The frequency a device sets for itself

A transistor has capacitance inside it, and one of the capacitances is in a much worse place than the other.

The capacitance across the base-emitter junction sits at the input, in parallel with everything else at the input, and does what a capacitance at an input does. The capacitance between base and collector sits across the stage, and the stage has a large inverting gain. Two picofarads connected between two nodes that move in opposite directions by a factor of a hundred and forty-five draws a current a hundred and forty-six times larger than its own admittance suggests, and the source has to supply all of it.

What the approximation says, and what it leaves out #

The Miller argument is one of the genuinely useful approximations in the subject, and it is short. The current through a capacitance between two nodes depends on the voltage across it. If one node carries v and the other carries −Av, the voltage across it is (1 + A)v, so the current it draws from the first node is (1 + A) times what it would draw if the far end were grounded. As far as the input is concerned, a capacitance $C_\mu$ between input and output is a capacitance $C_\mu(1 + A)$ to ground.

For the stage drawn here that turns 2 pF into 291 pF, which with the 20 pF already across the input gives 311 pF where a naive count would have said 22 pF. That is a factor of fourteen in the input capacitance and the single most consequential number in the stage.

What the argument then does — and this is where it stops being exact — is treat the resulting circuit as having one pole. It does not. The original network has capacitance at two nodes and a forward path through the capacitance, so it has two poles and a zero; the approximation has one node with capacitance and therefore one pole and nothing else.

Both as networks, measured the same way #

The comparison here is deliberately not between a circuit and an expression. Both the stage and its Miller approximation are built as netlists — the approximation is a real circuit with a 311 pF capacitor in it — and both are solved and bisected for their −3 dB points by the same routine.

The measured −3 dB point of the network is 503.89 kHz. The measured −3 dB point of the approximation is 642.84 kHz. The approximation is 21.6% optimistic.

That figure holds its character across the slider. At half a picofarad it is 18.5% optimistic, at two picofarads 21.6%, at eight 22.6% — growing slowly and never small. A designer who uses the Miller estimate and finds the measured bandwidth twenty per cent lower has not made a mistake; they have used a model whose error is twenty per cent.

The zero the approximation cannot have #

The network has a zero in the right half of the complex plane, at 3.078 GHz for the two-picofarad case, and it is at exactly $g_m/2\pi C_\mu$.

That equality is asserted rather than observed: the transconductance is 38.68 mS and the capacitance 2 pF, giving 3.0782 GHz, and the zero recovered from the network’s polynomial agrees to better than a part in 10⁹. It is worth knowing where it comes from, because a right-half-plane zero is a strange object and this one has a very concrete cause.

At any frequency the output current has two contributions: the transistor’s, $g_m v$, flowing from collector to emitter, and the capacitance’s, $sC_\mu v$, flowing forward from base to collector. They oppose. At low frequency the transistor wins by a wide margin. At $s = g_m/C_\mu$ they are equal and the output is zero. Above it the capacitance wins, and the signal that reaches the collector has gone straight through the capacitor rather than through the transistor — with the opposite sign, which is what “right half-plane” means for the phase.

A signal path through the component that was supposed to be a parasitic is a very literal description of what a right-half-plane zero is, and it explains why the Miller approximation cannot represent it: the approximation removed the component from between the two nodes and put it at one of them, which is exactly the step that deletes the forward path.

Two poles, and the trouble with finding the second #

The network’s poles come out at 503.9 kHz and 335.9 MHz — a factor of 667, which is 2.82 decades.

Recovering both took a change of units for as long as this page has existed, and the reason is worth recording because it looks like a numerical detail and was a real bug. This site’s transferFunction lifts the characteristic polynomial out of the matrix by sampling on a circle and then discards leading coefficients that are numerical dust rather than degree. The test for dust compared each coefficient against the largest in the same array — which sounds right and is not, because a polynomial in s whose roots are at 3×10⁵ has coefficients spanning (3×10⁵)ⁿ whether or not any of them is dust. With two poles 2.82 decades apart the s² coefficient is 1.5×10⁻¹⁶ of the constant term; the trim removed it, the recovery returned a first-order polynomial, and the stage appeared to have one pole — which is precisely the claim this page exists to contradict.

