Two parts that drift together

A tolerance is an arrow read a component’s tolerance as a displacement of a resonant network’s pole pair. One per cent more resistance moved the pair of a Q = 5 series RLC low-pass round its circle, away from the axis, and the response changed at the peak and only there, by 0.99 per cent. One per cent more capacitance moved the pair straight down, parallel to the axis; that changed the response hardly at all at the peak and by +1.78 and −3.20 per cent either side of it. The inductor did both. And the capacitor’s effect grew in proportion to the quality factor while the resistor’s did not: at Q = 20 a one per cent capacitor was a ten per cent change in the response.

Every one of those displacements was one part moving alone. Real parts rarely do. Two capacitors from one reel, two resistors on one substrate, an inductor and a capacitor in one enclosure at one temperature — all of them share part of their error, and what a ratio does not see found a divider cancelling the shared part of its resistors’ errors exactly. The arrow essay’s first open question was the reactive version: an inductor and a capacitor whose values move in opposite senses keep their product fixed, and should move the pair only round its circle, trading a frequency tolerance for a height tolerance. This essay measures how completely.

Holding the product #

The network is the arrow essay’s: 10 mH, 100 nF and a resistor chosen for the quality factor, here 63.2 Ω for Q = 5, so that $\omega_0 = 1/\sqrt{LC}$ is 5.03 kHz and the upper pole sits at $-0.1 + j0.995$ of $\omega_0$. The inductor is raised by one per cent. The capacitor is changed by $(1.01)^{-t} - 1$, so that at $t = 1$ the product $LC$ is exactly what it was, at $t = 0$ the capacitor does not move, and at $t = -1$ it rises with the inductor. Each pair of netlists is solved and the response compared, exactly as the arrow essay compared single parts.

With the capacitor held, one per cent more inductance changes the response by up to −2.73 per cent at 1.115 $\omega_0$ — the slide the arrow essay drew, the response moving down the frequency axis and its steep upper flank falling. With the capacitor tracking exactly, the same one per cent of inductance changes the response by 1.00 per cent at 1.000 $\omega_0$, at the peak, and nowhere else by as much. $\omega_0$ has not moved — to the last digit of the solve — and what is left is a dip centred on the peak, the shape the resistor alone made.

The other settings fill in between. Tracking half the inductor’s drift leaves a smaller slide, with its largest change, +1.44 per cent, below the peak at 0.927 $\omega_0$. The capacitor rising with the inductor doubles the slide: −5.82 per cent at 1.100 $\omega_0$. A pair of parts can therefore make the response anything from twice as sensitive as one of them to as insensitive as a resistor, depending on nothing but whether their errors move together or against each other.

What the arrow does #

The plane says why in one picture.

The inductor alone moves the upper pole 9.90 × $10^{-4}$ $\omega_0$ towards the axis and 48.88 × $10^{-4}$ down: it raises the quality factor, because $\sigma = R/2L$ falls, and it lowers the frequency, because $\omega_0$ falls. The capacitor alone moves it 49.88 down and not at all sideways, because $\sigma$ has no capacitance in it. Add the two with the capacitor’s sign reversed — the capacitor falling as the inductor rises — and the downward parts cancel: 9.90 towards the axis and 0.99 up, a displacement of 9.95 × $10^{-4}$ $\omega_0$ that is perpendicular to the radius to the precision of the solve. That is what holding $\omega_0$ means geometrically: the pole stays on its circle $|s| = \omega_0$ and can only move round it, and near the top of the circle, where a resonant pair sits, round is sideways.

The sideways move is what the arrow essay found felt only at the peak. The response is $\omega_0^2$ over the product of the two legs from the poles to the point on the axis, and a pole moving sideways changes the short leg’s length only where that leg is itself sideways — at the peak, where the point on the axis is level with the pole. Away from the peak the short leg tilts, the sideways move has a smaller component along it, and the effect fades. Moving together, the arrows add: 9.90 towards the axis and 98.51 down, the frequency shift doubled.

The arrow essay’s figure above is the reason tracking works. It drew the change in the response per unit of displacement in each of the two directions, and the two curves are felt in different places: a move towards the axis at the peak, with a weight of $1/\sigma$, and a move along the axis at the band edges, with a weight of $1/2\sigma$. Tracking leaves the first and removes the second.

The amplification it removes #

The arrow essay’s most consequential number was that a capacitor’s effect grows with Q while a resistor’s does not. Tracking moves an inductor’s error from the first category into the second.

