A ladder is not a cascade

Every filter in this field so far has been built the same way: one second-order section per conjugate pole pair, each section separated from the next by an ideal follower. That realisation was chosen deliberately and the reason is good — with buffers between them, each section’s response is its own, so a figure about a pole pair can draw a pole pair without the next section pulling it about.

It is also not what anybody builds. What gets built is a doubly terminated ladder: a source resistance, a chain of alternating series inductors and shunt capacitors with nothing between them, and a load resistance at the far end. Everything interacts with everything, no element belongs to any particular pole, and the whole thing is one network.

The foundation phase recorded the omission with a sentence that turns out to have been an understatement: “the filters here are cascades of isolated sections, which is a choice with a sensitivity consequence that is stated and not measured.” This essay is the measurement, and the consequence is not a percentage. It is a change in the order of the dependence.

First, that they are the same filter #

The comparison is worthless unless both realisations realise the same response, and “the same response” has to mean something stronger than “both were designed from the same poles”.

The cascade comes from filterOf, which this field has used since it was built. The ladder comes from a continued-fraction expansion, which is a completely different construction: the filter’s pole polynomial $E(s)$ and the polynomial $F(s)$ carrying its reflection zeros are combined into the input impedance

$$Z_\text{in}(s) = \frac{E(s) + F(s)}{E(s) - F(s)}$$

and that rational function is expanded about $s = \infty$, peeling off one element at a time. The first quotient is a series inductance, the remainder is inverted, the next quotient is a shunt capacitance, and so on until nothing is left but a constant, which is the load.

Then both networks are handed to the same nodal solver, and their responses are compared across the passband. They agree to 3 × 10⁻¹⁴ of a decibel. That is not a tolerance anybody chose; it is double-precision arithmetic, and it means the two networks are the same filter to the limit of the machine.

One step in getting there is worth recording because leaving it out does not produce a wrong ladder — it produces a correct ladder of a different filter. Prototypes are normalised to the band edge, which for a Chebyshev design is the edge of the ripple band; every other filter on this site is normalised to its measured half-power point, because that is the only normalisation under which four families can be drawn on one axis without slandering two of them. Compared without rescaling, the ladder and the cascade came out 4.36 dB apart at the band edge — a number large enough to look like a synthesis error and small enough to look like a tolerable one.

The even-order case, which falls out rather than being warned about #

The expansion terminates when the remainder is a constant, and at that point what is left is $as + b$ over that constant — which is two elements, the last reactance and the termination. Dropping the reactance is a silent error, and a nasty one: the values that survive are still every g-value except one, and they still look exactly like the table.

Getting it right produces a result that filter design texts usually give as a warning. For an even-order Chebyshev design the terminating resistance is not the source resistance: the expansion returns 1.984 at the fourth order. An even-order equiripple response cannot be realised between equal terminations at all, and here that is not a footnote but the last quotient of a division.

Why the ladder is stationary #

The result in the first figure has a one-paragraph explanation and it is worth following, because it explains not only that the ladder is better but exactly where.

A lossless doubly terminated network cannot deliver more power to its load than the source has available. At the frequencies where the passband response reaches its maximum, it is delivering exactly that maximum. So the response there is at a stationary point with respect to every element in the network: any change to any inductor or any capacitor, in either direction, can only take power away, because there is none to add. The first derivative of the response with respect to every reactance is zero at those frequencies, so the error from a component tolerance is second order rather than first.

Nothing in that argument mentions which filter, which order, or which element. It is a statement about lossless two-ports between fixed terminations, and this is the paragraph the whole measurement below is testing.

A buffered cascade has no such property. Each section’s natural frequency and quality factor depend directly on its own three components — $\omega_0 = 1/\sqrt{LC}$, $Q = \sqrt{L/C}/R$ — and a one per cent capacitor moves both by something first order in one per cent. There is no maximum being delivered and nothing to be stationary about.

The measurement, and the number that is actually the claim #

“The ladder is less sensitive” is a comparison, and a comparison could be luck. The claim worth making is about the shape of the dependence, and there is a direct test for it: change the tolerance and watch what the error does. A first-order sensitivity grows by ten when the tolerance grows by ten. A second-order one grows by a hundred.

