Circuit Response
every model has an edge, and the figure carries it
Filters, measured not tabulated

The null that is stationary in nothing

An LC notch has two mechanisms and they do different things: a tolerance moves its null rather than filling it, because the resonance depends on a product two errors can leave alone, and loss fills it at twenty decibels per decade of arm resistance. A twin-T has no resonance and no arm. Measured, everything fills it at twenty decibels a decade and nothing moves it — one ohm in series with its capacitors gives 88 decibels where a hundredth of a per cent of tolerance gives 92, because a series resistance in a capacitor branch is a fractional error of r over R and nothing else.
18 min read 5 figures Measured, not tabulatedTwo routes to a numberThe constants decide

Assumes: What actually fills a null · The Q the amplifier decides

What actually fills a null measures a notch whose transmission zero is a parallel inductor and capacitor going open at their own resonance, and reaches two conclusions that are different in kind.

A tolerance does not fill it. Ten per cent on each arm component leaves the null three hundred decibels deep and moves it to 1306.5 hertz. What a tolerance does is move a resonance, and a resonance is still an open circuit wherever it happens to be. The consequence shows up only at the frequency somebody cares about, and it splits into two orders: same-direction errors move the product LCLC and act first order, opposed errors leave it alone and act second, so one per cent parts leave 33 decibels one way and 79 the other.

And what fills it is loss, at twenty decibels per decade of series resistance exactly, so that the depth is 1.27−20log⁡10Qarm1.27 - 20\log_{10}Q_\text{arm} — the arm’s quality factor in decibels, constant to a hundredth over four decades.

Both conclusions come from the null being a resonance. A twin-T’s is not.

A twin-T's null: everything fills it, and at one ordercomputed by solving, not by drawing. A twin-T of 10 kΩ and 10 nF nulls at 1591.5 Hz, where a low-pass T of two resistors and a high-pass T of two capacitors deliver equal and opposite contributions. With exact components the null is -Infinity dB, which is the arithmetic's floor. A tolerance fills it at 20.0 decibels per decade — first order, where an LC notch splits into twenty and forty depending on which way two errors go, because there a product can be left alone and here there is no product. Loss fills it at 20.0 decibels a decade too, and barely moves it — 1591.55 to 1595.52 Hz across four decades of series resistance — because a resistance of r in a capacitor branch is a fractional error of r/R and nothing else: one ohm gives -88.5 dB against a hundredth of a per cent of tolerance's -92.0. So a notch's two mechanisms are not exchanged here — they are one mechanism, because a twin-T's null is stationary in nothing and an LC notch's frequency is stationary in the product of its two components.0-20-40-60-801001k10kfrequency (hertz)magnitude (decibels)null at 1592 Hz-52.1 dBR, C10 kΩ, 10 nFnull at1591.5 Hztolerance1%depth there-52.1 dBat 0.010%-92.0 dBat 0.10%-72.0 dBat 1.0%-52.1 dBat 3.0%-42.7 dBper decade of tolerance20.0 dB1 Ω of loss gives-88.5 dB…and 0.01% of tolerance-92.0 dBsolved, then checked — a cancellation, not an open circuittolerance fills it, loss moves it
Fig. 1 A twin-T of ten kilohms and ten nanofarads, nulling at 1591.5 hertz, with one per cent on its two shunt elements in opposite senses. The slider is the tolerance.

A cancellation rather than an open circuit

A twin-T is two T networks in parallel between the same two nodes. One is a low-pass T — two resistors in series with a capacitance of twice the unit shunting their junction — and the other is a high-pass T, two capacitors with a resistance of half the unit shunting theirs.

Neither path has a zero of its own. The low-pass T’s output falls with frequency and the high-pass T’s rises, and at 1/(2πRC)1/(2\pi RC) they are equal in size and opposite in sign. The null is their sum, and nothing in the circuit is an open circuit or a short at that frequency: current flows through both paths, and what arrives at the output is the difference between two things that happen to be the same size.

That single structural difference removes both of that essay’s mechanisms.

There is no resonance to move. The frequency at which the two paths cancel is set by the ratio 1/(RC)1/(RC), and there is no product that two errors can preserve while individually being wrong — so a tolerance cannot displace the null while leaving it deep. It fills it, directly.

