The same filter a thousand times larger

A filter’s response is a ratio of impedances. Multiply every impedance in a network by the same factor and every ratio is unchanged, so the response is unchanged.

Resistances scale by multiplying them; capacitances scale by dividing them, since a capacitor’s impedance is $1/sC$. Do both — every resistor times $k$, every capacitor divided by $k$ — and the transfer function is identical.

Identical is a strong word and this collection tries not to use it loosely, so: over six decades of scaling and every frequency in the passband, the solved magnitudes of the same design agree to $10^{-15}$ of each other. That is the last bits of a double, and it is what “identical” means here.

Which leaves a designer holding a parameter that the design has no opinion about at all.

The two quantities that do not scale #

Everything in the design scales. Nothing in the realisation does.

The amplifier has an output resistance. Fifty ohms, roughly, and it is fifty ohms whether the network around it is made of ten-ohm resistors or ten-megohm ones. At the small end of the range it is a substantial fraction of the network it is driving, and the section’s feedback has to work around it.

The board has a capacitance to everywhere. A couple of picofarads per node, from the pad, the track and whatever is underneath. At the large end of the range it is a substantial fraction of the design’s own capacitors — at ten megohms the design capacitors are 2.25 pF and 1.13 pF, so two picofarads of stray is not a perturbation of them, it is comparable to them.

Those two bind at opposite ends, which is what makes the answer a band rather than a direction. The same shape has appeared in this collection before: a switch is a switch only between 49.5 Ω and 1.01 MΩ, bounded below by its on-resistance and above by its off-leakage, and the reason is the same one — one real number cannot be small compared with one fixed quantity and large compared with another at the same time.

The band, measured #

Sweeping the impedance level and measuring the worst departure across the passband and the corner:

Inside a tenth of a decibel the band runs from 31.6 Ω to 31.6 kΩ — three decades, and symmetric about a kilohm to the precision the sweep resolves, which is a coincidence of the two parasitics’ sizes rather than a law.

The best point is a kilohm, at 0.013 dB, and it is in the interior. That is the second time in this phase that a boundary has turned out to have an interior optimum, and it means the same thing here as it did there: the failure cannot be avoided by pushing the parameter in either direction.

The comparison is made across the passband and the corner rather than into the stopband, deliberately. Reading the deviation eighty decibels down measures the amplifier’s own residual gain — seventy decibels of “error” in a place nobody designs — which is a true statement about somewhere that does not matter, and it would have put the low end of the band in the wrong place.

The consideration that is not on either axis #

There is a third quantity, and it points the opposite way from the tidy answer.

A resistor’s noise density is $\sqrt{4kTR}$, so scaling the impedance level up by a factor scales the noise by its square root. Across the six decades drawn that is a factor of a thousand, from 55 nV to 55 µV in the section’s own noise bandwidth.

So the three considerations do not agree:

• the amplifier wants the impedance high, so that it is not driving a load comparable to its own output resistance;
• the strays want it low, so that two picofarads is negligible;
• the noise wants it low, and keeps wanting it lower with no bottom, since the density falls all the way down.

The usual instinct — keep the capacitors small, they are the expensive part — is the one that loses on two of the three. Small capacitors mean a high impedance level, which means the strays matter and the noise is large. A kilohm-scale design has 22 nF capacitors, which are neither small nor expensive, and it is at the bottom of the deviation curve and 30 times quieter than the megohm version.

Why the invariance is exact and not approximate #

It is worth being precise about what kind of statement the invariance is, because “the response is unchanged” gets said about several things that are only nearly true.

Every element’s contribution to the nodal matrix is an admittance. Scaling every resistance by $k$ and every capacitance by $1/k$ multiplies every admittance in the matrix by exactly $1/k$, at every frequency, with no approximation anywhere — a capacitor’s admittance $sC$ and a resistor’s $1/R$ pick up the same factor. So the matrix is $1/k$ times what it was.

A transfer function is a ratio of two determinants of that matrix with one column replaced, and both determinants pick up $k^{-n}$ for the same $n$. The factor cancels identically.

Which is why the measured agreement is $10^{-15}$ rather than, say, $10^{-6}$: there is no approximation being made, only floating-point arithmetic being done in a different order. The number is a measurement of the arithmetic, exactly as it was for reciprocity.