Slope 2.00 against slope 0.99. The consequence is worth stating in the units a manufacturer thinks in: tightening every component from ten per cent to one per cent improves the cascade’s worst passband error by a factor of about ten, and the ladder’s by a factor of about a hundred. The two realisations respond to money spent on precision quite differently.

The third line on that plot is the one that stops the result from being an advertisement. It is the ladder’s own load resistance, perturbed by the same fractions, and its slope is 1.00 — first order, like the cascade. Orchard’s argument is about a lossless two-port between fixed terminations and says nothing whatever about the terminations themselves, so the ladder’s immunity covers its inductors and capacitors and stops there. At one per cent the load resistor alone moves the passband by 0.043 dB, which is fourteen times more than any of the reactances do.

Perturbing everything together and reporting one number would have averaged a specific, provable result into a vague one — and in this case worse than that: the terminations dominate, so the ladder’s whole advantage disappears from the summary.

Where the advantage is, and where it is not #

The first figure has one more thing to say and it is the honest caveat. Across the whole passband, the two realisations’ worst-case deviations are within three per cent of each other.

That is not a contradiction. Near the band edge the response is falling steeply, and any realisation whose corner frequency shifts by a fraction of a per cent produces a large decibel change there for reasons that have nothing to do with sensitivity to a particular element — it is the response’s own slope multiplying a small frequency error. The stationarity theorem says nothing about frequencies where the response is not at a maximum, and it should not be expected to.

So the honest summary is narrower than the usual claim and stronger where it applies: in the interior of the passband, at the frequencies where a doubly terminated ladder is delivering all the power available to it, its response is second-order insensitive to every reactance it contains. Near the band edge it is as sensitive as anything else.

The measurement almost measured something else #

Two things went wrong in building this and both are worth recording, because both produced answers that looked like results.

The ripple peaks have to be located properly. A first version found them by scanning the nominal response on the same 120-point grid the curves are drawn on and taking the local maxima. That locates a peak to one grid step, and the deviation at a peak is second order in the component and second order in how far off the peak the sample was taken — so a peak found to one grid step reports the second effect and calls it the first. The measured ladder sensitivity came out fifteen times too large. Locating each peak by golden-section search on the continuous response fixed it.

And direct current has to be excluded, although it is a maximum of every all-pole response here. It is the one frequency at which the comparison is rigged: each section of a buffered cascade has a gain of exactly one at direct current by construction — a resistor in series with a capacitor to ground divides nothing when nothing is flowing — so the cascade is trivially insensitive there whatever its components are. Measured at direct current a third-order Butterworth cascade moves by 9 × 10⁻¹⁰ dB for a one per cent part, which is the arithmetic’s floor and not a property of the realisation.

That exclusion has a consequence worth stating plainly rather than hiding: this argument is visible in a Chebyshev and invisible in a Butterworth, because a maximally flat response has no interior maxima to be stationary at. The effect is present in a Butterworth ladder — it is present in every lossless doubly terminated network — but the only frequency at which it can be demonstrated is the one where the cascade cheats.

What the ladder costs, which is not tolerance #

A realisation that is fifty times less sensitive and reproduces the design to fourteen digits sounds like it should have replaced the cascade everywhere, and it has not. The reasons are not about accuracy at all.

It needs inductors, and a cascade of this kind does too — but a ladder needs good ones. In a buffered cascade the damping resistor is a component the designer put there deliberately, and it is almost always much larger than the inductor’s own winding resistance, so the coil’s imperfection is a small correction to a number that was chosen. In a doubly terminated ladder there is no damping resistor at all: the only resistances are the source and the load, and every ohm of winding resistance is an unwanted addition to a lossless design. The response that results is not the design.

Its element values are not free either. The expansion returns what it returns — 0.618, 1.618, 2.000, 1.618, 0.618 for a fifth-order Butterworth — and scaling those to a real impedance and frequency gives inductances and capacitances that are whatever they are. A cascade lets the designer fix the capacitance at a convenient value and let the rest follow, which is why every filter in this field is built with 100 nF capacitors throughout. A ladder offers one scaling constant for the whole network.

