The device that never sees the swing

A cascode is one transistor’s collector driving another’s emitter, with the second one’s base held at a fixed voltage. It is one of the oldest arrangements in the subject and it is usually introduced with two claims: that it removes the Miller effect, and that it multiplies the output resistance by the intrinsic gain of the upper device.

The first is true and this essay measures how true. The second is not, for a bipolar device, and the expression that is usually given for it has no ceiling in it at all — which is a shape of error this collection is built to find, because an expression with no ceiling is a model with no stated edge.

Nothing about the lower device changes #

The first thing to establish is what the arrangement does not alter, because it is what makes the comparison fair.

The lower transistor carries the same current it did, so its transconductance is the same, its input resistance is the same, and its own output resistance is the same. The signal path through it is unchanged. Everything this essay measures is a consequence of what the lower device now sees at its collector, and of nothing else.

What it sees is an emitter. The upper device presents about $1/g_m$ there — 26 ohms at a milliamp — which is three thousand times smaller than the load the collector used to drive. So the voltage gain from base to the lower collector falls from about 1400 to about 1.8, and the stage’s gain reappears at the upper collector where it always was.

That is the whole mechanism, and both of the arrangement’s effects come from it.

The bandwidth, measured rather than quoted #

The base-collector capacitance is drawn into the input multiplied by one plus the gain across it. With a gain of 1400 across it, two picofarads is nearly three nanofarads at the base; with a gain of 1.8, it is 5.5 picofarads.

Measured — by bisecting each stage’s solved magnitude for its own half-power point, rather than by identifying a pole — the plain stage into a hundred kilohms manages 56.1 kilohertz and the cascode 787, a factor of fourteen. Into a kilohm the two are 504 kilohertz and 6.87 megahertz, a factor of 13.6. The factor is roughly constant because it is roughly the ratio of two Miller multipliers, and both scale with the load.

The reason to bisect rather than to root is the one this field established when it compared the Miller approximation against the network: the approximation predicts one pole and the network has two and a right-half-plane zero, so “the pole” and “the bandwidth” are different numbers and only the second is what a reader wants. Both stages here are measured the same way, so the comparison is between two circuits rather than between a circuit and a formula.

The output resistance, and the expression that has no ceiling #

Now the second claim.

Looking into the upper device’s collector, a change in the output voltage is fed back through that device’s own $r_o$ to its emitter, where the lower device’s $r_o$ resists it. The change in current is therefore smaller than it would have been by roughly $g_{m2}r_{o2}$, and the usual expression follows: $R_\mathrm{out} \approx g_m r_o^2$.

At a milliamp with an Early voltage of 80 volts that is 38.7 millisiemens times 80 kilohms squared — 248 megohms. The measurement, made by driving the output node with a current source and reading the voltage, is 11.5 megohms. The expression is out by a factor of twenty-one.

What is missing from it is the upper device’s base current. A bipolar transistor’s base is not an open circuit: $r_\pi$ — 3.9 kilohms at a milliamp — sits from the lower device’s collector to signal ground, because the upper base is held at a fixed voltage and a fixed voltage is a signal ground. That resistance shunts the very node the feedback has to act through. Instead of the feedback current meeting $r_{o1} = 80\,\mathrm{k}\Omega$, it meets $r_{o1} \parallel r_\pi = 3.7\,\mathrm{k}\Omega$, and the multiplication is by $g_m(r_{o1}\parallel r_\pi)$ rather than by $g_m r_{o1}$.

Since $g_m r_\pi = \beta$ by definition, the ceiling is

$$R_\mathrm{out} \to \beta r_o$$

which is 12.0 megohms for the numbers here, against 11.5 measured — a tenth. The multiplication stops at the current gain, and no Early voltage improves it past that.

Which ceiling binds, and when it does not #

The two expressions are $g_m r_o^2$ and $\beta r_o$, and their ratio is $g_m r_o/\beta$ — the intrinsic gain over the current gain. So the honest statement is that the output resistance is $r_o$ multiplied by whichever of $g_m r_o$ and $\beta$ is smaller, and for silicon bipolar devices that is almost always $\beta$: the intrinsic gain $g_m r_o = V_A/V_T$ is 3095 at these numbers and does not depend on the current at all, while $\beta$ is 150.

Two consequences, and one of them is why the usual expression survives.

For a field-effect device the base current is zero and $r_\pi$ is infinite, so nothing shunts the node and $g_m r_o^2$ is right. The expression is not wrong, it is written for a different device, and it travelled into bipolar textbooks by resemblance. Every number in this essay would be different for a transistor whose gate draws nothing.

