Resonance, and the bandwidth it sets exactly

An inductor’s reactance rises with frequency and a capacitor’s falls. Put them in series and there is exactly one frequency at which the two are equal, at which point they cancel — not partially, not approximately, but completely, because they are antiparallel in the complex plane. What remains is the resistance alone, and the circuit’s response there is at its extreme.

That is the whole content of resonance. Everything else in this essay is about the width of the region around it, which turns out to have one exactly true property and one that is universally drawn wrong.

Two ways to say the same thing #

There are two useful definitions of the quality factor, and they agree.

The first is the ratio of reactance to resistance at the resonant frequency. With 10 mH and 1 µF the reactance at resonance is 100 Ω; with 12.5 Ω of resistance, the quality factor is 8. The second is energy-based: 2π times the energy stored in the circuit divided by the energy dissipated per cycle. The first is easier to compute and the second is what makes the name meaningful — a resonator’s quality is how little it loses per cycle.

Both give the same number, and the reason they must is that the reactance at resonance measures how much energy sloshes back and forth per cycle while the resistance measures how much escapes.

The bandwidth, measured rather than derived #

The half-power points are the two frequencies at which the response has fallen to 1/√2 of its peak, which is half the power. In the figure they are found by bisection on the solved response — the same solve that produced the curve, asked where a specified value occurs — and they come out at 1.50 kHz and 1.69 kHz for a quality factor of 8, a width of 198.9 Hz.

The components predict f₀/Q, and f₀/Q is 1591.5/8 = 198.9 Hz.

What is worth pausing on is that those two numbers agree to every digit, at every quality factor from 1 to 64. This site is generally in the business of finding where a relation is approximate and saying by how much; here there is nothing to find. The relation is exact for this circuit, and that is unusual enough that assuming it without checking would have been a mistake of a different kind — several relations in this collection that look equally clean turn out not to be.

The reason it is exact is that the response has a particularly simple form. The magnitude depends on frequency only through the combination Q(f/f₀ − f₀/f), and setting that combination to ±1 gives the two half-power points immediately, with their difference falling out as exactly f₀/Q regardless of where they individually are. The exactness is structural.

The part that is not exact, and is drawn as though it were #

Setting that same combination to ±1 also says something the usual picture suppresses. The two half-power frequencies satisfy

f₁ · f₂ = f₀²

which makes the resonance their geometric mean, not their arithmetic one. The band is symmetric on a logarithmic axis and asymmetric on a linear one, and almost every drawing of a resonance curve is on a linear axis with the peak in the middle of the marked band.

How much does it matter? That depends entirely on the quality factor, and the figure reports it at every setting of the slider:

So the usual drawing is fine for the sharp resonators it is usually drawn for, and wrong by twelve per cent for a broad one. Twelve per cent of a centre frequency is a great deal — it is more than the whole passband of the same circuit at a quality factor of 8 — and it is exactly the region where a designer is most likely to be doing something casual, because a broad resonance feels like a forgiving one.

The general lesson is one this collection meets repeatedly: an approximation that vanishes at one end of a parameter range is usually taught as though it vanished everywhere. The high-quality case is the one that appears in textbooks, the low-quality case is the one that appears in circuits, and the difference has a number.

Three definitions of quality factor, and where they part #

The quality factor has been given two definitions above and there is a third in common use, so it is worth listing all three and saying where they disagree — because they are usually presented as interchangeable and they are not.

Reactance over resistance at resonance. The easiest to compute and the one used here. For a series circuit it is (1/R)√(L/C).

2π times stored energy over energy lost per cycle. The definition that gives the name its meaning, and identical to the first for this circuit.

Centre frequency over half-power bandwidth. The one a measurement produces, since both quantities can be read off an instrument.

For a simple series or parallel resonator all three agree exactly, which is the fact this essay measured. They part company as soon as the circuit is more complicated. A resonator with loss in several places has an energy-based quality factor that is a combination of the individual ones; a network with more than two reactive elements has a bandwidth that is not simply related to any of them; and a coupled pair of resonators has a response with two peaks, for which the third definition returns a number and the first two return two different numbers.

So the exactness measured here is a property of the simplest case, and reporting it as such rather than as a general law is the point of having measured it. The general statement that survives is much weaker and still useful: a sharper resonance stores more energy per cycle and takes longer to settle, in whatever circuit, with the constant of proportionality available only where the three definitions coincide.

