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

The band that does not close

A cascade of active sections can be built at three decades of impedance level at second order, one at sixth, and none at all at eighth — the band shuts by half a decade per order because two fixed quantities bind it from opposite ends. A doubly terminated ladder has only one of those quantities, because an inductor's loss is a fixed quality factor rather than a fixed resistance and therefore scales with the design. At order nine it still has four decades.
17 min read 8 figures Every model has an edgeThe parasitic is the componentMeasured, not tabulated

Assumes: The same filter a thousand times larger · What a steep skirt costs

Two rungs below this one, a filter design was shown to be exactly invariant under impedance scaling — multiply every resistance by a factor, divide every capacitance by it, and the solved response is unchanged to the last bits of a double across six decades. So the impedance level is free, and what decides it is the quantities that do not scale.

For a cascade of active sections there are two, and they bind at opposite ends. The amplifier has fifty ohms of output resistance whatever the network around it is, which hurts at the low end where the design’s own impedances are small. The board has two picofarads to everywhere, which hurts at the high end where the design’s own capacitances are small. Between them is a band, and the rung below this one measured what an order does to it: three decades at second order, two at fourth, one at sixth, and at eighth nothing at all — no impedance level whatever meets a tenth of a decibel, because the least departure any level allows has risen to 0.114 decibels.

That is a boundary in an order rather than in a frequency or an amplitude, and it left an obvious question. A cascade is one way to realise a response and a doubly terminated ladder is another, and the ladder is already known to be different in a way that ought to matter here: its component sensitivities are second order where a cascade’s are first. Does its band close too?

It does not.

At the order a cascade runs out, a ladder still has 4.2 decades of levelcomputed by solving, not by drawing. How many decades of impedance level each structure can be built at while staying inside 0.1 dB of its own design, against order. Both carry two picofarads of stray at every node. The cascade also carries fifty ohms of amplifier output resistance, a fixed resistance, which binds it from below; its floor rises 8.5× over the four orders drawn, to 0.114 dB at order eight. The ladder's inductors carry a fixed resistance per henry instead — a fixed quality factor, which scales with the design and bounds nothing — so what limits it from below is a few milliohms of track, and at order nine it still has 4.20 decades with a floor of 0.0337 dB, 3.3× its own floor at order three against the cascade's 8.5×.024623456789filter orderdecades of impedance level inside 0.1 dBthe ladder, and the cascade below ittolerance0.1 dBladder, order 35.32 decadesladder, order 94.20 decades…its floor there0.0337 dBcascade, order 23.00 decadescascade, order 61.00 decadescascade, order 8none at all…its floor there0.1138 dBfloor grows, cascade8.5× over 4 orders…ladder3.3×solved, then checked — two structures, one tolerancethe cascade is empty at order 8
Fig. 1 How many decades of impedance level each structure can be built at while staying inside a tenth of a decibel of its own design, against order. The lower line is the cascade and the upper one the ladder, and the ladder’s orders are two higher at every point on the axis.

What each structure has that does not scale

The measurement is only as good as the list of fixed quantities, so it is worth being explicit about what is in each realisation and why.

The cascade is unchanged from the rung below: equal-resistor Sallen-Key sections, each closed by a one-pole amplifier with a hundred thousand of open-loop gain, ten megahertz of gain–bandwidth and fifty ohms of output resistance, with two picofarads of stray hung on all three of each section’s nodes.

The ladder is the doubly terminated realisation whose element values come from a continued-fraction expansion. It carries the same two picofarads at every node. In place of the amplifier’s output resistance it carries two things: a winding resistance in series with every inductor, and a few milliohms of track and connector in series with the load.

Which of those bounds the low end is the whole result, and the surprise is that the obvious candidate does not.

An inductor’s loss is a quality factor, not a resistance

A winding resistance looks like exactly the kind of fixed quantity that should bind a ladder from below, in the same way an amplifier’s output resistance binds a cascade. It is not, and the reason is that it is not fixed.

