Circuit Response
every model has an edge, and the figure carries it
Filters, measured not tabulated

The load a filter is in its stopband

A fifth-order Chebyshev ladder filter can begin with a series inductor or, as its dual, with a shunt capacitor, and between its design resistances the two transmit identically. In the stopband they are opposite loads: at three times the corner the first is 5.24 R₀ of inductance and the second 0.191 R₀ of capacitance, their product R₀² at every frequency. That stops mattering only while the source is the design resistance. From a tenth of R₀ the shunt-first ladder filter rejects 55.9 dB at three times the corner instead of 64.0, draws 14 times the passband current at five times the corner against the series-first ladder filter's 0.24, and with a lead inductance of a tenth of R₀ over the corner's angular frequency grows a hole in its stopband — −33.8 dB where a matched source gives −54.0. A soft source does the mirror image to the series-first ladder filter. The first element has to be the one the source cannot short or open.
15 min read 6 figures The parasitic is the componentThe schematic is a labelTwo routes to a number

Assumes: The two resistors a ladder was designed between · Three families, one corner

What a filter does not pass, it sends back followed the power a passive ladder filter rejects and found that it is not removed but returned: at 1.4 times the corner a fifth-order Chebyshev reflects 96 to 99 per cent of the available power back to its source. Its open question was what the source makes of that. A signal generator with a resistive output absorbs it; an amplifier sees a nearly reactive load at its own harmonics, and whether that load is an inductance or a capacitance decides how the amplifier responds to it.

The heat a delay predicts then found the answer’s first half without looking for it. The same Chebyshev can be realised with a series inductor first or, as its dual, with a shunt capacitor first; the two transmit identically and store their stopband energy in opposite elements — the series-first ladder filter in its coils, 4.63 times as much magnetic energy as electric at 1.6 times the corner, and the shunt-first ladder filter in its capacitors by the same factor. A network storing magnetic energy at its input looks inductive from the outside. This essay measures what each looks like to its source, and what happens when the source is not the resistance the filter was designed for.

One filter, two loads

The filter is the 0.5 dB Chebyshev of the earlier essays, fifth order, its half-power corner at 1 kHz, designed between two 1 kΩ resistances. Built as a series inductor, a shunt capacitor and so on, it is one network; built from the same prototype values with every series inductor exchanged for a shunt capacitor and every shunt capacitor for a series inductor, it is another. Between the design resistances their transmissions agree to 10−1010^{-10} — the dual transmits exactly what the original does.

In the stopband the series-first ladder filter is an inductance — 5.24 R₀ at three times the corner — and its dual a capacitance, 0.191 R₀: the same filter, opposite loadscomputed by solving, not by drawing. The input impedance of the order-5 0.5 dB Chebyshev ladder filter realised with a series inductor first and with a shunt capacitor first, both terminated in the design resistance R₀, against frequency from a tenth of the corner to ten times it. In the passband both sit near R₀ and are nearly resistive. Above the corner they part: at 1.2 times the corner 1.33 R₀ at 89.4° against 0.749 R₀ at −89.4°; 2 times the corner 3.19 R₀ at 90.0° against 0.314 R₀ at −90.0°; 3 times the corner 5.24 R₀ at 90.0° against 0.191 R₀ at −90.0°; 5 times the corner 8.90 R₀ at 90.0° against 0.112 R₀ at −90.0°. The product of the two is R₀² at every frequency to 2e-15, which is what duality means for an input impedance. The slider is the order.100m110frequency ÷ half-power corner|input impedance| ÷ design resistance0.1110series inductor first+90.0° at 3 × cornershunt capacitor first−90.0° at 3 × cornercomputed by solving, not by drawingone filter, two loads
Fig. 1 The input impedance of the two realisations, as a multiple of the design resistance, from a tenth of the corner to ten times it. Near R0R_0 in the passband; in the stopband the series-first ladder filter rises as an inductance and the shunt-first ladder filter falls as a capacitance. The slider is the order.

