The straight lines, and where they are not the curve

Almost every response in this subject gets sketched before it gets computed. Flat, then falling at twenty decibels a decade past the corner; a second corner, forty; a zero, back to twenty. The construction is so cheap and so useful that it has outlived the slide rule it was invented for, and it is a model like any other model here, which means it has a range and the range has a number.

The number turns out to be exact, and the exactness is worth the essay on its own.

One real pole: three decibels, and never more #

Take a resistor and a capacitor. The asymptotic sketch is a horizontal line at unity up to $f_0 = 1/2\pi RC$ and a line falling at twenty decibels a decade after it. The true magnitude is $1/\sqrt{1 + (f/f_0)^2}$.

At the corner the true value is $1/\sqrt2$ and the sketch says $1$, so the sketch is high by

$$20\log_{10}\sqrt2 = 10\log_{10}2 = 3.0103\ \text{dB}$$

which the figure asserts against the solved network to nine decimal places. That is the whole of the one-pole answer, because the error is largest at the corner and falls away on both sides.

The measured departures are 0.969 dB at twice the corner, 0.458 dB at three times, 0.170 dB at five times and 0.0432 dB at ten. So a decade away from a corner the sketch is right to a twentieth of a decibel, which is better than most measurements of a real circuit, and the whole of the approximation’s badness is packed into about a decade either side of each corner.

The symmetry, which the construction does not suggest #

Look again at those numbers, and then at the same distances on the other side.

At half the corner the error is 0.969 dB. At a third, 0.458. At a fifth, 0.170. At a tenth, 0.0432. They are the same numbers, and not approximately — the figure asserts the pairs agree to $10^{-9}$ dB across four factors.

That is not obvious from the construction. Above the corner the sketch is a sloping line and the true curve is asymptotic to it; below the corner the sketch is a horizontal line and the true curve is asymptotic to that. Two different geometries, and the same error. The reason is that the magnitude in decibels,

$$-10\log_{10}\!\left(1 + (f/f_0)^2\right),$$

minus the sketch, is an even function of $\log(f/f_0)$: substitute $f \to f_0^2/f$ and the expression maps to itself. The corner is a mirror.

It has a practical consequence that is used constantly and derived rarely. Anyone reading a gain margin or a crossover frequency off a sketch is making an error that depends only on how many corners are nearby and how far, and the “how far” is symmetric — so a corner just below the frequency of interest is exactly as bad as one the same factor above it, and the two do not cancel, they add.

The phase asymptote is exact where the magnitude one is worst #

The phase gets a straight-line construction too — zero up to a tenth of the corner, ninety degrees past ten times it, and a straight line between them on a logarithmic axis — and it behaves in exactly the opposite way.

At the corner the true phase is $-45°$ and the straight line is at its midpoint, which is $-45°$. The construction is exact there, to the last bits of the arithmetic. Its worst error is at the two ends: $5.7106°$ at a tenth of the corner and the same at ten times it, with the sign reversed.

So the two constructions have their errors in disjoint places. The magnitude sketch is worst at the corner and negligible a decade away; the phase sketch is exact at the corner and worst a decade away. That is a useful thing to hold, because a stability reading takes the magnitude from one place and the phase from another, and knowing which of the two is trustworthy where is the difference between a margin read to a degree and a margin read to ten.

A pole pair has no bound #

Everything above concerns a real pole. The moment two poles become a conjugate pair the situation changes in kind, not in degree.

The asymptotic sketch of a second-order lowpass is flat, then falling at forty decibels a decade past $f_0$, and it says nothing about the damping because the damping does not appear in either asymptote. The true magnitude at $f_0$ is exactly $Q$. So the sketch’s error at the corner is

$$20\log_{10} Q$$

which the figure asserts against the solve at every value of the slider, and which is unbounded in both directions. At $Q = 10$ the sketch is twenty decibels low. At $Q = 0.3$ it is ten and a half decibels high. There is no worst case at all: the same two straight lines describe a response that peaks by forty decibels and one that is nearly critically damped, and the construction cannot tell them apart because neither asymptote contains the information.

The best a pair allows, which is not zero #

If the error at the corner is $20\log Q$, then $Q = 1$ makes it vanish, and it is tempting to stop there. The figure does not, because the corner is not where the worst error is once $Q$ is above about 0.93.

At $Q = 1$ the corner error is exactly zero and the worst error anywhere is 1.249 dB, at 1.413 times the corner. The curve crosses the sketch at the corner and departs from it on the skirt, where the true response is still falling more slowly than forty decibels a decade.