Scale a ladder to a lower impedance level and every inductance falls in proportion. An inductor of a tenth the value on the same core with the same wire has a tenth of the turns squared — which is a hundredth of the inductance — or, holding the geometry, its resistance falls in proportion to its inductance. So what is constant across a family of real inductors is not ohms but ohms per henry, and ohms per henry is a quality factor at a stated frequency: Q=ωL/RQ = \omega L/R, with the LL dividing out.

A fixed QQ scales with the design. Halve every impedance and every reactance halves, every winding resistance halves, and every ratio in the network is what it was. The inductor’s loss is invisible to impedance scaling, and a ladder whose only low-end fixed quantity was its inductors would have a band open at the bottom — which is a property of the model rather than of the circuit, and would have made this measurement meaningless.

So the ladder is given the thing that really is a fixed resistance: five milliohms of track and connector in series with the load. At a kilohm impedance level that is five parts per million and does nothing; at half an ohm it is one per cent of the termination and the design has stopped being the design.

A 0.5 mm conductor's resistance against frequency, exact and asymptotic. computed by solving, not by drawing. The exact ratio is computed from the Kelvin functions by their series; the dashed curve is the asymptote everybody quotes, which treats the current as flowing in one skin depth of the rim and is drawn only where that annulus is inside the wire. At 17.4 kHz, where the skin depth equals the radius and the rule of thumb says the effect "starts", the asymptote says 1.0000 — no effect at all — and the exact answer is already 1.0208. The rule of thumb names a frequency the effect has passed, which is the same shape as the tenth-of-a-wavelength criterion marking a point at which the lumped model is already 30% wrong. Two decades above, the two agree to 0.00%, which is what makes it an asymptote rather than a formula.
Fig. 2 Where the ohms-per-henry comes from, in the field about magnetics: a winding’s resistance against frequency, whose ratio to its own reactance is what a designer holds constant when choosing an inductor.
A resonator's Q against its inductor's, with a capacitor of Q 1581. computed by solving, not by drawing. The dashed line is what the resonator's Q would be if the inductor were its only loss; the solid one is what it is with a capacitor of Q 1581 beside it. They part company where the inductor stops being the worst component. At the marked point the inductor's Q is 79.06, the capacitor's is 1581.1, the reciprocals predict 75.2923 and the solved network measures 75.2923 — 2.2e-7% apart, by two routes that share only the element values. The resonance stays at 1/2π√(LC) to a part in a million throughout.
Fig. 3 And what that ratio does to a resonance, from the frequency field: the measured quality factor of a resonator against the two component qualities that set it, where the worst of the two decides. At four tenths of an ohm in the coil, a coil quality of 79.1 against a capacitor’s 1,581 gives a loop of 75.29 — within five per cent of the coil’s own, because a reciprocal sum is decided by its smallest term.

The numbers

At a tenth of a decibel, with the impedance level swept from about half an ohm to a hundred kilohms:

order decades of level least departure
cascade 2 3.00 0.0133 dB
cascade 4 2.00 0.0421 dB
cascade 6 1.00 0.0763 dB
cascade 8 none 0.1138 dB
ladder 3 5.32 0.0102 dB
ladder 5 4.76 0.0175 dB
ladder 7 4.48 0.0209 dB
ladder 9 4.20 0.0337 dB

Three things are worth reading out of that table rather than left in it.

The ladder’s band is wider at every order, and the comparison is unfair to the ladder: its orders are two higher in every pair. At order nine, where the cascade has already run out two orders earlier, it still has four and a fifth decades of level to choose from.

The floors tell the same story more cleanly than the bands do, because a band is a band only relative to a tolerance and a floor is a property of the structure. The cascade’s floor rises 8.5 times over the four orders drawn; the ladder’s rises 3.3 times. That ratio of 2.6 is the second-order sensitivity showing up in a quantity that has nothing directly to do with tolerance.

The ladder’s upper edge is what moves. Its lower edge sits at 0.48 ohms at every order — that is the track resistance, and it does not care how many elements there are — while the upper edge comes down from a hundred kilohms to 7.6, because a stray of fixed size is a growing fraction of design capacitances that get smaller as the order rises and the elements get more numerous.

