The gap a derivative needs

The derivative of a root computes how far a pole moves when one component moves, exactly, from two null vectors and a division. There is no approximation in it and no step size; it is the derivative rather than an estimate of one, and that essay checked it against re-rooting and against three closed forms.

Beside the formula sits a condition. The division is by a quantity that goes to zero when two poles coincide, so the derivative of a double root does not exist — the two members separate as the square root of a perturbation and a square root has no slope at the origin. That sentence was written down, and the separation between neighbouring poles was returned beside every answer so a caller could tell when the number was about a network and when it was about a coincidence.

Nothing measured it. A condition stated and never exercised is a caveat, and a caveat is what an argument looks like before it has a number in it.

The object, and why its damping is its geometry #

A series R–L–C is the whole apparatus. Ten millihenries and a thousand nanofarads put the natural frequency at 9999.689 radians a second; the resistance is $2\zeta\omega_0 L$, so the damping ratio is set directly and nothing else changes when it moves.

That is the reason for choosing it. The two poles sit at $-\zeta\omega_0 \pm j\omega_0\sqrt{1-\zeta^2}$, so the gap between them is $2\omega_0\sqrt{1-\zeta^2}$ — a damping ratio is a separation, read in the units where the behaviour is written down uses. Turning the resistance from 60.00 Ω to 199.99 Ω walks the pair along the circle of constant natural frequency until the two meet on the real axis, and every quantity in this essay is a function of where they are on that walk.

The poles are recovered rather than placed. The nodal matrix is affine in s, so its determinant is a polynomial in s, so sampling that determinant on a circle and transforming back gives the polynomial and its roots — the same machinery what a network answers is built on, with no transfer function typed in anywhere.

The reproduction, which is the instrument’s floor and not the network’s #

The first thing asked is the question the derivative is a statement about. Move the capacitor by a thousandth of a part per million, re-root the network, and compare the movement with the derivative times the change.

They agree to 1.56 parts in ten million. That is not the derivative’s accuracy — it is the re-rooting’s, and the way to tell is that the number barely moves across the slider: 2.62, 1.56 and 1.50 parts in ten million at dampings of 0.95, 0.99 and 0.999, and 1.22 in a million at 0.9999, while the sensitivities behind them rise by a factor of twenty-two over the same four. A floor that stays put while the physics moves belongs to the instrument, and it is the reason every departure quoted below is quoted where it is orders above this one.

The departure, and it is a reciprocal rather than a growth #

The sensitivity is not merely large near a collision. It is the reciprocal of the gap, and the fit says so to five figures: over the closest four dampings the exponent is −1.0000, and the product of the fractional sensitivity and the separation is 1.00000 times the natural frequency at every one of them.

That is the formula’s own denominator made visible, and nothing else in the network is doing it: the inductor’s curve lies exactly on the capacitor’s throughout, and the resistor’s runs at $2\zeta$ times both — below them at a damping of 0.3 and at twice them by the time the poles have met. A sensitivity that grew because the components were awkward would not do that. One that grows because of a single division by a single vanishing quantity does exactly that, and the next section is the arithmetic saying so.

The consequence for a tolerance table is immediate and is the wrong way round from the way such tables are read. A quality factor of 0.50505 — an ordinary, deliberately-almost-critical design — has poles 2821 radians a second apart and a capacitor sensitivity of 3.544. The same design pushed to 0.500050 has them 282.8 apart and a sensitivity of 35.36. Ten times tighter for a change in the damping ratio of a hundredth, and the two schematics differ by a resistor of 198.0 Ω against 199.97 Ω — parts a supplier would call the same value.

Why it is a reciprocal, and of what #

The formula divides by $v^{\mathsf{T}}(\partial M/\partial s)u$, and for a network with one denominator polynomial that quantity is the derivative of the polynomial at the root. A quadratic with roots $p_1$ and $p_2$ has $D'(p_1) = a(p_1 - p_2)$: the derivative of a polynomial at one of its roots is the leading coefficient times the distance to the others. The separation is therefore not merely correlated with the denominator, it is the denominator up to a constant, and every first-order statement built on that division inherits it.

For this network the whole thing closes in three lines. The denominator is $LCs^2 + RCs + 1$, so $\partial D/\partial C = Ls^2 + Rs$ and $\partial D/\partial s = 2LCs + RC$. At a root $Lp^2 + Rp$ is exactly $-1/C$, which turns the numerator into a constant; and $2Lp_1 + R$ is $L(p_1 - p_2)$, because the two roots sum to $-R/L$. What is left is

$$\frac{ds}{dC} = \frac{1}{C^2 L\,(p_1 - p_2)}$$

and the fractional form — the fractional move in the pole over the fractional move in the part — is $\omega_0$ divided by the gap, exactly, since $|p|$ is $\omega_0$ on the circle of constant natural frequency and $LC$ is $1/\omega_0^2$.

