The sign the null leaves behind

The null that is stationary in nothing found that a twin-T’s null is filled by everything and at one order. A tolerance in any element fills it at twenty decibels per decade of error, and so does loss, because a resistance in a capacitor branch is a fractional error like any other. It said a twin-T’s depth is bounded by how finely it can be adjusted rather than by anything else, and ended on the use that makes depth matter most. Placed in the feedback path of an amplifier, the null becomes a peak, and the peak’s sharpness is set by the loop’s gain. That is the standard high-Q selective amplifier and the standard route to a sine oscillator.

The question it left was one solve away: how precisely can that quality factor be set, given that the null it depends on is never exact? The expectation was a ceiling. A null of finite depth should cap the Q, and deeper parts should raise the cap. The solve says that is half of the answer, and the other half is an oscillator.

One null, reused as a peak #

The circuit is the twin-T of the previous essay, 10 kΩ and 10 nF with its null at 1591.5 Hz, in the feedback path of an amplifier of gain $A$. The amplifier drives the twin-T and amplifies the difference between the input and what comes back through it, so the closed-loop gain is $A/(1 + A\,T)$. Away from the null $T$ is near one and the gain is near one. At the null $T$ is zero, the loop is open, and the gain is the whole of $A$.

With exact components the peak is 1000 and its quality factor 250.2. That is a quarter of one more than the gain, $(A + 1)/4$, and both parts of that have a reason. A twin-T built from equal parts is a notch whose own quality factor is a quarter, a very broad notch. Feedback multiplies its sharpness by the loop’s return difference, $1 + A$. A gain of a thousand turns a broad notch into a sharp peak.

The hero figure’s other three curves are the same twin-T with its shunt elements moved by a tenth of a per cent — the resistance down and the capacitance up, or the reverse, which is the direction the previous essay found fills the null fastest. The two directions do not do the same thing. One lowers the peak to 800 and the Q to 200. The other raises them, to 1333 and 334, and three times that error raises the Q to 994, four times the design value. A null filled the wrong way has made the filter sharper, and four tenths of a per cent in that direction is not a sharper filter at all.

The residue is a real number #

The reason is what a filled null is. The previous essays measured null depth as a magnitude in decibels, which throws away the phase. Inside a loop the phase matters, because the loop adds $A\,T$ to one.

Solved at the design frequency, the twin-T’s residue is minus a quarter of the error in the shunt elements, to within two per cent over the whole range from −1 to +1 per cent, and it is real. Its imaginary part is second order, two hundred times smaller at one per cent. A filled null is not a small phasor of arbitrary phase: it is a small copy of the input, in phase or in antiphase, and the direction of the error decides which.

Multiplied by the loop’s gain, that real number adds to or subtracts from the one in $1 + A\,T$. In antiphase it subtracts. The loop’s return difference at the peak falls from $1 + A$ towards zero, the peak rises and narrows, and when $A\,t/4$ reaches one the return difference reaches zero at the null frequency. The loop then has a pair of poles on the imaginary axis at 1591.5 Hz, and beyond it a pair in the right half plane. It is the condition the gain that is exactly one measured for a Wien bridge, with the twin-T supplying the 180 degrees of phase that the amplifier’s inversion needs.

One curve for every gain #

If that is the mechanism, the quality factor should depend on the gain and the error only through their product. The exact-null Q is $(A+1)/4$, and the residue multiplied by the gain divides it.

All three gains fall on one curve, $1/(1 - A\,t/4)$, to within three per cent at every point, with thirty loops located on the solved response rather than read off a grid. To the left, errors on the damping side, the Q falls: at $A\,t/4 = -1$ it is halved and at $-3$ quartered, and as the gain grows without limit it approaches $1/|t|$. That is the ceiling expected at the start. An error of a tenth of a per cent caps the Q near a thousand whatever amplifier is behind it.

