The match with no upper edge

Several sections, and the band they buy turned impedance matching into filter design. A single quarter-wave section matches two impedances at one frequency; several sections whose reflections cancel over a band buy fractional bandwidth in proportion to their number, and the equal-ripple design, found by minimax search, buys a third more again at the same worst reflection. The number that was wrong corrected one of its rows — five sections cover 134 per cent of band, not 97 — and showed why: a band measured by bisection on a function that crosses its threshold five times. Both essays measured how wide the band is. Neither looked above it.

Above it, a stepped transformer is not matched at all. Its sections are a quarter of a wavelength long at the design frequency, so at twice that frequency each one is half a wavelength — and a half-wave line is transparent: it presents at its input exactly the impedance at its output. A stack of transparent sections connects the source to the load as though nothing were there, and the reflection is the bare mismatch the transformer was built to remove. The response repeats at every even multiple of the design frequency.

That periodicity is what the sections are, and it suggests what the limit is. Divide the same length into more and shorter sections and the frequency at which they all become half a wavelength moves up. In the limit of infinitely many, the sections become a continuous change of impedance — a taper — and there is no frequency at which the steps are transparent, because there are no steps.

A band-pass and two high-passes #

The hero figure compares three matching sections of the same physical length, half a wavelength at the design frequency $f_0$, between a 50 Ω source and a 200 Ω load. Every response is computed the same way: the load’s impedance is carried back towards the source through each uniform piece of line by the exact transmission-line formula, and the reflection is read at the source. The tapers are cut into four hundred thin uniform slices, enough that a thousand would change nothing on the plot.

The two-section binomial transformer is the earlier essays’ design: matched at $f_0$, rising either side, and at $2f_0$ back to 0.600 — which is $(200 - 50)/(200 + 50)$, the reflection of the load with no matching at all. It repeats there, and at $4f_0$, and at every even multiple. A stepped transformer is a band-pass filter for power, with passbands centred on the odd multiples of $f_0$.

The exponential taper — the characteristic impedance rising as a straight line in $\ln Z$ from one end to the other — is a high-pass. Below about $0.6f_0$ it is hardly a match at all, since a line much shorter than a wavelength cannot transform anything; at $f_0$ its reflection is 0.019; above $1.6f_0$ it never exceeds 0.131, and its sidelobes keep falling, below 0.082 past $2.6f_0$. The Klopfenstein taper, of which more below, is a high-pass with an equal ripple: 0.050 or less from $1.1f_0$ to the top of the figure and beyond.

So the comparison is not between two ways of doing the same job. A stepped transformer of a given length does the job over a band and fails above it; a taper of the same length does a slightly different job — matching from a lower edge upwards — and has no upper edge to fail at.

The staircase and its limit #

The connection between the two can be drawn directly, by building the exponential taper out of steps.

A staircase of $n$ equal steps spanning the same length has steps $1/n$ of the length, so each step is half a wavelength at $n f_0$. There the staircase is transparent and the bare 0.600 returns, exactly — computed, it does so to six decimal places at $n = 2$, 4, 8 and 32. Below that frequency the staircase follows the continuous taper, increasingly closely as the steps multiply: at 32 steps the two differ by under a hundredth at $1.5f_0$.

So a stepped transformer is a sampled taper, and its periodicity is aliasing in the most literal sense: the profile is sampled at $n$ points, and the response repeats at the frequency at which the sampling interval is half a wavelength. The filter that samples found the same thing in time — a switched-capacitor circuit’s response repeating at its clock — and the same cure applies: more samples push the image up. The taper is the limit in which the image has gone to infinity, and the price of having no upper edge is that the profile must be made continuously, which on a circuit board means a track whose width changes smoothly along its length.

Why the taper is a Fourier transform, almost #

The design theory for tapers is a single formula. A small change of impedance $dZ$ reflects $dZ/2Z = \tfrac12 d(\ln Z)$, and if every reflection is small enough that its own reflections can be ignored, they add with the phase of their round trip:

$$\Gamma(\beta) \approx \frac12\int_0^L \frac{d\ln Z}{dz}\, e^{-2j\beta z}\, dz.$$

That is a Fourier transform of the profile’s slope, and for the exponential taper, whose slope is constant, it is a sinc: $|\Gamma| = \tfrac12\ln(Z_L/Z_S)\,|\sin\beta L/\beta L|$. Every taper design starts from this formula. It is also the approximation every stepped design in the earlier essays was derived from, and several sections, and the band they buy reported it rather than assumed it small.