That expression never sees a matrix and the measurement never sees the expression. They agree at 9999.6894 against 9999.6894 at every damping drawn, which is a stronger claim than the fitted exponent: a fit over four points is consistent with a reciprocal, and an equality at ten dampings spanning three decades of separation is the reciprocal. The resistor’s number is the same quantity times $2\zeta$ — its own stamp arriving, and nothing else — and the inductor’s is the capacitor’s because the two reactances enter the natural frequency symmetrically. Three components, one shape, one cause.

The two exponents point in opposite directions, and that is the finding #

The derivative grows as one over the gap. The range over which the derivative is worth anything shrinks as the gap squared. Those are different powers of the same quantity, and their ratio — the absolute movement a first-order table can still describe — therefore falls as the gap itself.

Read as a design statement it is short. A one per cent capacitor is outside first order once the two poles are closer than 3194 radians a second; a tenth of a per cent one once they are closer than 1049.4. The ratio of those two boundaries is 3.044 against the square root of ten, which is what an exponent of two requires and is the check that the boundary is the second derivative’s rather than an artefact of the bisection.

Both of those separations are ordinary. 3194 radians a second is a damping ratio of 0.9872 — a quality factor of 0.5065, which is what a designer writes down when aiming at critical damping and accepting a per cent of component tolerance either way. So the case where a sensitivity table is most wanted, because the settling is most delicate there, is the case where it is answering a question about an infinitesimal. The cliff before the fastest settling is exactly that design, and its cliff sits at a damping where the gap is already closing.

This is the same shape as the boundary the derivative of a root found for a well-separated pole — first order missing a tenth of the movement at a change of 17.8 per cent — and at a damping of 0.9999 the same boundary is 72 parts per million. Three orders of magnitude tighter, with no component out of tolerance and nothing but the geometry moved.

What replaces it #

At the meeting point there is no derivative to be corrected. There is a different power law, and it is measurable to eight figures.

The coefficient is worth having because it is not a fitted constant. Two coincident roots of a quadratic whose constant term moves by $\delta$ separate by $2\omega_0\sqrt{\delta}$, and the measurement returns 0.99999960 of that over six decades of $\delta$ — which is the statement that the square-root law is exact rather than asymptotic, and that the departure at a tenth (0.9535) is the quadratic’s own next term rather than anything numerical.

The practical reading is the factor. A part that is one hundredth of a per cent out moves a simple pole by 1.1471 radians a second and splits a double one by 199.98. The ratio is $1.7437/\sqrt{\delta}$, so it grows without bound as the tolerance tightens: the better the part, the further out the first-order estimate is, which is the opposite of every intuition a tolerance table trains.

What the number is at the meeting point itself #

Ask the exact derivative for the sensitivity of a double pole and it returns a number. It is not a large number that should be read as a warning; it is a number with no content at all.

The two members of a conjugate pair are the same distance from the origin, are held by the same three components, and must therefore be held equally tightly. At a damping of exactly one the apparatus returns 2.85543 × 10⁷ for one of them and 4.22384 × 10⁷ for the other — forty-eight per cent apart, where symmetry requires equality to the last digit. Both are decided by where the rooting put two roots it cannot resolve, which is 2.9 × 10⁻⁴ radians a second apart when they are not apart at all.

So the pair is a check that costs nothing, and this collection was not making it. It is made now, at every damping the figure draws and at five more between them, to a part in ten to the ninth.

That assertion has already caught something. The null direction the whole derivative rests on is extracted by solving with the singular matrix itself and taking the direction the answer is dominated by, which works because the matrix is singular only to rounding. At two dampings in the ordinary range — 0.96 and 0.97, nothing exotic — it came out singular exactly rather than nearly, the solve declined, nothing was iterated at all, and the starting direction was handed back in place of an answer. The sensitivity reported for one member of the pair was then four orders of magnitude low and the other was right, and every assertion in the figure stayed green, because none of them compared the two. The repair is one shift of the diagonal by an ulp of the matrix’s own scale, which moves the eigenvalue and leaves the eigenvector where it was, and it is reached only after the plain solve has already refused.

That is the essay’s own subject arriving inside its instrument, and it is the reason every derivative, and the one that is zero insists on a second route for a quantity that has no obvious way of looking wrong.

The refusal, and how late it is #

There is a hard edge nearby that this collection already owns. A step response summed from residues refuses a network with a repeated pole by name, and the cliff before the fastest settling steps around that refusal and fills the point by marching instead.