To the right the curve has no ceiling. The Q is doubled at $A\,t/4 = 0.5$, ten times at $0.9$, and at 1 the loop oscillates. Past it the winding of $1 + A\,T(j\omega)$ about the origin counts two, which for a passive twin-T means two poles in the right half plane. The stability boundary sits at an error of $4/A$, so it moves with the gain: 0.4 per cent at a gain of a thousand, 0.04 at ten thousand, and 0.004 at a hundred thousand.

The practical meaning is that a designer who chooses a large gain to get a high Q, and buys good parts to protect it, has placed the design on this curve at a point that depends on an error nobody measured. The same batch of parts, built into two boards, can give one board a Q below target and the other a filter that rings for seconds, or an oscillator.

At a gain of a hundred the same pattern shows at a larger scale of error. The exact-null Q is 25.2. One per cent on the damping side lowers it to 20.2, one per cent the other way raises it to 33.5, three per cent to 94, and four per cent is an oscillator. A one per cent part is ordinary, so at a modest gain the twin-T loop is already sensitive to parts nobody would think of trimming.

What the Q is for #

A quality factor of 250 at 1591.5 Hz is a half-power bandwidth of 6.4 Hz, and that is the reason to build the loop: it passes one frequency and a few hertz either side of it and nothing else. The selectivity that is not free found that a filter’s selectivity is always paid for somewhere, in order, ripple or stopband, and here the payment is sensitivity. The 6.4 Hz band becomes 8.0 Hz with a tenth of a per cent of error one way and 4.8 Hz with the same error the other way, a factor of 1.7 between two boards built from one batch of parts. For a detector that integrates the energy in that band, the reading moves by the same factor. The frequency does not move: every peak in the hero figure sits at 1591.5 Hz, because the residue is real and only changes how much of the loop’s gain is left at the null, not where the null is.

That is the difference from the notch what actually fills a null measured. An LC notch’s tolerance moves its null in frequency rather than filling it, because the resonance depends on a product of two components that two errors can leave alone. Inside a loop, a moved null would be a moved peak with its Q intact. The twin-T’s tolerance cannot move its null, which is why its errors land entirely on the Q, and a designer choosing between the two for a selective amplifier is choosing which of the two quantities is allowed to wander.

The order of the null made the same kind of point about averaging: a null’s depth off its exact frequency depends on how the rejection was built, not on how deep it is at the centre. Here the centre itself is the whole story, because the loop spends all of its gain there.

What the null has to be #

Turned round, the curve says how deep a null has to be for a quality factor to be set to a stated accuracy. The binding direction is the one that raises the Q, since it has no ceiling. To hold the Q within ten per cent of its exact-null value, $A\,t/4$ must stay below $0.1/1.1$, so the residue must be smaller than that over the gain.

The null must be deeper than $20\log_{10}(44\,Q)$ decibels, which rises twenty decibels for every decade of quality factor: 42 dB for a Q of three, 62 for thirty, 82 for three hundred, 102 for three thousand. The twin-T’s null with one per cent parts in the worse direction is 52.1 dB, and it holds a Q to ten per cent only up to 9.6. With a tenth of a per cent the null is 72.0 dB and the Q can be 91.5; with a hundredth, 92.0 dB and 910. Each of those is bisected on the solved loop, raising the gain until the Q with the worse error leaves its band, and each agrees with the first-order rule to within six per cent.

So the null outranks the gain. The loop’s gain chooses the nominal Q, but only the null’s depth decides how well that Q is known, and a Q a hundred times larger needs parts a hundred times better matched or a trim to a part in ten thousand. The Q the components allow found an LC resonator’s quality factor capped by its components’ own loss, a ceiling a designer can read off a data sheet. The twin-T loop has no ceiling in the direction that matters. What its components set is the width of the band the Q falls in.

What a designer should take #

A twin-T selective amplifier sets its Q by its gain and its Q’s accuracy by its null. Choose the gain for the Q, $A \approx 4Q$, and then check that the null is deeper than $20\log_{10}(44Q)$ for ten per cent, or trim the twin-T to reach it. A trim should be judged by the residue it leaves, and its sign matters as much as its size: a trimmed twin-T left on the damping side is safe and slightly broad, while one left on the other side is sharp and close to oscillation. The prudent trim leaves a known residue on the damping side, deliberately, and raises the gain to compensate. That is the same trade stable, and unstable with less gain found in a different loop: a margin is a distance from a boundary, and here the boundary is set by a component tolerance, not by the amplifier.

