Several sections, and the band they buy

A quarter-wave and the path back built the transformer that matches two impedances with one length of line, and measured the thing about it that a formula does not say: it is exact at one frequency, and the band around that frequency narrows as the transformation grows. A 4:1 step holds the reflection below 0.1 over 17.1% of the centre frequency. A 20:1 step manages 6.0%.

The foundation phase recorded the gap that leaves as a stated one: “a single quarter-wave section is a narrow-band trick when the transformation is large, and the answer is several sections whose reflections cancel over a band. That is filter design with the reflection as the shaped quantity and it needs machinery this site does not have.”

The machinery exists now, and the sentence turns out to be more literally true than it was meant.

Why it is a polynomial #

Each junction between two lines contributes a reflection, and each contribution arrives back at the input having travelled down to that junction and back. So each carries a phase set by how far along the chain it was produced, and the phases are evenly spaced because the sections are all a quarter wave long at the centre.

The total is therefore a sum of terms with a common phase step — a polynomial in $e^{-2j\theta}$, where θ is the electrical length of one section. Choosing the section impedances is choosing that polynomial’s coefficients, and choosing the coefficients is the whole of filter design.

Which family of polynomial is then a familiar choice:

• Maximally flat at the centre — make the first N−1 derivatives of the reflection vanish there. The coefficients that do it are the binomial coefficients, and the design distributes the transformation as $\ln(Z_{n+1}/Z_n) = 2^{-N} \binom{N}{n} \ln(Z_L/Z_S)$. This is Butterworth, in the reflection.
• Equal ripple across a band — hold the worst reflection down over a stated width rather than making it perfect at a point. This is Chebyshev, in the reflection.

The filters field spent three essays measuring exactly this trade for a transfer function. Nothing new is being invented here; a different quantity is being pointed at.

What the sections buy #

The first column is the answer the stated gap asked for. One section holds |Γ| below 0.1 over 17.1% of the centre frequency; five sections take it to 92.6%, which is a band from about half the centre frequency to about one and a half times it.

The second column is what shaping rather than flattening is worth. At the same worst reflection, the equal-ripple design covers 35% more band at two sections, 44% more at three, 46% more at four and 45% more at five. That is the same trade the filters field measured between Butterworth and Chebyshev, in the same direction and of a similar size — a designer who accepts ripple where a competitor demands flatness gets a wider band for the same order.

The single-section row is the check that both designs reduce correctly: with one section there is no freedom at all, the minimax search has nothing to search, and both methods return the same line. The figure asserts that they agree to nine digits there rather than skipping the case.

The ripples are a result, not an assumption #

The binomial design comes from a closed form. The equal-ripple one does not: it is found by minimax search over the logarithms of the section impedances, minimising the worst |Γ| inside a stated band, with the band itself found by bisecting for the width whose best achievable ripple is exactly the specification.

That last step matters and it is easy to skip. The comparison worth making is not “which design is flatter” — the binomial is, by construction, at one frequency — but “which covers more band at the same worst-case reflection”. Comparing a design meeting one specification with a design meeting a different one is not a comparison at all, so the band is not given to the search. It is found.

And nothing in the search enforces equal ripple. The objective is the worst reflection in the band; the objective would be perfectly happy with a design whose peaks were unequal, as long as the largest of them was small. That the peaks come out level — to 2 × 10⁻¹³% at three sections and 0.08% at four — is a consequence of the minimax optimum being where it is, and it is therefore usable as the check that the search converged. A table has to be trusted. A search can be asked whether its answer has the property the theory says the optimum has.

This is the same construction the elliptic filter used in the scale phase, and for the same reason: an equiripple response is one whose ripples are equal, nothing in the code enforces that, and it falls out of the placement being right.

What the exact cascade does that the design did not know about #

The faint third curve in every figure here is the small-reflection sum — the polynomial the designs are derived from, evaluated directly. The heavy curves are what the cascade actually does.

They are not the same, and the difference is not negligible: up to about 0.021 in |Γ| for three sections, against a specification of 0.1. That is a fifth of the budget.

The reason is that the small-reflection approximation keeps only the first bounce at each junction. The real cascade has waves that reflect at one junction, travel back to a previous one, reflect again, and rejoin — an infinite set of multiple reflections that the polynomial does not contain. So the design procedure is derived from an approximation, and the response is computed here by cascading exact transmission-line input impedances from the load back to the source, which contains every multiple reflection there is.