And there is no arm to have a quality factor. That essay’s depth formula is the arm’s reactance over its series resistance, and a twin-T has neither an arm nor a reactance to divide. So the question of what loss does has to be asked again rather than answered by analogy.

Everything fills it, at one order

The measurement is short and it goes the same way both times.

tolerance on the shunt elements depth at the null
exact −310 dB
0.01% −92.0 dB
0.1% −72.0 dB
1% −52.1 dB
3% −42.7 dB
10% −32.7 dB

Twenty decibels per decade, to a tenth, over three decades of tolerance. First order, and only one column — where the earlier measurement has two columns twenty decibels a decade apart depending on which way two errors go. There is no second column here because there is no quantity for two errors to leave alone.

Now the loss.

series resistance in each capacitor as a fraction of R depth
0.01 Ω 10−610^{-6} −128.1 dB
0.1 Ω 10−510^{-5} −108.1 dB
1 Ω 10−410^{-4} −88.1 dB
10 Ω 10−310^{-3} −68.1 dB
100 Ω 10−210^{-2} −48.5 dB

Twenty decibels per decade again, and the two tables line up. One ohm is a fractional error of 10−410^{-4} and gives 88.1 decibels; a tolerance of 10−410^{-4} gives 92.0. Four decibels apart, which is the difference between perturbing one element and perturbing two in opposite senses, and is not a difference of mechanism.

A series resistance in a capacitor branch is a tolerance of r/Rr/R and nothing else. It is not a quality factor, it does not divide a reactance, and it does not have its own order. That essay’s two mechanisms are not exchanged in a twin-T; they are the same mechanism wearing two names.

A twin-T's null: everything fills it, and at one order. computed by solving, not by drawing. A twin-T of 10 kΩ and 10 nF nulls at 1591.5 Hz, where a low-pass T of two resistors and a high-pass T of two capacitors deliver equal and opposite contributions. With exact components the null is -Infinity dB, which is the arithmetic's floor. A tolerance fills it at 20.0 decibels per decade — first order, where an LC notch splits into twenty and forty depending on which way two errors go, because there a product can be left alone and here there is no product. Loss fills it at 20.0 decibels a decade too, and barely moves it — 1591.55 to 1595.52 Hz across four decades of series resistance — because a resistance of r in a capacitor branch is a fractional error of r/R and nothing else: one ohm gives -88.5 dB against a hundredth of a per cent of tolerance's -92.0. So a notch's two mechanisms are not exchanged here — they are one mechanism, because a twin-T's null is stationary in nothing and an LC notch's frequency is stationary in the product of its two components.
Fig. 2 A hundredth of a per cent: ninety-two decibels. Reaching that from the loss side needs a hundredth of an ohm in each capacitor, which no real capacitor has — so at the depths a twin-T is actually used at, it is the components’ tolerance that binds and not their loss.
A twin-T's null: everything fills it, and at one order. computed by solving, not by drawing. A twin-T of 10 kΩ and 10 nF nulls at 1591.5 Hz, where a low-pass T of two resistors and a high-pass T of two capacitors deliver equal and opposite contributions. With exact components the null is -Infinity dB, which is the arithmetic's floor. A tolerance fills it at 20.0 decibels per decade — first order, where an LC notch splits into twenty and forty depending on which way two errors go, because there a product can be left alone and here there is no product. Loss fills it at 20.0 decibels a decade too, and barely moves it — 1591.55 to 1595.52 Hz across four decades of series resistance — because a resistance of r in a capacitor branch is a fractional error of r/R and nothing else: one ohm gives -88.5 dB against a hundredth of a per cent of tolerance's -92.0. So a notch's two mechanisms are not exchanged here — they are one mechanism, because a twin-T's null is stationary in nothing and an LC notch's frequency is stationary in the product of its two components.
Fig. 3 A tenth of a per cent: seventy-two decibels. Reaching that from the loss side needs a tenth of an ohm in each capacitor, which is a fractional error of ten parts in a million — so at the depths a twin-T is used at, the two mechanisms ask for the same precision and neither is a quality factor.

And nothing moves it

The other half of that essay’s finding inverts too, and it is the half with the practical consequence.

Across four decades of series resistance the null moves from 1591.55 hertz to 1630 — one part in forty, and almost all of that at the last, absurd value. Across the tolerance range the null moves even less. The frequency is stable and the depth is what goes.

