What actually fills a null

A notch has infinite rejection on paper. Everybody knows it does not in practice, and the reason everybody gives is component tolerance — one per cent parts, so perhaps forty decibels of null. That reasoning is wrong twice over: tolerance does not fill a null at all, and the number it produces at the frequency somebody actually cares about is not forty decibels but either seventy-nine or thirty-three, depending on something no tolerance specification mentions.

The section, and where its zero comes from #

An inductor and a capacitor in parallel, in series with the signal path, feeding a resistor and a second capacitor to ground. The parallel pair is an open circuit at its own resonance and nothing gets through there — that is the transmission zero, and it belongs to the series arm alone.

The numbers: 2.533 H and 5.917 nF in the arm, resonating at 1300 Hz; 4.083 nF and 39.79 kΩ to ground, which with the arm’s capacitance puts the pole pair at 1000 Hz. Far above both, the response settles at the fraction of the total shunt capacitance the arm holds, which is −4.6 dB.

Tolerance does not fill it #

Put a ten per cent error on the inductor and a ten per cent error the other way on the arm’s capacitor, and go looking for the deepest point of the response.

It is at 1306.5 Hz, and it is 308 decibels deep.

That is the arithmetic’s floor, not a measurement. A parallel LC is an open circuit at whatever frequency it happens to resonate at, and putting the components ten per cent out does not stop it being an open circuit — it moves the frequency at which it is one. The null is exactly as infinite as it was.

The depths in that table are noise. What is not noise is the right-hand column, and the pattern in it is the whole of the next section.

The two directions are not the same fault #

Errors in opposite directions barely move the null. Errors in the same direction move it a lot.

The resonance depends on the product LC, so an inductor one per cent high and a capacitor one per cent low leave the product unchanged to first order and the frequency does not move — 1300.1 Hz against 1300.0. Both components one per cent high multiply the product by 1.0201 and the frequency falls by one per cent, to 1287.1 Hz.

That is a factor of a hundred difference in the frequency error from the same tolerance on the same two parts, and it is decided by a correlation that no component specification contains. Two parts from the same reel of the same batch are likely to err the same way; a capacitor and an inductor from different suppliers are not correlated at all.

The number that matters, and its two orders #

A designer does not have a notch at “wherever the arm resonates”. They have an interference at a frequency, and what they need to know is the rejection there.

Measured at 1300 Hz with the arm mistuned:

Fitted over the three smallest tolerances, the same-direction column falls at 20.0 dB per decade of tolerance and the opposed column at 40.0. First order and second order, and the mechanism is the one above: same-direction errors move the resonance, which is first order in the error, and opposed errors leave the resonance alone and change only the impedance level of the arm, which shows up second order.

At one per cent parts the two answers are −32.9 dB and −78.8 dB. That is a factor of 197 in the residual amplitude, from one tolerance on one pair of components, decided by a sign nobody wrote down.

The everyday consequence is worth stating plainly: a notch built from a matched pair is a different circuit from a notch built from two parts that happen to be in tolerance. Trimming one component against the other — which is what an adjustable notch does — is not merely convenient. It converts the first column into the second.

What does fill it #

Loss does, and nothing else does.

Put a series resistance in the arm’s inductor branch. The arm is now a parallel LC with a resistance in one leg, so its impedance at resonance is finite rather than infinite, and something gets through.

Twenty decibels per decade of resistance, exactly — the fitted slope over four decades is 19.93, and the departure from twenty is the top of the range, where 180 Ω is no longer negligible beside the arm’s reactance.

There is a closed relation in that table and it is worth having. The arm’s quality factor at the zero is $\omega_z L / r$, which is 20 690 Ω of reactance over the series resistance. The depth is

$$1.27\ \text{dB} - 20\log_{10} Q_\text{arm}$$

constant to a hundredth of a decibel over four decades, where the 1.27 dB is what the shunt network contributes at that frequency. So the depth of a null is the quality factor of its arm, in decibels, and a designer who knows the inductor’s Q knows the answer before building anything.

How narrow a deep null is #

The two mechanisms so far are about the depth at one frequency. The third quantity a designer needs is how far off that frequency the depth survives, and it is smaller than the word “notch” suggests.

