The band that closes with the order

The same filter a thousand times larger established two things. That a design is exactly invariant under impedance scaling — multiply every resistance and divide every capacitance by the same factor and the solved response agrees to a part in $10^{15}$, which is the last bits of a double — and that a realisation is not, because two of its quantities refuse to scale. Fifty ohms of amplifier output resistance binds the low end and two picofarads of stray capacitance to everywhere binds the high one, and between them a single Sallen-Key section is inside a tenth of a decibel over three decades.

That essay stopped at one section and named the obvious next question. A real filter is several sections, every one of them carrying its own strays and its own amplifier, so what happens to the band as the order rises?

It closes, and it closes fast enough to run out.

Four orders, four bands #

The measurement is the same one, run at four orders. Each section is an equal-resistor Sallen-Key with the pole pair a Butterworth of that order wants; each carries three stray capacitances, one per node, and one amplifier with a gain–bandwidth product and fifty ohms out. The comparison is against the same design realised with ideal amplifiers and no strays.

The pattern is as clean as anything in this collection: one decade of band lost per two orders, symmetrically from both ends, and at order eight the two edges have met and passed each other.

The last row is the finding. There is no impedance level at which an eighth-order Butterworth built this way is inside a tenth of a decibel of its own design. Not a bad one, not a marginal one: the best achievable anywhere on six decades of impedance is 0.1138 dB, and the free parameter the previous essay found has been spent.

Why the floor rises rather than the edges moving #

There are two ways a band could close and only one of them is what happens here, so it is worth separating them.

The edges could move towards each other because the mechanisms got stronger — a larger stray, a worse amplifier. They did not; the strays and the output resistance are the same two picofarads and fifty ohms at every order.

What happens instead is that the floor rises. The least departure achievable, in the middle of the range where neither mechanism dominates, goes 0.0133, 0.0421, 0.0763, 0.1138 dB across the four orders — and the band’s edges are simply where the rising curve crosses a fixed tolerance. The two edges appear to move because the whole curve has lifted.

The rise is close to proportional to the number of sections, which is what independent contributions adding would give: 0.0133 per section would be 0.053 at four sections against 0.114 measured, so the contributions are not equal and the high-Q section contributes more than its share. That is expected — a section’s sensitivity to a stray on its own nodes goes up with its quality factor, and an eighth-order Butterworth’s worst section has $Q = 2.56$ against a second-order’s 0.707 — but the counting argument gets the order of magnitude and the shape right.

The practical statement is therefore about accumulation rather than about any one section. Nothing is wrong with the fourth section that is not wrong with the first. The design has simply asked four imperfect things to agree, and their errors do not cancel.

The two mechanisms, per section #

The edges of the band are the same two mechanisms as before, and it is worth restating them in the form that says why each is per-section rather than per-filter.

At the low end, the amplifier’s output resistance is in series with the section’s feedback network. An equal-resistor Sallen-Key feeds its output back through $C_1$ to the node between the two resistors, so fifty ohms in series with that path is fifty ohms added to a resistance of $R$ — a fractional error of $50/R$, which is 5% at a kilohm and 0.05% at a hundred kilohms. Each section has its own amplifier and its own fifty ohms, so each has its own error of that size and they accumulate.

At the high end, two picofarads on each node is in parallel with the design’s own capacitances. The section’s $C_2$ at a hundred kilohms and ten kilohertz is 113 picofarads, so two picofarads is 1.8% of it; at ten kilohms it is 1.13 nanofarads and two picofarads is 0.18%. Three nodes per section, and again they accumulate.

Both errors are fractional and both are proportional to the number of sections, which is the whole content of the table above. Neither is a defect of the topology: any realisation with an amplifier per pole pair and a board under it has the same two numbers.

Where in the passband it goes wrong #

The table quotes a worst departure across the passband, and where that worst is turns out to move with the order in a way worth reading.

At order two the worst disagreement at the middle of the band is at 20.0 kHz — twice the corner, out on the skirt. At order four it is at 15.3 kHz, at order six 12.9 kHz, at order eight 11.8 kHz. It is walking down towards the corner as the order rises.

That is the high-Q section making itself felt. A Butterworth of order $2n$ has its highest-Q pole pair nearest the corner, and a stray on that section’s nodes moves that pair’s frequency and quality factor by more than it moves any other’s. So the error concentrates where that pair’s response is largest, which is just below the corner — and the higher the order, the higher that pair’s $Q$ and the more sharply the error localises there.

