The source that holds to the supply

The rung below this one measured a cascode as an amplifier and found the expression everybody gives for its output resistance to be out by a factor of twenty-one. The expression is $g_m r_o^2$, it has no ceiling in it, and the ceiling is $\beta r_o$ — because a bipolar transistor’s base draws current, and the upper device’s base resistance shunts the very node the feedback has to act through.

That was a stage, where the quantity being multiplied is a gain. This is the arrangement doing the other job it is used for, where the quantity being multiplied is a current source’s stubbornness: a mirror with a cascode on each branch. Three things come out of it, and only the first is the one the textbooks lead with.

The mirror, and the fourth transistor that is not obvious #

A plain mirror is two transistors: one diode-connected, carrying a reference current, and one sharing its base-emitter voltage and delivering a copy. A cascoded mirror is four, and the fourth one is worth a paragraph because it is easy to think three would do.

Q1 and Q2 are the pair. Q3 stands on Q2, its collector at the output and its base held at a fixed potential, which is the cascode of the rung below. Q4 stands on Q1 and is diode-connected, so the reference branch develops two base-emitter drops and its top node is the potential Q3’s base wants.

That is the obvious reason for Q4 — it makes the bias. The real reason is that it puts Q1 and Q2 at the same collector voltage, and that is what removes the largest error in the plain mirror. Every number below about the copy is a consequence of it.

Every device here carries its own Early conductance from collector to emitter, of value $I_C/V_A$, and that value is not known until the circuit has been solved. So the pair is iterated: solve, read four collector currents, restamp four conductances, solve again. It converges in ten passes to a part in $10^{13}$, with the residual rebuilt from the exponentials rather than from anything the iteration produced.

The output resistance, and where a fifth of it went #

Driving the output node a little either side of its operating point and reading the current back gives 81.9 kΩ for the plain mirror and 7.39 MΩ for the cascoded one, a factor of ninety.

The plain mirror’s number is $r_o = V_A/I_C$ and nothing else, which is what the cascode is multiplying. The cascoded one is not $\beta r_o$: that would be 13.3 MΩ, and the measurement is 55 per cent of it. The rung below measured 96 per cent of its own $\beta r_o$ on a stage, so something in the mirror is taking the other 45.

It is the upper device’s base current again, in a second place.

In the stage, Q3’s base is held by a voltage source that is a signal ground, and the base current is charged once: it flows in $r_\pi$, which shunts the inner node, and the multiplication stops at $\beta$. In the mirror, Q3’s base is held by Q4’s diode — which is fed from the reference branch. When the output rises, Q3’s collector current rises a little by its own Early effect, so its base current rises, so the current available to Q4 falls, so the potential at Q1’s base moves, so Q2’s collector current moves. The output has modulated the mirror’s own bias.

The measurement of that is the third curve. Replace Q4 with a perfect voltage source at exactly the potential Q4 was holding — nothing a data sheet would name, and not a circuit anybody builds — and the output resistance goes from 7.39 MΩ to 13.0 MΩ, which is $1.06\,\beta r_o$ and is the rung below’s result recovered. Forty-three per cent of the output resistance is spent in the reference branch, and the arrangement that spends it is the one that makes the bias.

The other ceiling, which is visible in the same sweep #

Sweeping the current gain from ten to a hundred thousand puts both of the rung below’s ceilings on one axis, which is worth more than either of them alone.

At small $\beta$ the measurement rises with $\beta$: 867 kΩ at ten, 7.39 MΩ at 150, 22.6 MΩ at 500. At $\beta = 100{,}000$ it stops rising and sits at 269.6 MΩ, which is $g_m r_o^2 = 268$ MΩ to within a per cent. So the honest expression is the one the rung below arrived at — the output resistance is $r_o$ multiplied by whichever of $\beta$ and $g_m r_o$ is smaller — and here both halves of it are measured on one circuit rather than one being inferred.

For silicon bipolar devices $g_m r_o = V_A/V_T$ is about 3,100 and $\beta$ is 150, so $\beta$ binds by a factor of twenty and the field-effect expression is never the right one. A designer who reads $g_m r_o^2$ off a page and builds a bipolar cascode current source has specified a part that is twenty times better than the one that arrives.

The copy, where the two errors move in opposite directions #

The cascode does something to the accuracy of the mirror as well, and it is two things at once with opposite signs.