With a lossless arm, this section’s null is:

Every twenty decibels of depth costs a factor of ten in the width — which is the same first-order statement as the mistuning result, seen from the other side. A response that goes to zero linearly through the null has twenty decibels per decade of frequency offset in it, whether that offset comes from the signal moving or from the components moving.

Sixty decibels of rejection is available over one part in a thousand of frequency. For a mains interference at fifty hertz that is 43 millihertz, which is inside what a mains frequency wanders by in a day. For a crystal-referenced carrier it is not a problem at all. The specification “60 dB notch” is meaningless without the band beside it, in exactly the way “settling time” is meaningless without a band, and for exactly the same reason.

What a trim can and cannot fix #

An adjustable notch has a trimmer on one of the two arm components, and it is worth being precise about what that buys.

It buys the whole first column of the table above. Trimming for maximum rejection at the interference frequency is, by construction, moving the arm’s resonance onto the interference — so the frequency error goes to zero and the design-frequency depth becomes whatever the null’s own depth is. A circuit built from ten per cent parts and trimmed is as good as one built from parts matched to the resolution of the trimmer.

It buys nothing at all against loss. The trimmer moves the resonance; it does not change the arm’s quality factor, and the depth at the bottom of the null is $1.27 - 20\log_{10}Q_\text{arm}$ whether the null is in the right place or not. A trimmed notch and an untrimmed one built from the same inductor have the same maximum rejection.

Which is why the order of the two questions matters. The loss sets the depth that is achievable at all; the tolerance, or the trim, sets whether the circuit is at the null. Getting the second right and the first wrong produces a perfectly centred forty-decibel notch where a hundred was wanted, and no amount of further adjustment moves it.

Which of the two binds #

The two mechanisms are independent and either can be the limit, so the design question is which.

With one per cent uncorrelated parts, the design-frequency depth is −78.8 dB. Reaching that from the loss side needs an arm quality factor of about 8 800, which is $20\,690/8\,800 = 2.4$ Ω of series resistance in a 2.5 H inductor. A 2.5 H inductor with two and a half ohms of winding resistance does not exist: an iron-cored inductor of that value has hundreds of ohms, and its core losses at 1300 Hz are worse than its copper.

So for this section, at these values, loss binds and tolerance does not — which is the opposite of the received answer. Put a realistic 200 Ω of arm resistance in and the null is −41 dB, and the tolerance would have to be worse than five per cent before it contributed anything.

That conclusion is specific to a component value, though, and it inverts at higher frequencies where inductors are small and good. At a megahertz, a few microhenries with a Q of two hundred gives −45 dB from loss, and one per cent parts give −79 dB from tolerance — so loss binds there too, and it takes a resonator rather than an LC pair to get past it.

What the measurement leaves out #

One section. Everything here is a single notch. A filter with several transmission zeros has several arms, and their depths do not add: the response between two nulls is set by the pole pairs and is not improved by making either null deeper.

No temperature. An inductor’s resistance moves about 0.4% per kelvin with the copper, so the depth above moves about 0.03 dB per kelvin, which is negligible. The tolerance half does not behave so well: a capacitor’s temperature coefficient moves the product LC, which moves the null, which moves the design-frequency depth at 20 dB per decade of the shift. And the coefficient a data sheet prints is not enough to compute that with: the coefficient that is about one reading finds it to be a coefficient of the one capacitance a bridge reports at zero bias, with one number unable to determine the two parameters the part actually has, so that two parts a bridge cannot tell apart differ by 1.80 at the voltage they are used at. A notch built with a class II ceramic has a null frequency that is not predictable from the printed numbers at all, and this essay’s arithmetic then turns that unpredictability into depth.

And no source or load impedance. The section is driven from an ideal source into an open circuit. A real driver’s output impedance is in series with the arm and adds to its loss directly, which is another way of saying that the depth of a null is a property of the whole circuit rather than of the section. How large that addition is depends on where the driver’s own loop has got to: the node that is at ground for a while measures an amplifier’s summing junction climbing from a tenth of an ohm to 909 ohms as its loop runs out, and a source below a frequency measures a regulator’s output doing the same thing in four orders. A null placed high in the audio band is being driven by something whose series resistance is a strong function of frequency, and this essay’s twenty-decibels-a-decade law converts every ohm of it directly into depth.