The workaround was to measure time in units of 1/ω₀ before asking, so that the roots land near one and the coefficients land within a few decades of each other. It works, it is still in the library, and it was the wrong place to fix it: the sampler already knows the radius it sampled on, and a term’s size there is exactly what says whether it is dust. The trim is told that radius now, and the plain call returns both poles unaided. The same repair moved this page’s own bandwidth from 503.13 kHz to 503.89 kHz, which is the next section.

Why the second pole matters anyway #

Three and a half decades is a long way, and it is tempting to conclude that the second pole can be ignored. For the magnitude it very nearly can, and the size of “very nearly” is now measurable rather than confounded: the measured −3 dB point of 503 886.1 Hz differs from the dominant pole’s 503 887.2 Hz by two parts in a million, which is the whole of what the second pole and the zero do to the corner frequency.

That number used to read 0.15%, and the difference between the two is a lesson rather than an improvement. The 0.15% was not the second pole. It was the trim of the previous section discarding the s² coefficient in one of the two routes and not the other, so the page reported a discrepancy between a measurement and a pole and then explained it with the physics that happened to be nearby. A gap of the right order of magnitude is the easiest kind of wrong number to keep.

For the phase it cannot. A pole contributes −45° at its own frequency and −5.7° a decade below it, and the feedback field is entirely about what an extra few degrees of lag at the crossover frequency does to a margin. Two measurements of one margin shows a phase margin from the loop gain and one from the measured overshoot agreeing to 0.1°, and what makes that a strong result is that a tenth of a degree is a quantity the circuit actually notices.

A stage whose second pole is at 336 MHz is contributing about 0.09° of lag at 504 kHz, which is nothing. Put three such stages in a loop that crosses over at 30 MHz and each contributes 5°, and the design has spent fifteen degrees of margin on poles that a single-pole model does not contain.

Where the twenty-two per cent comes from #

The approximation’s error is unusually stable across the slider — 18.5%, 21.6%, 22.6% for half, two and eight picofarads — and stability of that kind usually means a single mechanism rather than several.

It is one mechanism, and it is the load side. The Miller argument accounts for what $C_\mu$ does at the input and silently assumes it does nothing at the output. It does something: the same capacitance is also across the output node, contributing $C_\mu(1 + 1/A)$ there, which for a gain of 145 is very nearly $C_\mu$ itself. Two picofarads across a load of a few kilohms is a second pole in the hundreds of megahertz — the one recovered above at 336 MHz — and it is far enough away to move the −3 dB point by only a fraction of a per cent.

So the output side is not the explanation. The explanation is at the input, and it is that the gain used in the multiplication is not constant. The Miller factor (1 + A) uses the midband gain, and at the −3 dB point the stage’s gain has already fallen by a factor of √2 — so the actual multiplication near the corner is smaller than the one the approximation used, which should make the approximation pessimistic rather than optimistic.

The sign works out the other way because the approximation compares the wrong pair of circuits. Its single pole sits at $1/2\pi R_s C_\mathrm{in}$ with $R_s$ the source resistance in parallel with $r_\pi$, a combination the real network also has — and the real network’s dominant pole additionally sees the zero pulling the response down and the second pole starting to contribute. The measured pole of the network, 503.9 kHz, is below the approximation’s 642.8 kHz by very nearly the ratio of the two input capacitances as the frequency approaches the corner.

The useful conclusion is not the mechanism but its stability: an error that sits between eighteen and twenty-three per cent over a sixteen-fold variation in the parameter is a correctable one. The measured ratio of solved bandwidth to predicted is 0.811, 0.783 and 0.774 at half, two and eight picofarads, so multiplying the Miller estimate by about 0.78 lands within four per cent everywhere on the slider. That is what an experienced designer’s rule of thumb amounts to, and it is the kind of correction that never gets written down because the approximation it corrects is usually quoted without an error in the first place.

What the schematic is not telling anyone #

There is a drawing convention buried in all of this that is worth surfacing, because it is the reason the Miller effect surprises people who have looked at the schematic a hundred times.

$C_\mu$ is not on the schematic. It is inside the transistor symbol, along with $C_\pi$, and the symbol shows neither. A stage drawn with a source, a resistor, a transistor and a load looks like a circuit with no capacitors in it at all, and its bandwidth looks like something that ought to be infinite. The two picofarads that decide the answer are in the datasheet, three pages in, under a heading that does not mention bandwidth.