What breaks it at each end, in detail #

At the low end it is the amplifier driving its own feedback network. A unity-gain Sallen-Key’s amplifier drives the top of the first capacitor, and with ten-ohm resistors that capacitor’s branch is a low impedance. The fifty ohms in series with the ideal output is then a divider, the feedback cannot correct for what it cannot see beyond its own output node, and the section’s Q and corner both move. It shows up first as a level error and then as a shape error as the resistors fall further.

At the high end it is capacitance to ground. Two picofarads on each node adds directly to whichever design capacitor shares that node, so the section becomes a different section — one with the wrong Q and the wrong corner — and, worse, the amount added depends on the layout rather than on the design. Two builds of the same schematic then have different responses, which is the failure that is hardest to diagnose because the schematic is not wrong.

The gain–bandwidth product is a third mechanism at the low end and is not what binds there: at ten megahertz it is far above the ten-kilohertz corner, and its effect on the passband is smaller than the output resistance’s at every level swept.

What the section is, and why this one #

The section drawn is a unity-gain Sallen-Key low-pass with equal resistors: two resistors in series from the input, a capacitor from their junction to the output, a capacitor from the amplifier’s non-inverting input to ground, and the amplifier wired as a follower. Its natural frequency is $1/(R\sqrt{C_1C_2})$ and its Q is $\tfrac12\sqrt{C_1/C_2}$, so for a given corner and Q the two capacitors are fixed once the resistor is chosen — which is the free parameter this essay is about, arriving in the design equations as an underdetermined system rather than as an afterthought.

The equal-resistor form is chosen because it makes the free parameter a single number. The other common arrangement fixes the capacitors equal instead and puts the freedom in a resistor ratio, which scales identically and is harder to draw.

One property of this section matters for what follows: at direct current the amplifier’s output is connected to the input through the first capacitor’s branch, so the amplifier is inside the network rather than merely terminating it. That is why its output resistance appears in the response at all, and why a section that buffered its output separately would have a different low-end boundary.

The band is a property of the parts, not of the filter #

A natural question is whether a higher-order filter, or a different family, or a different corner frequency moves the window. Two of those do and one does not, and separating them says what the number is really about.

The corner frequency does not move it. Halving the corner doubles both capacitors, which is a scaling of exactly the kind the design is invariant under — so the window in ohms slides with the corner, and the window in impedance level relative to the reactance at the corner is fixed. The useful form of the boundary is therefore not “31.6 Ω to 31.6 kΩ” but a comparison between the design’s own impedance at its corner and the two parasitics.

The order does move it, because more sections mean more nodes, and each node carries its own two picofarads. A high-order active filter at a high impedance level accumulates stray at every stage.

And the amplifier moves it, which is the one a designer can actually buy their way out of. An output resistance of five ohms rather than fifty moves the low edge down by a decade, and costs current.

Where the free parameter is genuinely free #

It is worth saying that the invariance is not a curiosity; it is used constantly, and mostly without being named.

Integrated filters are scaled to what can be fabricated. On-chip capacitors are picofarads, so the resistors are megohms or the capacitors are switched — and a resistor made of a clock is exactly a way of realising a very large resistance at a small area, which is impedance scaling given a different implementation. A tenth of a picofarad shuttled at ten kilohertz behaves as a gigohm, which is how a filter with a one-hertz corner fits on a chip. What that technique substitutes for the invariance measured here is a different invariance: the corner becomes a capacitor ratio and a clock frequency rather than an R–C product, and a ratio is the one number a fabrication process reproduces.

The three conditions it pays for are all boundaries this essay does not have, and each binds a different quantity — a capacitor ratio under 0.0201, a signal below half the clock, and a clock below the frequency at which the charge stops arriving. The second of those is the one with no continuous analogue at all: the filter that samples finds an input at 992 kilohertz arriving at 7.8 kilohertz with the passband’s own gain, where the continuous model says it is 56 decibels down.

Discrete filters are scaled to what can be bought. A design that lands on 3.7 nF becomes one that lands on 10 nF and 3.7 kΩ, and the response is bit-for-bit the same one.

