The dual that is the same function

The residue that has only one sign put a bridged-T where a twin-T had been, in the feedback path of a selective amplifier, and found the one property that made it worth the change. Its residue at the null frequency is exactly $2R_2/(2R_2 + R_1)$: real, like the twin-T’s, but positive for any values the parts can take. So its loop can never oscillate, and its quality factor rises with the gain and stops at a ceiling near $\sqrt{R_1/R_2}/2$ — 5.00 for a ratio of a hundred — where, at the same Q, ±1% parts move the twin-T’s loop by 9.1% and the bridged-T’s by 2.0%.

It ended by naming the network it had not built. The bridged-T it used has two capacitors in series and resistors bridging and shunting them. Its dual has two resistors in series, a capacitor bridging them and a capacitor from their junction to ground. That essay’s question was whether the dual’s ceiling is the same function of its ratio, and whether its slope at the null falls differently — which would decide which of the two is the better low-Q network.

Two networks, one function #

The RC bridged-T here is the one that essay used: two 10 nF capacitors in series, a resistor $R_1$ bridging them, a resistor $R_2$ from their junction to ground, with $\sqrt{R_1R_2}$ = 10 kΩ so that the minimum sits at 1591.5 Hz whatever the ratio $n = R_1/R_2$. The dual is built to match: two 10 kΩ resistors in series, a capacitor $C_b$ bridging them and a capacitor $C_s$ from their junction to ground, with $\sqrt{C_bC_s}$ = 10 nF and the ratio now $n = C_s/C_b$.

The two transfers are the same function. Not approximately: over two decades of frequency the largest relative difference is $5.7\times10^{-8}$, and that is the solver’s own conditioning on a network with a teraohm load resistor in it, not a difference between the networks. The residue at the null is 0.0196 for both, $2/(2 + n)$ to six digits; the slope through the null is the same; and in a loop of gain a thousand both make a Q of 4.760.

That is what duality says, and it is worth seeing why it holds so exactly here. The transfer of a bridged-T depends on its four branch impedances only through two dimensionless combinations. With equal capacitors and resistors $R_1$, $R_2$ it is

$$T(s) = \frac{1 + 2sCR_2 + s^2C^2R_1R_2}{1 + sC(R_1 + 2R_2) + s^2C^2R_1R_2}$$

and the dual, with equal resistors and capacitors $C_b$, $C_s$, gives the same polynomials with $sRC_b$ and $sRC_s$ in the places of $sCR_2$ and $sCR_1$. Choose $\sqrt{C_bC_s}\,R = \sqrt{R_1R_2}\,C$ and $C_s/C_b = R_1/R_2$, and every coefficient matches. The answer to the earlier question is therefore yes, in every respect: the same residue, the same slope, the same ceiling as a function of the ratio. There is nothing in the response to choose between.

Four ratios, two bills of materials #

The slider moves the ratio, and at every setting the two curves stay one — a ratio of 10 gives a residue of 0.167 and a loop Q of 1.53 at a gain of a thousand, 30 gives 0.0625 and 2.69, 300 gives 0.0066 and 7.53. What the slider also moves is the parts list, and the two lists diverge.

At a ratio of ten both networks are ordinary: resistors within a factor of ten of each other, or capacitors within a factor of ten. At a hundred the RC network needs 100 kΩ and 1 kΩ, still ordinary, while the dual needs 1 nF and 100 nF — two capacitors whose values are a hundred apart, which in practice means two different constructions. At a thousand, where the ceiling reaches 15.8, the RC network needs 316 kΩ and 316 Ω and the dual needs 316 pF and 316 nF, three decades of capacitance in one network.

A resistor ratio of a thousand costs nothing: the two resistors are the same kind of part, from the same tolerance band, with the same temperature coefficient to within their own specification. A capacitor ratio of a thousand costs a great deal, because nothing that is a good 316 pF capacitor is also a good 316 nF one. That asymmetry is the whole of the difference between the two networks, and the next two figures price it.

At three hundred the pattern is plain. The RC network’s two resistors are still the same kind of part, a few hundred ohms and a couple of hundred kilohms from one tolerance band. The dual’s two capacitors are a few hundred picofarads and a couple of hundred nanofarads: the first is naturally a small ceramic of a stable class, the second is a film or a high-permittivity ceramic, and the two were made by different processes with different tolerances, temperature coefficients and ageing. The response the figure draws is exactly the same; the part each network hands to the ratio is not.

