The floor that outlives the arithmetic

The essay one rung down measured a band: how many decades of impedance level a filter can be built at while staying inside a tenth of a decibel of the response it was designed for. A buffered cascade’s band closes by half a decade per order and is empty at eight. A doubly terminated ladder’s does not close — at order nine it still had four decades to choose from, and the reason was a classification rather than a constant, because an inductor’s loss is a fixed quality factor and a fixed quality factor scales with the design.

A band is a consequence of a tolerance, though, and a tolerance is a choice. Ask for a hundredth of a decibel instead of a tenth and every band in that essay narrows; ask for a decibel and they all open. What a structure actually has, independent of what is asked of it, is a floor: the least departure any impedance level allows. This essay measures that, at six orders, and then finds out what stops it.

The floor is a minimisation, and it has to be found properly #

The floor is the minimum over impedance level of the worst departure over the passband, which is a minimax quantity and is easy to compute badly.

The band measurement one rung down sampled twenty-one levels across five and a half decades and reported the best of them. That is an over-estimate of the floor by however far the sample sat from the true minimum — up to a quarter of a decade of level, which at these orders is a few per cent of the answer. So the sweep here does the same scan and then refines it: a golden-section pass in the logarithm of the level, eighteen iterations, inside the bracket the grid provides. It costs a dozen more solves per order and it removes a bias that would otherwise have been in every number.

The refined floors, for a half-decibel Chebyshev at ten kilohertz, are 0.0087 decibels at order three and 0.0389 at order thirteen, at levels that fall from 3.80 ohms to 0.96 as the order rises.

That fall is worth a sentence. The best impedance level for a ladder of this kind is a few ohms, which is not where anybody builds one — it is a consequence of the two imperfections being balanced there rather than a recommendation — and it moves lower with order because the network has more inductors, each contributing loss, and the only quantity that pushes back scales as the load resistance.

It rises linearly, which is the answer to the question #

Fitted over six orders, the floor grows as the 1.06 power of the order. Not as a square, not exponentially — very nearly in proportion.

Which settles what the ladder’s band does in the limit, and the answer is that it does not close at any order a person builds. Extrapolating the fitted line to the tenth of a decibel that the band is drawn at puts the crossing at order thirty-two. At the winding quality on the slider’s quiet end — ten ohms per henry rather than forty — it is order one hundred and nineteen.

A linear rise is what an independent, similar error from each additional section would produce, and that is error, and that is close to what the decomposition below shows. It is emphatically not what a cascade does: the same measurement on a buffered cascade of Sallen-Key sections gives a floor rising eight and a half times over four orders, against the ladder’s four and a half over five, and the reason is the one the rung below identified — a fixed amplifier output resistance is a fixed resistance, which binds from below and gets relatively worse as the design’s own impedances are scaled to escape it.

The floor is not the sum of its causes #

The realisation carries three imperfections: two picofarads of stray capacitance at every internal node, a winding resistance of forty ohms per henry, and a few milliohms of track and connector in series with the load. Removing each in turn and re-measuring at the level that minimises the departure gives a result that is not a decomposition at all.

The stray capacitance contributes nothing measurable. At the level that minimises the departure — a few ohms — two picofarads is an admittance of nothing beside conductances of tenths of a siemens, and the departure with it removed is the same to four decimal places at every order.

The track resistance contributes negatively. Removing it makes the departure worse: at order thirteen the complete realisation is 0.0389 decibels from its design and the winding loss alone leaves 0.0625. The track resistance is in series with the load and the reading is taken across the pair, so the ladder is terminated in slightly more than it was designed for and the passband is lifted — exactly where the winding loss droops it.

So the floor is where two errors of opposite sign are least able to cancel, rather than where the smallest error is. That is a different thing from what the phrase “least departure” suggests, and it has a practical consequence: a designer who improves one of the two — better wire, or a shorter track — can make the realisation worse, and the improvement that helps is the one that reduces both.

What actually stopped the sweep #

The sweep in the figure stops at order thirteen, and the reason is neither the structure nor the parasitics nor the time it takes.

