The node the bandwidth belongs to

The frequency a device sets for itself built a common-emitter stage twice — as the network, with its two picofarads between collector and base, and as the Miller approximation, with that capacitance moved to the input and multiplied by one plus the gain — and found the approximation optimistic: 643 kilohertz predicted against 504 solved. It named what the approximation has no room for: a second pole, and a zero in the right half-plane at the transconductance over the collector-base capacitance.

It measured one stage driven from one source, a kilohm. The Miller estimate is an estimate of one node — the input — and a stage has two nodes with capacitance at them. Which of the two sets the bandwidth depends on how fast each one is, and the input node’s speed is set by the source driving it. This essay sweeps the source over five decades and watches the bandwidth move from one node to the other, and along the way finds that much of the earlier essay’s optimism came from the gain the multiplier was given rather than from the approximation itself.

Two netlists, and the gain the multiplier uses #

The stage is the earlier essay’s: a bipolar transistor at a milliamp, its small-signal model of $r_\pi$, $g_m$ and $r_o$, twenty picofarads from base to emitter, two from collector to base, a five-kilohm collector load. The network is that, solved at each frequency. The estimate is a second netlist in which the collector-base capacitance has been removed and an input capacitance of $C_\pi + C_\mu(1 + A)$ added at the base. Both bandwidths are found the same way, by bisecting the solved magnitude for its −3 dB point, so the comparison is between two circuits and not between a circuit and a formula.

The multiplier’s gain A deserves care, and getting it right is the first finding. The Miller effect is the current that flows through $C_\mu$ because the collector swings $A$ times the base, in the opposite direction; the gain that sets it is the gain from base to collector, $g_m$ times the collector’s total load — 182 here. The earlier essay used the stage’s midband gain from the source to the collector, 144.7, which is smaller by the divider between the source resistance and $r_\pi$. That shrinks the Miller capacitance by a fifth, and it is most of the reason the estimate came out 22 per cent optimistic: with the gain from base to collector, the same stage from the same kilohm is estimated at 523 kilohertz against 504 solved, 3.8 per cent high. Every figure below uses the gain from base to collector.

Close from a high source, far off from a low one #

From a high source the estimate is very good: within one per cent from ten kilohms up. From a low source it is very bad: 3.41 times the solved bandwidth from ten ohms. The transition is spread over two decades, from about thirty ohms — where the input and output nodes’ time constants are equal — to about a kilohm, where the input node’s has become much the larger.

The estimate is not wrong about the input node at any source. Its prediction is the input node’s time constant, $(R_s \parallel r_\pi) \cdot (C_\pi + C_\mu(1+A))$, and that is the input node’s speed. What it leaves out is the output node: the collector’s own resistance to ground, $r_o \parallel R_L$, charging the collector-base capacitance and whatever else sits at the collector. From a high source the input node is slow, its time constant dominates, and the output node’s contribution is a small correction. From a low source the input node is fast — ten ohms charges even 386 picofarads in four nanoseconds — and the output node, which the estimate cannot see, is the slower of the two.

Two poles that split as the source rises #

The network has two poles, and the source moves them very differently. The lower pole falls by two decades as the source resistance rises over four, because it is mostly the input node’s. The upper pole falls by less than half a decade, because it is mostly the output node’s, and the source reaches it only through $C_\mu$. So the poles separate as the source rises — 92 times apart from ten ohms, nearly three thousand from a hundred kilohms — and a stage with two widely separated poles has a bandwidth set by the lower one alone. The Miller estimate’s single pole lies on the lower pole once they are well apart, from about a kilohm, and above it where they are not.

This is pole splitting without a feedback capacitor added for the purpose: $C_\mu$ is itself a feedback capacitor, and its Miller multiplication is exactly the mechanism a compensation capacitor uses deliberately to split an amplifier’s poles. The source resistance is the knob that decides how strongly it splits them, because it sets how much the input node’s impedance lets the multiplied capacitance act.

A load the estimate cannot see #

If the output node sets the bandwidth from a low source, anything that slows the output node should matter there and not elsewhere — and the Miller estimate, which has no output node, should miss it entirely.