The Q follows the ratio, halved #

Near the ceiling the loop’s Q is about $\sqrt n/2$, and a square root halves a fractional error: a ratio that is one per cent off moves the Q by half a per cent. So the Q near the ceiling inherits the tolerance of whichever two parts carry the ratio, halved, and it is nearly indifferent to the other two.

With resistors of ±1% and capacitors of ±5% — a common pairing, and a generous one for capacitors of a hundredfold ratio — the RC network’s Q spreads 2.13 per cent across every corner of its parts and the dual’s 10.02. Two resistors at ±1% bound the ratio to about ±2% and the Q to about ±1%; two capacitors at ±5% bound the ratio to about ±10% and the Q to about ±5%. The figure’s numbers are exactly that arithmetic, confirmed by locating the loop’s peak and edges at each of the sixteen corners. The RC network’s capacitors, at five times the resistors’ tolerance, add about a tenth of a per cent to its spread; the dual’s resistors add even less to its own. In each network the two parts that do not carry the ratio are nearly invisible to the Q, and the section on the other two parts says why.

The twin-T’s spread was the earlier essay’s reason for preferring the bridged-T at all. The shape a tolerance does not print is a reminder that a corner analysis is the worst case and a production spread is narrower, but the ratio between the two networks survives any distribution: the dual’s spread is five times the RC network’s because its ratio’s parts are five times looser.

The Q follows the drift, halved #

The same square root applies to anything that moves the ratio after the network is built, and temperature is the one that does it continuously.

The coefficients in that figure are stated rather than attributed: ±25 ppm/K between two resistors of one type is a typical tracking figure, and 0 against +400 ppm/K is the kind of mismatch two capacitors of different dielectrics have. With them, the dual’s Q moves by half of 400 ppm/K over a hundred kelvin — 2.00 per cent, which the figure requires to within a tenth — and the RC network’s by half of 50 ppm/K, 0.25 per cent.

A drift is harder to live with than a tolerance, because a tolerance can be trimmed out once at the bench and a temperature coefficient returns every time the room warms. A selective amplifier whose Q is set to five at 20 °C and reads 5.05 at 70 °C has not failed any specification a tolerance band would catch, and it has changed its bandwidth by a per cent.

That is the practical form of the answer. The bridged-T and its dual are the same filter, and the RC version is the better component. Its ratio lives in the kind of part that can hold a ratio.

Why the other two parts hardly matter #

The spread figure varies all four parts, and the Q near the ceiling moves by the ratio’s error and almost nothing else. It is worth saying why, because it is a statement about which derivatives of the loop are zero, and a zero derivative is the most useful thing a tolerance analysis can find.

Each network has four parts and the response has two properties worth tracking: where the null is, and how deep. The null’s frequency is set by the geometric means — $\sqrt{R_1R_2}\,C$ in the RC network, $R\sqrt{C_bC_s}$ in the dual — and the depth by the ratio. A part that changes the geometric mean and not the ratio moves the peak’s frequency and leaves its Q; a part that changes the ratio moves the Q. In the RC network the two capacitors appear only in the product with the resistors’ geometric mean, so their tolerances move the frequency and barely touch the Q; in the dual the two resistors play that role. It is the same division of labour what actually fills a null found in an LC notch, where a tolerance on the arm’s two parts moves the null and only loss fills it.

The three tolerances that do nothing is another network in which some tolerances turn out to move nothing that is being asked about, and every derivative, and the one that is zero is the method that computes them. Here the partition is exact at the ceiling and approximate below it: at a gain of 10⁵ the Q is 4.996 against a ceiling of 5.00, and the tiny remainder is where the non-ratio parts still have a say.

So the spread figure’s lesson is sharper than “capacitors are looser”. It is that each network assigns the Q to one pair of its parts and the frequency to the other pair, and the two networks make the opposite assignment. The RC bridged-T puts its Q in its resistors and its frequency in its capacitors; the dual puts its Q in its capacitors and its frequency in its resistors. A designer who needs a stable Q and can tolerate a drifting frequency wants the first; one who needs the opposite wants the second.

Where the dual still earns a place #

The asymmetry reverses in one circumstance, and it is worth stating so that the rule does not become a reflex.

