Two parasitics, and the resonance neither of them has

The Q the components allow derives a ceiling and states it in one line: the losses sit in series around the loop, so the reciprocals of the component quality factors add, and the total is therefore below the smallest of them. A coil of 79.06 beside a capacitor of 1 581.1 gives a resonator of 75.29. The worst component decides and the best one cannot help.

That essay also names, in a paragraph about why it measures rather than evaluates, the thing it did not do. Both of its components carry a second reactance as well as a resistance — the coil’s turn-to-turn capacitance sits across the coil, the capacitor’s lead and plate inductance sits in series with it — and both were left at values small enough to do nothing. The claim made about leaving them there is that the measurement would not care: golden section finds whatever maximum is there, bisection finds whatever half-power points are there, and the division is the same division whether the network has three elements or thirty.

Switched on together, both halves of that turn out to be worth measuring. The ceiling is passed rather than approached, and the bisection finds half-power points that are not the ones it is looking for.

What is being solved #

One loop: a source, the coil as an inductance and a winding resistance with a capacitance across the pair, and the capacitor as a capacitance, a series resistance and a series inductance. Six elements rather than three. The current round it is read as the voltage across a sense resistance of a nanoohm, which is what a network answers asked once at each of several hundred frequencies and nothing more.

Beside that solve there is a second reading of the same six values, and it is not the reciprocal sum. The loop’s impedance is written out directly — the coil in parallel with its own capacitance, plus the capacitor’s three elements in series — which gives a reactance and a resistance separately at every frequency. A series resonance is a zero of the first. Its quality factor is

$$Q = \frac{\omega_0}{2R(\omega_0)}\frac{dX}{d\omega}\bigg|_{\omega_0}$$

which collapses to ωL/R for a plain three-element loop, because dX/dω is exactly 2L there. The general form is what this essay needs, since the whole subject of it is a loop whose reactance is not ωL − 1/ωC, and a quality factor read off such a loop as reactance over resistance is wrong by whatever the parasitic did to the slope. The two routes share the six element values and nothing else: one is a search on a solved network, the other is a derivative of an algebraic expression.

The capacitor’s inductance goes the wrong way #

Take the coil’s capacitance out and give the capacitor an inductance of its own. Nothing about the arrangement is unusual — a capacitor’s terminals, plates and internal connection are a loop of wire like any other, which is the whole content of the capacitor that is an inductor — and the only question is what that inductance does when the part is inside a tuned circuit rather than alone.

The result is not a penalty. The quality factor rises, past the ceiling the coil’s own quality is supposed to set, and it does so along a curve with a closed form.

The mechanism is a single sentence and it is the reason the reciprocal sum has nothing to say. A capacitor’s series inductance is loss-free reactance placed in the loop. It is in series with the coil’s inductance and it adds to it, so the loop’s total reactance grows while its total resistance does not move at all; the resonance falls from 5 032.92 Hz to 4 109.36 with half a millihenry of it, and the stored energy per cycle grows faster than the dissipation.

That gives the loop’s quality factor in closed form as √((L+ESL)/C) over DCR+ESR, and it gives something sharper as a ratio. The resonator’s Q divided by its inductor’s own is

$$\frac{Q}{Q_L} = \left(1 + \frac{\mathrm{ESL}}{L}\right)\frac{\mathrm{DCR}}{\mathrm{DCR}+\mathrm{ESR}}$$

which is above one exactly when ESL/L exceeds ESR/DCR. There is no component value in that condition — it is a ratio of ratios, and it says the resonator is sharper than its own coil whenever the capacitor contributes a larger share of the loop’s inductance than of its resistance. At the default parts that crossing is at ESL/L = 5.00 per cent, and the solved network puts the resonator at 77.15167 against a coil of 77.15168 there. At half a millihenry the ratio is 10/7 exactly: 92.21 against 64.55, with the reciprocal sum 47.6 per cent below the measurement.

The other parasitic breaks the same ceiling and hides it better #

The coil’s own capacitance sits across the coil rather than in the loop, which is a different circuit and not the mirror image of the first. It magnifies both of the things the coil presents. The resistance it shows at the terminals goes up because the shunt current and the winding current are no longer the same current; the reactance goes up because the pair is on its way to a parallel resonance; and the two do not go up by the same amount.

