The Q the components allow

Resonance and its bandwidth produced one of this site’s cleanest results. The half-power bandwidth of a series resonant circuit is exactly f₀/Q at every Q tested, to the precision of the solve, and the band is not centred on the resonance — at Q = 1 its arithmetic middle sits 11.8% above. Both are statements about a circuit with three elements, and the resistance in it was the one put there on purpose.

That circuit does not exist. An inductor is a coil of wire with resistance, a capacitor has resistance in its leads and its dielectric, and both arrive with their losses whether or not a designer wants them. The question this essay answers is what those losses do to the Q that can be obtained — and the answer has a shape that is worth more than the number.

The reciprocals add #

A component’s quality factor is the ratio of the reactance it presents to the resistance it comes with — how much energy it stores per cycle against how much it dissipates. For a coil at resonance that is ωL/R, for a capacitor 1/(ωCR), and both are dimensionless numbers that describe the component rather than the circuit.

In a series resonator the losses are in series around the loop, so the resistances add, and because each Q is a reactance over a resistance the reciprocals add:

$$\frac{1}{Q} = \frac{1}{Q_L} + \frac{1}{Q_C} + \frac{1}{Q_\text{ext}}$$

That is the arithmetic of parallel resistors, and it has the consequence parallel resistors have: the total is below the smallest term. A perfect component contributes a reciprocal of zero and changes nothing; a bad one dominates completely.

For the default resonator here — a 1 mH coil with 0.4 Ω of winding resistance and a 1 µF capacitor with 20 mΩ of equivalent series resistance, resonating at 5.033 kHz — the inductor’s Q is 79.06, the capacitor’s is 1 581.1, and the resonator’s is 75.29. Replacing the capacitor with a perfect one would give 79.06: an improvement of five per cent, for a component made infinitely better.

That is the sentence the figure exists to make unavoidable. Almost all of the loss in almost every LC resonator is in the inductor, and almost all effort spent on the capacitor is wasted.

Two routes to the number #

The measured Q is read off the solved network and never from the expression above.

The resonance is located by golden-section search on the loop current — not assumed to be 1/2π√(LC), because losses move it and the only way to find out by how much is to look. The half-power points are then found by bisection either side of it, and the Q is f₀ divided by their difference. That is three searches on a solved network and no algebra at all.

The predicted Q is the reciprocal sum, evaluated from the element values at the measured resonance.

They agree to 2 × 10⁻⁹ per cent, which is the solver’s own noise floor, at every position of the slider. The two share the element values and nothing else: one is a search for a maximum and two bisections for half-power points, the other is three divisions and an addition.

The second thing the search settles is how far the losses move the resonance, and the answer is almost nowhere: the measured f₀ agrees with 1/2π√(LC) to better than a part in a million even at the lossiest setting on the slider. That is asserted rather than assumed, because it is the kind of thing a reader is entitled to be suspicious of — the losses have visibly ruined the Q and might be expected to have moved the frequency too, and they have not.

Which component decides, and when it changes hands #

The figure’s two curves are the whole argument, and the gap between them is a story about design effort.

The dashed line is $Q_L$: what the resonator would do if the coil were its only loss. The solid line is the reciprocal sum with the capacitor included. At low coil quality they lie on top of each other, because a reciprocal of 1/20 swamps a reciprocal of 1/1581. As the coil improves the two separate, and the solid curve bends away toward a horizontal asymptote at the capacitor’s own quality.

Everything to the left of the bend is coil-limited and everything to the right is capacitor-limited, and the bend is where the two reciprocals are comparable rather than where the two Qs are. That distinction matters: at a coil Q of 158 and a capacitor Q of 1 581 — a factor of ten apart — the capacitor is already costing nine per cent of the answer, which is more than most people would guess for a component ten times better than the one beside it.

The practical reading is a rule for where to spend. Improving the worst component moves the answer almost one for one; improving the best one moves it by the ratio of their reciprocals, which is usually negligible; and once the two are within a factor of about three of each other, neither can be improved usefully on its own.

What sets an inductor’s Q, and why it is the bad one #

The asymmetry in the numbers above is not an accident of this example. Inductors are worse than capacitors, essentially always, and the reason is geometric.

A capacitor stores energy in a field between two conductors separated by a dielectric, and the loss mechanisms — lead resistance, dielectric absorption — are small and getting smaller with materials work. An inductor stores energy in a field around a conductor, and to get useful inductance the conductor has to be long. Long means resistive. A coil’s Q is ωL/R where both L and R grow with the number of turns, and the ratio improves only as the wire gets thicker or the core better — both of which cost volume.