From ten ohms, twenty picofarads at the collector cuts the bandwidth eightfold, from 12.1 megahertz to 1.49. From a hundred kilohms the same twenty picofarads cuts it by six per cent. The Miller estimate predicts the same bandwidth for every load. A designer who estimates a fast stage driven from a low impedance by Miller’s rule, and then connects it to a cable or the next stage’s input capacitance, has estimated the wrong node, and the error is not a percentage but a factor.

The output node’s time constant is $(r_o \parallel R_L)(C_\mu + C_L)$, and from a low source it is the bandwidth to within the small interaction of the two nodes. That is the estimate the open-circuit time-constant method makes systematically — every capacitor’s time constant summed — and it is why that method stays accurate where Miller’s does not: it includes the output node by construction.

The load resistor moves the gain and the node at once #

A larger collector load raises the gain, which the estimate counts — the Miller capacitance grows with it — and raises the output node’s resistance, which the estimate does not count. From a high source the first effect dominates and the estimate tracks the network across gains from 19 to nearly twelve hundred. From a low source the second matters too, the output node slows faster than the estimate expects, and its optimism grows from 12 to 26 per cent across the same loads.

The two curves also say what a designer gains by raising the load to raise the gain. From a high source, gain and bandwidth trade as Miller’s estimate says they do, in near-constant product, because the input node carries the multiplied capacitance and it is the slower node. From a low source the product falls as the load rises, because the output node slows in proportion to its resistance while its gain-bandwidth benefit goes to a node that was not limiting anything. A higher-gain stage driven from a low impedance is therefore slower by more than the gain it adds, and the measured optimism of 26 per cent at the largest load is the size of that extra cost.

So the rule for using the estimate reads off the source resistance alone. When the source resistance is much larger than the collector’s load divided by the gain — roughly when the input node is more than a few times slower than the output node — Miller’s estimate is accurate to a few per cent at any gain. When it is not, the estimate is an input-node number for a stage whose bandwidth belongs to the output, and it is optimistic by however much slower the output node is.

The estimate that includes both nodes #

The open-circuit time-constant method sums, over every capacitor, that capacitor times the resistance it sees with every other capacitor removed, and takes the reciprocal as the bandwidth. A sum that is exact, and the estimate that is not found that the sum is exactly the first coefficient of the network’s characteristic polynomial — the sum of the reciprocals of the poles — and that the bandwidth estimate built on it is therefore a statement about all the poles at once. For this stage the sum has three terms: $C_\pi$ through the input node’s resistance, $C_\mu$ through a resistance that includes the Miller multiplication, and any load capacitance through the output node’s resistance.

The Miller estimate is the first two of those terms with the output node’s contribution to the $C_\mu$ term dropped. That is why it agrees with the time-constant sum when the input node dominates and departs when the output node does: the term it drops is the one that grows as the source falls. Where the estimate stops being a bound found the time-constant estimate itself becoming optimistic when a network’s poles are complex or close; the poles here are always real and always at least 92 times apart, which is the regime in which the time-constant sum is reliable, and so the regime in which it is the right replacement for Miller’s number from a low source.

What drives the stage from a low source #

A source of ten or a hundred ohms is not exotic. It is what an emitter follower presents, and the classic way to drive a common-emitter stage hard is to put a follower in front of it. The buffer that is not a buffer found a follower’s output impedance rising with frequency, so the source it presents is low at low frequency and becomes inductive near the stage’s bandwidth; the input that pushes back found a capacitive load at a follower’s output reflecting into its input as a negative resistance. A follower driving a common-emitter stage’s Miller capacitance is exactly that load, and the combination can peak or ring at the frequency where the two interact.

So the low-source regime this essay maps is the regime of a stage driven by a follower, and in that regime the stage’s bandwidth belongs to its own collector — the output node — while the interaction at its base belongs to the follower. Neither is described by the Miller estimate, which in that configuration reports a bandwidth several times too high and says nothing about the peaking.