Both are returned, and the figure asserts that they differ. That assertion is the interesting one: if the exact cascade ever agreed with the approximation it was designed from, one of the two would be computing the other, and the check would be vacuous.

The practical consequence is a design margin. A multi-section transformer designed to hit exactly 0.1 by the closed form will miss, in a direction the closed form cannot predict, and the amount is a substantial fraction of the specification. Designing to the approximation and verifying on the cascade is the correct order of operations, and it is what this figure does.

The same trade, in two fields #

It is worth putting the two tables side by side, because the correspondence is exact and neither field would have noticed it alone.

What a steep skirt costs measured, for a transfer function: Butterworth is flat and Chebyshev is steep, and at a given order the ripple bought is the skirt gained. The zeros that buy an order measured what adding transmission zeros does to that trade. A ladder is not a cascade measured what the realisation costs in sensitivity.

Every one of those statements has a translation here.

And the numbers translate too. Chebyshev’s advantage over Butterworth in the filters field grows with order and is worth a factor rather than a percentage at high order; here the equal-ripple design’s advantage is 35% at two sections, 44% at three and 46% at four — growing, and of the same character.

The one place the correspondence breaks is instructive. A filter’s stopband is where the response is wanted to be small, and a designer cares how deep it goes. A transformer’s out-of-band region is where the match is simply given up, and nobody cares how bad it gets — so there is no equivalent of stopband attenuation and no equivalent of the elliptic family’s trade, which spends stopband depth to buy skirt steepness. The design space here is genuinely smaller, and that is a fact about the problem rather than about the machinery.

Why the band is measured and not derived #

Every band in the table is found by scanning the exact cascade and locating the two frequencies where |Γ| crosses the specification, rather than by evaluating a closed form for the bandwidth of an N-section binomial transformer — which exists, and which this essay deliberately does not use.

The reason is the section above. The closed form is a statement about the small-reflection polynomial, and the polynomial is not what the cascade does. A bandwidth derived from it would inherit the same 0.021 of error the response has, in a direction that depends on the design, and the table would be reporting the approximation’s band rather than the transformer’s.

The scan costs nothing — a few hundred impedance cascades per curve — and it has the property that matters here: it does not know how the sections were chosen. The same measurement is applied to the binomial design, to the minimax design, and to the single section, and it would be applied unchanged to a design arrived at by any other means. That is what makes the two columns of the table comparable.

The five-section row, which was wrong here for a phase #

This table read 97.3% in the five-section row when it was first written, which is less than the 119.6% at four, and the essay said so and blamed the search: coordinate descent stopping early on a flattened objective, with a Remez exchange recorded as the fix.

The number was wrong and so was the diagnosis. The search had converged to eight digits. What had failed was the measurement of the band and the objective the search was given, and the number that was wrong, and the reason that was wrong too is the rung above this one, which is about exactly that.

Two sentences of it belong here. The objective was the worst |Γ| over a grid of eighty-one frequencies rather than the worst over the band, so the optimiser drove the grid down and left the true peaks 1.4 parts in ten thousand above it. And the band was found by bisecting between the centre and twice the centre — a bisection on a function that crosses its threshold five times — so peaks a hair above the specification returned an interior crossing instead of the edge.

Both are repaired, the row now reads 134.0%, and the sentence this essay ended on is worth keeping exactly as it was written: an assertion that passes is not the same as a result that is right. The assertion here was that the equal-ripple design covers more band than the binomial one, and 97.3% is more than 92.6%, so it passed for a phase while the figure printed a number that was thirty per cent short.

What is outside this #

Three assumptions bound every number above, and all three are the ones the lines field has been careful about elsewhere.

The lines are lossless. Real line has attenuation, which lifts the reflection floor and — more awkwardly — makes the ideal section impedances complex rather than real. For short sections at these frequencies the effect is small; over a wide band it is not uniform, because attenuation grows with frequency.

The sections are exactly a quarter wave at the centre. They are, by construction, at one frequency; the whole design is about what happens away from it, and the electrical length is proportional to frequency only if the propagation velocity is. Dispersion in a real dielectric makes the sections slightly the wrong length at the band edges, which is exactly where the design has the least margin.

Every junction is abrupt and reflectionless apart from the impedance step. A real junction between two line widths has a discontinuity capacitance, which adds a reflection the polynomial does not contain and which grows with frequency — the same shape of parasitic the frequency field’s capacitor essay is about, in a different guise.