Compare that with the LC notch, where ten per cent of same-direction error moves the null from 1300 hertz to 1181.8 — nine per cent — while leaving it three hundred decibels deep. The two circuits fail in opposite coordinates: one keeps its depth and loses its frequency, the other keeps its frequency and loses its depth.

Which decides what a trimmer buys, and the answer reverses.

On an LC notch a trimmer is transformative. The earlier measurement is explicit: trimming for maximum rejection at the interference frequency moves the arm’s resonance onto the interference, so the first column of its table becomes the second, and a circuit built from ten per cent parts becomes as good as one built from matched ones. What it cannot buy is depth, because the depth at the bottom of the null is the arm’s quality factor whether the null is in the right place or not.

On a twin-T a trimmer is the whole thing. There is nothing else limiting the depth, so an adjustment against a null goes as deep as the adjustment’s own resolution allows — and the null it trims to is at the frequency it was already at, so the trim is one-dimensional rather than a search in two.

That is why an adjustable twin-T reaches sixty or seventy decibels on ordinary components and an adjustable LC notch built from a wound inductor does not. That essay’s own arithmetic says why the second cannot: a 2.5 henry inductor with a quality factor good enough for seventy decibels would need two and a half ohms of winding resistance, and an iron-cored inductor of that value has hundreds.

The only thing that fills a null is loss. computed by solving, not by drawing. With a lossless arm the deepest point of this notch is -313 dB, which is the floating-point floor rather than a depth. Put resistance in series with the inductor and the depth falls at 19.93 decibels per decade of it, over four decades — so a null is a statement about the inductor's quality factor and about nothing else. Forty decibels is all that is left by 180 Ω, which on this 2.53 H arm is a quality factor of 115.0 at the notch.
Fig. 4 That essay’s own loss measurement, for the contrast: an LC notch’s depth against four decades of series resistance in its arm, at twenty decibels a decade, with the lossless case off the bottom of the axis at the arithmetic’s floor. The slope is the same twenty; the quantity it is a slope in is a quality factor there and a fractional component error here.

Why one circuit has a stationary quantity and the other does not

The difference between the two nulls is worth stating once in the language the rest of this field uses, because it is the same distinction that decides how a filter’s passband responds to its components.

A quantity is stationary in a parameter when its first derivative with respect to that parameter is zero, so an error in it costs second order rather than first. That essay’s LC notch has one: the null’s frequency is 1/LC1/\sqrt{LC}, so an inductor one per cent high with a capacitor one per cent low leaves the product unchanged to first order and the frequency does not move — 1300.1 hertz against 1300.0, where two errors in the same direction give 1287.1. A factor of a hundred in the frequency error from the same tolerance on the same two parts, decided by a correlation no specification contains.

A twin-T’s null frequency is 1/(2πRC)1/(2\pi RC) and depends on a product too, so its frequency is stationary in the same way. What is not stationary is its depth, and that is the quantity this essay is about. The depth is the residual of a cancellation, and a cancellation’s residual is first order in everything by construction: it is a difference of two numbers, and the derivative of a difference is the difference of the derivatives, which is not zero unless the two paths respond identically.

The LC notch’s depth, by contrast, is not a residual at all. It is an impedance being infinite, and an infinity is not perturbed into a finite number by a small error — only by a mechanism that introduces a different kind of quantity, which is what loss is.

So the general rule the pair gives is worth having, because it applies past both circuits. A zero produced by an element becoming infinite or vanishing is robust to the tolerances of everything around it and vulnerable only to whatever breaks that element’s own idealness. A zero produced by two paths cancelling is vulnerable to everything, at one order, with no exceptions.

The same distinction shows in a passband, among the same inductors and capacitors: A ladder is not a cascade finds a doubly terminated ladder’s response stationary in every one of its element values, because at maximum power transfer the network is already doing the best it can — so its error grows as the 2.00 power of a tolerance where a buffered cascade’s grows as the 0.99. The exponents rather than the ratio are the claim there, and they are the claim here.

What the twin-T pays instead

Nothing is free and the twin-T’s cost is somewhere else, so it is worth saying where before this reads as an argument for one circuit over the other.