And a filter is scaled to what will drive it. Which brings the argument back to the two resistors a ladder was designed between: a doubly terminated ladder’s impedance level is not free at all, because its terminations are part of the design, and 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. The free parameter here belongs to the active realisation, and it is one of the things a designer buys when they give up the ladder’s insensitivity.

The word “free” then has a range of its own, and it shrinks with the order. The band that closes with the order sweeps exactly this measurement over section counts and finds one Sallen–Key section inside a tenth of a decibel across three decades of impedance level, four sections across one decade, and five sections across none at all — because 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. That is a boundary in the order rather than in the impedance, and it is the sharpest qualification this essay’s free parameter has. The band that does not close then puts the passive realisation beside it: the ladder has only one of the two binding quantities, because an inductor’s loss is a fixed quality factor rather than a fixed resistance and therefore scales with the design, and at order nine it still has four decades.

The number a designer should carry #

Reducing all of it to one rule: make the design’s own impedance at its corner frequency about a kilohm, or equivalently make the reactance of the smaller capacitor at the corner a few kilohms. That is a decade and a half above the amplifier’s output resistance and three orders below the reactance of two picofarads at the same frequency, and it is where the deviation curve bottoms out.

The rule has an obvious failure case, and it is the reason it is a rule of thumb rather than an answer: at a corner of ten hertz, a kilohm needs microfarads, and microfarad capacitors with the stability a filter needs are large and expensive. There the trade is made deliberately in favour of the strays, the impedance level goes to megohms, and the design has to live with a response that depends on the layout. Which is why very-low-frequency active filters are built with switched capacitors or as digital realisations instead — both of which are ways of getting a large impedance without a large resistor.

What breaks the invariance at each end #

An exact invariance broken by absolute quantities at both ends is measured in three places. The resistor that is only a resistor and The capacitor that is an inductor are the components’ own parasitics, which do not scale. The floor a circuit has is the noise, which scales as the square root of the impedance level and so is not invariant either. Three families, one corner is the response being held fixed while everything else moves.

What is checked #

Three assertions, and the first is the one everything else stands on.

The invariance, at $10^{-12}$ or better, across all seven scalings and every frequency in the comparison band. If the scaling were implemented wrongly — a factor applied to the wrong element, or applied once instead of inversely — this fails immediately and loudly, and no amount of agreement downstream would matter.

That the realisation is inside a tenth of a decibel over a band and not everywhere, which is the essay’s claim, stated as a count: at least two of the levels swept are inside and at least one is outside. A version of this figure with the parasitics omitted would pass every other check and fail this one, which is the point of writing it as a two-sided condition.

That the best level is in the interior, which is what makes it a band. If either parasitic were removed the optimum would run to one end of the sweep, and the assertion would catch it.

Why an exact invariance is worth measuring #

There is an objection to this essay that is worth answering, because it applies to several measurements in this collection: the invariance is algebra, it is provable in three lines, and checking it to a part in $10^{15}$ proves nothing about the world.

The answer is that the check is not of the algebra. It is of the implementation — the netlist builder, the scaling code, the solver and the comparison — against a statement whose exact form is known independently. A quantity that must be exactly invariant is the most sensitive detector of a mistake that exists anywhere in this site’s machinery, because there is no tolerance to hide inside: any error of any kind in scaling an element, in stamping it or in solving the result shows up as a departure many orders above the arithmetic’s floor.

That is the same reason the essays in this field check a matched ladder’s loss against exactly $20\log_{10}2$ and an all-pass section’s magnitude against exactly one. None of the three is a measurement of a circuit. All three are measurements of the instrument, made on the one kind of input whose answer is not in doubt — and they are what entitles the numbers that are in doubt, three decades of impedance level and a tenth of a decibel, to be quoted at all.

The two parasitics are the opposite kind of number and are stated as such. Fifty ohms and two picofarads are plausible values for an ordinary part and an ordinary layout, they are not measured here, and every number in this essay’s band moves with them. What does not move with them is the shape: a band with an interior optimum, bounded below by something that does not scale down and above by something that does not scale up. A reader with better parts has a wider band centred somewhere else, and the argument is unchanged.