The termination an even order cannot have

The two resistors a ladder was designed between established that a passive ladder is a two-port designed between two stated resistances rather than a transfer function with resistors attached, and measured what happens when the source is wrong: an order-five Butterworth driven outside 0.886 to 1.137 times its design resistance is more than half a decibel out on the passband.

It also said, in one sentence and without a number, that the load resistance falls out of the same expansion as its last quotient — “which is why an even-order equiripple design cannot be realised between equal terminations at all”. A ladder is not a cascade said the same thing and printed the value: 1.984 at the fourth order.

A prohibition with no cost attached is an invitation. What happens if it is ignored, and the same reactances are simply built between two equal resistors, is a question neither essay asks and the synthesis is one substitution away from answering.

What the last quotient is #

A ladder is not derived from its transfer function. It is derived from its reflection, and the element values are the successive quotients of a continued-fraction expansion of

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

where $E$ carries the filter’s poles and $F$ its reflection zeros — the frequencies at which the doubly terminated network reflects nothing at all. Each step peels off one reactance and hands the remainder to the next, alternating series and shunt, and the expansion ends when the remainder is a constant.

That constant is the load resistance. It is not chosen, it is not looked up, and it is not adjustable: it is what is left of a division. The design procedure’s input is “how much shall be reflected, and at which frequencies”, and its outputs are the elements and the two resistances together.

There is a second ladder hidden in the same expansion and it is worth naming, because it is where the other number a reader may have seen comes from. $Z_\text{in} = (E+F)/(E-F)$ is one of two sign choices; the other gives the dual ladder, shunt capacitor first, which is the same filter built the other way up. Its expansion ends at a shunt position, where the constant left over is a conductance rather than a resistance, so it returns the reciprocal: 0.504018 rather than 1.984056. The two are one statement, and which of them a table prints depends only on which arm the table’s ladder starts with.

The two are not separable, and the reason the last number looks optional is typographic rather than mathematical. Four of the five quotients have units of henries and farads and one has units of ohms, so the one that is a resistance gets printed somewhere else — and once it is somewhere else it looks like an assumption rather than an answer.

Why zero frequency decides it #

The alternation with parity has a one-paragraph explanation, and it is worth following because it also says why a Butterworth never has the problem.

At zero frequency a lossless low-pass ladder is transparent. Its series inductors are short circuits, its shunt capacitors are open circuits, and what is left between source and load is two resistances in series and nothing else. A resistive divider between $R_s$ and $R_L$ delivers the maximum power the source has available if and only if the two are equal — which is the statement terminated at both ends makes about a transmission line from the other direction, and which costs exactly half the voltage.

So a ladder terminated in its own source resistance necessarily reflects nothing at direct current. That is a constraint on the design, not a bonus: it says zero must be one of the frequencies the reflection polynomial has a zero at. A Butterworth puts all of its reflection zeros at the origin — that is what maximally flat means, and it is why three families, one corner can describe its passband as having one point of perfection rather than several. An odd-order Chebyshev’s reflection zeros are the roots of an odd Chebyshev polynomial and one of them is zero, which the synthesis returns as $6\times10^{-17}$. An even-order Chebyshev’s are the roots of an even one, and none of them is.

An even-order equiripple response therefore cannot have its maximum at direct current, and a network whose direct-current transmission is not the matched maximum cannot be terminated equally. The prohibition is not about ladders. It is about the response, and the ladder is merely honest enough to report it.

This collection has already met the consequence from the far side without recognising the cause. The ratio that does not walk to one measures the noise bandwidth of thirty-two realised filters and finds the Chebyshev branch splitting by parity, with the two branches separating above 0.1968 decibels of ripple, because an even-order response starts at the bottom of its ripple band and comes up while an odd one starts at the top and dips. The bandwidth noise sees reports the same split from the integral. Both describe it as a property of the shape.

It is the same fact as the termination, seen in a different quantity. Direct current at the bottom of the ripple band is direct current below the maximum; direct current below the maximum is a network not delivering the available power where it must be two resistances; and a network not delivering the available power there has terminations that are not equal. The correctly terminated order-four design measured here sits at −3.5451 dB at direct current against maxima of −3.0451, and the difference is 0.5000 dB, which is the ripple to four decimals.

So a shape fact, a noise fact and a termination fact are one fact with three faces, and only the last of them ever appears in the design procedure.

