A sum that is exact, and the estimate that is not

There is a shortcut in amplifier design that everybody who has ever sized a compensation capacitor has used. For each capacitor in a circuit, work out the resistance seen looking into its own two terminals with every other capacitor removed and the independent sources set to zero. Multiply by that capacitor’s value. Add up the results. Then the upper corner frequency is about one over two pi times the sum.

It is normally introduced as an approximation with a rule of thumb attached — good to within a few tens of per cent, conservative, useful for finding which element is limiting the bandwidth. All of that is true. What is not usually said is that the shortcut has an exact theorem inside it and an approximation wrapped round the outside, and the two are cleanly separable. This essay separates them and measures each.

The network, and what the sum is taken over #

A resistor–capacitor chain with no buffers in it: a kilohm from the input to the first node, a hundred nanofarads from that node to ground, another kilohm to the second node, another capacitor, a third kilohm to the output, a third capacitor. Three poles, and — importantly — three poles that are not the poles of three isolated sections, because each stage loads the one before it.

The open-circuit time constants are read off the netlist rather than written down. For each capacitor in turn, a resistor-only version of the network is assembled — the other two capacitors simply absent, the source present but set to zero, so that its terminals are a short and the topology is unchanged — and the resistance between the nodes that capacitor occupied is measured with the same impedanceBetween used everywhere on this site.

The resistances come out as 1, 2 and 3 kΩ, and it is worth seeing why the second is two kilohms rather than one. With the first capacitor removed, the first node is connected to the shorted source through a kilohm and to nothing else — so looking into the second node one sees the second kilohm in series with the first, and the third kilohm leading to an open circuit. The chain’s own loading is already in the number.

Their sum is 600.00 µs.

Three routes to that sum, and none of them shares arithmetic #

The sum has a second identity that is exact, and it is the half of the shortcut that is a theorem. Writing the network’s transfer function as $N(s)/D(s)$ with $D(s) = a_0 + a_1 s + a_2 s^2 + \dots$,

$$\sum_i R_i C_i = \frac{a_1}{a_0} = \sum_i \left(-\frac{1}{p_i}\right)$$

where the $p_i$ are the poles. The first equality is the open-circuit time constant theorem; the second is elementary once the denominator is factored. Three quantities, three completely different computations.

The site recovers $D(s)$ by sampling a determinant on a circle in the complex plane and taking a discrete Fourier transform of the samples — no symbolic algebra anywhere — and then roots it by Durand–Kerner. So the three routes are: a set of resistances measured on a resistor-only netlist; a coefficient of a polynomial recovered from complex determinants; and a sum over the roots of that polynomial. They share the netlist and nothing else.

They agree to $2.0\times10^{-9}$ at the worst point of a sweep covering three decades of capacitor spread, and to a part in $10^{14}$ at the near end. The tolerance is the recovery’s rather than the theorem’s: at a spread of a thousand the three poles span six decades, which is where sampling a determinant on one circle starts to lose digits. The theorem is exact and the arithmetic is not, which is the ordinary situation and worth saying so it is not mistaken for the other one.

And now the part that is not a theorem #

The bandwidth. One over $2\pi \times 600\ \mu s$ is 265.3 Hz. The −3 dB frequency measured on the solved response is 309.2 Hz. The estimate is 14.2% low.

Fourteen per cent is not a disaster and the direction matters more than the size. Across every setting of the slider — three decades of capacitor spread, and therefore three decades of pole separation — the estimate is below the measurement, every time. It is a conservative bound in practice, and the figure asserts that rather than merely noting it: an estimate that were sometimes optimistic would be a different and much less useful object.

The trend is the one the shortcut’s reputation rests on: as one time constant grows past the others, the response becomes single-pole and the estimate converges on the truth. Bisecting for where it comes within ten per cent puts that at a spread of 7.48 — one capacitor about seven times the others is enough to make the shortcut a good number, and three equal ones are not.

Two circuits with the same time constants #

The second row of that table is the interesting one, and it arrived by accident.

At a spread of one, the three time constants are 100, 200 and 300 µs. At a spread of three, the capacitors are 300 nF, 100 nF and 33.3 nF, and the three time constants are 300, 200 and 100 µs. The same three numbers. Their sum is identical to the last bit, so the estimate is identical — 265.3 Hz both times.