So the honest question is the minimax: over all $Q$, what is the smallest the worst error can be made? The figure golden-sections it on the solved response and gets

$$Q^\star = 0.9152, \qquad \text{worst error } 0.7698\ \text{dB at } 1.576 f_0$$

and the two errors are equal there — $20\log_{10}(0.9152) = -0.770$ dB at the corner, and $+0.770$ dB at 1.576 times it. An equal-ripple condition, arrived at by a search that knew nothing about equal ripple.

There is therefore no damping at which a two-slope sketch of a pole pair is inside three quarters of a decibel. The Butterworth pair that every reader has met, at $Q = 0.7071$, is 3.0103 dB out at its own corner — numerically the same as one real pole, and for a different reason, which is a coincidence worth noticing and not reading anything into.

The band of damping over which the sketch is inside two decibels everywhere is $0.794 < Q < 1.129$, and inside three decibels, $0.708 < Q < 1.305$. Outside that the sketch is not a description of the response; it is a description of where the response is going.

Several corners at once, and why the errors add #

A real response has more than one corner, and the sketch is built by adding straight-line segments, so the errors add too — but only in the ordinary sense of adding, which is worth saying because there is a tempting wrong version.

Two well-separated real poles, a decade apart, give a sketch that is 3.0103 dB out at the first corner plus whatever the second corner contributes there, which by the numbers above is 0.0432 dB. So the error at the first corner is 3.054 dB rather than 3.010. A decade of separation is enough to treat the corners independently to better than a twentieth of a decibel.

Two coincident real poles are the other extreme: the sketch is flat then falling at forty decibels a decade, and the error at the corner is exactly $2 \times 3.0103 = 6.021$ dB, because the magnitudes multiply and the decibels add. That is also, exactly, $20\log_{10} Q$ for $Q = 0.5$ — which is what a coincident real pair is. The pair formula and the repeated-pole formula are the same statement, and the figure runs continuously between them as the slider crosses 0.5.

The tempting wrong version is to suppose that many corners make the sketch worse without limit. They do not, in the place it matters: on a falling skirt far from every corner the sketch converges on the response from both sides, and an eighth-order filter’s stopband asymptote is as good as a first order’s. The error lives at corners and stays there.

Where this costs something #

The asymptotic sketch is used for three things, and it is dangerous in exactly one of them.

Reading a bandwidth. Safe. The corner is read as the intersection of the two lines, which is $f_0$ by construction, and the 3.01 dB is the difference between the intersection and the half-power point rather than an error in locating it. The two definitions coincide for a single real pole and the sketch gets the right one.

Reading a roll-off. Safe, and this is what the sketch is genuinely for. Twenty decibels a decade per pole is exact asymptotically and the approach is fast.

Reading a stability margin. Not safe, and this is where the whole of the error lands. A phase margin is read at the frequency where the loop gain crosses unity, and that frequency is read off the sketch’s sloping line. With a real pole a factor of two above crossover the lines cross unity at 500 Hz where the loop crosses at 455, 0.041 of a decade apart, and the margin they report is 2.10° short; with the pole a factor of two below, 2.00 kHz against 1.88 and 1.46° short. The worst a single real pole does is with crossover on the corner itself, where the lines report 45.00° for a loop that has 51.83°. For real poles the error is always on the safe side, for a reason the margin the straight lines report makes exact — and it stops being safe the moment a pole pair peaks.

And that is with a real pole. With a resonant pair near crossover the sketch’s error is $20\log Q$ and the crossover frequency it predicts can be wrong by a factor, which is not a correction but a different circuit.

What a sketch is still for #

None of this is an argument against the construction. It is an argument for knowing its number, and the number is good news twice over.

Away from corners the sketch is excellent — a twentieth of a decibel a decade out — so for the question it is usually asked, which is “what is the slope here and roughly how much gain is left”, it is right to a precision nothing downstream can use. The subject’s habit of sketching first is well founded.

Near corners it is wrong by an amount that is completely known: 3.01 dB at a real one, $20\log Q$ at a pair, and an even function of log frequency about the corner in the first case. Those are corrections a reader can carry in their head, and carrying them turns a sketch that is wrong by an unknown amount into a sketch that is right to a fraction of a decibel.

What the construction cannot be repaired for is the resonant case, because the missing information is not a correction, it is a parameter. The two lines are the same for every $Q$, and no annotation of them recovers a number they never contained. That is the boundary in its sharpest form: the asymptotic sketch is a model of the poles’ magnitudes and not of their angles, and everything it gets wrong is a consequence of the angle.