In the passband both input impedances stay near the design resistance and nearly resistive, which is what a doubly-terminated ladder filter is for. Above the corner they part, and quickly. At 1.2 times the corner the series-first ladder filter is 1.33 R0R_0 at +89.4° and the shunt-first ladder filter 0.749 R0R_0 at −89.4°; by twice the corner they are 3.19 and 0.314 R0R_0 and both phases are ninety degrees to the first decimal; at three times, 5.24 and 0.191 R0R_0; at five, 8.90 and 0.112. A source looking into the series-first ladder filter in its stopband sees an inductor, and one looking into the shunt-first ladder filter sees a capacitor.

The two impedances are not merely opposite; they are reciprocal. Their product is R02R_0^2 at every frequency, to two parts in 101510^{15}, which is what duality means for an input impedance: the dual network presents the inverse of the original’s impedance, scaled by the square of the reference resistance. So at every frequency where one realisation is a high impedance to its source, the other is the corresponding low one, by exactly the same factor.

Which loss matters in the stopband depends on the first element: at 1.6 times the corner the series-first ladder filter stores 4.63 times as much magnetic energy as electric, and its dual 4.63 times as much electric. computed by solving, not by drawing. The ratio of the power absorbed with lossy inductors to the power absorbed with lossy capacitors, at the same small loss rate, which is the ratio of magnetic to electric stored energy, for the fifth-order Chebyshev realised with a series inductor first and with a shunt capacitor first. The two transmit identically. Through the ripple band both ratios stay near one — 0.95 and 1.06 at half the corner, 1.02 and 0.98 at 0.95. Away from it they part and mirror each other: 2.37 and 0.42 near direct current, 4.63 and 0.22 at 1.6 times the corner.
Fig. 2 From the essay on heat: the ratio of magnetic to electric stored energy in the two realisations. Where the series-first ladder filter’s input is inductive its energy is magnetic, and the dual mirrors it.

The energy split from the heat essay is the same fact seen from inside. The series-first ladder filter’s stopband energy is in its first inductor, because that inductor carries the input current and the shunt capacitor behind it, nearly a short at these frequencies, leaves the rest of the ladder filter almost undriven. The shunt-first ladder filter’s is in its first capacitor, because that capacitor holds the input voltage and the series inductor behind it, nearly an open, blocks the rest.

A source that is not the design resistance

All of this would be a curiosity if sources were always the design resistance, because between matched resistances the two realisations are indistinguishable from outside: they transmit the same, reflect the same magnitude, and present impedances whose reactive nature only the phase of the reflection reveals. Sources are rarely the design resistance at harmonics. An amplifier’s output impedance is a few ohms when it is built as a voltage source and kilohms when it is a current source, and either way it is the design resistance only at the frequency someone matched it at, if at all.

The two resistors a ladder filter was designed between found the passband’s shape surprisingly sensitive to that: half a decibel of shape error for a source only 12 per cent off. The stopband is a different question with a different answer, and the realisation decides it.

At three times the corner both realisations reject 64.0 dB from a matched source; from a tenth of R₀ the shunt-first ladder filter rejects only 55.9, from ten times R₀ the series-first ladder filter only 55.9. computed by solving, not by drawing. The fifth-order 0.5 dB Chebyshev ladder filter, both realisations, driven at three times its corner from a source resistance between a hundredth and a hundred times the design value, the load fixed. The response at three times the corner relative to the same source's response near direct current: 0.01 R₀: −69.8 dB series-first, −55.6 dB shunt-first; 0.1 R₀: −69.1 dB series-first, −55.9 dB shunt-first; 1 R₀: −64.0 dB series-first, −64.0 dB shunt-first; 10 R₀: −55.9 dB series-first, −69.1 dB shunt-first; 100 R₀: −55.6 dB series-first, −69.8 dB shunt-first. A stiff source drives the first shunt capacitor directly and leaves it nothing to divide against, and a soft one feeds the first series inductor a current it cannot change, so each realisation loses the use of its first element at one end of the range. Duality makes the two curves exact mirror images about the design resistance, to 1e-12 dB.
Fig. 3 The response at three times the corner, relative to the same source’s passband response, against the source resistance from a hundredth to a hundred times the design value. Each realisation loses rejection at one end of the range: the shunt-first ladder filter to a stiff source, the series-first ladder filter to a soft one. The two curves are mirror images.