And it has no internal buffering, which is the whole point and also a cost. Every element sees every other, so there is nowhere to take an intermediate signal from, nowhere to insert gain, and no way to tune one pole pair without moving the others. The cascade’s isolation is what makes it adjustable.

What this changes about the rest of the field #

Nothing in the earlier essays is wrong, and one thing in them is now placed. The comparison between Butterworth, Chebyshev, Bessel and the elliptic family is a comparison of approximations, and every result in it — the ripple, the skirt, the delay, the ringing — is a property of the transfer function rather than of the circuit. Those results survive this essay untouched.

What does not survive is the implicit assumption that a filter’s tolerance sensitivity follows the same ordering. It does not follow the family at all, in the first instance; it follows the realisation, and the realisation is a free choice made after the family is picked. A fifth-order elliptic ladder is less sensitive in its passband than a third-order Butterworth cascade, and the elliptic design has the higher pole Q of the two.

There is a real ordering by family underneath — higher pole Q does mean higher sensitivity, all else equal — and it is a second-order effect sitting on top of a first-order choice. Measuring the family ordering properly means measuring it within one realisation, which is what the figures here do and what the earlier essays, built entirely on cascades, could not have done.

It also settles a question the previous essay left hanging. An elliptic design is the most tolerance-sensitive of the four families as an approximation, and it is also the one whose specification is tightest — a stopband floor computed to a hundredth of a decibel is not much use if a five per cent inductor moves it by two. Both statements are true, and together they say something specific rather than discouraging: an elliptic filter is a design that needs the ladder. Built as a buffered cascade it is a precise answer realised by the sensitive method, which is the combination with nothing to recommend it.

The last thing worth saying is about what this measurement is made of. Everything above came out of one function that perturbs each element of two netlists in turn and re-solves — about forty lines, sitting on the same nodal analysis that draws every other figure on this site, with no derivative of anything taken symbolically and no sensitivity formula anywhere. The theorem it tests is fifty years old and is usually presented as a result to be accepted. Testing it needed no more machinery than changing a number and solving again, which is the whole argument for having built the solver first.

That instrument was later replaced by a better one, and the comparison is worth having because it changes what can be asked rather than what was answered. Every derivative, and the one that is zero transposes the matrix and solves once more, returning the derivative with respect to every element at once and exactly, against a difference quotient’s best possible accuracy of four parts in a hundred million. Pointed at the claim this essay makes, it returns two parts in ten billion for the ladder where the cascade realising the identical response returns 0.72 — so the stationarity measured here as a small number is confirmed as a zero, which a perturbation could never have done.

Two later measurements then bound the claim rather than confirming it. The tolerance that can only take away asks what a first derivative of zero leaves, and finds every second derivative negative: of six hundred ladders built from one per cent components not one is above nominal, the mean has shifted rather than the spread having grown, and doubling the tolerance quadruples the damage instead of doubling it. So stationary does not mean insensitive, it means insensitive in one direction and second order in the other. And the derivative of a root finds the advantage very much smaller when the quantity asked about is a pole rather than a magnitude: a factor of 2.17 rather than eight orders, because what is stationary is the magnitude at one frequency and that says nothing about where the poles are.

The condition under which any of it holds is the one this field measures separately. The two resistors a ladder was designed between shows that stationarity is a property of a ladder at maximum power transfer, and that driving an order-five Butterworth from anything outside 0.886 to 1.137 times its design resistance puts more than half a decibel on the passband. So the factor of fifty-four measured here is bought with a requirement on a component that is not part of the filter, and the two results are the same theorem read in opposite directions: the elements interact, which is why their errors cancel and why the terminations cannot be got wrong.

Where the sensitivity result is spent #

A second-order insensitivity at the ripple peaks is what four later essays in this field are built on. Every derivative, and the one that is zero turns the comparison into an exact derivative and finds the stationarity rather than inferring it. One inductor, and ten components and Eight amplifiers, and what they add are what it costs to have the ladder’s sensitivity without the ladder’s inductors. The band that does not close is where the same comparison is read as a range of impedance levels rather than as a decibel. And The floor that outlives the arithmetic is the ceiling the ladder has anyway, which no amount of insensitivity reaches past.