A capacitor ratio is set by areas and dielectric thicknesses; on an integrated circuit two capacitors of the same dielectric, drawn as arrays of one unit, match to a tenth of a per cent and track with temperature almost exactly, while two resistors of very different values drawn in the same material match less well and resistors of a thousandfold ratio are expensive in area. Switched-capacitor filters exist for that reason — a resistor made of a clock is the arithmetic — and in that technology the dual’s parts list is the easy one. The rule is not “resistors carry ratios better”; it is “the ratio belongs in whichever pair of parts the technology matches best”, and the two networks let a designer choose which pair that is without changing the response at all.

The other place the dual differs is at direct current, and it matters in a feedback loop. Both networks pass DC — the RC bridged-T through its bridging resistor, the dual through its two series resistors — so both close the amplifier’s DC loop at unity, which keeps its operating point defined. They differ in the resistance they present at DC: the RC network’s $R_1$, 100 kΩ at a ratio of a hundred, against the dual’s 20 kΩ. An amplifier’s input bias current flows through whichever it is, so the dual’s lower resistance is a smaller offset, by the same factor of five the tolerance cost.

What duality does not reach #

Duality exchanges the network’s own elements and nothing else. The source that drives the network and the load that reads it are the same ohms and picofarads whichever network sits between them, and they do not transform. An amplifier’s output resistance in series with the RC network’s input is in series with a capacitor; in series with the dual’s input it is in series with a resistor. The two respond to it differently, and for the same reason the two respond differently to a stray capacitance across the bridging element: in one network that stray is in parallel with a capacitor that dwarfs it, in the other with a resistor it eventually shorts.

At the frequencies here — a null at 1.6 kHz with a gain-of-a-hundred-thousand amplifier — those effects are far below the tolerances above: a ten-picofarad input capacitance at the network’s output moves either loop’s peak by less than a part in ten thousand, which is what locating the loop with it in place finds. They would matter for a null in the megahertz range, where the network’s impedances are a thousand times lower and the strays are not.

How the comparison was made #

Both networks are solved as netlists at every frequency drawn, with the same source and load, and their transfers compared point by point; the worst relative difference over two decades is required to be below a part in a million and is 5.7×10⁻⁸.

The loops are located, not read from a grid: stability by the winding of $1 + A\,T(j\omega)$, the peak by golden section, the two edges by bisection, the same routine the earlier essays used. The two networks’ Qs are required to agree to a part in a million at a gain of a thousand.

The spreads are corner analyses over all sixteen combinations of the four parts at their tolerance limits, each corner’s loop located separately. The temperature sweep moves the stated parts by their stated coefficients at six temperatures and locates each loop; the dual’s swing is required to equal half the capacitors’ relative drift to within a tenth.

Capacitors that are more than a value #

Real capacitors are not only a value. A capacitor of a hundred nanofarads in a dielectric chosen for size has loss and absorption as well as a temperature coefficient, and the capacitor that remembers is what the second of those does to a network that settles. The dual carries its ratio in exactly the parts that have them.

The amplifier is ideal. A real amplifier’s own bandwidth adds phase at the null, and the peak the amplifier moves finds that this moves the peak away from the null frequency by an amount that depends on the amplifier and not on which network is in its feedback path — so it is the same for both.

And the corners are worst cases. A production spread is a distribution inside them, and it is narrower for both networks by the same factor.

Still open: the trim that lowers the residue, the integrated dual, and the ceiling’s drift in a loop #

Trimming the residue towards zero. Both networks have the same positive residue, and the earlier essay asked whether a small change — a resistance in series with the shunt capacitor, or a fraction of the output returned to the junction — could move it towards zero from the positive side. If it could, the ceiling would rise with it; whether the ceiling rises, or whether the slope at the null falls with the residue and leaves the ceiling where it was, is the measurement that decides whether a trimmed bridged-T is ever worth building.

The dual on an integrated circuit. With capacitors matched as unit arrays, the dual’s ratio is set by a count of units and is exact at build, while its absolute values track together; the Q’s spread then comes from the resistors, which on silicon match poorly. The integrated version reverses this essay’s conclusion, and by how much depends on a process’s matching figures.

The ceiling’s drift inside a working loop. Near the ceiling the Q is the ratio’s; below it, the Q depends on the gain as well. A loop run at half its ceiling, where the gain’s own drift matters, has a temperature coefficient that is a mixture of the amplifier’s and the ratio’s, and the mixture depends on how close to the ceiling the design sits.