At the order a cascade runs out, a ladder still has 3.6 decades of level. computed by solving, not by drawing. How many decades of impedance level each structure can be built at while staying inside 0.05 dB of its own design, against order. Both carry two picofarads of stray at every node. The cascade also carries fifty ohms of amplifier output resistance, a fixed resistance, which binds it from below; its floor rises 8.5× over the four orders drawn, to 0.114 dB at order eight. The ladder's inductors carry a fixed resistance per henry instead — a fixed quality factor, which scales with the design and bounds nothing — so what limits it from below is a few milliohms of track, and at order nine it still has 3.64 decades with a floor of 0.0337 dB, 3.3× its own floor at order three against the cascade's 8.5×.
Fig. 4 The same construction asked to hold inside 0.05 dB. The ladder covers 3.64 decades at order nine and the cascade covers nothing at all at order eight — there is no band anywhere in which the buffered realisation is inside a twentieth of a decibel of its own design.
A ladder's band is wider at every order, at two orders more, and its floor rises 2.6× more slowly. computed by solving, not by drawing. How many decades of impedance level each structure can be built at while staying inside 0.2 dB of its own design, against order. Both carry two picofarads of stray at every node. The cascade also carries fifty ohms of amplifier output resistance, a fixed resistance, which binds it from below; its floor rises 8.5× over the four orders drawn, to 0.114 dB at order eight. The ladder's inductors carry a fixed resistance per henry instead — a fixed quality factor, which scales with the design and bounds nothing — so what limits it from below is a few milliohms of track, and at order nine it still has 4.76 decades with a floor of 0.0337 dB, 3.3× its own floor at order three against the cascade's 8.5×.
Fig. 5 Two tenths of a decibel, and the cascade appears: 1.50 decades at order eight against the ladder’s 4.76. The cascade’s band is not zero because it is impossible; it is zero at tighter tolerances and small at looser ones, which is a different statement and the one the numbers support.

Why the sensitivity result reappears here

The ladder’s advantage in this measurement has the same cause as its advantage in the tolerance measurement, and it is worth connecting the two explicitly because they look like different subjects.

A doubly terminated ladder designed for maximum power transfer is stationary with respect to its element values at the frequencies where it delivers that maximum: the first derivative of the transmission with respect to any lossless element is zero there, so an error in an element moves the response by a second-order amount. The essay that measured it found the cascade’s passband error growing as the 0.99 power of a component tolerance and the ladder’s as the 2.00 power — first order against second, fitted over two decades.

A stray capacitance is an element error like any other. It is not random and it does not average out — every node gets one, all in the same direction — but the stationarity does not care about the sign of the perturbation, only about its size. So the ladder’s response moves by an amount second order in the stray, and the cascade’s by an amount first order in it, and that is the 2.6 in the floor growth.

What the stationarity does not protect is the terminations, which is the same exception the tolerance essay found: the load resistor’s sensitivity slope is 1.00, because the theorem covers the lossless two-port and stops. That is why the track resistance bounds the low end at all, and it is why lumping the terminations in with the reactances would have erased the whole effect.