A doubly terminated ladder is synthesised by expanding the input impedance $(E+F)/(E-F)$ as a continued fraction, peeling off one reactance at each step. Every step divides two polynomials and subtracts, and the subtraction cancels the leading coefficients — that is what makes the degree drop, and it is what the method is for. When the polynomials are long enough, the cancellation removes more digits than double precision has, the remainder is dust rather than a polynomial, and the expansion stalls: the next quotient has no positive degree and there is no reactance to peel.

For a Chebyshev prototype this happens at order fourteen. For a Butterworth it happens at order nine, and the reason it happens five orders sooner is worth stating because it is a property of the filter rather than of the code. A Butterworth’s reflection zeros are all at the origin, so $F$ is exactly $s^n$ and the leading term of $E-F$ cancels exactly — the very first step is catastrophic cancellation by construction, and each subsequent one starts with fewer significant digits than the last. A Chebyshev’s reflection zeros are spread along the imaginary axis, so the same subtraction is between polynomials that differ everywhere rather than agreeing in their first term.

Which means the sweep in this figure ends because of the arithmetic and not because of the circuit. The element values for a nineteenth-order ladder exist; this method cannot compute them in double precision.

What the winding buys, measured #

The slider on the figure is the inductor technology rather than the tolerance, and that choice is deliberate: the tolerance moves nothing in the drawing, because the floor is what it is at every level whatever band is asked of it. A slider that moves only the labels is a fault this collection has already shipped once and now has a gate against.

What the winding loss does move is everything. Measured at five, ten, twenty, forty and eighty ohms per henry, the floor at order three runs 0.0011, 0.0022, 0.0043, 0.0087 and 0.0174 decibels — exactly proportional, to three figures, over sixteen times in the loss. The same proportionality holds at every order.

That is a stronger statement than it looks. A departure proportional to a loss means the loss is acting as a small perturbation on a lossless design and nothing about the network is being driven into a different regime by it: doubling the resistance doubles the error and does not, for instance, damp a resonance that was making the error large. It is the linear-response result, and it is the justification for quoting a single floor per technology instead of a curve.

It also converts the extrapolation into a design statement. An order-thirteen ladder built with wire good enough for ten ohms per henry sits at 0.0097 decibels — four times better than the forty-ohm one and a tenth of the tolerance — and the order at which that technology reaches the tolerance is one hundred and nineteen. Whether a ladder is realisable at a given order is therefore a question about the inductors, answered in ohms per henry, rather than a question about ladders.

A passband is not a stopband, and the choice is stated #

Every departure in this essay is measured over the passband and the corner, from a hundredth of the corner to twice it, and the stopband is deliberately excluded.

The reason is that a departure in the stopband is a large number about a place nobody designs in. A realisation whose stopband floor is at −70 decibels instead of −90 has a departure of twenty decibels, which would dominate any minimax measure and would say nothing about whether the filter works. The two essays below this one made the same choice and it is repeated here so that the three numbers are comparable.

It has a cost, and the cost belongs in the same sentence as the choice. The stopband is exactly where the stray capacitance this essay reports as contributing nothing would contribute: two picofarads at every node is a path from input to output that does not go through the ladder at all, and it is what puts a floor under a real filter’s rejection. So “the stray contributes nothing” is a statement about the passband, not about the component — and the essay in this field about what actually fills a null is where the same two picofarads is the whole subject.

Two boundaries of different kinds, on one axis #

This essay has ended up with two edges and they are not the same kind of thing, which is the reason to say both.

The structural edge is order thirty-two: the order at which the floor, extrapolated along a fitted line, reaches the tenth of a decibel that this field’s band measurements use. It is a statement about inductors and tracks and it would move if either got better. It is also an extrapolation, and is labelled as one.

The numerical edge is order fourteen: the order at which the synthesis stops returning element values. It is a statement about double precision and the conditioning of a polynomial subtraction, it would move if the arithmetic changed, and it is measured rather than extrapolated — the routine is asked, and it refuses.

The numerical edge is the smaller of the two, which is the uncomfortable part. A collection that draws every model with the frequency, amplitude or size at which it stops being true has to be willing to report that one of its own tools stops sooner than the physics it was built to study. This is such a case: everything in this field above order fourteen is unmeasured here not because it is uninteresting but because the machinery cannot reach it.

