Shorted instead of opened, and the error changes sign

A sum that is exact, and the estimate that is not separated two claims that usually travel together. Add each capacitor’s value times the resistance seen at its own terminals with the other capacitors removed, and the total is the ratio of the first two coefficients of the denominator polynomial — a theorem, holding to nine digits at every capacitor spread tested. Divide one by two pi times that total and what comes out is an estimate of the upper corner frequency, which is fourteen per cent low with three equal capacitors and never once optimistic.

That essay ended with a single sentence about the other construction: every capacitor shorted except one, the resistance measured across the one left, and the sum taken over the reciprocals instead. It said the result gives the lower corner of a band-pass response and left it there.

This essay takes that sentence apart, and the interesting thing about it is not the symmetry. The two constructions are duals and they behave as duals, which is unsurprising. What is worth having is that the estimate’s error changes sign with the construction, and that the reason it does is exactly the reason the first one is conservative. A designer who uses both ends of the shortcut on one amplifier gets a predicted passband that is narrower than the real one at both ends, which is a much stronger statement than two independent rough numbers.

The network, and why every resistance in it is smaller than it looks #

Three series capacitors and three shunt resistors: a hundred nanofarads from the input to the first node, a kilohm from that node to ground, another capacitor to the second node, another kilohm, a third capacitor to the output, a third kilohm. At high frequencies the capacitors are shorts and the output is the input; at low frequencies they are open and nothing arrives — which is the arrangement the corner that says nothing about an edge measures with a pulse rather than with a sinusoid. Three zeros at the origin and three poles, and the poles are not the poles of three isolated sections, because each section is loaded by the one after it.

The short-circuit time constants are read off the netlist rather than written down. For each capacitor in turn, the other two are replaced by voltage sources of zero volts — which is what a short is when the network is assembled as a set of nodal equations, and leaves the topology untouched — the input source is zeroed for the same reason, and the resistance between the two nodes that capacitor spans is measured with the same probe-current solve used everywhere else.

Every one of those resistances is smaller than the kilohm the corresponding open-circuit construction sees, and that is the whole structural difference between the two. Opening the other capacitors disconnects parts of the network and leaves each capacitor looking into whatever is left; shorting them connects parts of the network together and puts every other resistor in parallel with the one being measured. The first capacitor, with the other two shorted, looks into three kilohms in parallel — 333.3 Ω — because the shorts have tied all three shunt resistors to the same pair of nodes. The third looks into one kilohm, because the shorts put the first two resistors on the grounded side of it where they do nothing.

The sum that matters is not of these products but of their reciprocals: 30 000 + 20 000 + 10 000, which is 60 000 per second. Two pi of that is 9549 hertz.

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

The sum of reciprocals has a second identity that is exact, and it is the half of the shortcut that is a theorem. Writing the denominator as $D(s) = a_0 + a_1 s + \dots + a_n s^n$,

$$\sum_i \frac{1}{R_i C_i} = \frac{a_{n-1}}{a_n} = \sum_i (-p_i)$$

where the $p_i$ are the poles. The first equality is the short-circuit time constant theorem; the second is elementary once the polynomial is factored, since the sum of the roots of a polynomial is minus the ratio of its two leading coefficients. Note where the coefficients are: the open-circuit sum is $a_1/a_0$, the two lowest, and this one is $a_{n-1}/a_n$, the two highest. One polynomial read from each end, and the two readings answer different questions.

The three computations share nothing, which is the standard a number is held to here and the reason one step, computed twice exists at all. The first is a set of resistances measured on a netlist that has no capacitors in it at all. The second is a coefficient of a polynomial recovered by sampling a determinant on a circle in the complex plane and taking a discrete transform of the samples. The third is a sum over the roots of that polynomial, found by Durand–Kerner.

They agree to one part in $10^{12}$ at the worst point of a sweep covering three decades of capacitor spread. That is better agreement than the open-circuit construction manages, and for a reason worth knowing: the leading coefficients of a recovered polynomial are the ones the sampling gets most accurately, because they dominate the determinant at the sampling radius. The open-circuit sum reads the two coefficients the recovery holds least well, which is why it is quoted to nine digits and this one to twelve. Neither is a statement about the theorem.

And the part that is not a theorem errs upward #

The corner. One over $2\pi$ times 60 000 per second is 9549 Hz. The −3 dB frequency measured on the solved response is 8192 Hz. The estimate is 16.6 per cent high.

Sixteen per cent is about the same size as the fourteen the open-circuit construction gives at the other end, and the direction is the thing. Across every setting of the slider the estimate sits above the measurement, every time, exactly as the open-circuit estimate sits below it. The figure states that rather than noting it, because an estimate that were sometimes on the other side would be a different and much less useful object.

