The corner error a filter hides in its sections

A filter is usually drawn as one set of straight lines: flat to the corner, then falling at twenty decibels a decade for every pole. It is usually built as something else. An eighth-order lowpass on a board is four second-order sections in a row, each with its own two components setting its frequency and its own ratio setting its quality factor, and each tuned, checked and toleranced on its own.

The straight lines describe the first object well. The straight lines, and where they are not the curve measured their error on a single pole — 3.0103 dB at the corner and never more — and on a pair, where it is $20\log_{10}Q$ at the corner and unbounded in both directions. The gain margin the straight lines get exactly wrong then found that for a loop the pair’s corner error is not merely an error but the correction that turns a reading into the truth.

This essay asks what the sketch says about the second object. A filter’s sketch is one set of lines; the circuit is several sections, each with a sketch of its own. The two descriptions have to agree somewhere, and where they disagree is a statement about which section a designer should worry about.

Each section’s lines, and the whole filter’s #

The filter is a Butterworth lowpass with its corner at 1.00 kHz, built as buffered sections so that each section’s response is unaffected by the next. The poles are computed from the definition — equally spaced on a semicircle — and each conjugate pair becomes a series resistor, inductor and capacitor at the quality factor $-1/(2\,\mathrm{Re}\,s_k)$ of its pole, with a real pole as a resistor and a capacitor when the order is odd. Every section shares the corner, because a Butterworth filter’s poles are all the same distance from the origin.

Each section’s straight lines are flat to 1.00 kHz and then fall at forty decibels a decade, or twenty for the real pole. The whole filter’s straight lines are flat to 1.00 kHz and then fall at twenty decibels a decade per pole. Since the sections are buffered, the whole filter’s response in decibels is the sum of the sections’, and since the lines of the whole are the lines of the sections added, the whole sketch’s error is the sections’ errors added. The figure checks that at all 241 frequencies it draws.

At the corner the four sections are out by −5.852, −4.418, −0.915 and +8.175 dB. The first three sit below their lines, as any pair with a quality factor under one does, and the fourth — at a quality factor of 2.5629 — sits above them by more than the other three combined. Added, they give −3.0103 dB. The whole filter, drawn in the heavy line, is 3.0103 dB below its sketch at the corner and closer to it everywhere else, exactly as a single real pole is.

Away from the corner the sections’ curves separate further. The high-quality section’s error peaks at +8.343 dB, a little above the corner at 1.04 kHz, which is where its response has already begun to fall while its flat asymptote has not. The three low sections’ errors are worst at the corner itself. Every curve converges on zero a decade either side. The sum of the four stays below 3.0103 dB in magnitude at every frequency drawn, and the figure checks that too.

Why the sum is fixed #

The −3.0103 dB is not a property of this filter. It is the definition of a Butterworth filter at its corner, which is the frequency at which its power response has fallen to half, at every order. Write that at the corner, section by section. A pair’s gain at its own frequency is its quality factor; a real pole’s is $1/\sqrt2$. So the magnitude of the whole filter at the corner is the product of the sections’ quality factors, times $1/\sqrt2$ if the order is odd, and it must equal $1/\sqrt2$:

$$\prod_k Q_k = \frac{1}{\sqrt2}\ \text{(even order)}, \qquad \prod_k Q_k = 1\ \text{(odd order)}.$$

That is an identity about the angles of points on a semicircle, and it is one that the sketch turns into decibels without anyone noticing. The figure checks the product to twelve places at every order it draws. The sketch of the whole is always 3.0103 dB out at the corner because the sections’ quality factors always multiply to the number that makes it so.

An odd order makes the arithmetic visible. A third-order Butterworth filter is a real pole and a pair at a quality factor of exactly one. The pair’s sketch is right at the corner, since $20\log_{10}1 = 0$, and the real pole carries all −3.0103 dB of the whole filter’s error by itself. The pair is not exact elsewhere: its worst error is +1.249 dB at 708 Hz, where its response has risen above its flat line. That is the same 1.249 dB at 0.708 of the corner that the margin the straight lines report found for the pair that made a phase margin over-report, arrived at inside a filter instead of inside a loop.

At fifth order the sections are a real pole and pairs at 0.6180 and 1.6180, whose corner errors are −4.180 and +4.180 dB: they cancel exactly, because the two quality factors are the golden ratio’s reciprocal and the golden ratio, whose product is one. The real pole again carries the whole error.