Two routes to the same number #

The load’s value has a closed form, $R_L = (\sqrt{1+\varepsilon^2} + \varepsilon)^2$, and the agreement between it and the synthesis is the calibration everything below is quoted against.

Those are two genuinely independent computations, which is the thing the site’s habit of demanding two routes is usually unable to guarantee: the expansion never mentions $\varepsilon$ after building the polynomials, and the expression never mentions a polynomial at all. Agreement at a part in $10^{15}$ is the arithmetic’s floor rather than a tolerance, which matters here because the numbers under discussion are differences of quantities that nearly cancel — the failure the digits the arithmetic did not have tracks through this same expansion until it stalls.

The sweep also shows the load running from 1.10075 at a hundredth of a decibel of ripple to 5.80890 at three. A tenth of a decibel — which is the ripple most designs actually specify — wants 1.35536, and half a decibel wants 1.98406. The mismatch is not a rounding error at any of them.

What insisting costs, in the visible currency #

The substitution is one element. The reactances are the ones the expansion returned; only the load is changed, from the value it demanded to the source resistance, which is exactly what a designer reaching for a table of g-values and a matched pair of terminations builds.

Two and two-thirds times the design ripple, on a second-order filter, from changing one resistor to the value everybody assumes it has. The direction is worth stating: the forced response is not merely a different equiripple filter with more ripple in it. It is not equiripple at all.

The peak that sits above half #

One number in both of those pictures looks like a free lunch and is not, and it has to be dealt with before the sensitivity argument will read correctly.

A matched two-port loses exactly half its voltage, 6.0206 dB, and the two resistors a ladder was designed between identified that half as what buys the insensitivity rather than as waste. The correctly terminated even order reaches −3.0451 dB at its ripple peaks, which is three decibels less loss. Nothing has been got for nothing: the load is 1.984 times larger, so the same power arrives as a larger voltage across it, and $\tfrac{1}{2}\sqrt{R_L/R_s}$ is the voltage ratio that corresponds to delivering all the available power into an unequal load. The figures assert that identity rather than assuming it — the measured maxima land on −3.0451 dB to within a thousandth of a decibel.

Which is the point about maximum power transfer that is easy to lose. “All the power the source has” is not a statement about a voltage ratio; it is a statement about a power, and the voltage ratio that realises it depends on what the power is being delivered into. A designer who compares the two insertion losses and concludes the unequal termination is better has made the mirror image of the mistake this essay is about.

The count is the sharper of the two readings and it holds at every order measured: correctly terminated, an even-order design of order $n$ has $n/2$ interior maxima; forced, it has $n/2 - 1$. One maximum does not move, it disappears. The response leaves direct current already falling instead of touching its ceiling there, and the peak that would have been the first one is gone.

The maximum that goes missing #

A passband maximum on a doubly terminated ladder is not a feature of the curve. It is a frequency at which the network is delivering all the power the source has available, and there is no more, so the response cannot go higher there whatever happens to any element.

That is the whole of the ladder’s celebrated insensitivity, and the tolerance that can only take away measured its two halves: at a ripple peak the first derivative of the magnitude with respect to every reactance is zero to ten digits, and every second derivative is negative, so of six hundred ladders built from one per cent components not one came out above nominal.

Forcing equal terminations is exactly the change that stops the surviving maxima being available maxima. With a load of 1000 Ω against a source of 1000 the most a lossless two-port can deliver is half the voltage, −6.0206 dB, and the forced response reaches that only at direct current, where the reactances have no say. Everywhere else it is below its own ceiling by an amount that the elements can change in either direction.

The fitted powers are the content and the ratio is the consequence. Two against one is the difference between a quantity whose first derivative vanishes and one whose first derivative does not, measured over four tolerances rather than asserted at one point — the same test a ladder is not a cascade uses to establish that the ladder’s advantage over a cascade is structural rather than lucky. And because the two grow at different rates, the advantage is worth most to whoever is buying the tightest components: at half a per cent the forced realisation is a hundred and eighty-two times worse, and at four per cent only twenty-nine, by which point neither is any good.

Order six behaves the same way with more corners in it — sixty-four rather than sixteen, the 2.000 power against 1.178, and 1.60×10⁻² dB against 3.98×10⁻⁴ at one per cent, a factor of forty. The exponents are the invariant; the ratio depends on where the surviving peak happens to land.

And the stopband, by a fixed amount #

The ripple and the sensitivity are the passband’s two currencies. The stopband pays as well, and by an amount that turns out to be the same one at every order.