A ladder's band is wider at every order, at two orders more, and its floor rises 2.6× more slowly. computed by solving, not by drawing. How many decades of impedance level each structure can be built at while staying inside 0.5 dB of its own design, against order. Both carry two picofarads of stray at every node. The cascade also carries fifty ohms of amplifier output resistance, a fixed resistance, which binds it from below; its floor rises 8.5× over the four orders drawn, to 0.114 dB at order eight. The ladder's inductors carry a fixed resistance per henry instead — a fixed quality factor, which scales with the design and bounds nothing — so what limits it from below is a few milliohms of track, and at order nine it still has 5.32 decades with a floor of 0.0337 dB, 3.3× its own floor at order three against the cascade's 8.5×.
Fig. 6 Half a decibel: ladder 5.32 decades, cascade 2.50. Read the four tolerances together — 0.03, 0.05, 0.2 and 0.5 dB — and the ladder’s band grows steadily while the cascade’s goes from nothing to nothing to 1.50 to 2.50 decades. The gap between the two realisations is not a constant number of decibels; it is a threshold the cascade has to clear before it has a band at all.
At the order a cascade runs out, a ladder still has 4.2 decades of level. computed by solving, not by drawing. How many decades of impedance level each structure can be built at while staying inside 0.1 dB of its own design, against order. Both carry two picofarads of stray at every node. The cascade also carries fifty ohms of amplifier output resistance, a fixed resistance, which binds it from below; its floor rises 8.5× over the four orders drawn, to 0.114 dB at order eight. The ladder's inductors carry a fixed resistance per henry instead — a fixed quality factor, which scales with the design and bounds nothing — so what limits it from below is a few milliohms of track, and at order nine it still has 4.20 decades with a floor of 0.0337 dB, 3.3× its own floor at order three against the cascade's 8.5×.
Fig. 7 An elliptic prototype rather than a Chebyshev, at the same tenth of a decibel. The ladder holds 4.20 decades and the cascade again holds nothing at order eight. The family changes the shape of the response and not the shape of this result, which is what makes the finding about realisation rather than about approximation.

Where the departure is largest, and why it is not the edge

A worst-case departure taken over a band of frequencies hides where it happened, and where it happens is different for the two structures in a way that says what each is failing at.

For the cascade at a high impedance level the worst departure is at the corner, and it is the stray capacitance adding to the smaller of each section’s two capacitances. A high-quality section is one whose two capacitances are far apart — Q=12C1/C2Q = \tfrac12\sqrt{C_1/C_2} — so a stray of fixed size is a larger fraction of the smaller one, and the sections with the highest quality factors are hurt worst. That is why the floor rises with order: the pole pairs a high-order design needs are the high-quality ones.

At a low level the worst departure is also at the corner, and it is the amplifier’s output resistance in series with the feedback network, which is a divider that gets worse as the network’s own impedances approach fifty ohms.

For the ladder at a high level the worst departure moves to the middle of the passband rather than the corner, and it is the stray adding to shunt capacitors that are themselves getting small. At a low level the departure is at direct current, and it is the track resistance in a divider with a termination — which is a flat error rather than a shape, and is the one error in this essay that a calibration could remove.

That last point is worth keeping. Three of the four mechanisms here change the shape of the response and one changes its level, and a designer who can measure the passband gain has a repair for one of them and not for the other three.

The measurement’s own limits

Two things about how this was measured are worth stating, because both bound what the numbers mean.

The comparison band is the passband and the corner, from a hundredth of the corner to twice it. Extending it into the stopband would measure something else entirely — the amplifier’s own gain floor, seventy decibels of “error” at eighty decibels down — which is a true statement about a place nobody designs in. The choice is the same one the rung below this one made and it is what makes the two comparable.

The impedance levels are a grid, twenty-one of them over five and a half decades, so a band edge is located to about a quarter of a decade and the tabulated widths carry that granularity. The floors do not: a floor is a minimum over the grid, and a minimum found on a coarse grid is an over-estimate of the true one by an amount that shrinks with the grid. Both structures are measured on the same grid, so the comparison is unaffected and the absolute widths are the ones that carry the error.

What a ladder costs, which is not nothing

None of the above is an argument that ladders are better, and the honest comparison needs the other column.

A ladder needs inductors. At audio frequencies and a kilohm impedance level those are henries, they are physically large, they pick up magnetic fields, and their quality factors are what this essay has been treating as a constant. The Q the components allow is what that constant actually is: the reciprocals of the component quality factors add, so the total sits below the smallest of them — an inductor of 79 beside a capacitor of 1 581 giving a resonator of 75 — and the worst component decides while the best cannot help. Every ladder in this essay’s comparison is therefore carrying a loss the cascade does not have, and it is the inductor’s loss and nothing else’s.