It adds base current. Q3’s base current is taken out of the output — the emitter current of the upper device is the collector current of the lower one, and only $\alpha$ of it reaches the collector — and Q4’s is taken out of the reference. So the cascoded mirror’s copy is short by about $4/\beta$ where the plain mirror’s is short by $2/\beta$.

It removes the Early error entirely, because Q1 and Q2 now sit at the same collector voltage whatever the output does. That is the 11 per cent the rung below found on a plain mirror with its output five volts above its reference, and it is gone.

Which of the two wins is a question about $\beta$, and the answer is measurable across the whole sweep:

Two things in that table are worth more than the numbers. The sign flips: above a current gain of about forty the plain mirror delivers more than the reference and the cascoded one less, because one is dominated by an Early effect that adds and the other by a base current that subtracts.

And the last row is the whole argument for the arrangement. As the current gain goes to infinity the cascoded mirror’s error goes to nothing, and the plain mirror’s goes to 5.31 per cent and stays there. One of the two errors is curable by process and the other only by topology, and the plain mirror is stuck with the one that is not.

What it costs, which is not what it is said to cost #

Now the headroom, which is the sentence every reference finishes with: a cascode costs a base-emitter drop of output swing.

It does, and the amount is measured rather than assumed. The upper device saturates when its collector falls to about 0.2 V above its emitter, and its emitter sits where the lower device’s collector is — which the solve puts at 0.713 V. So the floor is 0.913 V against the plain mirror’s 0.200, and the cascode has taken 0.71 V it will not give back.

The sentence is also, for a current source, the wrong way round.

A floor is not what a designer has. What a designer has is the range over which the current is what it was set to, and that range has a top as well as a bottom, because a plain mirror’s current never stops climbing: its output resistance is $r_o$, so it rises about one per cent per volt for ever.

Bisecting both ends against the current at five volts, at a band of one per cent:

• the plain mirror holds from 4.15 V to 5.85 V — 1.70 volts of a ten-volt supply;
• the cascoded mirror holds from 0.913 V to the supply — 9.09 volts.

Five point three times the usable swing, from an arrangement whose floor is 0.71 V higher. The headroom sentence is true and it is an accounting of the wrong quantity.

The two spans do not scale together and that is the sharpest way to put it. Ask the plain mirror for a tighter band and its range shrinks in proportion — 8.50, 3.40, 1.70, 0.85, 0.34 volts for bands of 5, 2, 1, 0.5 and 0.2 per cent. Ask the cascode for a tighter band and its range does not move at all, because what bounds it is not the Early effect but the voltage at which a transistor stops being a transistor.

Why the measurement is a solve and not a formula #

Everything above is a direct-current measurement on a nonlinear netlist, and it is worth saying what that buys, because the same numbers can be written down from a small-signal model in four lines.

The four lines require choosing what to keep. The base currents of Q3 and Q4 are exactly the terms a hand derivation drops — they are $1/\beta$ of something — and they are 43 per cent of the output resistance and the entire sign of the copy error. The Early conductance is the term that has to be iterated because its value depends on the answer, and a hand derivation fixes it at a plausible number and moves on.

A netlist has no such decisions in it. What comes back includes every path the circuit has, whether or not the person who assembled it was thinking about that path, and the two findings here are both paths nobody was thinking about.

Where this ladder can go next #

The rung below named three questions and this one answers the first. Two remain, and a third opens.

The folded arrangement puts the upper device on a separate current path so the drops are not stacked. It should give the same output resistance with the floor back, at the price of a second bias current, which would turn this essay’s trade into a question about power.

The wide-swing mirror is the more interesting one, because it attacks the number this essay measured. The 0.913 V floor is a full base-emitter drop plus a saturation voltage, and most of that drop is not needed: the lower device only requires its own 0.2 V. Biasing Q3’s base one $V_{BE}$ plus one saturation voltage above ground instead of two $V_{BE}$ would put the floor near 0.4 V and give back half a volt of a supply that may only be 1.8.

And the noise is still unmeasured, from the rung below. The standard argument is that a cascode device contributes nothing because it is a common-base stage in a low-impedance node. This essay has now found two places where a standard argument about this arrangement omitted a base current, so that one is worth asking rather than repeating.