Measured against each realisation’s own passband maximum, the correctly terminated order-four design is 30.60 dB down at twice the ripple band edge, 56.53 at four times and 88.84 at ten. The forced one is 28.82, 53.85 and 85.91. The loss grows from 1.78 dB to 2.93 dB and then stops, and order six does the identical thing — 1.87, 2.71, 2.93 — against much larger absolute numbers.

The limit is not a coincidence. Far into the stopband the load resistance has almost no influence on the transmission, so both realisations converge on the same absolute attenuation and the difference between them is entirely the difference between their two passband ceilings: −3.0451 dB against −6.0206, which is 2.9755. The measured 2.93 at ten times the edge is that number arriving.

Which means the forced realisation is worse in every quantity a filter is bought for, by amounts that have to be read separately: three and a half times the ripple, a factor of ninety-five in sensitivity, and 2.93 dB of stopband. The last of those is small next to the 30.60 dB the order-four design has at twice its edge, and it is not small next to what it costs to buy back — the selectivity that is not free measures the three-way exchange between ripple, order and stopband on this same family, and decibels of stopband are the expensive corner of it.

A tighter ripple does not rescue it #

The obvious hope is that a design with less ripple has a load nearer one and therefore suffers less. The load does move nearer one — 1.35536 at a tenth of a decibel against 1.98406 at a half — and the damage does not shrink in proportion.

So the penalty is roughly a fixed multiple of the ripple rather than a fixed number of decibels, and a design bought for its flat passband loses the same proportion of its flatness as one bought for selectivity. Halving the specification halves the error and the specification together.

This matters because a tenth of a decibel is where the interesting designs are. The selectivity that is not free prices the exchange between ripple, order and stopband and finds relaxing the ripple worth 9.12 dB of stopband wherever it is spent; a design that has paid two extra orders to keep its ripple at a tenth and then triples that ripple in the assembly has thrown the purchase away.

The termination that can be chosen #

There is one more reading in the ripple sweep and it is the practical one. The load is a continuous function of the ripple, and the ripple is a specification rather than a constant, so the load can be chosen.

Bisecting the synthesis for the ripple whose load is exactly two gives 0.511525 dB, which $10\log_{10}(9/8) = 0.511525$ predicts to six decimal places, and which is the same at orders two, four, six and eight. A 0.511525 dB even-order Chebyshev terminates in exactly twice its source resistance — fifty ohms into a hundred, or a two-to-one transformer, realised exactly rather than approximately.

That is a better answer than either accepting the mismatch or ignoring it, and it costs a fiftieth of a decibel of ripple against the half-decibel design it replaces. It also generalises: any load that can be built corresponds to a ripple that can be computed, and the computation is a bisection on a continued fraction that takes no measurable time.

What it does not say #

It does not say that an even-order Chebyshev is a bad choice. Its stopband at a given order is what what a steep skirt costs says it is, its element spread is unremarkable, and nothing above touches the response the synthesis actually returns — which is equiripple to 0.5000 dB and stationary to the 2.000 power, exactly as advertised.

It says that the last quotient is part of the answer and that dropping it is not a small simplification. The failure it produces is not the shape of a component tolerance and would not be found by tightening one: 1.8123 dB of ripple on a design specified at 0.5 looks like a synthesis error or a modelling error, and it is neither.

Nor does it say the odd orders are safe from everything. They terminate in one, so this particular question does not arise for them, and the rung below measured how narrow the source window is even then. An odd-order Butterworth built between the resistances it asks for and driven from a fifth of one of them is 8.279 dB out on shape, which is a much larger number than anything here.

The number worth carrying #

An even-order Chebyshev terminates in $(\sqrt{1+\varepsilon^2}+\varepsilon)^2$, which is 1.984056 at half a decibel of ripple and depends on the ripple alone. Forcing it to one costs three and a half times the ripple, one passband maximum, and a factor of ninety-five in component sensitivity at one per cent parts.

The habit that goes with it is about what a design procedure returns. A synthesis that produces both the elements and the resistances has said something about the resistances, and a number that arrives by the same arithmetic as the element values has the same standing as they do. The temptation is always to treat the resistances as environment — as what the filter happens to be plugged into — and the whole of the ladder’s advantage over a cascade of active sections is that they are not. They are the two components that make every other component’s error second order, and an even order is the case where the arithmetic says so loudly enough to be heard.