What the transformation ratio does #

The slider in this figure is the section count, and the transformation is held at 4:1 — from 50 Ω to 200 Ω, which is a common enough step. The other axis is worth describing even though it is not on a slider here, because it is the axis the stated gap was originally about.

The bare mismatch between 50 Ω and 200 Ω has |Γ| = 0.600. One quarter-wave section takes that to zero at the centre and holds it under 0.1 over 17.1%. For a 20:1 step the bare |Γ| is 0.905, one section holds 0.1 over only 6.0%, and the number of sections needed for a given band grows accordingly.

The reason is visible in the design formula. The total transformation is distributed as $\ln(Z_L/Z_S)$ shared out among the junctions, so each junction’s individual reflection is proportional to the log of the total ratio divided by the section count. A larger transformation means larger individual reflections, which means the polynomial has larger coefficients and reaches the specification sooner as θ moves away from 90°.

The practical rule that falls out is that sections trade against transformation ratio roughly logarithmically, which is why a very large step is done with a taper — a continuously varying impedance, which is the limit of infinitely many infinitesimally different sections — rather than with a discrete cascade. A taper is outside what this machinery computes; it needs a differential equation rather than a polynomial, and it is a genuine extension rather than a special case.

What is outside this, continued: the lumped alternative #

There is one comparison this essay owes the reader, because the whole field it sits in exists to mark where a wire stops being a wire.

A transformation between two impedances can be done with a lumped network — two reactances in an L, three in a pi or a T — and below the frequency at which Kirchhoff’s laws give out that is what everybody does. It is smaller, it is cheaper, and its band is set by the loaded Q of the network rather than by any length.

The distributed answer in this essay is for the region above that frequency, which the lines field established at 3.97 MHz on ten centimetres of track — and the honest statement of the trade is that the two approaches do not compete over any wide range. Below the lumped limit the sections are absurdly long; above it the lumped network’s own parasitics have stopped being parasitics and it has become a distributed structure whether or not it was designed as one.

What is genuinely shared is the arithmetic. An L-network’s band is set by its transformation ratio in exactly the way a single section’s is, a cascade of L-networks widens it in exactly the way a cascade of sections does, and the polynomial being shaped is the same polynomial. That is the third time in this essay that the answer to “what is this really” has been “filter design”, and it is the reason this rung sits where it does in the ladder.

The rule this model stops being true under #

17.1%, and 92.6%.

The first is what one section buys for a 4:1 transformation and the second is what five buy, and the distance between them is the content of the stated gap this essay closes. What the sections cost is length: five quarter waves is more than a wavelength of line at the centre frequency, in a technology where physical size is the reason anybody wanted a transformer rather than a lumped network in the first place.

That is the trade this rung adds to the field, and it is the same one what a steep skirt costs states in another currency: order buys attenuation at twenty decibels per decade per pole and no arrangement of components changes that, while what varies between families is how quickly the slope is reached and the currency it is paid in is delay — with the steepest of three families distorting delay eight hundred times more than the gentlest. Order buys performance, order costs something, and which currency the cost is paid in is what distinguishes one realisation from another.

The correspondence with filter design is closer than an analogy and is worth stating exactly, because it says where to look for the next result on this ladder. A multi-section transformer’s reflection $|\Gamma|$ against frequency is a filter response with the reflection as the shaped quantity: the binomial design is a Butterworth, the equal-ripple design found here by minimax search is a Chebyshev, and the two resistors a ladder was designed between is the same synthesis run in the other direction — a ladder filter designed between two stated resistances by placing reflection zeros.

Which means two of that field’s results are waiting here. The tolerance sensitivity a ladder is not a cascade measures — first order in a cascade, second order in a doubly terminated ladder, because at maximum power transfer the response is stationary in every element — should apply to a section’s characteristic impedance exactly as it applies to an inductor. And the equal-ripple design’s own ripples being level to four decimals is the same statement as a Chebyshev’s passband, which is what makes the minimax search’s result checkable against something other than itself.

Where the band is spent, and what hides it #

A reflection coefficient held under a tenth over a decade of band is only worth what the rest of the field lets it be worth. A quarter wave, and the path the current takes back is the single section this design is an answer to, and its band narrows as the transformation grows. The mismatch that the cable hides is the measurement problem: through twenty metres of ordinary cable a 4:1 mismatch reads as 1.02:1, so the design’s whole merit is invisible from the instrument end. The delay that is not one number is the assumption the synthesis makes and does not test, and A ladder is not a line is what a lumped approximation to any of it would be worth. The number that was wrong is the correction this page’s own measurement produced.