At three times the corner both realisations reject 64.0 dB from a matched source. From a tenth of the design resistance the shunt-first ladder filter rejects 55.9 dB and the series-first ladder filter 69.1; from a hundredth, 55.6 and 69.8. From ten times the design resistance the roles exchange exactly: the series-first ladder filter rejects 55.9 and the shunt-first 69.1. The two curves are mirror images about the design resistance to 10−1210^{-12} dB, which is duality again — a shunt-first ladder filter driven from mR0mR_0 is, in every ratio a load can measure, the series-first ladder filter driven from R0/mR_0/m.

The reason is the first element, and it is simple once seen. A stiff source holds the input voltage whatever current is drawn, so a shunt capacitor across it has no effect on the voltage at that node: the filter’s first pole has nothing to divide against and the ladder filter is, to a stiff source in its stopband, one element shorter. A soft source holds the current whatever voltage is needed, so a series inductor in its path changes nothing: the series-first ladder filter loses its first element to a current source in the same way. Losing one element of five costs about eight decibels at three times the corner, and the curves level off once the source is stiff or soft enough that the first element is entirely wasted.

The ladder filter that keeps its rejection against a source is the one whose first element the source cannot neutralise: a series inductor facing a voltage source, a shunt capacitor facing a current source. That is the rule a power-amplifier designer learns from harmonic filters, here measured as a number of decibels and shown to be exact duality rather than a rule of thumb.

The current a stiff source must deliver

The rejection is the load’s view. The source has its own: what current it must supply at frequencies it was never meant to deliver.

From a stiff source, at five times the corner the shunt-first ladder filter draws 14.2 times the passband current and the series-first ladder filter 0.239 — a ratio of 59. computed by solving, not by drawing. The current drawn from a 1 V source with a tenth of the design resistance in series, by each realisation of the fifth-order 0.5 dB Chebyshev ladder filter, against frequency, in units of the matched passband current. The series-first ladder filter's input is an inductance in the stopband and its current falls as one over frequency (slope −1.04); the shunt-first ladder filter's is a capacitance and its current rises with frequency, towards the ceiling of 21 passband currents that the source's own resistance sets. 2 times the corner: 0.668 and 6.47; 3 times the corner: 0.406 and 9.88; 5 times the corner: 0.239 and 14.16; 10 times the corner: 0.118 and 18.61.
Fig. 4 The current drawn from a source with a tenth of the design resistance by each realisation, in units of the matched passband current, against frequency. The series-first ladder filter’s current falls as one over frequency in the stopband; the shunt-first ladder filter’s rises towards the ceiling the source’s own resistance sets.

From a source of a tenth of the design resistance the series-first ladder filter draws less current the further into the stopband the frequency goes — 0.668 passband currents at twice the corner, 0.406 at three times, 0.239 at five and 0.118 at ten — falling as one over frequency because its input is an inductance. The shunt-first ladder filter draws more: 6.47 passband currents at twice the corner, 9.88 at three times, 14.2 at five and 18.6 at ten, rising towards the twenty or so that the source’s own resistance allows once the first capacitor’s reactance has fallen below it. At five times the corner the ratio between them is 59.

For a signal source that is a curiosity. For an amplifier whose output contains harmonics — any switching stage, any class B output, any saturating driver — it is the harmonic current the output devices must carry and the harmonic power they must dissipate in their own resistance, and a shunt-first harmonic filter makes those harmonics larger than they would be with no filter at all. A capacitor across a stiff output is a short circuit at the harmonics it was meant to remove.