And for a bipolar device the ceiling moves with $\beta$, which the figure sweeps: from 810 kilohms at $\beta = 10$ to 220 megohms at $\beta = 3000$. It is worth having as a design statement because $\beta$ is the least controlled parameter a bipolar transistor has — a three-to-one spread within a batch is ordinary — so a cascode’s output resistance has a three-to-one spread too, where the same arrangement in a field-effect process does not.

Why the measurement is a current source and not an expression #

The output resistance in this essay is measured by putting a one-ampere current source at the output node and reading the voltage. That is worth a paragraph because it is the difference between this collection and a textbook derivation, and because it is what caught the missing term.

An expression for an output resistance is derived by drawing the small-signal circuit, choosing which elements to keep, and solving the reduced circuit by hand. Every step is a decision about what to neglect, and the decision that produces $g_m r_o^2$ is neglecting $r_\pi$ of the upper device — which is entirely reasonable when the derivation is being done for a field-effect transistor and is silently carried over when it is not.

A netlist has no such decision in it. The elements are stamped, the matrix is solved, and what comes back includes every path the network has whether or not the person who wrote it was thinking about them. The $r_\pi$ that costs a factor of twenty-one is present because the model of a bipolar transistor has one, not because anybody remembered it.

That is the general case for measuring rather than substituting, and it applies to the two other quantities here as well: the bandwidth is bisected on a solved magnitude rather than read from a pole because the stage has two poles and a right-half-plane zero, and the inner node’s gain is solved rather than approximated as $-g_m/g_m = -1$ because it is 1.76 and not 1.

What it costs is supply, and that is not in either expression #

Neither of the two effects above appears in the third quantity, which is the one that decides whether a cascode can be used at all.

The upper device needs its own collector-emitter voltage. It is biased with its base at a fixed potential — two volts here — and its emitter sits a base-emitter drop below that, so the lower device’s collector is at about 1.3 volts and the upper one’s collector cannot go below about 2.2 without saturating. On a five-volt supply that leaves 2.8 volts of headroom against the plain stage’s 4.8: the arrangement has taken two volts of signal swing, permanently, in exchange for the bandwidth and the output resistance.

At five volts that is a forty per cent reduction in the largest signal the stage can produce. At the 1.8 volts a modern process runs on it is most of the supply, which is the single reason cascodes went from being the default analogue arrangement to being a thing that has to be justified — and why the folded-cascode arrangement, which puts the upper device on a different current path so that the headroom is not stacked, exists at all.

So the trade is three-sided and only two of the sides are usually stated. More output resistance and more bandwidth, for less swing, and the exchange rate is set by the supply rather than by the device.

The gain–bandwidth product, which does not behave as expected #

One measurement here contradicted the guess that produced it, and the result is better than the guess.

A plain common-emitter stage’s gain–bandwidth product is nearly a constant of the device: measured across three decades of load resistance it moves by 1.27 times. That is the Miller effect being exactly self-cancelling — the input capacitance is $C_\mu$ multiplied by the gain, so a load that doubles the gain halves the bandwidth and the product does not move.

A cascode’s is not constant. It rises by 9.6 times across the same three decades, because there is no multiplication to cancel against: the input capacitance is fixed at roughly $C_\pi + 2C_\mu$ whatever the load, so gain bought at the load is not paid for in bandwidth.

That is the sharpest statement of what the arrangement is for, and it is the opposite of the intuition that a cascode “trades” anything in the frequency domain. It does not trade; it removes a coupling. A designer who needs both gain and bandwidth from one stage has, in a plain common-emitter, a fixed budget to divide — and in a cascode, two independent quantities.

Where this ladder can go next #

This is a first rung and it opens onto three questions the collection has the machinery for.

The cascode current mirror replaces the resistance the copy is compared against with one of these, and should therefore move the mirror’s output impedance from $r_o$ to $\beta r_o$ — which is a fourteen-fold improvement in the compliance-limited region and a claim this field’s mirror essay could be checked against directly.

The folded arrangement puts the upper device on a separate current path, so the headroom is not stacked. It costs a second bias current and it should give the same bandwidth and output resistance with most of the swing back, which would make the three-sided trade of this essay into a two-sided one at a price in power.

And the noise is the unmeasured one. The standard argument is that a cascode device adds nothing because it is a common-base stage in a low-impedance node, and this collection has been shown enough standard arguments that fall over under measurement — the recovery loss in the wrong component, the gain-bandwidth product that rises rather than trading — to want that one checked rather than repeated.