The spread, and what grows with order #

The whole sketch is 3.0103 dB out at every order. The sections are not.

At fourth order two sections are out by −5.333 and +2.323 dB. At sixth, three are out by −5.719, −3.010 and +5.719. At eighth, the spread is −5.852 to +8.175, and at tenth, where the figure stops, it is −5.913 to +10.093 dB.

The two ends of the spread behave differently, and the asymmetry is exact rather than empirical. The lowest section’s quality factor tends to one half as the order grows, because its poles approach the real axis, so its error can never be worse than $20\log_{10}\tfrac12 = -6.0206$ dB. The highest section’s poles approach the imaginary axis, and its quality factor is $1/(2\sin(\pi/2n))$, which grows without bound and approaches $n/\pi$: 3.1962 at tenth order against $10/\pi = 3.1831$. Its error grows as $20\log_{10}(n/\pi)$. Double the order and the highest section’s corner error rises by six decibels while the whole filter’s sketch does not move.

So the more sections a filter has, the better its sketch describes the whole and the worse it describes the part that dominates it. A sketch drawn to design a tenth-order filter is 3 dB out at the corner and ten decibels wrong about the section that sets the corner’s shape.

The headroom the sketch leaves out #

A corner error is a statement about a drawing, but the high section’s is also a statement about a voltage. A pair at a quality factor of 2.5629 has a gain peak of its own, and a closed form says where: $Q/\sqrt{1 - 1/4Q^2}$ at $\sqrt{1 - 1/2Q^2}$ of its frequency, which is a gain of 2.613 — +8.34 dB — at 961 Hz. A sine at 961 Hz and one volt in amplitude leaves that section at 2.61 volts, whatever the whole filter does to it afterwards, and the whole filter passes it at about −1 dB.

The sketch of the whole filter has no way to show this. Flat to the corner means unity to the corner, and the lines of every section say the same. So a designer working from lines sizes every section for unity gain, and a section built on an amplifier with three volts of swing and driven with a signal that already uses a volt and a half of it will clip at 961 Hz on a signal the filter was meant to pass untouched. The clipping is inside the filter, and at its output it appears as distortion on a tone that the whole response says is in the passband.

At tenth order the highest section’s quality factor is 3.1962 and its peak is 10.19 dB, a factor of 3.23. The internal gain grows with order at the same rate as the sketch error, because the two are nearly the same number: for a quality factor well above one the peak and the corner gain differ by the square root of $1 - 1/4Q^2$, which at 2.5629 is under two per cent. The large corner error is the headroom warning written in the only form a sketch allows, and it is the warning the whole filter’s lines erase.

That is the reason the order of the sections matters to anyone who reads it off the sections rather than the whole. Which section goes first measured an eighteen-decibel spread in the largest internal level across the twenty-four orderings of a four-section cascade, and the section responsible for the top of that spread is the one this essay singles out.

Which section’s tolerance moves the filter #

The obvious question about the high section is whether its large corner error means anything for the circuit, or only for the drawing. A sketch’s error is not a tolerance, and a section that the lines misdescribe is not, on that fact alone, a section that must be built more carefully.

The way to find out is to move one section at a time and watch the whole filter. The next figure makes one section’s natural frequency one per cent high, leaves the other three exact, and plots the change in the whole filter’s response from an octave below the corner to three octaves above it.

Far above the corner every curve converges on +0.173 dB. That is $40\log_{10}1.01$: a second-order section moved up by one per cent is, in its stopband, simply one per cent higher, and every section contributes the same forty decibels a decade. The straight lines say this and only this, since a section’s lines move with its frequency and carry nothing else.

Near the corner the sections part company. The three with quality factors below one rise towards +0.173 dB more or less smoothly, reaching +0.170, +0.172 and +0.180 dB at their worst. The section at 2.5629 overshoots to +0.312 dB at 1.23 kHz, 1.8 times what any section’s stopband shift gives, and does it in the band just past the corner where a filter’s specification is usually tightest. At tenth order the same one per cent on the highest section, at a quality factor of 3.1962, moves the filter by +0.366 dB at 1.18 kHz; at fourth order, +0.209 dB at 1.56 kHz. The overshoot grows with the section’s quality factor, as the sketch error did.