Why a floor is the right quantity to end a ladder on #

Four rungs of this ladder have measured the same trade in four different currencies, and this one is the last because it is the one with no choice left in it.

The same filter a thousand times larger asked what happens when a design is scaled, and found the invariance exact to a part in $10^{15}$ — the last bits of a double — with a departure appearing at both ends of the range for different reasons: fifty ohms of amplifier output resistance at one end and two picofarads of stray at the other. The band that closes with the order turned that into a band and watched it close — three decades at second order, one at sixth, and no impedance level at all at eighth — because every added section brings three more nodes each carrying their own stray and one more amplifier carrying its own output resistance. The band that does not close repeated it for a ladder and found four decades still open at order nine, with the reason being a classification rather than a number: a fixed quality factor scales with the design and a fixed resistance does not.

Each of those three has a tolerance in it. Change the tolerance and the band changes; change it enough and any structure is realisable at any order. The floor is what is left when the tolerance is removed, and it is the only one of the four numbers that a specification cannot argue with: at order thirteen, with forty ohms per henry of wire, no impedance level whatever will get this filter closer than 0.039 decibels to the response it was designed for.

That is the shape of statement this collection exists to produce, and it is why the ladder stops here. The remaining questions are about the inductors and about the arithmetic, and both are asked elsewhere with answers this essay can borrow.

On the inductors: the Q the components allow is where the forty ohms per henry becomes a ceiling rather than a parameter — the reciprocals of the component quality factors add, so the total sits below the smallest of them and the best component cannot help — and the resistance that grows with frequency says the figure is not even constant across a passband, a conductor’s resistance being already 2.05 per cent up at the frequency the rule of thumb names as the point where the effect begins. The floor measured here is therefore a floor computed at one frequency with one quality factor, and the real one is a curve.

On the arithmetic: the digits the arithmetic did not have is the essay that separates the two kinds of stall this measurement runs into, and it does so by the one experiment that distinguishes them — adding digits. An ill-conditioned problem stays ill-conditioned however many digits are used and a well-conditioned one computed badly gets better. Its verdict on this synthesis is the uncomfortable one: the ladder’s loss is mostly the data’s, and doubling the digits makes it worse. So the extrapolation to order thirty-two is not waiting on a better implementation, and the note above about extended precision has an answer, which is that it would not help.

What the floor bounds #

A floor that rises linearly with the order is the ceiling on three later results. The band that does not close is the band this floor closes. The band that closes with the order is the cascade’s own version. A ladder is not a cascade is the insensitivity that does not reach past it, and What a steep skirt costs is the decision this floor bounds from above. The Q the components allow is the same loss read as a resonator’s quality factor rather than as a filter’s floor.

What is checked #

The floor is asserted to rise at every order, to rise as a power of the order within a third of one, and to reach the drawn tolerance at an order above eight — which is the order at which the cascade’s band was empty, so the comparison the rung below made survives at every tolerance the slider offers.

The decomposition is asserted in the direction that is surprising: the winding loss alone must leave a larger departure than the complete realisation, at every order, and the stray capacitance must change the answer by under two per cent. And the synthesis is asked where it stalls rather than told: the assertion is that Butterworth stalls before Chebyshev, which is the numerical statement about reflection zeros written as a comparison.

What is not measured: whether an extended-precision synthesis would show the floor continuing along the same line past order fourteen. It would take arithmetic this collection does not have, and the extrapolation to order thirty-two is stated as an extrapolation everywhere it appears.

That last point is the one to carry out of this ladder, because it is what makes a floor worth measuring at all. What a steep skirt costs prices a family’s passband ripple as a design choice — half a decibel bought deliberately in exchange for a steeper skirt — and the floor measured here is spent from the same account without anybody choosing it. A half-decibel Chebyshev realised at order thirteen with ordinary wire is a design whose ripple specification is met by its poles and missed by its inductors, and the two contributions are indistinguishable in a measurement of the built filter.

Which is the practical reading. A ripple figure quoted for a family is an upper bound on what the design contributes and says nothing about what the realisation adds; the floor is the part the realisation adds that no impedance level, no tolerance and no care removes; and the sum of the two is what a bench sees.