The escape is to build the inductance out of amplifiers, and this field has measured what that substitutes rather than removes. The inductor that is an amplifier presents one henry from four resistors, a capacitor and two amplifiers, to within one per cent over three and a half decades — and finds its series resistance going negative at 63 hertz, well inside the band where it is still an excellent inductor. One inductor, and ten components then measures the thing that matters to this essay’s whole argument: a floating gyrator reproduces the ladder’s response to a hundredth of a decibel and does not reproduce its stationarity, each of its ten components moving the response by exactly a half where the inductor it replaced moved it by 10−810^{-8}. And eight amplifiers, and what they add prices the arrangement in the two quantities the previous rung had not measured, which turn out to be one quantity: the resistors that buy the accuracy are the noise, and the amplifiers inside the gyrators carry the inductor’s own current, so the floor rises as the square root of the impedance scale and the ceiling falls as the scale.

Which is the sharpest statement of what this essay’s comparison is about. A gyrator-realised ladder has the topology of a ladder and the sensitivity of a cascade, so it inherits the band that closes rather than the one that does not. At a tenth of a hertz an inductor does not exist at all, which is precisely why active filters were invented — and the active realisation is the one this essay finds has no impedance level at all at order eight.

A ladder is harder to design, and the difficulty is structural rather than arithmetic: every element loads every other, so there is no section-by-section decomposition and no adjusting one pole pair without moving the others. That is the same fact as the insensitivity — the errors cancel because the elements interact — and it cannot be had one way round.

A ladder is not tunable. A cascade’s sections can be trimmed one at a time because they are independent; a ladder cannot, because they are not.

A ladder is also sensitive to what it is connected to, in a way a cascade is not. The two resistors a ladder was designed between measures 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 — a window narrower than the tolerance a resistor is usually specified to, and one whose two sides are not alike. That is a sensitivity to a component the cascade does not have at all, and it is not in the impedance-level sweep this essay makes, because every point of that sweep scales the terminations along with everything else.

And the whole comparison here is at a fixed tolerance on the realisation, which is not the only thing a designer cares about. A cascade’s amplifiers give gain, drive a load and isolate the source; a ladder gives none of those and its passband loss is six decibels by construction.

A ladder's band is wider at every order, at two orders more, and its floor rises 2.6× more slowly. computed by solving, not by drawing. How many decades of impedance level each structure can be built at while staying inside 0.03 dB of its own design, against order. Both carry two picofarads of stray at every node. The cascade also carries fifty ohms of amplifier output resistance, a fixed resistance, which binds it from below; its floor rises 8.5× over the four orders drawn, to 0.114 dB at order eight. The ladder's inductors carry a fixed resistance per henry instead — a fixed quality factor, which scales with the design and bounds nothing — so what limits it from below is a few milliohms of track, and at order nine it still has 0.00 decades with a floor of 0.0337 dB, 3.3× its own floor at order three against the cascade's 8.5×.
Fig. 8 The same comparison at three hundredths of a decibel, where the ladder’s own band has closed at order nine as well — its floor there is 0.0337 — so the shape the whole essay is about is a matter of where the tolerance sits and not a difference in kind.

Where the two realisations are compared

A ladder that holds four decades where a cascade holds none is the sharpest form of a comparison this field makes four times. A ladder is not a cascade is the sensitivity, measured element by element. Every derivative, and the one that is zero is where that becomes an exact statement about a derivative. The floor that outlives the arithmetic is the ladder’s own ceiling, and The band that closes with the order is the cascade’s version of the same accounting on the axis this page uses.

What is checked

Four assertions, and all four are written to hold at every tolerance the slider offers rather than at the one this essay quotes.

That the ladder’s floor is below the cascade’s at every order — the least departure any impedance level allows, which is a property of the structure and not of a tolerance.

That its band is wider at every one of them, although its order is two higher in every pair.

That the cascade’s floor rises with order more than twice as fast as the ladder’s, which is the second-order sensitivity in a quantity that is not a tolerance.

And that both bands narrow with order rather than one of them being independent of it — because the claim being made is that one shuts and the other does not shut as fast, and an assertion that let the ladder’s band be constant would have allowed a bug that ignored the order entirely.