The arithmetic behind that gap is worth writing out, because it is what decides which ceiling holds. With half a microfarad across the coil, at the resonance of 4.11 kHz, the coil’s terminal resistance is 2.2501 times its winding resistance and the loop’s reactance slope is 2.2496 times the 2L a three-element loop would have. Those two numbers are within a part in four thousand of each other, and the quality factor is one divided by the other, so almost all of the magnification cancels and the loop’s Q falls only from 75.292 to 63.135.

The coil’s own quality factor does not have that cancellation available. It is a reactance over a resistance rather than a slope over a resistance, and the reactance is up by only 1.4998 against the resistance’s 2.2501 — so the coil reads 43.028 where its nameplate ωL/DCR says 64.55, a fall of a third. A loop can be sharper than its own coil for this reason as well, and it is a quieter reason than the first: nothing has been added to the loop and no reactance has moved anywhere, the coil’s own reading has simply stopped being the right thing to divide.

That leaves the reciprocal sum in a peculiar position. Fed the coil’s honest reading of 43.028 it predicts 42.09 against a measured 63.14 — 33.3 per cent low, which is the same order as the capacitor’s inductance managed. Fed the nameplate ωL/DCR instead, which is what the machinery here computes, it predicts 62.471 and lands within 1.06 per cent of the answer. The expression that looks right is right by cancellation: the nameplate reading is too optimistic about the coil by exactly the factor the slope is too optimistic about the loop, and the two errors are the same size because they are the same term.

This is the parasitic that would be tested first, because the inductor that is a capacitor is the more famous of the two failures and a coil’s self-resonance is a number data sheets print. It is also the one that gives no warning, and now for two reasons rather than one — the sum survives it numerically, and the component reading that would have exposed the survival is not the one anybody feeds in.

What the two of them build together #

Put both in. The loop now has a reactance that crosses zero twice, and a picture with three features in it where the rung below has one.

The third feature is the one worth naming, because neither parasitic produces it alone. A capacitor’s series inductance on its own simply adds to the coil’s, as the closed form above says, and a loop with one inductance and one capacitance has one resonance. A coil’s capacitance on its own gives a null and nothing above it, because above the null the loop is two capacitances in series and two capacitances do not resonate.

Together they do. Above the coil’s own self-resonance the coil is a capacitance, that capacitance is in series with the tuning capacitor, and the capacitor’s series inductance resonates against the pair. The frequency is therefore 1/2π√(ESL·Cpar·C/(Cpar+C)), and at these values the series pair is 333.3 nF.

That last figure carries its own boundary, and it is the shape every model has an edge collects. The approximation is a statement about being far above the coil’s self-resonance, which here is 7.12 kHz. When the capacitor’s inductance is a microhenry the second resonance is at 275 kHz, thirty-nine times that, and the coil really is a capacitance there. When it is half a millihenry the second resonance has come down to 13.8 kHz, less than twice the null, and the coil’s residual inductance is a term that matters. The expression does not fail; it drifts, in one direction, at a rate the figure shows.

What size these two things really are #

The sweeps above run to half the partner’s value in each direction, and that is far past anything a catalogue part carries. It is worth saying where a real one sits, because the two ratios that decide everything here are ratios rather than values and a reader is entitled to know whether they are ever reached.

The condition for the first result is ESL/L above ESR/DCR, and neither side of it is a component value: it compares the capacitor’s share of the loop’s inductance with its share of the loop’s resistance. That is why the crossing on the second figure moves by a factor of twenty — from 1.25 per cent of the coil’s inductance at 5 mΩ of capacitor resistance to 25.00 at 100 mΩ — while no inductance in the circuit changes at all. A low-loss capacitor is the one that breaks the ceiling soonest, which is the opposite of what a ceiling is normally sensitive to, and it is worth noticing that improving the capacitor moves the crossing down.

The condition for the second result is that both parasitics are non-zero, which is met always, because the alternative is a component that does not exist. What varies is where the resonance it produces lands, and that is set by the coil’s own self-resonance and by nothing in the tuned circuit. The slider on the loop figure walks it: 50 nF across the coil puts the null at 22.5 kHz and the second resonance at 76.2, and 500 nF puts them at 7.12 and 28.2 — the null down by 3.2 and the second resonance by 2.7, while the first resonance falls only from 4.70 kHz to 4.02, which is 14.5 per cent. That ratio of movements is the practical signature: a feature that moves a great deal when the tuned circuit has hardly moved at all belongs to the parasitics, and it is how to tell one apart from anything the design put there.

Where the measurement stops being a measurement #

The rung below reports its quality factor by finding the peak, bisecting outward for the two frequencies at which the current is one part in √2 of it, and dividing. That procedure assumes the current falls monotonically away from the peak on both sides. With one parasitic it does. With two it does not, and the failure is not subtle.