Where this sits among the stage’s other limits #

The cascode removes one of the four things that bound a common-emitter stage and leaves the other three where they were. The frequency a device sets for itself measures the one it removes — two picofarads between collector and base, multiplied by the gain — and is the reason this arrangement exists at all. The copy, and its two errors supplies the current the stage is biased with, and its compliance is the reason the upper device has a volt to spare rather than three. The buffer that is not a buffer is what usually follows, and it undoes some of what was gained here by presenting a capacitance of its own. And How small is small signal is the boundary none of them moves: the cascode holds one node still, and the exponential at the other is exactly as nonlinear as it was.

What the arrangement is worth elsewhere #

Two devices in series is used twice in this field for two different reasons. The source that holds to the supply uses it for output resistance and compliance rather than for bandwidth. The exponent that is a square is where the same arrangement on a square-law device loses one of its two errors entirely.

What is checked #

The plain stage’s output resistance is asserted to be exactly $V_A/I_C$, which is the number the cascode is multiplying and which anchors everything else. The cascode’s is asserted to be at least five times below $g_m r_o^2$ — so that the essay’s central claim would fail loudly if the shunting mechanism were removed — and to equal $\beta r_o$ to within a tenth. The ceiling is asserted to move by more than twenty times across the swept range of current gain, which is what makes it a ceiling set by $\beta$ rather than by the Early voltage.

The bandwidth is asserted to be larger for the cascode at every load, and the inner node’s gain to stay in single figures across a range over which the stage’s gain spans two orders — which is the mechanism, stated as a measurement. The headroom is asserted to be smaller for the cascode, so that the cost appears in the same figure as the benefit. And the two gain–bandwidth products are asserted in the direction the measurement found rather than the direction the first version of this essay guessed.

What is not modelled: the upper device’s own base-collector capacitance to the output node, which is in the netlist but is not multiplied by anything and therefore does not carry an argument; the finite impedance of the bias source holding the upper base, which if it is not a signal ground converts some of the output swing back onto the inner node; and the noise, where a cascode’s contribution is usually argued to be negligible and this collection has not checked it.

Where the two volts are spent, and where they cannot be #

Two volts of a five-volt supply is the number that decides whether this arrangement is available, and the collection has two circuits where it decides it in opposite directions.

What gets through from the rail is where it is refused. A regulator’s rail rejection improves by exactly its pass device’s intrinsic gain — sixty decibels for the device measured there — and a cascode would raise that gain and therefore the rejection. It cannot be fitted, because the whole specification a low-dropout regulator is sold on is the difference between its rail and its output, and two volts of it is not available to spend.

The source that holds to the supply is where a smaller version of the same cost is paid willingly: cascoding a current mirror raises the floor by 0.71 volts and takes the output resistance from 82 kΩ to 7.39 MΩ, with the range over which the current is actually what it was set to going from 1.70 volts to 9.09. Seventy centivolts of headroom for a factor of ninety in resistance and a factor of five in useful range is a trade almost any bias network takes.

The difference between the two is not the size of the improvement but whose budget the headroom comes out of. A mirror inside an integrated circuit spends headroom that nothing else wanted; a regulator’s pass device spends the quantity the part exists to minimise. Which is the general form of what $\beta r_o$ against $g_m r_o^2$ costs: the expression everybody quotes has no ceiling in it and no supply voltage in it, and the second omission binds before the first.

The bandwidth half of the result has no such ceiling, and that asymmetry is what decides where a cascode is actually used. Fourteen times is the whole of the Miller multiplication removed, and the Miller multiplication is a product of a capacitance and a gain rather than a device parameter — so the improvement is available in full whatever $\beta$ does, and it is available at the input of a stage whose output swing is small. Which is exactly the arrangement inside an integrated amplifier’s first stage, where the swing is a few millivolts and the headroom is spent on something with a return.

Which is why the two results on this page should be quoted separately rather than as one improvement. The resistance gain is real, smaller than the expression says, and unavailable wherever headroom is the specification; the bandwidth gain is real, exactly what the expression says, and available almost everywhere. A summary that offers both together will be right about one of them in any given application and wrong about the other. Which is the ordinary fate of a technique described by what it does rather than by what it costs: two independent improvements with two different prices, sold as one. Splitting them is what this essay is for, and the split is available because the two are measured separately on the same netlist rather than inferred from one expression.