The reflection the source sees completes the picture. A stiff source of a tenth of the design resistance has a reflection coefficient of −0.818 against R0R_0, so most of what the filter returns is sent back to the filter again, and in the stopband, where the filter returns nearly everything, the wave rattles between two near-total reflectors. What leaks through the filter on each pass is small, but the passes add, and how they add depends on the round-trip phase — which is set by the filter’s input reactance and the source’s. The shunt-first ladder filter’s capacitive input and a stiff source make the round trip a short circuit’s; the series-first ladder filter’s inductive input makes it nearly an open circuit’s, and nothing builds up. That is the same statement as the rejection curves, in the language of waves rather than of elements.

Choosing the realisation from the source

The mirror-image curves turn into a procedure. Find the source’s impedance at the frequencies the filter must reject — not at the passband, where it was presumably matched, but at the harmonics. If it is below the design resistance, the filter should begin with a series inductor; if above, with a shunt capacitor. The gain from choosing right is not small: from a tenth of the design resistance the right realisation rejects 69.1 dB at three times the corner and the wrong one 55.9, a difference of thirteen decibels between two filters with identical data-sheet curves.

The choice costs nothing in the passband, because the two realisations transmit identically between the design resistances and have input impedances of the same magnitude. It costs nothing in components either: the dual of a five-element ladder filter is another five-element ladder filter with the same prototype values, three capacitors and two inductors instead of three inductors and two capacitors. What it can cost is size, since inductors are usually the larger parts, and that is the trade a designer actually faces — the realisation with fewer inductors is the one with its first element in shunt, and it is the wrong one for a stiff source.

Matching the source to the filter at the harmonics is the other way out, and it is rarely available. A resistor in series with a stiff source restores the shunt-first ladder filter’s rejection by giving its first capacitor something to divide against, and it dissipates the passband power the filter was meant to deliver. A transformer changes the source’s level but not its character: a stiff source behind a transformer is still stiff.

A lead inductance and a first capacitor

Real stiff sources are not purely resistive either. An amplifier’s output reaches the filter through a bond wire, a trace or a lead, and a few nanohenries in series with a low resistance is the ordinary case. For the series-first ladder filter that inductance simply adds to its first inductor, which is harmless. For the shunt-first ladder filter it is a series resonator. And when the source is an emitter follower, a capacitive load is the one it handles worst: the input that pushes back found a nanofarad on a follower’s emitter putting a negative resistance at its base, and the shunt-first ladder filter’s stopband is exactly such a load.

A stiff source with 0.1 R₀/ωc of series inductance opens a hole in the shunt-first ladder filter's stopband at 2.42 times the corner: −33.8 dB where a matched source gives −54.0. computed by solving, not by drawing. Transmission of both realisations of the fifth-order 0.5 dB Chebyshev from a source with a tenth of the design resistance, with and without a series inductance of 0.1 R₀/ωc, against frequency from the corner to ten times it; the matched response is the reference. The inductance resonates with the shunt-first ladder filter's first capacitor at 2.353 of the corner, where the stiff source has nothing to damp the resonance but its own resistance, and the transmission there is −33.8 dB against −54.0 matched and −42.3 without the inductance. For the series-first ladder filter the same inductance adds to its first inductor and costs no rejection anywhere. The slider is the source inductance.
Fig. 5 Transmission of the two realisations from a source of a tenth of the design resistance with a series inductance of a tenth of R0/ωcR_0/\omega_c, against the matched response. The inductance resonates with the shunt-first ladder filter’s first capacitor and opens a hole in its stopband; the series-first ladder filter is unaffected.

With a source inductance of a tenth of R0/ωcR_0/\omega_c the shunt-first ladder filter’s first capacitor resonates with it at 2.35 times the corner. At resonance the source drives the capacitor through nothing but its own small resistance, the voltage on the capacitor rises by the resonance’s quality factor, and the stopband has a hole: −33.8 dB at 2.42 times the corner, where a matched source gives −54.0 and the stiff source without its inductance −42.3. The series-first ladder filter from the same source with the same inductance never loses rejection anywhere.