The band above the half-power level is two intervals, so the width of it is not a number. That is a statement about the circuit and not about the search: a half-power bandwidth is the width of one interval, and there are two here because there are two resonances and the taller one is the second. The quantity a bisection outward from the peak is looking for does not exist, so what a bisection returns instead is decided by where its probes happen to land — 0.1597 at a hundred microhenries of capacitor inductance and 65.02 at a hundred and thirty, on a circuit whose actual selectivity moved by 0.70 per cent between the two.

This is the one place where the rung below’s claim about measuring rather than evaluating needs qualifying, and the qualification is worth stating precisely because the claim is otherwise right. Golden section does find whatever maximum is there; bisection does find a frequency at which the current is one part in √2 of the peak. What neither can supply is the assumption that makes the division meaningful, which is that there is exactly one such interval. A search is not more robust than a formula when the quantity being searched for has stopped being well defined — it is less so, because a formula that no longer applies is usually visibly wrong and a search always returns something.

The reactance slope is the route that survives. It is local to the resonance it is evaluated at, it knows nothing about what the loop does two decades higher, and it agrees with the bandwidth measurement to 0.01 per cent wherever the bandwidth measurement is defined at all.

What it does not say #

It does not say the reciprocal sum is wrong. Inside the circuit it describes — two components each with one resistance and no second reactance — it is exact to the solver’s noise, and the figure at the top of this essay is that agreement. What it has is a range, and the range is set by the two ratios above rather than by a frequency: the sum is within about one per cent while the capacitor’s inductance is under a twentieth of the coil’s, and it is 47.6 per cent low at a half.

It does not say a resonator built this way is better. A quality factor of 92.21 where the reciprocal sum predicted 62.47 is a sharper resonance, and it is sharper at 4.11 kHz rather than at the 5.03 kHz the two nameplate values name — so a filter designed around it is tuned somewhere it was not meant to be, by a parasitic nobody specified, in a way that the tolerance that is not on any part is about. A selectivity that arrives from a lead length is not a selectivity anybody can hold.

And it does not repair the report an instrument of this kind gives. The verdict on which component sets the ceiling still names the inductor in every case above, including the one where the resonator measures 92.21 against an inductor allowing 64.55, because that verdict is reached by comparing the two component quality factors and the measurement is not one of them. The two ceilings it prints are still each component’s own; what has changed is that the loop’s quality factor is no longer bounded by either. Nothing that reads only the ceilings can see this, which is much the same failure as the one the edge that is a region collects — a number that is right about the object it names and silent about the circuit it is in.

What this is a case of #

Two of these essays already meet the pattern from other directions. The reactance cancelled, and the resonance it buys puts a capacitor in series with a line to cancel its inductance and finds a resonance nobody asked for; the capacitor that is not where the load is finds one between a decoupling capacitor and the inductance of the track to it. In every case the new resonance is between a reactance somebody chose and a reactance nobody drew, and in every case it is above the band the design is about and therefore outside the range anybody looked at.

What is different here is that the third resonance requires both parasitics. The two previous cases are one specified reactance meeting one parasitic; this is two parasitics meeting each other, with the specified components acting only as the coupling between them. That is why it survives every test that switches one parasitic on at a time, and it is the reason to distrust a sweep that varies one thing.

The other half of the pattern is about quality factors specifically, and the Q the amplifier decides is the sibling: there a filter’s selectivity comes from an amplifier’s closed-loop gain rather than from a component, so the two parts that appear to set it do not. A quality factor is a property of a loop and it keeps being attributed to the parts in it, which is the same mistake made twice in two fields.

The number worth carrying #

92.21 against a coil that allows 64.55, and a second resonance at 28.2 kHz that neither component has. The first happens above ESL/L = ESR/DCR and the second happens whenever both parasitics are non-zero at all.

The habit that goes with it is about how a parasitic is tested. Switching one on, measuring, switching it off and switching the next on is the obvious sweep and it is the one that misses this: each parasitic alone leaves a loop with one resonance, and one resonance is what every quantity in the rung below is defined against. The configuration worth solving is the one where every part is as imperfect as it will actually be, because the interesting object here is not either imperfection — it is the resonance between them, and it exists only in the arrangement nobody swept.

That is the same argument one solve, read four ways makes about readings and this essay makes about elements. A network with everything in it answers one question; four networks each missing something answer four easier questions, and their answers do not add up to the first.