With the capacitor held, one per cent of inductor drift changes the response by at most 0.74 per cent at Q = 1, 2.73 at Q = 5, 5.11 at Q = 10 and 10.20 at Q = 20. With the capacitor drifting the same way: 2.04, 5.82, 10.36 and 20.66. With it tracking: 1.00 per cent at every quality factor in the sweep, to two decimal places.

The reason is the length of the short leg. A frequency shift moves the pole by a fixed fraction of $\omega_0$, and the short leg it is measured against is $\sigma = \omega_0/2Q$ long, so the response’s change is the shift times $2Q$. The tracked displacement is not a shift; it is a change in $\sigma$ itself, a one per cent change in the short leg whatever its length, and so a one per cent change in the response at the peak whatever the quality factor. A Q = 20 filter whose inductor and capacitor track has the tolerance of its resistor. One whose capacitor and inductor drift the same way has twice the tolerance of either part, and at Q = 20 that is a fifth of the response for one per cent of drift.

Temperature, which makes parts drift together anyway #

The most common reason two parts move together is that they share a temperature, and each has a temperature coefficient. Ceramic capacitors are sold with specified coefficients — zero within a few tens of parts per million per kelvin, or deliberately negative, minus 150, minus 750 — precisely so that they can be paired with a part that drifts the other way. Inductors, especially those with magnetic cores, usually drift upward with temperature by an amount that depends on the material.

With an inductor of +150 parts per million per kelvin and a capacitor of zero coefficient, the Q = 20 network’s response at its lower half-power frequency moves by up to 9.13 per cent over −40 to 85 °C, and at the peak by 2.30. With a capacitor of −150 ppm/K the band edge moves by 0.565 per cent and the peak by 0.970 — the band edge sixteen times steadier. The peak’s change is the inductor’s drift in $\sigma = R/2L$, the one part of its effect that no capacitor can cancel, and it is now the larger of the two; with the unmatched capacitor it was buried under the frequency shift.

The arithmetic behind the match is the arrow essay’s. A temperature change $\Delta T$ moves the inductor by $\alpha_L\Delta T$ and the capacitor by $\alpha_C\Delta T$, and $\omega_0$ by half their sum with the sign reversed; choosing $\alpha_C = -\alpha_L$ makes the sum zero. What the figure adds is the size of what is left. From 25 °C down to −40 the inductor moves by 0.98 per cent, and a 0.98 per cent change in $\sigma$ is a 0.97 per cent change at the peak — exactly the resistor-like dip the hero figure drew, scaled to the drift. The band edge’s remaining 0.565 per cent is the same change in $\sigma$ seen from the flank, where a sideways move of the pole is felt at a little over half its weight. A designer who matches the coefficients has turned a band-edge problem of nine per cent into a peak-height problem of one, and if the specification is on the band edges, as it usually is for a filter, the match is nearly the whole of the cure.

The match is never exact in practice: a coefficient is specified with a tolerance of its own, and the inductor’s is rarely a straight line over 125 kelvin. A residual mismatch of thirty parts per million per kelvin moves $\omega_0$ by up to 0.1 per cent across the range, which at Q = 20 is about 1.9 per cent at the band edge on top of the residual above. That is still a fifth of the unmatched figure, and it is set by how well two coefficients agree rather than by how small either one is.

Correlation as a tolerance #

Temperature is one shared cause; manufacture is another, and its sharing is statistical rather than exact. Two parts from one batch have errors that are correlated by some coefficient ρ, and the population of builds a factory produces has a spread that depends on ρ as much as on either part’s tolerance. The draws here are Gaussian; the shape a tolerance does not print found that the shape inside a tolerance can move a spread by a factor of two, and a correlated pair inherits whatever shape its parts have.

Two hundred builds of the Q = 10 network, with each part’s error drawn with a standard deviation of one per cent, spread the response at the lower band edge by 6.75 per cent when the two errors are independent. Correlated at +1, the capacitor’s error copying the inductor’s, the spread is 9.32 per cent. Correlated at −1, the capacitor’s error mirroring the inductor’s, it is 0.50 per cent — fourteen times smaller than independent parts of the same tolerance. The same two hundred pairs of draws are used at every correlation, so the curve’s shape is the correlation’s and not the sampling’s.

The curve is first order. The band edge moves by $g_L\delta L + g_C\delta C$, with the two sensitivities solved from small changes in each part — 4.738 and 4.263 per unit — so its spread is one per cent times $\sqrt{g_L^2 + g_C^2 + 2\rho\,g_Lg_C}$, and the builds sit on it. The two sensitivities are nearly equal because both parts move the frequency, which dominates at the band edge; their difference, 0.475, is the inductor’s extra effect through $\sigma$. At ρ = −1 the shared part cancels and only that difference survives, which is why the spread there is not zero but half a per cent.