Where the ladder’s own band ends

This essay stops at order nine, and the reason is not that the band is still open there. The floor that outlives the arithmetic pushes the same measurement further and finds two quite different things stopping it. The structural one is a floor rather than a band: a doubly terminated ladder’s least achievable departure rises as the 1.06 power of the order and would not reach a tenth of a decibel until order thirty-two, which is an order nobody builds. The other is arithmetic, and it arrives first — the continued fraction that turns a reflection polynomial into element values loses its leading coefficients to cancellation and stalls at order nine for a Butterworth and fourteen for a Chebyshev.

So the honest boundary on the ladder column of this comparison is a property of the synthesis rather than of the filter, and it is worth knowing which, because the two have different remedies. The digits the arithmetic did not have is the essay that tells them apart, by the one experiment that can: an ill-conditioned problem stays ill-conditioned however many digits are used and a well-conditioned one computed badly gets better. Its verdict is that the ladder synthesis’s loss is mostly the data’s, and that doubling the digits makes it worse — so the stall is not waiting on a better implementation.

Which leaves the comparison in this essay standing and bounds it. Between order two and order nine the cascade’s band shuts and the ladder’s does not; past order nine there is no ladder to compare with, for a reason that has nothing to do with impedance levels, tolerances or parasitics.

Part 3 on impedance scaling

One argument about Impedance scaling, and one of 4 essays on it so far, each part numbered by how much of the idea it assumes. What sits either side of it:

An order-6 cascade at 10 kΩ: a band 10× wide. computed by solving, not by drawing. A 10.0 kHz unity-gain Butterworth of order 6, 3 Sallen-Key sections in cascade, realised at seven impedance levels three decades apart, with every resistance multiplied and every capacitance divided by the same factor. The design is exactly invariant: the solved magnitudes agree to 4.0e-15 of each other across all seven and every frequency in the passband, which is the last bits of a double rather than a good approximation. The realisation is not, because two of its quantities do not scale — 50 Ω of amplifier output resistance and 2 pF of stray capacitance to everywhere — and they bind at opposite ends. Inside 0.1 dB the band runs 316 Ω to 3.16 kΩ, with the least departure of 0.0763 dB at 1000 Ω; at this setting it is 0.182 dB at 12.9 kHz. The consideration that does not appear on either axis is noise: the resistors' density goes as √R, so across the six decades drawn the noise moves by 1000 times and points at the low end of the band. The band that closes with the order Part 2 — One Sallen-Key section is inside a tenth of a decibel of its own design over three decades of impedance level, bounded below by fifty ohms of amplifier output resistance and above by two picofarads of stray. Give it three more sections and the band is one decade; give it four and there is no impedance level at all that meets a tenth of a decibel. Every section brings three more nodes each carrying their own stray and one more amplifier carrying its own output resistance, so the floor rises with the order until it crosses the tolerance — a boundary in the order rather than in the impedance. The floor rises as the 1.06 power of the order, and the arithmetic gives out first. computed by solving, not by drawing. The least departure a doubly terminated ladder can achieve at any impedance level, against order — the floor, which is what is left when the tolerance the band is drawn at is taken away. It rises from 0.0087 dB at order three to 0.0389 dB at order 13, as the 1.06 power of the order, so the order at which it reaches the 0.1 dB the band is drawn at is near 32. Drawn over it are the same networks with one imperfection removed at a time, and they do not add: the winding loss alone leaves a larger departure than the complete realisation does, because the track resistance is in series with the load and lifts the passband exactly where the winding loss droops it. The stray capacitance contributes nothing at all, since the level that minimises the departure is a few ohms. What ends the sweep is neither: the continued-fraction synthesis loses its leading coefficients to cancellation and stalls at order 9 for a Butterworth and 14 for a Chebyshev. The floor that outlives the arithmetic Part 4 — A band of impedance levels is what a tolerance allows; a floor is what a structure has. Measured at six orders, a doubly terminated ladder's floor rises as the 1.06 power of the order and would not reach a tenth of a decibel until order thirty-two — an order nobody builds. What stops the sweep is neither the structure nor the parasitics: the continued fraction that turns a reflection polynomial into element values loses its leading coefficients to cancellation and stalls at order nine for a Butterworth and fourteen for a Chebyshev.