Then two things make it worse with frequency. Skin effect confines the current to the surface of the wire, so R grows as √f above a corner set by the wire diameter, and Q stops rising linearly with ω the way the simple expression suggests. And a core, if there is one, has losses of its own that grow with frequency and with flux.

None of that is in this model. The coil here is a fixed inductance with a fixed series resistance, and the honest statement is that its Q is therefore proportional to frequency, which no real coil’s is. That is the model’s boundary and it is stated here because the site’s rule requires it — the figure’s Q is right at the resonance it is measured at and increasingly optimistic above it.

The other ceiling, which is not a Q at all #

There is a second thing that limits a resonator and it does not appear in the reciprocal sum, because it is not a loss.

The capacitor that is an inductor measured it: a 100 nF capacitor with 1.2 nH of lead inductance stops being a capacitor at 14.5 MHz and is 99 times its nominal impedance a decade above that. Above its self-resonance the component is an inductor, and a resonator built from it is not the circuit that was designed.

The two limits are different in kind and it is worth being precise about the difference. The Q ceiling is a limit on how sharp the resonance is; the self-resonance is a limit on whether the component is the component. A design can trade against the first — accept a broader response, use a better coil, add gain — and cannot trade against the second at all.

What Q is, before it is a bandwidth #

The word carries three definitions that are usually introduced as though they were obviously the same thing, and this figure measures one of them. It is worth separating them, because which one is being used decides what a measurement means.

Energy stored over energy dissipated per radian. This is the definition, and it is the one that makes the reciprocal sum above obvious rather than surprising: the stored energy is shared by the whole loop, the dissipated energies add, so the reciprocals of the ratios add.

The reactance over the resistance of a component. This is the definition a datasheet uses, and it is the first one specialised to one element at one frequency. It is a property of a component and it depends on where it is measured.

The resonance over the half-power bandwidth. This is the definition the previous rung measured, and it is a property of a circuit. It is the one an instrument reports, because it is the one that can be observed without knowing anything about the components.

The three agree for a simple series resonator, and the agreement is a theorem rather than a convention. This figure uses the third — golden-section search for the peak, bisection for the half-power points, a division — and compares it with the second, summed by the first. That the comparison is meaningful at all is the content of the theorem; that it agrees to 2 × 10⁻⁹ per cent is the check that both are implemented correctly.

Where they part company is worth knowing about, because it is not exotic. A circuit with more than one resonance, a circuit whose response is not symmetric about its peak, or a component whose loss changes appreciably across the passband will give three different numbers. The previous rung already found the second of these: at Q = 1 the half-power band’s arithmetic middle sits 11.8% above the resonance, so even for this circuit the third definition needs care about which frequency is being divided.

What an external load does #

The third term in the reciprocal sum is the one a designer controls, and it is the one this essay’s figure holds at zero.

Anything connected to the resonator — a source resistance, a load, the input of the next stage — appears in the loop and adds its reciprocal like any other loss. The distinction between the unloaded Q, which is the components alone, and the loaded Q, which is what the circuit in place actually achieves, is exactly this term.

It is also the term that makes the ceiling matter. A filter designed for a loaded Q of 20 out of components allowing 75 has three quarters of its loss deliberately placed and is insensitive to the components; the same filter asked for a loaded Q of 70 is being asked for almost everything the components have, and its response then depends on the coil’s winding resistance, which is neither controlled nor stable. The unloaded Q is not a target; it is a budget, and a design that spends most of it has made the components’ tolerances into the response’s tolerances.

That is the same shape of statement a ladder is not a cascade made about filter realisations, and it is why the two essays are neighbours in a way their fields do not suggest.

Why the search rather than the formula #

There is a shorter way to write this figure. The Q of a series RLC is (1/R)√(L/C), the component Qs are two divisions, and the whole essay could be three lines of arithmetic with no solve in it at all.

It is not written that way, and the reason is the same one that runs through this collection. The closed form is a statement about a circuit with exactly three ideal elements in exactly one arrangement. The moment a real component is used it is four elements — a coil is an inductance and a resistance, a capacitor is a capacitance and a resistance — and the moment a parasitic is added it is six, and the closed form has to be rederived each time or quietly assumed to still apply.

The search does not care. Golden section on the current 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. When the capacitor’s own parallel capacitance and lead inductance are switched on — which resonatorQ accepts and this figure leaves at their vanishing defaults — nothing about the measurement changes and the closed form is no longer applicable.

