Circuit Response
every model has an edge, and the figure carries it
Frequency, which is the same solve

Resonance, and the bandwidth it sets exactly

The half-power bandwidth of a resonant circuit is f₀/Q — not approximately, but to every digit the arithmetic has, which is rare enough to be worth checking. What is not exact, and is drawn as though it were, is the idea that the band sits centred on the resonance. At a quality factor of one its middle is twelve per cent above.
13 min read 4 figures Two routes to a numberThe constants decideEvery model has an edge

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.

A resonant circuit of Q = 8, and its measured bandwidthThe half-power points are 1.50 kHz and 1.69 kHz, a bandwidth of 198.9 Hz. The components predict f₀/Q = 198.9 Hz. They differ by 0.000%.00.200.400.600.8011001k10kfrequency (hertz)fraction of the source across the resistorhalf the power198.9 Hz measuredresonance 1.59 kHzsolved, then checked — half-power points by bisectionf₀/Q predicts 198.9 Hz — exactly
Fig. 1 The fraction of the source appearing across the resistor of a series resonant circuit, against frequency. At resonance it is all of it. The marked span is the half-power band, found by bisecting the solved response rather than read off a formula. The slider is the quality factor, and the width tracks f₀/Q at every setting to every digit the arithmetic carries.

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:

Quality factor Arithmetic centre, relative to resonance
1 11.80% above
2 3.08% above
4 0.78% above
8 0.20% above
16 0.05% above
64 0.00% above

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.

Impedance of a series RLC of Q = 4, measured by driving itOne ampere is forced into the terminals at each frequency and the resulting voltage is the impedance. The minimum is 7.91 Ω at 5.03 kHz.1101001k10k1001k10k100k1Mfrequency (hertz)impedance magnitude (ohms)reactances cancel at 5.03 kHz7.91 Ωsolved, then checked — one ampere in, 201 frequenciesnot a component value: what the pair does
Fig. 2 The impedance of a series resonant circuit, measured by forcing one ampere in at each frequency and reading the voltage. The V of the minimum here is the peak of the previous figure turned upside down: same circuit, same solve, different quantity read off it, with the depth of the minimum set by the same resistance.

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.

A second-order step at ζ = 0.3Overshoot measured off the curve is 37.2%, and it settles inside 2% after 1.12 ms. Inverting the standard relation on that overshoot returns a damping ratio of 0.300 against the 0.3 the components were built for.00.50011.5001234time (milliseconds)output (volts), for a 1 V step inthe final value37.2% oversolved, then checked — overshoot read off the curvesettles inside 2% after 1.12 ms
Fig. 3 The same statement from the other side. A step applied to a second-order circuit, with the ringing that follows: a lightly damped circuit — a sharp one — rings for many cycles, and a heavily damped one settles almost at once and has almost no resonance to speak of. Damping ratio and quality factor are reciprocals of each other, up to a factor of two.

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 poles at ζ = 0.3, recovered from the matrixThe poles are at -477.4 ± j1518 hertz. Their distance from the origin is the natural frequency to six digits; the cosine of their angle from the negative real axis is the damping ratio. The step response beside them follows.the step this produces00.50011.5001234σjωζ = 0.3000ω₀ = 1592 Hzsolved, then checked — poles by rooting the determinantnatural frequency recovered to 6 digits
Fig. 4 The same resonance as a pair of poles. The quality factor is the angle: the cosine of the angle from the negative real axis is the damping ratio, and the damping ratio is 1/(2Q). Everything in this essay about width and about settling time is one statement about how far those two points sit from the imaginary axis.

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 this goes next

realimaginaryacross Racross Lacross Cthe source, 1 Vmagnitudes|v_R| = 1.000 V|v_L| = 2.128 V|v_C| = 2.128 Vsum 5.255 Vvector sum 1.000 Vsolved, then checked — one solve at 1.59 kHzsteady state only: 3 cycles to settle Three voltages that close on one, and the steady state they assume Kirchhoff's voltage law drawn as a polygon in the complex plane. The three element voltages of a series circuit add head to tail to the source exactly — while their magnitudes add to five times it. And the whole picture is a statement about a settled circuit, which takes a computable number of cycles to arrive. 10m1.0e+2m1101001k10k10k100k1M10M100M1Gfrequency (hertz)impedance magnitude (ohms)1/(ωC), the symbol's promise10% off above 4.69 MHzinductive above 14.5 MHz30 mΩ — the floor the resistance setssolved, then checked — the part as three elementsa capacitor below 14.5 MHz, an inductor above The capacitor that is an inductor A hundred-nanofarad capacitor follows 1/(2πfC) for four decades and then turns round and climbs. Above 14.5 MHz it is an inductor, and a decade past that its impedance is ninety-nine times what its capacitance predicts — all of it caused by about a nanohenry of lead and via that nobody chose and nobody drew. the step this produces00.50011.5001234σjωζ = 0.3000ω₀ = 1592 Hzsolved, then checked — poles by rooting the determinantnatural frequency recovered to 6 digits Where the behaviour is written down Two numbers in the complex plane contain everything a second-order circuit will ever do. Their distance from the origin is the natural frequency, the cosine of their angle is the damping — and the fastest-settling circuit is not the critically damped one, which is the case the textbooks name.