The trend is the mirror image of the other end’s. As one time constant becomes much smaller than the others its reciprocal dominates the sum, the response becomes single-pole near its lower corner, and the estimate converges on the truth from above. Bisecting for where it comes within ten per cent puts that at a spread of 8.54, against 7.48 for the open-circuit construction — the same statement about the same kind of dominance, arrived at through a different arithmetic.

Why one construction is high and the other low #

Both estimates make the same substitution and it fails in opposite directions because the two corners are approached from opposite sides.

At the top of the band, $1/\Sigma\tau$ weights every pole equally and the magnitude response is dominated by the slowest one, so the sum over-counts the total time constant and under-states the frequency. At the bottom, $\Sigma 1/\tau$ weights every pole equally and the response is dominated by the fastest one, so the sum over-counts the total rate and over-states the frequency. One quantity is summed and one is summed in reciprocal, and in both cases the sum is larger than the term that is actually doing the work.

The worst case can be computed in closed form, and it is the same expression twice. For $n$ coincident poles, the open-circuit estimate is low by a factor $n\sqrt{2^{1/n}-1}$ — 1.287 at second order, 1.529 at third. For $n$ coincident high-pass poles the true corner is $\omega_0/\sqrt{2^{1/n}-1}$ against an estimate of $n\omega_0$, so the estimate is high by $n\sqrt{2^{1/n}-1}$. The same number. The two ends of the theorem are not merely analogous; the worst case one of them can produce is the worst case the other can produce, and it is the same function of the same integer.

The chain measured here is not three coincident poles — its loading spreads them — which is why 16.6 per cent rather than 52.9.

Two circuits the sum cannot tell apart, again #

The second row of the table is the one that settles what the shortcut can ever be.

At a spread of one, the three short-circuit time constants are 33.33, 50 and 100 µs. At a spread of three, the capacitors are 300 nF, 100 nF and 33.3 nF, and the three time constants come out at 100, 50 and 33.33 µs — the same three numbers, in a different order. The sum of their reciprocals is identical to the last bit, so the estimate is identical: 9549 Hz both times.

The measured corners are 8192 Hz and 8030 Hz.

That is a two per cent gap between two circuits the sum cannot distinguish, and it disposes of the idea that a better constant than $2\pi$ could ever make the shortcut exact. Whatever constant is chosen, one of these two is wrong by two per cent, and a wider spread of examples widens the gap. The theorem says $\sum 1/R_iC_i = a_{n-1}/a_n$. It says nothing whatever about $a_{n-2}$, and the −3 dB point depends on it.

The coincidence is not one. The three resistances the capacitors see are in the ratio 1 : 1.5 : 3, so spreading the three capacitances by a factor of three reverses the multiset of products instead of changing it. A sum that is exact found the identical accident at the other end of the polynomial with resistances in the ratio 1 : 2 : 3, and both times it arrived by accident and both times it is the sharpest thing in the essay. A chain of equal resistors has ratios in it, and a slider that scales capacitances by a ratio will eventually hit one.

What the decomposition is for at this end of the band #

The same thing it is for at the other end, with the signs reversed and one difference in emphasis.

A lower corner estimate that is sixteen per cent high is worth very little on its own. What the sum gives is a breakdown — 30 000 per second here, 20 000 there, 10 000 at the output — which says immediately that the first coupling capacitor is half the problem and the last one is a sixth of it. That attribution is not available from the poles at all, for the reason Where the behaviour is written down sets out: a pole belongs to the whole network and cannot be assigned to a component, while a term in this sum is attached to exactly one capacitor by construction.

The difference in emphasis is which way the design question points. At the top of the band the dominant term is the one to attack and the attack is usually on a resistance rather than on a capacitor, because a resistance can be changed by moving a node. At the bottom the dominant term is the largest reciprocal, which is the smallest product, and the fix is almost always to make that capacitor bigger — which costs money and board area in proportion, so knowing which one to enlarge is worth more here than it is at the other end. The frequency a device sets for itself is the same ranking used at the top of the band, where the term that dominates is a capacitance multiplied by a stage gain. Enlarging all three by a factor of three moves the corner by a factor of three; enlarging the first one alone by a factor of three moves it by 1.67, and the sum says so before anything is built.

It is also why the two constructions use opposite treatments of the other capacitors. Opening them is what makes each open-circuit term a property of one capacitor’s own neighbourhood at the top of the band, where the other capacitors are not yet doing anything. Shorting them is what makes each short-circuit term a property of one capacitor’s neighbourhood at the bottom, where the other capacitors are already doing everything. Each construction puts the other elements into the state they are actually in at the corner it is about, and that is the whole reason both work at all.

The same construction has a use outside a chain of stages, and one figure is enough to say where.