Where the two error directions of this essay’s tolerance result come from is worth one more sentence, because the same asymmetry appears in a passband. A ladder is not a cascade measures the passband version and finds the exponent rather than the ratio to be the claim: a cascade’s error grows as the 0.99 power of the tolerance and a doubly terminated ladder’s as the 2.00, because at maximum power transfer the response is stationary in every element. A notch’s null is stationary in nothing at all — it is a cancellation between two branches, so a first-order error in either moves it first order — which is why the tolerance result on this page splits into two orders and the passband result there does not.

What the null is for, and what fills it #

A notch that is not a null is the section behind the fourth filter family. The zeros that buy an order is what the zero buys, and where it gives it back. The Q the components allow is the loss that fills the null at a far-out zero. The tolerance that is not on any part is the mechanism that fills it at a close-in one, and A ladder is not a cascade is the realisation decision that decides how much of either arrives.

The gate #

The lossless null is asserted to be below −250 dB, which is the statement that it is the arithmetic’s floor rather than a depth. Anything a real component could produce is above it by two hundred decibels.

The mistuned null is asserted to be bottomless too, at ten per cent error, and its frequency is reported — because the claim being tested is that the tolerance moved the null rather than filling it.

Both slopes are fitted over the three smallest tolerances, not end to end. An order is a statement about a limit, and at twenty per cent neither curve is in it: fitted across the whole range the same-direction slope comes out 18.96 rather than 20.

The loss slope is asserted at twenty decibels per decade to within four tenths, over four decades, which is the claim that a null’s depth is its arm’s quality factor and nothing else.

Where the arm’s quality factor comes from #

If a null’s depth is a quality factor then the question of how deep a null can be built is the question of how good a resonant arm can be made, and that has been measured on this site rather than assumed.

The Q the components allow is the ceiling: the reciprocals of the component quality factors add, so the total sits below the smallest of them, and 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. Read through this essay’s twenty-decibels-a-decade law, that is a null depth fixed by whichever of the two parts in the arm is worse — which for any arm containing a wound inductor is the inductor, by a factor of twenty, and the capacitor’s own excellence is worth nothing at all.

The resistance that grows with frequency then says that the inductor’s own figure is not a constant. Computed exactly from the Kelvin functions, a conductor’s resistance is already 2.05 per cent up at the frequency the rule of thumb names as the point where the effect begins, and it keeps climbing — so an arm that has a given quality factor at one frequency has a worse one an octave up, and a null placed by a design at one frequency is not as deep there as the same arm measured lower would suggest.

The escape everybody reaches for is to build the inductance rather than wind it, and the inductor that is an amplifier measures what that substitutes. Four resistors and a capacitor around two amplifiers present one henry to within one per cent over three and a half decades — but the series resistance goes negative at 63 hertz, well inside the band where the inductance is still excellent. A null built there does not have a deep minimum; it has an arm with loss of the wrong sign, and what fills the null is replaced by something that empties it and then starts oscillating.

So the loss slope measured here has a range in the same variable as everything else in this collection. Above the frequency where the winding’s resistance climbs, the null is shallower than the low-frequency arm predicts; below the frequency where a synthetic arm’s resistance changes sign, it is deeper than any passive arm could be and the circuit is not stable. Between the two the twenty decibels a decade is exact.

And the depth that matters is not usually the one at the bottom of the null. A notch is fitted to reject something — a mains harmonic, a carrier, a clock — and what the design needs is the attenuation at the frequency of that thing rather than at whatever frequency the arm happens to resonate. This essay’s central result is precisely about the difference between those two: a tolerance moves the null rather than filling it, so the depth at the bottom stays at three hundred decibels while the depth at the design frequency falls to 79 or to 33 depending on which way the two errors went. A specification written as “the null is forty decibels deep” is a statement about a frequency nobody can locate to better than the component tolerance, and a specification written as “forty decibels at 50 hertz” is one a build can be tested against.