The inductance moves the hole. A third as much, 0.03 R0/ωcR_0/\omega_c, puts the resonance higher, at 4.3 times the corner, where the hole is −60.2 dB against −80.2 matched; three times as much, 0.3, pulls it down to 1.53 times the corner, where it is −10.0 dB against −30.9 — a stopband with a twenty-decibel hole in it, just past the band edge, where a power amplifier’s second harmonic often falls. Every one of these is a filter that measures correctly between 50-ohm instruments and fails in its circuit, and the failure is the source’s lead and the filter’s first capacitor together, neither of which is wrong alone.

The hole’s depth is the resonance’s quality factor, and the quality factor is set by the very thing that made the source stiff. At resonance the lead’s reactance is a tenth of R0/ωcR_0/\omega_c times 2.35, which is 0.235 R0R_0, and the source’s resistance is 0.1 R0R_0, so the resonance’s Q is 2.35 and it lifts the capacitor’s voltage by about 7.4 dB over the stiff source alone — against the 8.5 dB the solve measures, the rest being the ladder filter behind the capacitor. A stiffer source makes a sharper and deeper hole, and an ideal voltage source with any lead inductance at all makes an infinitely deep one: the better the amplifier’s output stage by the usual measure, the worse this filter behaves behind it.

One input impedance predicts the filter's response from any source: 5 source resistances, each 1 − |ρ|² to 10⁻¹². computed by solving, not by drawing: the fifth-order 0.5 dB Chebyshev ladder driven from 500 Ω, 800 Ω, 1000 Ω, 1250 Ω, 2000 Ω, its response across the design's ripple band drawn as the power delivered to the load over the power that source makes available. Each curve is computed twice: by solving the circuit with that source, and by taking the input impedance of the design once and forming 1 − |ρ|² against the new source resistance. They agree to 10⁻¹² everywhere. At ×0.5, the least delivered inside the ripple band is 0.6487, against a bound of 0.6431 from the filter's reflection and the source's composed at their worst phase. At ×0.8, the least delivered inside the ripple band is 0.8221, against a bound of 0.8191 from the filter's reflection and the source's composed at their worst phase. At ×1, the least delivered inside the ripple band is 0.8913, against a bound of 0.8913 from the filter's reflection and the source's composed at their worst phase. At ×1.25, the least delivered inside the ripple band is 0.8205, against a bound of 0.8191 from the filter's reflection and the source's composed at their worst phase. At ×2, the least delivered inside the ripple band is 0.6459, against a bound of 0.6431 from the filter's reflection and the source's composed at their worst phase. The filter does not change with its source; only the reference its impedance is compared against does.
Fig. 6 From the reflection essay: the passband response from five source resistances, each predicted from the design’s input impedance alone. The passband is decided by the input impedance’s magnitude near R0R_0; the stopband, by its reactance far from it.

The reflection essay’s figure above makes the contrast. In the passband, the response from any source follows from one input impedance through 1−∣ρ∣21 - |\rho|^2, and the two realisations, having input impedances of the same magnitude, behave the same from any source. In the stopband the input impedance is far from the source resistance in opposite directions, and what the source does with a load that is nearly a pure reactance depends on which reactance it is.

How the numbers were obtained

Both realisations are netlists built from the same prototype values: the series-first ladder filter from the synthesis, the shunt-first ladder filter by exchanging every element for its dual at the same impedance scale. The input impedance is read from the solved node voltage behind the source resistance; the rejection is the load voltage at the stopband frequency over the load voltage at a hundredth of the corner, from the same source; the source current is the current through the source itself. The source inductance is a series element between the source resistance and the filter, given in units of R0/ωcR_0/\omega_c, which for this filter is 159 millihenries per unit and for a 50-ohm filter at 100 MHz is 80 nanohenries per unit.