At the corner itself the order is reversed. There the four changes are +0.086, +0.086, +0.085 and +0.075 dB, and the high section is the least sensitive. To first order every section is equally sensitive there: the slope of a pair’s gain against its own frequency, taken at that frequency and in logarithmic units, is exactly one at any damping, which is the 0.0864 dB — $20\log_{10}1.01$ — that the low sections show. The high section falls short of it because its response bends sharply around its peak, and over one per cent the bend already takes back an eighth of the first-order change. Test a filter at its corner and the high section’s tolerance hides.

Five per cent makes the same picture larger: +0.835, +0.842 and +0.882 dB at worst for the low sections, +1.529 dB for the high one at 1.26 kHz, and at the corner +0.413, +0.409, +0.390 and +0.160 dB, where the high section’s change is now less than half of the others’.

A quality factor out, which every section feels alike #

The other tolerance a section has is on its quality factor, set by a ratio of components rather than by a product. It behaves in a way that looks, at first, as though the sketch had been right after all.

Every section moves the filter by +0.0864 dB at the corner, and that is where each is worst. The figure checks that the four corner changes equal $20\log_{10}1.01$ to nine places, and it holds at every order drawn. The reason is short: a pair’s gain at its own frequency is its quality factor, so one per cent more quality factor is one per cent more gain there, at any quality factor. A reader who wanted to know which section’s resistor ratio to trim more tightly gets no preference out of the corner.

So the two tolerances point different ways. A section’s quality factor matters equally in every section, at the corner, and the high section’s is no more critical than the low one’s. A section’s frequency matters most in the section with the highest quality factor, above the corner, and least in that section at the corner. Neither fact is in the sketch. The lines of the four sections are the same two slopes through the same corner; they contain neither the quality factor nor any way for a frequency tolerance to show as anything but a uniform shift.

That is the practical content of the sections’ corner errors. The section that the sketch misrepresents by +8.17 dB is the section whose frequency tolerance moves the filter by 1.8 times what any other does, and the sketch cannot show either fact because the quantity that produces both is the one it leaves out. The Q the amplifier decides found a real section’s quality factor 1.97 per cent from its design value, and where the Q comes from traces what sets it; the frequency of a section is set by a product of two parts, and a product accumulates both parts’ tolerances.

What the measurement does and does not cover #

Every section here is buffered, so the filter is a true product of its sections and the corner-error arithmetic is exact. A filter built as a doubly terminated LC ladder of inductors and capacitors is not a product of sections at all. A ladder is not a cascade found that one per cent on a component moves that LC ladder’s passband fifty-four times less than it moves a cascade of sections’, and its error grows as the square of the tolerance rather than in proportion to it; there is no section to sketch, and the comparison in this essay does not apply to it.

The filters are Butterworth. Other families put their poles at different distances from the origin, so their sections do not share a corner and the whole sketch’s error is not simply the sum of the sections’ corner errors — three families, one corner sets out how differently their poles sit. The product identity is specific: a Chebyshev filter’s quality factors do not multiply to $1/\sqrt2$, and its sketch is not 3.0103 dB out at its nominal corner.

And the tolerances are applied one at a time. A real filter has every section’s parts off at once, and the tolerance that can only take away is a reminder that tolerances combined are not tolerances added.

Still open: the zeros, the other families, and the order of the sections #

Loops and filters with zeros. Every section here is all-pole, and every error on a low section lay below its lines. An elliptic filter has zeros in its stopband and a lead-compensated loop has one near crossover, and a real zero’s response lies above its lines everywhere. The zero that lifts the lines measures what that does to a margin read off a sketch, which is the case where the safe side of every error here stops being guaranteed.

A family whose sections have different corners. A Chebyshev filter’s poles lie on an ellipse, so its sections’ natural frequencies differ and each section’s sketch has its own corner. The sections’ errors then add at different frequencies rather than at one, and the whole sketch’s worst error is no longer at a single place. Measuring where it lands, and whether it is still bounded by anything like 3 dB, is a distinct question.

Which section to put first. Which section goes first found that the order of a cascade’s sections changes its internal signal levels by eighteen decibels without changing its response. The section this essay singles out — high quality factor, large sketch error, large frequency sensitivity — is also the one whose internal peak is largest, and whether it belongs at the front or the back once its tolerance is part of the question is unmeasured here.