The refusal fires when the closest pair is nearer than a ten-millionth of the largest pole, which on this network is a thousandth of a radian a second. The question this essay is in a position to ask is whether that is where the answer stops being right.

It is not. The residues themselves grow as one over the gap, exactly as the sensitivity does: 3.544 at a separation of 2821 radians a second, 3.58 million at a separation of 0.00279. Two exponentials of that size have to cancel to nine digits to leave a step response inside the unit interval, and the roots they sit on are located to a fraction of the gap that is itself growing. The product is an answer forty-four times too large, returned without complaint, one decade before the routine declines to answer at all.

The marched route has no separation anywhere in it and is unmoved throughout, which is what makes the comparison a measurement of the exact route rather than of both. That is the reversal worth carrying out of this section: one step, computed twice set the two routes against each other to measure the marched one, because the residue expansion is exact and the trapezoidal rule is not. Everything above is the same pair with the roles exchanged. Exact is a statement about algebra and not about arithmetic, and the route with a vanishing denominator in it is the one that fails first however exact it is on paper.

What this does not say #

It does not say the derivative is wrong. It is exact, it costs two null vectors and a division against a re-rooting’s whole determinant sample, and away from a collision its usable range is generous — 17.8 per cent for a well-separated pair. What has a range is the use of it, and the range is geometric rather than numerical.

It does not say a critically damped design is unanalysable. It says the analysis is a square root rather than a straight line, and the square-root law is exact, cheap and has a coefficient. A sensitivity table that returns a large finite number there has understated the problem and misdescribed its shape at the same time.

And it says nothing about a repeated pole in a real circuit. Two poles coincide exactly only when two component values are in exact ratio, which no part is. What a real design has is a pair a few per cent from meeting, which is precisely the region measured above and precisely the region where neither the derivative nor the square root is the whole answer.

Nor is any of this about conditioning in the usual sense. The digits the arithmetic did not have separates a problem that is ill-conditioned from one computed badly by adding digits and seeing which improves. Every exponent above would survive that test unchanged: the reciprocal and the square root are properties of the algebra, not of double precision. Only the two floors — the re-rooting’s part in ten million and the residue sum’s collapse — belong to the arithmetic.

What to do instead, which is not to stop differentiating #

The derivative is still the cheapest thing in sight and its condition is returned beside it, so the whole of the repair is to read the condition and act on the two exponents rather than on the one number.

Three moves follow from the measurements above and none of them is expensive. Read the gap first and the sensitivity second: it is returned with every answer, it costs nothing, and the ratio of the gap to the natural frequency is the only thing that decides whether the sensitivity beside it is a fact about the network or about an infinitesimal. Compare it against the tolerance rather than against a threshold, because the boundary is a square root of the tolerance — the same design that a one per cent part takes outside first order at 3194 radians a second is inside it for a tenth of a per cent part until 1049.4, and no fixed limit can express that. And where the gap has closed, re-root rather than differentiate. One perturbed solve is a determinant sample and a rooting; it is several times a derivative and a small fraction of anything a designer does afterwards, and it is the only route above that returns the right answer at every separation drawn.

The one thing that does not work is tightening the parts. The ratio between the true movement and the first-order estimate at a collision is $1.7437/\sqrt{\delta}$, so halving the tolerance makes the estimate worse by a factor of 1.41, and a design specified into a double pole with laser-trimmed components has bought itself a table that is further out than it was before. That is the shape of every result on this page: what has failed is not the accuracy of a number, it is the applicability of a whole class of statement, and no amount of the first repairs the second.

Where it points #

The registry of what a tolerance does to a filter is built on first-order sensitivities, so the boundary above is a boundary on that whole apparatus. The tolerance that is not on any part draws the distribution rather than the derivative and is unaffected; the direction a response is most sensitive to diagonalises a curvature matrix and is the natural place to ask what the second derivative does when the poles close, since the second derivative is exactly what the range measured here is set by.

And three cliffs, and where they are has three discontinuities in the ordinary range of damping, all of which move when a third pole does. A third pole is a third separation, and every number on this page is about one separation only.

The number worth carrying #

One over the gap for the sensitivity, one over the gap squared for the range it is usable over, and a square root instead of a straight line once the gap is gone.

The habit that goes with it is a question to ask of any exact derivative. A derivative that carries a condition carries a range, and the range is not the derivative’s accuracy — those are different quantities and they move in opposite directions here, which is why quoting the first says nothing about the second. The way to find the range is to make the finite change the derivative is standing in for and re-solve, and the way to know the answer means anything is to have a second route with none of the same denominators in it.