That turns a correlation into something a designer can specify. A filter that must hold its band edge to two per cent at Q = 10 cannot do it with independent one per cent parts, whose band edge spreads by nearly seven, but can do it with the same parts if their errors are correlated at about −0.9 — which is what a matched pair, or a capacitor selected against its inductor, provides. The tolerance on the band edge is a tolerance on the correlation as much as on either part. The same distinction runs through a resistor divider: the small leg of a large ratio found that two parts of different values share a process error only if they share it by the same fraction, and an inductor and a capacitor, made by different processes, share it by whatever fraction a selection imposes.

What tracking cannot do #

Tracking cancels whatever the two parts share and nothing else. It cannot touch the resistor, which sets $\sigma$ by itself, and it cannot touch the part of the inductor’s drift that acts through $\sigma$, because the capacitor is not in $\sigma$ at all. Those two are what the tracked network’s tolerance is made of, and they are the same size as any resistor’s: a one per cent change in the short leg, felt as a one per cent change in the peak.

That sets a floor that no amount of matching passes. A Q = 20 filter with perfectly tracking reactive parts and a one per cent resistor has a one per cent tolerance on its peak and almost none on its band edges; the same filter with its reactive parts drifting independently has a ten per cent tolerance on its band edges. The arrow essay’s advice — that a filter specified by its band edges should spend its money on its reactive parts — becomes more precise: it should spend it on their agreement, which can be bought more cheaply than their individual accuracy, by matching coefficients or selecting pairs.

There is also a limit on what the first-order picture says. A one per cent displacement is small against the long leg and not quite small against the short one, and at Q = 20 the short leg is a fortieth of $\omega_0$. Tracking defined by equal and opposite fractional changes is not quite tracking defined by a fixed product — $(1 + \alpha\Delta T)(1 - \alpha\Delta T)$ falls short of one by $\alpha^2\Delta T^2$ — but for coefficients of a few hundred parts per million that shortfall is a few parts in a hundred thousand, and nothing in the figures above can see it.

How the numbers were obtained #

Every response is the ratio of two solved netlists — the nominal series RLC low-pass and the same network with its parts changed — exactly as the arrow essay computed its single-part changes; the poles are the roots of the solved transfer function. Exact tracking changes the capacitor by $(1.01)^{-1} - 1$ for a one per cent inductor, which holds $LC$ to the last digit. The temperature sweep changes the inductor by +150 ppm/K and the capacitor by zero or −150 ppm/K about 25 °C and reads the response at the nominal network’s lower half-power frequency, $\omega_d - \sigma$, and at its peak. The two hundred builds draw pairs of standard normal numbers once, with a fixed seed, and form each correlation from the same pairs; the first-order law uses sensitivities solved from changes of $10^{-5}$ in each part.

What it leaves out #

More than one pair. A ladder filter has several poles and several parts in each, and its tolerance is the sum of many arrows. Whether a filter can be designed so that its parts’ dominant shared drift moves every pole round its own circle at once is the general question; the single pair is the case where it can be done with one choice.

Ageing and the shared history it brings. Parts age together as well as heat together, and an inductor’s core and a capacitor’s dielectric age differently. Tracking that holds at installation may not hold after ten years, and a correlation measured on new parts is not the correlation the filter will have.

Still open: the notch’s zero, a ladder filter’s many arrows, and a match bought by trimming #

A zero’s arrow. At the bottom of a notch the response is a short leg in the numerator, and the arrow picture applies to the zero as it does to the poles. A zero moving along the axis moves the null without filling it; one moving off the axis fills it to a depth set by how far. Whether a notch’s parts can be paired so that their shared drift moves its zeros only along the axis — keeping the null at full depth over temperature — is the tracking question asked of a numerator.

A ladder filter’s arrows. A fifth-order ladder filter has five poles, and each part moves all of them. The famous insensitivity of doubly terminated ladder filters should appear as arrows lying in directions the response does not feel, and a tracking pair in a ladder filter moves several poles at once. Whether the right pairing makes a ladder filter’s band edge as steady as this single pair’s is a question about eigenvectors of the sensitivity matrix.

A match bought by trimming. A trimmer capacitor adjusted at one temperature sets the product $LC$ there and does nothing for its drift. Whether trimming plus matched coefficients is worth more than either, and how the residual mismatch in coefficient compares with the trimming resolution, is the production version of this essay’s temperature sweep.