This site’s habit is that a schematic is a label rather than a drawing — the layout carries no information, so it is put small and in a corner and the canvas is given to the response. The corollary is on this page: a schematic that omits the components that decide the answer is not a compressed version of the circuit, it is a different circuit. The figures here show the netlist that was actually solved, which has every capacitance in it explicitly, and that is the only version in which the answer is derivable from the picture.

The transit frequency, and what a stage actually gets #

The device’s own figure of merit is the transit frequency: the frequency at which its current gain falls to one, $g_m/2\pi(C_\pi + C_\mu)$. For the parameters here that is 279.8 MHz.

The stage delivers a gain of 144.7 and a bandwidth of 504 kHz, whose product is 72.9 MHz. So the circuit gets about a quarter of the device’s transit frequency as a gain-bandwidth product, and where the rest went is the substance of the stage’s design: the source resistance drives the input capacitance, and that combination decides the pole rather than anything internal to the device.

The consequence is that a stage’s bandwidth is not a device property at all. Change the source resistance and it moves; change the load and both the gain and the Miller multiplication move with it. The transit frequency is an upper bound the arrangement cannot exceed, and the arrangement usually falls a long way short.

What is being claimed #

The stage has a bandwidth of 504 kHz, and that number came out of a solved network rather than an expression.

The Miller approximation predicts 643 kHz. It is a good approximation in the sense that matters most — it identifies the right mechanism, gives the right dependence on gain and capacitance, and is wrong by a fixed twenty per cent rather than by an amount that varies wildly — and it is a bad one in the sense the site cares about, which is that it is normally quoted with no error attached.

The two things it cannot represent at all are the second pole and the right-half-plane zero, and both are consequences of the same step: moving a component from between two nodes to one of them deletes the path through it. That step is what makes the approximation tractable and it is what makes it incomplete, and the two facts are the same fact.

The site’s rule asks every figure to carry the frequency at which the model in it stops being true, and this page has three answers rather than one, which is unusual and worth ending on. The bandwidth — 504 kHz — is where the stage’s own flat-gain model gives out. The second pole — 336 MHz — is where its single-pole model gives out, and that is 2.82 decades higher and matters only to phase. The zero — 3.08 GHz — is where the transistor stops being the thing that carries the signal, and above it the circuit is a capacitive divider with a transistor attached.

Three boundaries, spread over 3.79 decades, all belonging to the same two picofarads. Which one is “the” limit depends entirely on what is being asked, and a single number quoted without that context is the thing this collection exists to avoid.

Twenty-two per cent, and the term it has no room for #

An approximation that is twenty-two per cent optimistic is a usable approximation, and the reason this essay is not a footnote is the second half: the Miller expression has no place at all for the second pole or for the right-half-plane zero the network also has.

The zero is the one worth following, because it is invisible in every measurement that would normally be made. The phase the magnitude already knows establishes what a right-half-plane zero costs: for a minimum-phase network the phase is fixed everywhere by the magnitude, so a magnitude sweep is a complete measurement — and a network with such a zero is not minimum phase, so a phase margin computed from a magnitude sweep of this stage is computed for a different circuit. That failure mode is the worst kind, a comfortable-looking margin on a loop that oscillates.

The device that never sees the swing is the repair and prices it in the currency this collection insists on: the bandwidth really is fourteen times better, and what it costs is two volts of a five-volt supply. What the cascode removes is the gain across the capacitor, which removes the multiplication, the zero and the second pole together — so the repair is not an improvement to the approximation on this page but a circuit in which the approximation was never needed.

Where the two picofarads are paid for #

The Miller capacitance is the reason three later arrangements exist. The device that never sees the swing removes it by holding one node still, and pays a volt of headroom for the privilege. The buffer that is not a buffer avoids it by having no voltage gain at all, and acquires an inductive output instead. What a pair cancels, and what it only halves halves it by splitting the swing between two devices. All three are compensations for a number on a data sheet, and The ideal amplifier, and where it stops being one is what happens when the same capacitance is inside a loop rather than in front of one. Where the behaviour is written down is where the pole this page computes joins the pair every second-order result in the collection is about.