Its null is wide. That essay’s LC notch has a pole pair at a kilohertz and a zero at 1300 hertz, so its response returns to the passband quickly on both sides. A twin-T’s does not: it is a single real-pole shape on each side of the null, so it attenuates over a broad band — which is a disadvantage when the interference is close to signal that matters. The two circuits’ widths differ even where their depths do not.

Its source and load matter. A twin-T’s cancellation is between two paths whose outputs are summed at a node, and anything else at that node joins the sum. The measurement above loads the output with a gigohm; loaded by anything comparable to RR, the two paths’ contributions are re-weighted and the null moves and fills. So a twin-T must be buffered, where an LC section’s zero belongs to its arm alone and is indifferent to what follows it.

And its depth is a difference of two large numbers, which is the arithmetic’s own warning. At the null, each path delivers a signal of the order of the input and the output is a part in a million of it — so the circuit is a subtraction with a condition number of a million, and everything that perturbs either path appears amplified by that. That is the same shape the tolerance that is not on any part measures for a divider’s ratio, and it is the general reason a cancellation is a fragile way to get a zero.

Which is the honest summary of the pair. An LC notch gets its zero from a component being an open circuit and pays in that component’s quality factor; a twin-T gets it from two paths cancelling and pays in everything’s tolerance. Neither is better; they fail in different coordinates, and the choice is decided by whether a trim is available and by whether the inductor is.

What the two circuits are actually chosen between

The arithmetic above is about depth, and a real design chooses between the two for reasons that mostly are not depth — so it is worth setting them out, because the depth result changes the weighting rather than the list.

Whether an inductor exists. At mains frequencies the LC notch’s inductor is 2.5 henries, which is an iron-cored component with a quality factor of tens, a saturation current, a self-resonance and a microphonic response to being knocked. The twin-T is four resistors and three capacitors. That single fact settles most low-frequency designs before anything else is considered, and that essay’s own conclusion — that loss binds and tolerance does not, at these values — is the arithmetic of why.

Whether the frequency is known. A mains notch has to sit on a frequency that wanders, and neither circuit tracks it. The order of the null measures what that costs and finds a first-order null holding forty decibels over two per cent of frequency — so both circuits here are narrow in the same way and both need either a trim or a wider arrangement.

Whether a buffer is available. The twin-T needs one at its output and an LC section does not, which on a discrete design is a real cost and on anything with an amplifier in it already is not.

And what happens either side of the null. The LC section’s pole pair brings the response back up quickly; the twin-T’s does not. For a notch inside a measurement path that is the difference between removing an interferer and removing an octave of signal with it.

Read against that list, the depth result this essay measures moves one item. It says that a twin-T’s depth is limited by its adjustment rather than by its parts, so if a trim is available the twin-T reaches depths the LC section cannot — and if one is not, the twin-T built from one per cent parts reaches 52 decibels where the LC section built from the same parts reaches 79 or 33 depending on a correlation nobody controls.

So the presence of a trimmer decides which circuit is better, and it decides it in opposite directions at the two depths. Below about forty decibels the LC section is adequate untrimmed and the twin-T is not; above about sixty the twin-T is achievable trimmed and the LC section is not, because its own inductor’s quality factor has become the limit. That crossing is the practical content of both essays together and neither of them alone contains it.

The order of the null, which neither of them has

There is a third quantity that neither circuit’s null has and it is worth naming, because this field’s neighbouring field has just measured it.

The order of the null, not the number of them measures the width of a rejection trough rather than its depth, and finds the governing quantity to be the order of the zero: a first-order null’s band at a stated depth scales as that depth’s amplitude and a second-order null’s as its square root, so the ratio is 10−dB/4010^{-\mathrm{dB}/40} — a factor of ten at forty decibels and a hundred at eighty.

Both nulls on this page are first order. A parallel LC going open is a simple zero on the imaginary axis; a twin-T’s cancellation is a simple zero too. So both have the narrow trough the first-order law gives, and both would be widened by exactly the same arrangement — two of them, at the same frequency.

That is buildable in both cases and it is cheap in one of them. Two twin-Ts in cascade is two more amplifiers and no more inductors; two LC sections in cascade is two more wound components at the quality factor the depth already depends on. So the field that has no inductors is also the one where the order of a null is affordable, which is a second reason the twin-T is the circuit that gets adjusted and the LC section is the one that gets accepted.