What it leaves out

Sources that are reactive in other ways. An amplifier’s output capacitance is the dual of a lead inductance, and it does to a series-first ladder filter what the lead inductance does to a shunt-first one: resonates with the first inductor, in parallel, and opens a hole. The rule is the same and so is the cure.

Nonlinear sources. The harmonic current a stiff source must deliver into a capacitor changes the amplifier’s own operating point and its harmonic content, so the currents drawn here are the first step of a calculation that, for a real power stage, has to be iterated.

Elliptic filters. A filter with transmission zeros in the stopband, built as a ladder filter with resonant branches, has a stopband input impedance that changes sign between zeros, so neither realisation is purely inductive or capacitive across the whole stopband, and the rule has to be applied band by band.

Still open: equal ripple in reflection, a source that changes with level, and the filter that absorbs

Equal ripple in reflection rather than transmission. A diplexer splits a band between two filters sharing an input, and its specification is a return loss across the whole band at the common port. Each filter’s stopband reactance is then the other’s load, and the pair is matched only if the two reactances cancel in parallel — a condition the duality found here should make exact for one choice of realisations and impossible for the other.

A source whose impedance moves with its output level. A saturating amplifier is a stiff source at low level and a soft one as it limits, so the realisation that suits it changes with drive. Whether a ladder filter can be built whose stopband is indifferent to the source’s resistance — by placing the first element so that neither extreme neutralises it — is the design question this essay’s mirror-image curves raise.

A filter that absorbs its stopband instead of reflecting it. A ladder filter with resistive terminations on a diplexer’s second arm, or a lossy element placed where the stopband energy is stored, can turn the reflected harmonics into heat inside the filter. The heat essay’s delay law says how much a given loss absorbs and the energy split says which element should carry it; together they price an absorptive filter before it is built.

Part 5 on filter termination

One argument about Filter termination, and one of 5 essays on it so far, each part numbered by how much of the idea it assumes. What sits either side of it:

With inductors of Q 100 the filter absorbs up to 10.5% of the available power, and the loss rate times its delay predicts the curve's shape — 11.2% at the same frequency. computed by solving, not by drawing. The fifth-order 0.5 dB Chebyshev ladder filter of the reflection essay, each inductor given a series resistance ωL/Q at the corner, so that each loses energy at the rate σ = 2πfc/Q. The power absorbed in the inductors, as a fraction of the available power, against frequency; beside it the lossless ladder filter's delay, D = |t|²τ₂₁ + |ρ|²τ₁₁, times σ — the stored energy over the available power, times the loss rate. At Q 100 the absorption peaks at 10.53% at 0.957 of the corner, where σ·D is 11.24%; between 0.3 and 0.95 of the corner the two agree to 7%, the rest being the split between magnetic and electric energy and the terms of second order in the loss. The slider is the inductors' Q. The heat a delay predicts Part 4 — A lossy LC ladder filter absorbs power where it stores energy, and a lossless ladder filter stores energy in exact proportion to its delay — the transmitted delay weighted by what it transmits plus the reflected delay weighted by what it reflects. So the droop of a lossy filter is its loss rate times that delay: with equal losses in inductors and capacitors, to 0.09% at Q 10,000. With inductors of Q 100 in a fifth-order Chebyshev, 10.5% of the available power is absorbed at the band edge against 11.2% predicted. Inductor loss alone misses only by the split between magnetic and electric energy — near one in the passband, 4.6 to one at 1.6 times the corner — and the dual ladder filter mirrors the split. The law is first order, and it overstates the band-edge peak by 17.5/Q.