Loading, which moves the quality factor and not the frequency #

A resonator is almost never used alone, and connecting anything to it changes the number that matters most.

Adding resistance in parallel with a parallel resonator — which is what a load does — lowers its quality factor, because it adds a route for energy to escape. It does not change the resonant frequency, because the frequency is set by the two reactances and the resistance is at right angles to both. The result is a broader, lower peak in the same place.

The distinction between the resonator’s own quality factor and the one it has once loaded is standard enough to have names — unloaded and loaded — and the ratio between them is the fraction of the stored energy that reaches the load rather than being wasted internally. A resonator with an unloaded quality factor of 200 loaded down to 20 delivers ninety per cent of what it stores; one with an unloaded factor of 25 loaded to 20 delivers twenty per cent.

That is the same loading argument as the first field of this collection, in its natural home. A divider’s ratio is spoiled by a load in proportion to how its resistance compares with the load’s; a resonator’s quality factor is spoiled in proportion to how its own losses compare with the load’s. In both cases the number that decides it is an impedance the simple description throws away.

The dual, and why the same figure describes both #

Everything above concerns a series circuit, where the impedance is minimum at resonance and the current is maximum. An inductor and capacitor in parallel behave in the mirror image: their impedance is maximum at resonance, and a current forced into them produces the largest voltage there.

The two are not different phenomena. The quality factor of a parallel circuit is the ratio of resistance to reactance rather than reactance to resistance — the reciprocal — but the bandwidth relation is the same f₀/Q, the geometric-mean asymmetry is the same, and the shape of the curve is the same. The only thing that changes is which quantity is being plotted.

Selectivity, and what it costs #

The practical use of a resonance is selectivity: passing one band and rejecting others. The quality factor is the knob, and the essay’s central relation says exactly what turning it buys — a factor of two in quality factor is a factor of two in narrowness.

What it costs is time, and the cost is not separately negotiable. A circuit’s bandwidth and its settling time are two readings of the same pole. The transient in a resonant circuit decays with a time constant of 2L/R, and expressed in cycles of the resonant frequency that is Q/π — so about 1.47 Q cycles for it to fall below a hundredth of its initial size.

A quality factor of 8 therefore needs about twelve cycles, which at 1.59 kHz is seven milliseconds. A crystal with a quality factor of 50,000 needs about seventy-five thousand, which is why a sharp filter cannot also be a fast one. The relation is not a limitation of any particular technology; it is the same pole, read once in frequency and once in time.

The measurement, done properly #

Since this essay’s headline is a measurement, it is worth saying how the half-power points were actually found, because the obvious method is not good enough for the claim being made.

The obvious method is to sweep, find the two samples nearest 1/√2 of the peak, and interpolate. At sixty points per decade that locates each point to about two per cent of a decade, which on a bandwidth of 199 hertz out of 1592 is an uncertainty of several hertz — larger than the difference the essay is claiming to be zero.

What is used instead is bisection on the solved response: bracket the crossing, halve repeatedly, and stop when the interval is at the level of floating-point resolution. Two hundred halvings is far more than needed and costs a few hundred matrix solves, which is nothing. The result locates each half-power point to fourteen digits, so the claim that the width is exactly f₀/Q is a claim about the first fourteen and not about the first three.

That is the general reason bisection appears throughout this collection wherever a specific value of a response is wanted — a corner frequency, a crossover, a departure point. A sweep is for drawing; a bisection is for measuring, and confusing the two is how a figure comes to assert more precision than it has.

Where the model runs out #

The resonance discussed here is an idealisation in two directions, and both boundaries are numbers.

The components are not what they are called. An inductor has winding resistance, which lowers the quality factor, and turn-to-turn capacitance, which gives it a resonance of its own — usually well above the one being designed for, but not always. A capacitor has series resistance and series inductance, and above its own self-resonant frequency it is an inductor. A resonant circuit built near either of those frequencies is not the circuit that was drawn.

Resonance itself has an upper frequency. Every statement here treats the circuit as a lumped network — three elements at three points, with the signal arriving everywhere simultaneously. That assumption has a frequency of its own, set by the physical size of the circuit, and above it a resonant circuit stops being an LC pair and becomes a resonant structure whose modes are set by its dimensions. A cavity resonator and a wire loop are the same physics at different scales.