The measured bandwidths are 309.2 Hz and 315.4 Hz.

That is a two per cent gap between two circuits that the sum cannot tell apart, and it settles the question of whether the shortcut could ever be made exact by a better constant than $2\pi$. It could not. A single number computed from a multiset of time constants cannot determine a bandwidth, because here are two circuits with the same multiset and different bandwidths. Whatever constant is chosen, one of the two is wrong by two per cent, and a wider spread of examples widens the gap.

The theorem says $\sum R_iC_i = a_1/a_0$. It says nothing whatever about $a_2$, and the −3 dB point depends on $a_2$.

Why it is conservative, and by how much at worst #

The worst case for the estimate is all the poles in the same place, and it can be computed in closed form. For $n$ identical poles at $\omega_0$, the sum of time constants is $n/\omega_0$, so the estimate is $\omega_0/n$; the true −3 dB point is $\omega_0\sqrt{2^{1/n} - 1}$. The ratio is $n\sqrt{2^{1/n}-1}$: 1 at $n = 1$, 1.287 at $n = 2$, 1.529 at $n = 3$.

So the estimate can be pessimistic by about a third at second order and by half at third order, and never in the other direction. The chain measured here is not three coincident poles — its loading spreads them — which is why 14% rather than 53%.

The bound holds for the same reason that made the first equality true. $\sum(-1/p_i)$ weights every pole equally, and the magnitude response is dominated by the slowest one, so summing the reciprocals always over-counts unless a single pole is doing all the work.

What the shortcut is actually for #

Not the number. The decomposition.

A bandwidth estimate that is fourteen per cent low is worth very little on its own; the corner can be measured exactly and in less time than the estimate takes to set up. What the sum gives is a breakdown — 100 µs here, 200 there, 300 at the output — which says immediately that the output node is half the problem and the first node is a sixth of it. That is not available from the poles at all. A pole is a property of the whole network and cannot be attributed to a component; a term in this sum is attached to exactly one capacitor by construction.

So the design question — which element should be made smaller — has an answer here and does not have one in the pole locations. That is the whole value of the theorem, and it survives the estimate being mediocre, because the ranking is what is used and the ranking is exact.

It is also why the theorem is stated with the other capacitors open rather than shorted. Opening them is what makes each term a property of one capacitor’s own neighbourhood; the short-circuit version of the same construction gives the lower corner of a band-pass, by the same argument run upside down.

Three places the ranking decides something #

Sizing a compensation capacitor. The term that dominates the sum is the one to attack, and the attack is usually not on the capacitor. A term is a product, so halving the resistance seen by the dominant capacitor is worth exactly as much as halving the capacitor — and the resistance is often the thing a designer can actually change, by moving a node rather than by buying a part.

Deciding whether buffers are worth it. Buffering the three sections apart removes the loading, which changes the second capacitor’s resistance from two kilohms to one and the third’s from three to one. The sum falls from 600 µs to 300, so the estimate doubles, from 265.3 Hz to 530.5 Hz. The measured bandwidth goes from 309.2 Hz to 811.4 Hz — a factor of 2.62 rather than 2.00. The shortcut got the direction right and the magnitude wrong by a third, and it got it wrong in the predictable direction: three buffered sections are three identical poles, which is the worst case for the estimate, and the ratio there is exactly $3\sqrt{2^{1/3}-1} = 1.529$.

Explaining a bandwidth that came out wrong. A stage that measures at half its predicted corner has one term in this sum that was underestimated, and the sum says which node to go and probe. Nothing about the pole locations offers that, because a pole belongs to the network and not to a component.

The construction also runs the other way. Every capacitor shorted except one, and the resistance measured across the one left — the short-circuit time constants — sums to the reciprocal of the lower corner of a band-pass response, by the same argument applied to the other end of the denominator. Coupling and bypass capacitors are sized that way, and the two constructions are one theorem read from each end.

What this essay does not claim #

That the theorem needs the poles to be real. It does not. $\sum(-1/p_i)$ is real whenever the coefficients are, because complex poles arrive in conjugate pairs, and the theorem holds for a resonant network exactly as it does for a chain. What fails for a resonant network is the estimate, badly, because a pair of poles with a high quality factor has a bandwidth far above $1/\Sigma\tau$ — the one direction in which the shortcut is not conservative, and the reason it is stated for amplifiers rather than for filters.