Splitting the poles on purpose #

The source resistance splits the stage’s poles by controlling how much the multiplied $C_\mu$ acts on the input node. An amplifier designer splits poles deliberately by adding capacitance across a gain stage — Miller compensation — so that one pole falls to become dominant and the other rises out of the way. The figures here are that mechanism with the knob turned by the source rather than by an added capacitor: from a high source the poles are three thousand times apart and the stage has a clean dominant pole; from a low source they are less than a hundred apart and the second pole is near enough to add phase at the bandwidth.

The gain added to buy a margin and the tee that charges for its own compensation both measured what a loop pays to keep its second pole away from its crossover. A common-emitter stage inside such a loop contributes its own pair of poles, and how far apart they are is set by the impedance driving its base — a fact that shows up in a loop’s margin as a second pole that moves when the stage before it is changed.

Where this reading is used #

The practical cases sort themselves by source. A stage driven from a signal source or a sensor of a kilohm or more — an audio preamplifier, an instrumentation front end — is the regime where Miller’s estimate works, and where a cascode is the standard remedy: the noise the cascode device does add measured what a cascode costs in noise, and what it buys is the removal of the Miller multiplication by holding the collector still. A stage driven from the low-impedance output of a previous stage — the inside of a multi-stage amplifier, a driver following an emitter follower — is the regime where the output node decides, and there the remedy is not a cascode but less capacitance at the collector or a lower collector resistance.

The same split governs an operational amplifier’s input stage driven from different sources. The current the instrument draws found the input current mattering only from high source impedances; the input node’s bandwidth matters in the same regime, and from a low source the amplifier’s own internal nodes set the response regardless of how the input is driven.

What the measurement leaves out #

The transistor is its small-signal model at one bias point. Its capacitances depend on the bias — $C_\pi$ grows with the collector current, $C_\mu$ shrinks with the collector-base voltage — and a stage biased differently has different time constants at both nodes; the crossing between the two regimes moves with them. The base resistance $r_b$, absent here, adds in series with the source and sets a floor under the effective source resistance, so a real stage never reaches the ten-ohm regime however low its source: its input node is never faster than $r_b$ allows.

The right-half-plane zero at $g_m/C_\mu$, which the earlier essay measured, is present in every network here and contributes a little phase lag at the bandwidth; it is far above the bandwidth at every source, at about 3 gigahertz, and moves none of the bandwidths measurably.

And the earlier essay’s figures still carry its 22 per cent, computed with the source-referred gain. Its argument — that the approximation has no room for a second pole or a zero — stands; its headline number overstates how far the approximation misses at a kilohm, and the right comparison from that source is 523 against 504 kilohertz.

The number to keep #

The Miller estimate is the input node’s time constant, and it is the stage’s bandwidth only when the input node is the slower one: within one per cent from ten kilohms of source at any gain from 19 to 1,190, 24 per cent optimistic from 100 ohms and 3.41 times from ten. The stage’s two poles split from 92 to 2,993 times apart as the source rises, and from a low source the output node sets the bandwidth — 20 picofarads there cuts it from 12.1 megahertz to 1.49, while from 100 kilohms it moves it six per cent. The multiplier is the gain from base to collector, 182, not the 144.7 from the source; with it the estimate from a kilohm is 3.8 per cent high, not 22.

Still open: the base resistance, the cascode’s two nodes, and the stage that drives the next #

The base resistance as a floor. Adding $r_b$ in series with the source bounds how fast the input node can be, so a real stage’s bandwidth from a low source is set by a competition between the output node and $r_b$’s input time constant. Where that crossing falls for a stated $r_b$ is the practical version of the ten-ohm regime.

A cascode’s two nodes. A cascode removes the Miller multiplication at the input by holding the first collector at a low impedance, and adds a node at the second collector. Measuring which of its three nodes sets the bandwidth, against source resistance, extends this map to the stage that was invented to change it.

A chain of stages. In a multi-stage amplifier each stage’s source is the previous stage’s output resistance, so the regime is set stage by stage by the circuit, not by the designer’s choice of source. Solving a chain would show how often the output-node regime is the one a real amplifier is in.