Which is why the parameter it discards is the one three fields of this collection spend their time measuring. Where the behaviour is written down is the statement in its general form: two numbers in the complex plane contain everything a second-order circuit will ever do, their distance from the origin is the natural frequency and the cosine of their angle is the damping — so a sketch that keeps the distance and discards the angle has kept the half that every design decision is made about. The cliff before the fastest settling shows how sharply that half matters: settling time against damping is not a smooth curve with a minimum but a step, falling by a third in one increment of five thousandths, with a design a hundredth of a damping ratio to the left of the optimum settling forty-eight per cent slower and looking no different. Two measurements of one margin is the same parameter read from a loop, where it is called a phase margin and inverted out of an overshoot to a tenth of a degree.

And the frequency field’s own resonance essay puts a number on the width of the region the sketch gets wrong. Resonance, and the bandwidth it sets exactly measures the half-power bandwidth as $f_0/Q$ to every digit the arithmetic has, so the band over which the two asymptotes are seriously in error is $1/Q$ of a decade wide and $20\log Q$ deep — narrow and tall at high Q, broad and shallow at low. A sketch is therefore worst exactly where it is smallest, which is why the error is easy to overlook on a plot and expensive in a design.

The other constructions quoted instead of solving #

A straight-line construction 3.01 dB wrong at every corner it describes is one of three shortcuts this collection measures rather than repeats. A sum that is exact, and the estimate that is not is exact about one quantity and an estimate about another. The one current a constant is right at is a constant standing in for a logarithm. Where the behaviour is written down is where the curve this construction approximates is a pair of numbers, and The phase the magnitude already knows is the reading the construction handles worst.

What is checked #

Four assertions, and each is made against the solve rather than against the expression it quotes.

That the one-pole sketch is 3.0103 dB above the response at the corner, to a part in a billion, computed as $10\log_{10}2$ and compared with a network solved at that frequency.

That the error is even in log frequency about the corner — the departures at $k$ times and $1/k$ times the corner agreeing to $10^{-9}$ dB at four values of $k$. That is the claim of this essay that is least likely to be believed on sight and easiest to check.

That the pair’s corner error is $20\log Q$ at every setting of the slider, which is what makes the error unbounded rather than large.

And that no damping brings the worst error below three quarters of a decibel: the minimax found by golden section on the solved response, asserted to lie between 0.7 and 0.85 dB. That is the refusal that keeps the essay from being read as “and $Q = 1$ fixes it”.

The phase sketch, which is not a second approximation #

Every reader who draws the two-slope magnitude sketch also draws the phase sketch beside it — nought degrees a decade below the corner, 45° at it, ninety a decade above, joined by a straight line on a logarithmic axis — and treats the two as independent approximations to two independent curves. They are not independent, and the phase the magnitude already knows is why: for a minimum-phase network the phase is fixed everywhere by the magnitude, through an integral that returns 39.289 degrees at a lead network’s corner against a solved 39.289 and tracks the whole curve to two hundredths of a degree.

Which means the magnitude sketch’s error and the phase sketch’s error are one error with two consequences. A sketch that is 3.0103 decibels high at the corner is a sketch of a magnitude curve that does not exist, and the phase belonging to the curve it does sketch is not the phase belonging to the real one. The two cannot be corrected separately, and improving the magnitude sketch — by rounding the corner, say — implies a particular improvement in the phase sketch whether or not anybody draws it.

That matters wherever the sketch is used to reason about stability rather than about response, which is most of the time. A phase margin is read at the frequency where the magnitude passes unity, so a sketch used for that purpose is being asked for both curves at once, at a frequency the sketch places by its own asymptotes. What is left at crossover is the reason the collection cuts the loop and measures instead: the definition is computable where the sketch is three decibels and some tens of degrees adrift, and the two errors reinforce rather than cancel — the sketch puts the crossover at the wrong frequency and reports the wrong phase there.

The pole pair is worse in the same way and for the same reason. Its magnitude error is $20\log Q$ and therefore unbounded, so its implied phase error is unbounded too, and the phase of a resonant pair turns through 180° over a band whose width is $f_0/Q$ — resonance, and the bandwidth it sets exactly measures that width as exact to every digit the arithmetic has. A sketch that draws the phase transition over two decades is describing a pair with a quality factor of about a third, whatever the magnitude sketch above it says.