What the branch depends on elsewhere in the collection #

The forty-three per cent this page ends on is a property of three things measured separately, and the compliance it turns on is measured three more ways in the field. The copy, and its two errors is where the current comes from and where its own compliance limit is set — a limit that contains neither the current nor a resistor — and the two errors it names are both present in this branch. The device that never sees the swing is the same two devices used for bandwidth rather than for output resistance, and it spends the headroom this page is counting. The exponent that is a square is the same arrangement on a square-law device, where one of the two errors disappears entirely. A bias point is a solution, not a choice is why the operating point here is a root rather than a number somebody chose, and it is the same damped Newton that finds it. And Exact outside and wrong within is the warning that goes with reducing this branch to a source and a resistance: the reduction is exact at the terminals and says nothing at all about what is dissipated inside it, which for a branch nobody looks at is the quantity that matters. What all five are distances from is The source that is not a source, which is the model this arrangement exists to approach and the one the solver has to refuse.

What is checked #

The plain mirror’s output resistance is asserted to be $r_o$ at its own collector current — not at the cascode’s, because at a small current gain the two branches do not carry the same current and comparing against the other circuit’s $r_o$ compares two operating points.

The lift the cascode gives is asserted against whichever of $\beta$ and $g_m r_o$ is smaller, rather than against a number, so the check carries the mechanism and holds at every setting of the slider rather than at the one it was written at. At the top of the sweep the measurement is asserted to land on $g_m r_o^2$ to within a quarter, which is the other ceiling appearing.

The held-base experiment is asserted to give more output resistance than the diode stack, which is the claim that the reference branch is where the missing resistance goes. The copy error is asserted to go to nothing for the cascode and to a floor above four per cent for the plain mirror as $\beta$ rises, and to have opposite signs at ordinary $\beta$ — three statements that between them say which error is which.

The compliance figure asserts that the cascode’s floor is higher — the cost, kept in the same figure as the benefit — and that its span is wider anyway at every band on the slider. That the plain mirror’s span is proportional to the band while the cascode’s is not is a claim across settings, so it is in the site’s gate rather than in the figure.

What is not modelled: the base-collector capacitance, so nothing here is a bandwidth; the mismatch between the four devices, which in a real mirror is a larger error than either of the two measured here and is a statistical statement rather than a solved one; the temperature, since every number is at 300 K; and the noise.

Forty-three per cent, in the branch nobody looks at #

The result worth carrying out of this essay is not the factor of ninety in output resistance; it is that forty-three per cent of the resistance which should have been there is missing, and that it is missing in the reference branch rather than in the output one.

That is a mistake with a shape the collection has met before. The device that never sees the swing is the same device in the same arrangement, and its finding is the same kind: the output resistance is not $g_m r_o^2$ — 248 megohms for the case measured there — but $\beta r_o$, 11.5 megohms, because the upper device’s base draws current and shunts the very node the improvement was supposed to appear at. An expression with no ceiling in it, and a real ceiling set by a quantity the expression does not contain.

In both cases the correction comes from asking where the current actually goes rather than from refining the expression, and in both cases the answer is a node the small-signal picture draws as an open circuit. That is the argument for solving the netlist rather than evaluating the formula, made twice on one device — and it is why this essay reports the range over which the current is what it was set to (1.70 volts to 9.09) alongside the resistance, since a resistance is a slope and a range is what a bias network is actually bought for.

The range being the more useful number is worth one more sentence, because the two are not the same kind of quantity. An output resistance is a derivative at a point and says how much the current moves for a small change in voltage; a range is a statement about a tolerance being met across an interval, and it is what a specification is written in. A plain mirror’s current never stops climbing, so it has no range at any tolerance until one is stated — which is why 82 kΩ and 7.39 MΩ understate the improvement and 1.70 volts against 9.09 does not.

That distinction is the same one the source that is not a source draws for a voltage source, where an internal resistance is a slope and the edge is the current at which a stated tolerance is exceeded. A mirror is the current-source half of the same argument, and it has the same two ways of being described.

The symmetry between the two is exact and worth stating once. A voltage source is characterised by a resistance that should be zero and has an edge on the current axis; a current source by a resistance that should be infinite and has an edge on the voltage axis. Every result in this collection about one of them has a counterpart about the other, and the headroom this essay spends is the current-source version of the dropout voltage a regulator spends.