The consequence for a measurement is direct. A realisation check made in the middle of the passband, where an eyeball would put it, sees a fraction of the error; the number that matters is at the corner, which is also where the design is least forgiving and where a specification is normally written. This is why the figure compares over the passband and the corner rather than over the passband alone, and why reading the stopband instead would measure the amplifier’s own gain floor rather than the filter.

Which repairs work, and which only move the edge #

Four things are usually suggested when a high-order active filter does not meet its specification, and they are not equivalent.

Choose the impedance level well. Worth one decade of margin at best, because the band is a decade wide at order six and gone at order eight. It is the free parameter the previous essay found and it is the first thing to spend, but it does not extend the range of orders this construction can reach.

Use a better amplifier. Moves the low edge only. Halving the output resistance moves the low edge down by a factor of two and does nothing to the high edge — which the previous essay measured directly, finding that a tenth of the output resistance moves the low edge by 0.100× and the high edge by 1.000×. Against a floor that has risen above the tolerance, moving one edge does nothing at all.

Reduce the strays. Moves the high edge only, symmetrically.

Use a different realisation. This is the one that changes the answer rather than the edges. A doubly terminated passive ladder has a response that is stationary with respect to its element values — the error goes as the square of a component error rather than as the first power — which is this field’s own measurement in a-ladder-is-not-a-cascade. A cascade of biquads has no such property, and this essay is a second measurement of the same fact, in a different currency: the cascade’s errors add up per section, and there is nothing in its structure that makes them cancel.

What the floor is worth in decibels of specification #

A tenth of a decibel is a stringent passband tolerance and not an absurd one — it is roughly what a 0.1 dB-ripple Chebyshev design is asking for, and asking for a realisation error smaller than the design’s own deliberate ripple is a reasonable demand.

Relaxing it changes the arithmetic and not the shape. At a tolerance of 0.25 dB, order eight has a band again; at 0.5 dB, a wide one. So the boundary this essay draws is a boundary jointly in the order and the tolerance, and the useful form of it is: the achievable passband accuracy of a buffered cascade is about 0.013 decibels per section, on a well-chosen impedance level with these parasitics.

That is a number a designer can carry. Six sections is 0.08 dB and twelve is 0.16, before any component tolerance at all — and component tolerances then add on top, at their own first-order rate. It also says which specifications are simply not available from this construction: a twelfth-order filter flat to a hundredth of a decibel is not a matter of buying better parts, it is a request the topology cannot meet.

The other quantity that does not scale, and points the other way #

There is a third consideration that appears on neither axis and gets stronger with order, which is worth putting beside the two that bound the band.

Noise. The floor a resistor sets is where the density comes from — 4.00 nanovolts per root hertz for a kilohm at room temperature, because it is warm rather than because of anything about how it was made — and it goes as $\sqrt R$, so the low end of the impedance band is the quiet end and the high end is 1000 times noisier across the six decades drawn, which the figure asserts to a part in a billion. A cascade adds the noise of each section, filtered by the sections after it, so an eighth-order filter’s output noise is larger than a second-order’s at the same impedance level.

Two refinements of that arithmetic are measured elsewhere and both matter at the order where the band shuts. The bandwidth noise sees says what the filtering is worth: a single pole passes π/2 times as much noise power as a brick wall at its own corner, so a figure computed from the −3 dB point is twenty-one per cent low — while a five-pole Chebyshev’s ratio is 0.964, less than one, so a high-order filter is quieter than its corner frequency suggests and a first-order one is noisier. And the floor a circuit has says the amplifiers are not free either: each contributes a voltage generator and a current generator, 4 nV/√Hz and 0.6 pA/√Hz for an ordinary part, with a source resistance of 6.67 kΩ at which their sum is least. So the noise is not monotone in the impedance level after all — it is monotone in the resistors’ own contribution and has a minimum once the amplifiers’ current noise into those resistors is included, and the minimum moves with the number of amplifiers, which is the order.

Which does not change the shape of the design problem stated below, and does move its answer. The noise-optimal level is at or near the low edge of the accuracy band rather than exactly at it, and how near depends on a part parameter the filter design says nothing about.