At zero frequency any taper is no match at all, and the exact reflection is the bare one, $(r - 1)/(r + 1)$ for an impedance ratio $r$. The small-reflection formula gives $\tfrac12\ln r$ there. The two are related exactly — $(r-1)/(r+1) = \tanh(\tfrac12\ln r)$ — so the formula is the first term of the hyperbolic tangent’s series, and it overstates the reflection by 4.0 per cent at 2:1, by 15.5 per cent at the 4:1 of this essay’s examples, and by 41 per cent at 10:1, where it predicts a reflection of 1.151, more than total. The sinc’s nulls are where its error is most visible in shape: the exact cascade’s nulls are not zeros, because the multiple reflections the formula drops do not cancel where the first-order ones do. At $f_0$ the exponential taper’s first-order null is 0.019 in the exact line.

None of that undermines the design method — the formula still says where the sidelobes go and how they scale — but it does say what the method’s numbers are worth. A taper designed for a stated reflection from the Fourier formula, at a large impedance ratio, meets a slightly different reflection, and the difference has to be measured on the exact line, as every figure here does.

The optimum taper, and its steps #

The exponential taper’s sidelobes are a sinc’s, falling only as one over the length, so its first sidelobe is the worst and the rest are wasted margin. The equal-ripple taper levels them, exactly as the Chebyshev stepped design levels its in-band ripples, and it has a closed form found by Klopfenstein: the profile whose reflection above its band edge is $\Gamma_0\cos\sqrt{(\beta L)^2 - A^2}/\cosh A$, with $\Gamma_0 = \tfrac12\ln(Z_L/Z_S)$ and $A = \operatorname{acosh}(\Gamma_0/\Gamma_m)$ for a ripple $\Gamma_m$. It is the limit of the Chebyshev stepped transformer as its sections multiply.

Drawn along its length, the Klopfenstein profile is S-shaped: flatter near its ends than the exponential, steeper through its middle, and symmetric about the geometric mean, 100 Ω, at its centre. And its ends do not meet the lines they join. It starts at 52.56 Ω rather than 50 and finishes at 190.25 Ω rather than 200, so a Klopfenstein taper has a small step at each end — a reflection of 0.0250 at each, half its design ripple. The optimum continuous match is not continuous. The steps are part of the equal-ripple design: the reflection they make is what the rest of the profile cancels, and smoothing them away raises the ripple.

What equal ripple buys in length #

The useful comparison is length. Given a reflection to stay below, and a lowest frequency above which it must hold, how long must each taper be?

The lengths were found by scanning each taper’s exact reflection down from high frequency until it first exceeds the limit, and they are given in wavelengths at that band edge. For a reflection of 0.2 the two tapers are nearly the same length, 0.393 and 0.315 wavelengths. At 0.1 the Klopfenstein needs half the exponential’s length, 0.430 against 0.866. At 0.05 it needs 0.543 against 1.846, and at 0.02, 0.692 against 5.295 — an eighth. The Klopfenstein’s band edge sits at its design point, $\beta L = A$, to within the scan’s resolution at every ripple: its length grows only as the inverse hyperbolic cosine of the ratio $\Gamma_0/\Gamma_m$, which is to say logarithmically in the reflection demanded, while the exponential taper’s grows as one over it, because its sinc sidelobes fall only as one over the length.

That is the stepped design’s lesson carried to its limit. Several sections, and the band they buy found the equal-ripple stepped design buying 35 to 46 per cent more band than the maximally flat one at the same reflection. In length rather than band, and in the continuous limit, the equal-ripple taper’s advantage grows without bound as the demanded reflection falls.

The filter families, read as tapers #

The correspondence the earlier essays drew — reflection as the designed response, sections as poles — runs all the way through, and it is worth spelling out because it says which taper to reach for without a search. Three families, one corner built Butterworth, Chebyshev and Bessel filters from their definitions and found each one optimal for a different statement of what a good response is. The stepped transformers inherit the first two directly: the binomial transformer is maximally flat at its centre, which is the Butterworth statement, and the equal-ripple one is the Chebyshev statement. Their continuous limits are, respectively, a taper whose reflection is flat to high order at one frequency and the Klopfenstein taper, whose reflection is equal-ripple above an edge.

