collector-emitter voltages in the reference and output branches — about 0.7 V against 0.3 V — which the Early effect turns into a systematic error the stacked mirror’s matched voltages do not have, and at an Early voltage of 80 V it is half a per cent; the ratio above shows it as a slight change from the stacked mirror’s.

And the folded arrangement, which puts the upper device on a separate current path so that the drops are not stacked at all. It removes the floor problem differently, at the price of a second bias current, and it is not measured here.

Still open: the margin across temperature, the folded cascode, and the reference branch’s own Early error #

The margin across temperature. Building the bias as a second junction at a lower current density and solving the mirror from −40 °C to 125 °C would say whether a margin designed at 300 mV stays above the 150 mV where the lower device saturates, and how large a current-density ratio the generator needs for it to do so.

The folded cascode. Putting the upper device on its own current path removes the stack from the floor altogether, and costs a second current and that current’s noise. Measured on the same saturating model, it would give a third point on the floor-against-resistance plane and say what the second current buys.

The branches at different voltages. The wide-swing mirror’s reference transistor runs at one drop and its output transistor at the margin, so the Early effect copies the current with an error the stacked mirror’s matched branches do not have. Adding a matching device to the reference branch, so that both lower transistors sit at the margin, should remove it, and whether the added device costs back any of the floor is the measurement that would settle the arrangement.