Put that beside the accuracy floor and the design problem has a shape. Accuracy is best in the middle of the impedance range and gets worse at both ends; noise is monotone and best at the low end. So the noise-optimal choice is always at the lower edge of the accuracy band, never in its middle — and at the order where the accuracy band closes, there is no longer a lower edge to sit on and the two requirements have nothing to trade against each other.

That is the sense in which the closing of the band is a real design boundary rather than an arithmetic curiosity. Below it there is a region and a choice within the region; above it there is neither.

What a section is, and why every one costs the same #

It is worth saying plainly what is being cascaded, because the counting argument above depends on each section being an independent object and that is a property of this realisation rather than of filters in general.

An equal-resistor Sallen-Key section is two resistors, two capacitors and a unity-gain amplifier. Its pole pair has $\omega_0 = 1/(R\sqrt{C_1C_2})$ and $Q = \tfrac12\sqrt{C_1/C_2}$, so the frequency is set by the geometric mean of the two capacitances and the quality factor by their ratio. A high-$Q$ section is one with a large capacitance ratio, which is exactly why a stray of fixed size hurts it more: two picofarads is a larger fraction of the smaller of two capacitances that are far apart than of two that are close.

The amplifier at the end of each section is what makes the sections independent. Its output drives the next section and its input draws nothing, so no section loads the one before it and each realises its own pole pair exactly. That independence is what makes a cascade easy to design and is also, precisely, what stops the errors from cancelling: a ladder is not a cascade measures the consequence and finds the exponent rather than the ratio to be the claim — a cascade’s passband error growing as the 0.99 power of a component tolerance and a doubly terminated ladder’s as the 2.00, because at maximum power transfer the response is stationary in every element it contains. An error in one section is not seen by any other section, so there is nothing in the structure that could correct it.

The amplifier is not free in the other direction either, and the free parameter this essay sweeps is not the only thing it constrains. The Q the amplifier decides finds a section built with a part a hundred times its corner — the usual rule — coming out with its quality factor 1.97 per cent high and its pole frequency 1.97 per cent low, the same number in both directions, and the number being the designed Q divided by the ratio. A high-$Q$ section is therefore hurt twice by the same design choice: the stray hurts it more, as the paragraph above says, and the finite gain–bandwidth hurts it more, in proportion to the very Q that made it sensitive.

The ladder alternative is the opposite arrangement in both respects. Every element loads every other, which makes it far harder to design and gives it the stationarity that makes its errors second order. Those two facts are the same fact.

What closes the band at each end #

A band of impedance levels that closes as the order rises is bounded by two quite different things. The same filter a thousand times larger is the invariance being tested, exact in the ideal network and broken by absolute quantities. The floor that outlives the arithmetic is the passive structure’s own floor, which no impedance level reaches past. A ladder is not a cascade is the realisation this one is measured against, and What a steep skirt costs is the decision that puts the order up in the first place.

What is checked #

Three assertions, and the third exists because the second stops being true.

That the design is invariant to $10^{-12}$ across six decades of scaling and every frequency in the passband — the claim the whole essay stands on, unchanged from the previous rung and re-checked at every order.

That the realisation is inside the tolerance over a band rather than everywhere, with the least departure in the interior rather than at either end. That is what makes it a band and not a direction, and it is asserted at every order where a band exists.

And that where there is no band, it is because the least departure any level allows is already above the tolerance — asserted at order eight, which is the case the previous rung’s version of this assertion would have failed on. A gate that had been written to require a non-empty band would have called this essay’s finding a bug.

That is worth one more sentence because it is a general hazard rather than an incident. An assertion written while a phenomenon is being discovered tends to encode the shape the phenomenon had at the time — here, that a sweep of impedance levels has a best value with worse values on both sides of it. The finding at order eight is that the shape itself changes: there is a best value and it is not good enough, so the curve has a minimum and no band. An assertion requiring a non-empty band would have been red on the run that produced this essay’s result, and the natural reading of a red gate is that the code is wrong rather than that the claim has been extended past where it held.

The repair is to assert the measurement rather than the conclusion — the least departure any level allows, and separately whether it is inside the tolerance — which is what the two clauses above do. The band that does not close then reuses the same two clauses on a structure where the answer is different at every order, and the floor that outlives the arithmetic reuses the first of them alone, having dropped the tolerance entirely.