That the resistance seen by a capacitor is always obvious. In this chain it is; in a circuit with feedback round it, the resistance seen at a node depends on the loop and can be very much smaller than any resistor in the netlist. That is what the feedback field’s virtual-earth essay measures, and it is exactly the quantity this sum wants.

That the recovered denominator is exact. It is a polynomial fitted to samples of a determinant, and at a pole spread of $10^6$ it holds nine digits rather than fifteen. The theorem is asserted to $10^{-7}$ for that reason and not because anything about the mathematics is approximate.

That this replaces solving. It does not, and on this site nothing does — every number above came from a solve. The sum is a way of attributing a bandwidth to components, and the attribution is what a solve does not give.

Where the shortcut is used, and where it fails #

The sum of open-circuit time constants is the one estimate this collection quotes without solving, and the essays either side of it say what that is worth. One step, computed twice is the exact route it is measured against. The frequency a device sets for itself is where the shortcut is actually used, on a stage whose dominant time constant is one capacitance times a Miller-multiplied resistance. The cancellation that leaves a tail is the case it cannot see at all, because a doublet moves a residue and not a time constant. The cliff before the fastest settling is the other quantity that refuses to be estimated, and Where the behaviour is written down is the pair of poles the sum is a statement about.

The gate #

Three routes to the sum, required to agree. Resistances off a resistor-only netlist; the ratio $a_1/a_0$ of a denominator recovered from complex determinants; and the sum of the negated reciprocal poles of that denominator. Asserted to $10^{-7}$ and coming out at $2\times10^{-9}$ at the worst setting of the slider.

The estimate is asserted to be conservative at every setting, not merely at the one drawn. That is the property a designer relies on and it is the property that would be silently lost if the sign of the error ever changed.

The two circuits with identical time constants are asserted as a pair — the same three values to a part in $10^{12}$, and measured bandwidths differing by more than 1.5%. It is the sharpest statement in the essay and it would be worth nothing as an observation about a picture.

And the ten per cent crossing is bisected on the measured response, not read off the sweep the figure draws, so the 7.48 does not depend on how many points the sweep happens to have.

An exact theorem underneath an inexact estimate #

The structure of this essay — a sum that is exactly the ratio of two polynomial coefficients, and a bandwidth estimate derived from it that is never better than fourteen per cent low — is worth naming, because it is the most useful shape a design tool can have.

The exactness is what makes the sum diagnostic. Each term is a capacitance times a resistance seen at its own terminals, so the largest term names the element to attack, and that identification is exact however wrong the bandwidth estimate is. The estimate’s error comes from the step after — turning a sum of time constants into a single corner frequency, which is exact only for a single pole — and it is one-sided, never optimistic, which makes it a bound rather than a guess.

Two other results in this collection have the same two-layer shape. The three tolerances that do nothing computes an exact second-derivative matrix and uses its eigenvalues to say which combinations of component variations matter, without the eigenvalues themselves being a design target. And every derivative, and the one that is zero is the exact instrument underneath both: a transposed solve giving the derivative with respect to every element at once.

Which suggests where this essay’s tool is best pointed. Not at “what is the bandwidth” — for which the solver already has an exact answer — but at “which of these eleven capacitances is costing it”, which is a question the exact answer does not decompose into and which the sum answers in one pass.

The two settings of the slider with identical time constants and bandwidths two per cent apart are the refusal that keeps that claim honest. A sum of time constants is a projection of a pole configuration onto one number, and two different configurations can share it — which is the same defect a floor, or a line finds in a jitter total and the constant that is a window in an ideality factor. The difference here is that the projection is exactly the ratio of two coefficients rather than a fit, so what is lost is known precisely: everything about the denominator beyond its first two terms. Which is a rare luxury for a summary statistic — most of them lose something nobody can name, and this one loses exactly the higher coefficients, so a circuit whose behaviour depends on them is identifiable in advance rather than after the estimate disappoints. The tell is two settings with the same sum and different bandwidths, which is a comparison a designer can make on their own netlist in two solves.

And the estimate being never optimistic is what makes it usable without that check. A bound that errs in one direction is a bound; a bound that errs in both is an estimate, and the difference decides whether a design can be signed off on it. Fourteen per cent low with three equal capacitors is a margin, and it is a margin in the direction a bandwidth requirement wants.