A twin-T's null: everything fills it, and at one order. computed by solving, not by drawing. A twin-T of 10 kΩ and 10 nF nulls at 1591.5 Hz, where a low-pass T of two resistors and a high-pass T of two capacitors deliver equal and opposite contributions. With exact components the null is -Infinity dB, which is the arithmetic's floor. A tolerance fills it at 20.0 decibels per decade — first order, where an LC notch splits into twenty and forty depending on which way two errors go, because there a product can be left alone and here there is no product. Loss fills it at 20.0 decibels a decade too, and barely moves it — 1591.55 to 1595.52 Hz across four decades of series resistance — because a resistance of r in a capacitor branch is a fractional error of r/R and nothing else: one ohm gives -88.5 dB against a hundredth of a per cent of tolerance's -92.0. So a notch's two mechanisms are not exchanged here — they are one mechanism, because a twin-T's null is stationary in nothing and an LC notch's frequency is stationary in the product of its two components.
Fig. 5 Ten per cent: thirty-three decibels, which is what an untrimmed twin-T of ordinary components gives. The essay below’s LC notch at one per cent gives thirty-three the same way — the two circuits reach the same depth from tolerances two orders apart, because one of them has a resonance that absorbs most of the error and the other does not.

What is not here

The amplifiers are not in the netlist. A twin-T is almost always used inside a feedback loop — its null placed in an amplifier’s feedback path to make a band-reject or, with positive feedback, a high-quality-factor notch. Everything above is the passive network, so the loop’s own gain and its finite bandwidth are absent, and the q the amplifier decides is where that kind of error is measured for a different section: two per cent high in quality factor and two per cent low in pole frequency for an amplifier a hundred times the corner.

The capacitors are capacitances. Their own dielectric absorption, which the capacitor that remembers measures, adds slow time constants in parallel with the intended one — a set of extra poles and zeros near the null, at a level this figure does not resolve.

And the tolerance is applied in the worst direction. The two shunt elements are moved in opposite senses, which is what breaks the cancellation hardest. A real set of components is a distribution, and the depth a production run achieves is the tail of that distribution rather than its worst case — the tolerance that is not on any part is where the difference between a worst case and a measured spread is priced, at 0.29 per cent for one per cent components.

Still open: the trim’s own resolution, and the twin-T inside a loop

How deep a trim reaches. The essay’s conclusion is that a twin-T’s depth is bounded by its adjustment rather than by its components, which turns the question into one about the trimmer: its resolution, its temperature coefficient and its own stray capacitance. A depth of eighty decibels needs the ratio held to a part in ten thousand, and whether a mechanical trimmer holds that over a working temperature range is a measurement nobody on this page has made.

The null inside a loop. Placed in a feedback path a twin-T’s null becomes a peak, with a quality factor set by the loop’s gain — which is the standard high-Q notch and the standard sine oscillator. The essay’s finding that everything fills the null at one order ought to translate directly into a statement about how precisely that quality factor can be set, and the translation is one solve away.

And the same question for a bridged-T. A bridged-T reaches a null with one fewer component and a different cancellation, and whether its depth is first order in everything too, or whether its structure restores something for errors to leave alone, is the natural next case. That essay’s split into two orders came from one product being preserved; the question is whether any three-element cancellation has one.

What is checked

The exact null is required below −250 dB, which is the statement that it is the arithmetic’s floor rather than a depth — the same discriminator the essay before it uses and for the same reason.

Both slopes are required at twenty decibels a decade, to a decibel and a half, over three decades each. The pair is the finding: one slope alone would be a measurement of a circuit and two together are a measurement of a mechanism.

And the equivalence between them is required directly — a series resistance of rr giving the same depth as a tolerance of r/Rr/R, to a few decibels — because that is the sentence the essay rests on and a pair of equal slopes does not establish it.

The loss is required NOT to move the null, to five per cent across four decades of it. That requirement is the one that failed first: the figure as written expected loss to do nothing at all, on the reasoning that a twin-T has no resonant arm to have a quality factor. It has no arm and it has a cancellation, and a cancellation is broken by anything — which is the essay.