Where this goes next

With inductors of Q 100 the filter absorbs up to 10.5% of the available power, and the loss rate times its delay predicts the curve's shape — 11.2% at the same frequency. computed by solving, not by drawing. The fifth-order 0.5 dB Chebyshev ladder filter of the reflection essay, each inductor given a series resistance ωL/Q at the corner, so that each loses energy at the rate σ = 2πfc/Q. The power absorbed in the inductors, as a fraction of the available power, against frequency; beside it the lossless ladder filter's delay, D = |t|²τ₂₁ + |ρ|²τ₁₁, times σ — the stored energy over the available power, times the loss rate. At Q 100 the absorption peaks at 10.53% at 0.957 of the corner, where σ·D is 11.24%; between 0.3 and 0.95 of the corner the two agree to 7%, the rest being the split between magnetic and electric energy and the terms of second order in the loss. The slider is the inductors' Q. The heat a delay predicts A lossy LC ladder filter absorbs power where it stores energy, and a lossless ladder filter stores energy in exact proportion to its delay — the transmitted delay weighted by what it transmits plus the reflected delay weighted by what it reflects. So the droop of a lossy filter is its loss rate times that delay: with equal losses in inductors and capacitors, to 0.09% at Q 10,000. With inductors of Q 100 in a fifth-order Chebyshev, 10.5% of the available power is absorbed at the band edge against 11.2% predicted. Inductor loss alone misses only by the split between magnetic and electric energy — near one in the passband, 4.6 to one at 1.6 times the corner — and the dual ladder filter mirrors the split. The law is first order, and it overstates the band-edge peak by 17.5/Q. A 0.5 dB Chebyshev ladder of order five passes what it does not reflect: the two powers sum to one to 2e-15. computed by solving, not by drawing: a fifth-order Chebyshev ladder of 0.5 dB ripple between 1 kΩ terminations, solved at 481 frequencies. The transmitted power is the transducer gain, read from the load voltage; the reflected power is |ρ|², read from the impedance the filter presents to its source. Their sum is one to 2.0e-15 everywhere. The reflection reaches 0.1087 at each ripple trough, against ε²/(1 + ε²) = 0.1087 with ε² = 10 to the power ripple/10, less one, a return loss of 9.64 dB, and falls to zero at direct current and at 0.555 and 0.898 of the corner, where the transmission is at its maxima. The ripple band ends at 0.944 of the corner. The slider is the ripple. What a filter does not pass, it sends back A passive LC ladder built from lossless reactances has nowhere to put power except the load and the source. So the power it transmits and the power it reflects sum to what the source makes available — to 2 × 10⁻¹⁵ across a fifth-order Chebyshev, read from two different ports — and every passband specification is a return-loss specification in disguise: half a decibel of ripple is 9.64 dB of return loss at every order, a hundredth of a decibel is 26.4. The same identity predicts the filter's response from any source resistance out of one input impedance, and bounds the damage a wrong source can do by composing two reflections. An order-5 ladder driven from 2× the 1 kΩ it was designed between. computed by solving, not by drawing. A doubly-terminated Butterworth ladder is a two-port designed between two stated resistances, and the resistances are part of the design rather than the environment it happens to be used in. At match it loses 6.021 dB — exactly half the voltage — and its passband has no peak anywhere in it. Driving the same five reactances from 2× that resistance moves the shape by 2.345 dB and the insertion loss to 9.542 dB. The band inside which the shape is right to 0.5 dB runs 0.8857× to 1.1371× — a window of 25 per cent on a quantity usually written down as a round number. And the two sides are not alike: at twenty times the design resistance the departure has settled at 5.33 dB, while at a twentieth of it the passband is 15.51 dB out with 7.21 dB of peaking on it, so driving a ladder from too low an impedance is worse than driving it from too high a one. The two resistors a ladder was designed between A passive ladder filter is not a transfer function with some resistors attached; it is a two-port designed between two stated resistances, and the resistances are as much part of the design as the inductors. Driving an order-five Butterworth from anything outside 0.886 to 1.137 times its design resistance puts more than half a decibel of error on the passband — a tolerance tighter than the resistor is usually specified to — and the two sides of that window are not alike.

The objects named here

The third axis, after the field and the idea: the things themselves, and every essay that touches each one.

Doubly terminated ladder filterDualityInput impedanceLead inductanceReflection coefficientSeries resonanceSource impedanceStopband attenuation