What this does not say about the other end #

That the two estimates are ever both tight. They converge under opposite conditions. The open-circuit estimate is good when one time constant is much larger than the others; the short-circuit estimate is good when one is much smaller. A three-stage amplifier whose coupling capacitors are all the same value and whose internal capacitances are all comparable gets the worst of both, and the honest reading of its predicted passband is that both ends are inside the truth by about fifteen per cent.

That the sum of reciprocals is the lower corner of anything with a zero in the wrong place. The theorem is about the denominator and holds whatever the numerator is. The estimate additionally needs the numerator to be $s^n$ — every zero at the origin — which is what an all-capacitor-coupled network has and what a network with a bypassed emitter resistor does not. A stage with a zero part-way up its high-pass skirt has a corner the sum knows nothing about, and a pair of poles with a quality factor above one has a corner where the behaviour is written down rather than where any sum of time constants puts it, in the same way that the cancellation that leaves a tail has a settling time the poles know nothing about.

That shorting a capacitor is the same as deleting it. It is the opposite operation and the figure does it with a zero-volt source rather than by merging nodes, so the network the resistance is measured on has the same node count as the network it came from. A first version of this measurement merged the nodes instead and produced a netlist one node short, which the assembler accepted and which measured a different resistance for the middle capacitor.

That the twelve-digit agreement is a tighter theorem than the nine-digit one. It is a tighter recovery. Both identities are exact. The leading coefficients of a polynomial fitted to samples of a determinant are held better than the trailing ones, and that is a statement about sampling a circle rather than about circuits.

Three routes to one sum, and the two circuits that share it #

Three routes to the sum, required to agree. Resistances off a netlist with no capacitors in it; the ratio $a_{n-1}/a_n$ of a denominator recovered from complex determinants; and the negated sum of that denominator’s roots. Checked to $10^{-7}$ and coming out at $1.4\times10^{-12}$ at the worst setting of the slider.

The estimate is checked to be above the measurement at every setting, which is the property a designer relies on and the one that would be silently lost if the sign of the error ever changed.

The two circuits with identical time constants are checked as a pair — the same three values to a part in $10^{12}$, and measured corners differing by more than 1.5 per cent.

And the ten per cent crossing is bisected on the measured response rather than read off the sweep the figure draws, so the 8.54 does not depend on how many points the sweep happens to have.

Both ends of one polynomial #

The structure worth carrying out of this pair of essays is that a denominator polynomial has two ends and each of them is a different design tool.

$a_1/a_0$ is the sum of the open-circuit time constants and estimates the top of the band, low. $a_{n-1}/a_n$ is the sum of the reciprocal short-circuit time constants and estimates the bottom, high. Each is exactly a ratio of two coefficients and each throws away everything else about the polynomial, so each is a projection of the pole configuration onto one number — and the two projections lose different information. The first knows nothing about $a_2$ and the second knows nothing about $a_{n-2}$, which is why the same network can be well estimated at one end and badly at the other, and why the three tolerances that do nothing is the right shape of question to ask about either: what exactly does the summary statistic not contain?

Between them they bracket the passband from the inside. That is worth more than either number is worth alone, and it is not a coincidence that has to be checked in each case: it follows from both sums over-counting whichever pole is doing the work, which follows from both of them weighting every pole equally. The same sentence explains the sign at both ends.

What neither of them offers is a route to the shape between the corners, which is the property one solve, read four ways gets for nothing by not summarising anything, and which is why a solve remains the answer to “what does this do” while these two sums answer “which component decides it”. The cliff before the fastest settling is the other quantity here that refuses to be estimated, and for the same reason: it is defined by a threshold and every estimate of it is derivative-shaped.

Still open: the sum with an inductor in it, and the sum inside a loop #

The same construction where the reactance is an inductance. Every term above is a capacitance times a resistance. An inductor’s open-circuit term is $L/R$ with the other reactances shorted rather than opened, which is the treatment this essay gave the capacitors — so a network with both kinds of element has a sum in which half the terms are measured one way and half the other, and the two halves are not obviously separable. Working it through on a network with one of each would say whether the theorem survives the mixture intact or only in one of its two forms.

The resistance a feedback loop shows a capacitor. Both constructions here measure a resistance on a passive netlist. In a circuit with feedback round it, the resistance seen at a node depends on the loop and can be orders of magnitude smaller than any resistor present — which is exactly the quantity the sum wants and is not the quantity a resistor-only netlist reports. Measuring the open-circuit terms of an amplifier with its loop closed, and again with it cut, would put a number on how much of a compensated amplifier’s bandwidth the shortcut can actually account for.

And whether the bracket is ever tight enough to be a specification. The two estimates bound the passband from the inside by about fifteen per cent each on the network here. Whether that margin grows or shrinks with the number of stages is a question about how pole multiplicity behaves as a chain lengthens, and the closed form for $n$ coincident poles — $n\sqrt{2^{1/n}-1}$, which reaches 1.529 at three and keeps rising — suggests it grows. A chain of six would settle it.