Part 2 on null depth

One argument about Null depth, and one of 2 essays on it so far, each part numbered by how much of the idea it assumes. What sits either side of it:

What a mistuned arm leaves at 1300 Hz. computed by solving, not by drawing. Neither curve is the null's depth — the null is still bottomless, it has simply moved — but the depth at the frequency the notch was designed for, which is the number a filter is bought for. One per cent components leave -32.9 dB if both errors go the same way and -78.8 dB if they oppose, a factor of 197 from the same tolerance on the same two parts. The slopes are 20.0 and 40.0 decibels per decade: first order in the error on the product LC, second order in the error on the impedance level. What actually fills a null Part 1 — A ten per cent error in the two components of a notch's arm leaves the null three hundred decibels deep — it moves it rather than filling it. What fills it is loss, at twenty decibels per decade of arm resistance exactly. And the depth at the frequency the notch was designed for splits into two orders depending on which way the two errors go: one per cent parts leave 79 decibels one way and 33 the other.

Where this goes next

What a mistuned arm leaves at 1300 Hz. computed by solving, not by drawing. Neither curve is the null's depth — the null is still bottomless, it has simply moved — but the depth at the frequency the notch was designed for, which is the number a filter is bought for. One per cent components leave -32.9 dB if both errors go the same way and -78.8 dB if they oppose, a factor of 197 from the same tolerance on the same two parts. The slopes are 20.0 and 40.0 decibels per decade: first order in the error on the product LC, second order in the error on the impedance level. What actually fills a null A ten per cent error in the two components of a notch's arm leaves the null three hundred decibels deep — it moves it rather than filling it. What fills it is loss, at twenty decibels per decade of arm resistance exactly. And the depth at the frequency the notch was designed for splits into two orders depending on which way the two errors go: one per cent parts leave 79 decibels one way and 33 the other. A resonator's Q against its inductor's, with a capacitor of Q 1581. computed by solving, not by drawing. The dashed line is what the resonator's Q would be if the inductor were its only loss; the solid one is what it is with a capacitor of Q 1581 beside it. They part company where the inductor stops being the worst component. At the marked point the inductor's Q is 79.06, the capacitor's is 1581.1, the reciprocals predict 75.2923 and the solved network measures 75.2923 — 2.2e-7% apart, by two routes that share only the element values. The resonance stays at 1/2π√(LC) to a part in a million throughout. The Q the components allow Resonance and its bandwidth measured a half-power width of exactly f₀/Q at every Q tried, for a circuit whose only resistance was the one deliberately put there. Real components arrive with resistance of their own, and the consequence is a ceiling rather than a penalty. The reciprocals of the component quality factors add, so the total sits below the smallest of them: an inductor of 79 beside a capacitor of 1 581 gives a resonator of 75. The worst component decides and the best one cannot help. What widens a null: the order of the zero, not the placement of two. Integrated by quadrature on the union of both windows' nulls, at a fixed total length of 40 ms so that every arrangement settles in the same time. One mean of 40 ms holds -40 dB over 2.002 per cent of 50 Hz and passes 12.50 Hz of noise. Two means of 20 ms in cascade — a second-order null in the same place — hold it over 20.59 per cent — a factor of 10.3, which is 10^(−dB/40) because a first-order null rises linearly through the target and a second-order one quadratically — for 1.333 times the noise bandwidth. Splitting the two lengths puts two first-order nulls either side instead, and the widest trough is at a split of 10.12 per cent — 17.98 and 22.02 ms — holding 29.93 per cent for 16.53 Hz — which is exactly √2 times the equal pair's band, at every depth, for slightly LESS noise. It is also a cliff: a split of 10.118 per cent — 0.01 per cent further — leaves no band at 50 Hz at all, because the peak between the two nulls has climbed through the target. The order of the null, not the number of them A mean rejects the mains exactly and rejects it one per cent off frequency by forty decibels, whatever the window length. Two means in cascade of the same total length settle in the same time, cost a third more noise bandwidth and hold that forty decibels over ten times the band — the factor being ten raised to minus the depth in decibels over forty, because a first-order null rises linearly through a target and a second-order one quadratically. Splitting the two lengths to straddle the frequency buys a further factor of exactly root two, for no noise at all, and puts the design two parts in ten thousand from a cliff.

The objects named here

The third axis, after the field and the idea: the things themselves, and every essay that touches each one.

Component toleranceModel rangeParasiticsThe quality factorStopband attenuationTransmission zero