Where this goes next

An order-6 cascade at 10 kΩ: a band 10× wide. computed by solving, not by drawing. A 10.0 kHz unity-gain Butterworth of order 6, 3 Sallen-Key sections in cascade, realised at seven impedance levels three decades apart, with every resistance multiplied and every capacitance divided by the same factor. The design is exactly invariant: the solved magnitudes agree to 4.0e-15 of each other across all seven and every frequency in the passband, which is the last bits of a double rather than a good approximation. The realisation is not, because two of its quantities do not scale — 50 Ω of amplifier output resistance and 2 pF of stray capacitance to everywhere — and they bind at opposite ends. Inside 0.1 dB the band runs 316 Ω to 3.16 kΩ, with the least departure of 0.0763 dB at 1000 Ω; at this setting it is 0.182 dB at 12.9 kHz. The consideration that does not appear on either axis is noise: the resistors' density goes as √R, so across the six decades drawn the noise moves by 1000 times and points at the low end of the band. The band that closes with the order One Sallen-Key section is inside a tenth of a decibel of its own design over three decades of impedance level, bounded below by fifty ohms of amplifier output resistance and above by two picofarads of stray. Give it three more sections and the band is one decade; give it four and there is no impedance level at all that meets a tenth of a decibel. Every section brings three more nodes each carrying their own stray and one more amplifier carrying its own output resistance, so the floor rises with the order until it crosses the tolerance — a boundary in the order rather than in the impedance. One per cent on one component, in two realisations of the same order-5 filter. computed by solving, not by drawing. The two realisations agree to 3e-14 dB before anything is moved. Moving each element in turn by 1%, the worst deviation at the 2 ripple peaks is 0.1621 dB for the cascade and 0.0030 dB for the LC ladder — 54 times smaller. Across the whole passband the two are within 3% of each other, because near the band edge both are dominated by the response's own steepness rather than by the realisation. A ladder is not a cascade The same fifth-order Chebyshev, built two ways. A one per cent component moves the buffered cascade's passband by 0.162 dB at the ripple peaks and the doubly terminated LC ladder's by 0.003 dB — fifty-four times less. The number is not the claim: the claim is the exponent. Fitted over two decades of tolerance the cascade's error grows as the 0.99 power and the LC ladder's as the 2.00 power, because at maximum power transfer the response is stationary in every element it contains. One Sallen-Key design at 10 kΩ, and the band of impedance levels it survives. computed by solving, not by drawing. A 10.0 kHz unity-gain Sallen-Key section realised at seven impedance levels three decades apart, with every resistance multiplied and every capacitance divided by the same factor. The design is exactly invariant: the solved magnitudes agree to 1.0e-15 of each other across all seven and every frequency in the passband, which is the last bits of a double rather than a good approximation. The realisation is not, because two of its quantities do not scale — 50 Ω of amplifier output resistance and 2 pF of stray capacitance to everywhere — and they bind at opposite ends. Inside 0.1 dB the band runs 31.6 Ω to 31.6 kΩ, with the least departure of 0.0133 dB at 1000 Ω; at this setting it is 0.036 dB at 20.0 kHz. The consideration that does not appear on either axis is noise: the resistors' density goes as √R, so across the six decades drawn the noise moves by 1000 times and points at the low end of the band. The same filter a thousand times larger Multiply every resistance by a thousand and divide every capacitance by a thousand and the response does not change — not approximately, but to a part in ten to the fifteenth, which is the last bits of a double. So a designer has a free parameter that the design says nothing about, and what decides it is the two quantities that refuse to scale: fifty ohms of amplifier output resistance at one end and two picofarads of stray at the other. Between them the realisation survives over three decades of impedance level and nowhere else.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down, the 8 sharing most with it of 11.

The objects named here

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

Component sensitivityDesign tradeoffImpedance scalingModel rangeParasiticsQuality factorRealisation