Both boundaries are the subject of essays elsewhere in this collection, and both are the same shape of statement as the ones above: the model is excellent inside a range, the range has an edge, and the edge is computable from the model’s own parameters.

Two settings at the other end of the slider say where the usual picture becomes safe, which is the useful form of the criticism. An asymmetry of a tenth of a per cent is invisible on any plot anybody draws; an asymmetry of twelve per cent is the difference between a filter that meets its specification and one that does not.

What the quality factor is a property of #

One clarification is worth making explicitly, because the quality factor is often spoken of as a property of a component and it is a property of a circuit.

An inductor has a quality factor, and it is a statement about that inductor’s own losses at a stated frequency. A resonant circuit has one too, and it is a statement about the whole loop — the inductor’s losses, the capacitor’s, and above all whatever resistance was put there deliberately. The two are related and they are not the same number, and a resonant circuit built from a high-quality inductor and a deliberate twelve-and-a-half-ohm resistor has the quality factor the resistor gives it.

That is why the slider on this page’s figure moves a resistance rather than a component’s grade. The resonant frequency is set by the reactances and the bandwidth by the resistance, and those are two independent choices — which is the whole reason a resonant circuit is a useful design element rather than a fixed property of the parts available.

A last note on the exactness #

It is worth restating the essay’s first finding, because a reader who skims will take away the opposite of the intended lesson.

The point of measuring f₀/Q rather than quoting it was not to catch the relation out. It was to find out whether it holds, which is a different thing, and it does — exactly, at every quality factor tested, on a response solved by matrix inversion at several hundred frequencies with the half-power points located by bisection to fourteen digits.

A check that confirms is not a wasted check. The habit this collection runs on is to compute the same quantity twice by routes that share no arithmetic, and to accept the answer only when they agree; the value of the habit does not depend on the agreement failing. It happens that this particular relation is one of the few in the subject that is true without qualification, and knowing which relations those are is precisely what makes the qualifications elsewhere worth taking seriously.

Where the exact relation gets used #

$f_0/Q$ being exact rather than approximate is what allows several later measurements to treat a quality factor as a width rather than as an adjective, and three of them are worth naming because the number does different work in each.

The q the components allow is the immediate one, and it turns this essay’s exactness into a ceiling. The reciprocals of the component quality factors add, so the total sits below the smallest of them — an inductor of 79 beside a capacitor of 1 581 giving a resonator of 75 — which through the relation above means the achievable bandwidth is decided by the worst component and cannot be improved by the best.

What the limiter charges for uses it as a filter rather than as a bandwidth. An oscillator’s harmonics are divided by whatever selectivity its loop has, and a Wien network’s equivalent quality factor is a third — so the loop removes two and a half decibels of its own third harmonic where a resonator of Q = 100 would remove forty-eight and a half. The whole difference between a Wien bridge’s distortion figure and an LC oscillator’s is that number read through this relation.

And what actually fills a null uses it in the stopband, where the same arithmetic decides a depth: a notch’s null is filled by its arm’s loss at twenty decibels per decade exactly, which is the arm’s quality factor and therefore the arm’s bandwidth, restated.

The twelve per cent this essay measures — the band’s centre sitting above the resonance at Q = 1 — matters in none of those, because all three are at quality factors where the asymmetry is negligible. It matters exactly where a quality factor is low and a band is being placed by its edges rather than by its centre, which is a lightly damped supply decoupling network or a broad matching section rather than a filter.

The same two numbers, elsewhere in the collection #

A resonance is a pair of poles, and this field reads that pair four ways. Where the behaviour is written down is the pair itself, as a radius and an angle. Three voltages that close on one, and the steady state they assume is the same circuit drawn as arrows, where the element voltages exceed the source and still close. One solve, read four ways is the discipline behind both: one matrix, evaluated once, read as a magnitude, a phase, an impedance and a power. And the exactness this page is about has a boundary of its own — The capacitor that is an inductor and The resistor that is only a resistor are where the components stop being the reactances the formula assumes, and The same part written two ways is where the quality factor that sets the bandwidth stops being one number.