That is the argument for measuring rather than tabulating, made concretely: the formula is right about the circuit it describes, the search is right about the circuit that is there, and the difference between those two only shows up when something has been added that the formula did not know about.

Where the slider stops #

The slider walks the coil’s resistance over a factor of thirty-two, from 50 mΩ to 1.6 Ω, and both ends are chosen rather than round.

At 1.6 Ω the resonator’s Q is 19.5 and the capacitor costs 1.2% of it. There is nothing wrong with the model at that end — it is simply the uninteresting regime, where one component dominates so completely that the reciprocal sum has nothing to say.

At 50 mΩ the coil’s Q is 632 and the resonator manages 452, so the capacitor is now costing 28%. That end is where the figure’s argument is strongest and also where its model is weakest, and the two facts are connected. A coil with 50 mΩ at 5 kHz is a physically large object with thick wire, and thick wire is exactly where the skin effect this model omits arrives soonest: the skin depth in copper at 5 kHz is about 0.93 mm, so a wire much thicker than two millimetres is already carrying its current in a shell rather than in its cross-section.

So the left-hand end of the figure describes a coil that could be built and whose resistance would be higher than the model says. The figure is right about the arithmetic — reciprocals add, and the capacitor becomes the ceiling once the coil is good enough — and optimistic about how easily the coil gets there. Stating that is cheaper than modelling it, and it is the boundary this site’s rule asks for.

The same arithmetic, three fields along #

The reciprocal sum in this essay is the third time this collection has met the same shape, and the recurrence is worth naming because it is not a coincidence of algebra.

The divider and its load found that a divider is 1% accurate only into a load about 49.5 times its own resistance — a statement about two conductances adding, with the larger one deciding. The floor a circuit has, in the noise field, found a source resistance at which two noise contributions are equal and their sum least. And here, two losses in series around a loop whose reciprocals add.

In each case the quantity a designer wants is bounded by a combination in which the worst participant dominates and the best cannot compensate, and in each case the practical consequence is the same: effort spent on anything but the limiting element is close to wasted, and knowing which element is limiting is most of the design work.

What differs is the direction. The divider and the resonator are both bounded from above by their worst part. The noise optimum is a genuine minimum with a best answer in the middle, because its two terms move in opposite directions rather than the same one. Telling those two situations apart — a ceiling to be approached, or a trade with an interior optimum — is worth doing before reaching for either intuition, and this site now has a measured example of each.

The rule this model stops being true under #

75.29 against 79.06 and 1 581.1. The resonator is below both components and closer to the worse one, because reciprocals add — and the whole of design practice for resonant circuits follows from that one arithmetic fact.

The frequency at which this stops applying is the coil’s own: the model here has a resistance that does not change with frequency, and the resistance that grows with frequency is where real windings are measured — already 2.05 per cent up at the frequency the rule of thumb names as the point where the effect begins, computed exactly from the Kelvin functions. So every Q quoted here is an upper bound that gets looser the further from the measured resonance it is read.

Where a ceiling on Q decides something #

The reciprocals adding is the kind of result that sounds like bookkeeping until it is the binding constraint, and three measurements in this collection are bound by it.

What actually fills a null is the most direct: a notch’s depth is its arm’s loss at twenty decibels per decade exactly, so the null a design can achieve is this essay’s ceiling read in decibels — and for any arm containing a wound inductor the ceiling is the inductor’s, by a factor of twenty, with the capacitor’s excellence worth nothing.

What the limiter charges for uses it to explain why a Wien-bridge oscillator’s distortion is what it is: harmonics are divided by whatever selectivity the loop has, and a resonator of Q = 100 would remove 48.5 decibels of third harmonic where the Wien network removes 2.5. The ceiling measured here is what says whether the resonator is buildable at audio frequencies.

And the same part written two ways is why the component figures going into the reciprocal sum need their frequencies attached: the conversion between a series loss and a parallel one is exact at one frequency and at no other, with the width of the band around it set entirely by the quality factor being converted.

The three ceilings on a resonance #

A resonator’s quality factor is bounded three ways and only one of them is on this page. This one is the components’ own loss, and it is nearly always the inductor’s. The capacitor that is an inductor and The resistor that is only a resistor are the second: above the parts’ own self-resonances there is no reactance left to divide by, so the quality factor is not merely low but undefined. The same part written two ways is where the loss this page uses stops being one number. Resonance, and the bandwidth it sets exactly is what the quality factor is spent on, and The bandwidth noise sees is the same integral asked a different question.