The exponential taper is neither. Its profile has a constant slope, so its reflection is the transform of a rectangle — a sinc, which is what a filter with no design at all would give. It is the easiest taper to describe and draw, and the figures show what that convenience costs: twice the length at a reflection of 0.1, eight times at 0.02. A triangular profile — the slope rising linearly then falling — gives a sinc squared, whose sidelobes fall twice as fast, and a cosine-squared profile faster still; they are the taper’s equivalents of the window functions a spectrum analyser uses, and they trade the first sidelobe’s height against the main lobe’s width in exactly the way a window does.

The zeros that buy an order found the elliptic filter beating every all-pole family by putting zeros in its stopband. A matching section has no stopband to put zeros in — its job is to pass power, not stop it — so there is no elliptic taper, and the Klopfenstein is where the sequence ends. That is a statement about what the transform of a real, finite profile can do: its reflection can be made equal-ripple above an edge and no better, because a finite length cannot make a response exactly zero over a band.

The lower edge, which every taper keeps #

A taper has no upper edge, but it has a lower one, and nothing removes it. Below the frequency at which the taper is a fraction of a wavelength long, the source simply sees the load through a short piece of line, and the reflection rises towards the bare mismatch. The length figure puts a number on it: to hold a reflection of 0.1 a Klopfenstein taper must be 0.430 wavelengths long at the lowest frequency it is to match, so a board that has to match down to 100 MHz on a substrate where a wavelength is about 1.5 metres needs a taper some 65 centimetres long. That is why tapers are used where the lower edge is high — microwave connectors, antenna feeds, the transition from a coaxial line to a waveguide — and why a quarter-wave section, whose band can be put anywhere, is used where it is not.

A quarter wave, and the path the current takes back built the single section and found its band, 17 per cent wide at a reflection of 0.1, to be the whole of what it is worth. The taper turns that statement around. Its band has no top, but its bottom is set by length, and length on a board is the scarcest thing there is.

Choosing between a staircase and a slope #

The practical choice is between a stepped transformer — easy to build, each section a uniform track — and a taper, which needs a track of smoothly varying width. The figures say when the taper is worth the trouble. If the signal occupies a band around one frequency, a few sections cover it, and their return at twice the frequency does not matter because nothing is there. If the signal is broadband — a fast edge, whose spectrum runs from its repetition rate to well beyond its rise time’s reciprocal — then the stepped transformer’s return at even multiples of its design frequency reflects part of the edge’s spectrum however well the band was designed. The staircase in time drew what a line’s reflections do to a fast edge; a stepped matching section adds a reflection at each step to that staircase, and a taper replaces them with one smeared reflection whose size falls with frequency.

A digital designer matching a line to a receiver, then, wants a taper or nothing, and an RF designer matching an amplifier over an octave wants sections. The same filter theory describes both, and the difference is whether the passband is bounded above.

The arithmetic #

Matching 50 Ω into 200 Ω over half a wavelength at $f_0$: two quarter-wave sections are matched at $f_0$ and reflect the bare 0.600 at $2f_0$ and every even multiple; a staircase of n steps does so at $n f_0$; an exponential taper is 0.019 at $f_0$ and below 0.131 above $1.6f_0$, and a Klopfenstein taper holds 0.050 from $1.1f_0$ up. The small-reflection formula’s ½ ln r is the first term of the line’s tanh(½ ln r) — 41 per cent high at 10:1. The Klopfenstein profile starts and ends with steps of Γm/2, and for |Γ| ≤ 0.1 above a band edge it needs 0.430 wavelengths against the exponential’s 0.866; for 0.02, 0.692 against 5.295.

Still open: a taper with loss, a taper on a real board, and a taper that must also be short #

Loss along the taper. A real taper on a board has conductor and dielectric loss that varies with the track’s width, so the narrow end loses more per unit length than the wide one. The mismatch that the cable hides found loss improving a measured match by twice the loss; on a taper the loss is distributed through the very profile that does the matching, and whether it improves the far-end match or merely hides it is a cascade with a loss per slice.

The track’s own discontinuities. A width change on a board is not a pure change of characteristic impedance: a step in width has a fringing capacitance of its own, and a smooth taper has a distributed version of it. How much that shifts the Klopfenstein profile’s equal ripple, and whether the end steps should be designed with it in mind, is a question for a field solve, and one that has not yet been made.

A taper that must also be short. The Klopfenstein taper is optimum for its length in the sense of the smallest ripple above a band edge. A match for a fast edge also cares about the delay the taper adds and about its reflection below the band edge, which no high-pass design minimises. The trade between length, ripple and the low-frequency reflection a short taper cannot avoid is the design question a short interconnect actually poses.