The oscillator is the same analysis read from the other side. A twin-T oscillator is this loop with the error deliberately on the oscillating side, and its gain need only exceed $4/t$. How fast it grows and at what amplitude it settles are what the gain that is exactly one and its successors measured for another network. The twin-T supplies the frequency and the sign, and the null depth sets how much gain is needed to cross.

How the numbers were obtained #

The twin-T is solved as a netlist at each frequency: two 10 kΩ resistors with a 20 nF shunt, two 10 nF capacitors with a 5 kΩ shunt, the two shunt elements moved by $t$ in opposite senses, and a gigohm load. The loop is an ideal amplifier of gain $A$, so the closed-loop gain is $A/(1 + A\,T)$ computed from the solved $T$. The peak is found by golden section in a window of ±10 per cent and both half-power edges by bisection, so each quality factor is located rather than read off a grid. Stability is decided by the winding of $1 + A\,T(j\omega)$ about the origin over twelve decades, with six thousand extra points in the six per cent around the null. The permitted Q for each tolerance is bisected in log gain, twenty-two iterations, with its bracket checked to straddle the answer.

What it leaves out #

The amplifier’s own bandwidth. A real amplifier’s gain falls with frequency and its phase turns, so at the null it contributes a phase of its own, which rotates the residue’s effect off the real axis. For an amplifier whose gain-bandwidth product is a thousand times the null frequency the rotation is small; for one closer to it, the Q and the stability boundary both move, and an amplifier’s phase can push a safe twin-T towards oscillation.

The load on the twin-T. The gigohm here is an ideal input. A finite input resistance at the amplifier loads the twin-T’s output and is itself a fractional error in the shunt path, so it moves the residue by an amount comparable to a tolerance.

Temperature, which is less of a threat than it looks. A twin-T’s null depends only on the ratios within each kind of component: the null exists for any resistance and any capacitance so long as the shunts are half and twice the series values. A drift common to all three resistors, or to all three capacitors, moves the null frequency and leaves the residue at zero, and with it the peak’s Q. What fills the null with temperature is mismatch in the coefficients within one kind, which for resistors from one network or capacitors from one reel is a small fraction of the coefficient itself. The peak frequency does drift, by the sum of the two coefficients, and a narrow peak makes that drift visible: at a Q of 250 a drift of 6.4 Hz is a whole bandwidth, which is 0.4 per cent of the frequency, and at a hundred parts per million per kelvin between the two kinds of part that is forty kelvin of temperature change.

And the other errors. The figures move the two shunt elements in opposite senses, which is the direction the previous essay found fastest. Errors in the series elements produce residues with their own signs, and a real twin-T carries all six errors at once, which sum as signed quarters.

Still open: the amplifier’s phase, the trim that leaves a margin, and the bridged-T #

The amplifier’s phase at the null. With a finite gain-bandwidth product the loop’s gain at the null is complex, and its phase rotates the real residue. The stability boundary then depends on both the residue’s size and the amplifier’s phase, and there is a gain-bandwidth product below which even an exact twin-T oscillates or never reaches its Q. Finding it would put the amplifier on the same axis as the parts.

The trim that leaves a margin. The argument above suggests trimming a twin-T to a deliberate residue on the damping side rather than to the deepest null. The trimmer’s own resolution and temperature coefficient set how close to zero that residue can be held, and so how much gain can be spent on Q before the margin is used up. That is the measurement the previous essay’s still-open section asked for, now with a sign attached.

The bridged-T in the same loop. A bridged-T nulls with one fewer component. Whether its residue is also real and signed, and whether its loop falls on the same curve with a different constant, is the natural comparison, and it would say whether the $1/(1 - A\,t/4)$ law belongs to the twin-T or to every resistor-capacitor null.