Five readings of one Q

Resonance, and the bandwidth it sets exactly measured a series resonator’s half-power bandwidth and found it to be $f_0/Q$ not approximately but to every digit — rare enough to be worth checking, it said — and found the band’s centre sitting above the resonance at low Q. The Q the components allow and two parasitics, and the resonance neither of them has then measured how real components set that Q and how a capacitor’s own inductance can raise it past the ceiling the components seem to impose.

All three used one reading of Q, the bandwidth. Engineers use at least four others, and use them interchangeably: the slope of the phase at resonance, the ratio of energy stored to energy lost per cycle, the rate at which free ringing dies away, and the height of the resonant peak. Each is a definition of Q in some textbook, and each is what a particular instrument measures — a network analyser, a calorimeter, an oscilloscope watching a ring-down, a Q meter reading a voltage magnification.

They agree when Q is large. This essay reads all five on one resonator, at every Q from three-quarters to fifty, and finds which agree exactly at every Q, which do not, and why the split falls where it does.

One resonator, five instruments #

The resonator is a ten-millihenry inductor and a one-microfarad capacitor in series with a resistor chosen to set Q, driven by a voltage source — a resonance at 1.592 kilohertz with a characteristic impedance of 100 ohms. It is solved as a netlist, and each reading is taken from the solution the way the corresponding instrument would take it, never from the formula it will be compared with.

The bandwidth reading bisects the current’s magnitude for the two frequencies at which it is $1/\sqrt2$ of its value at resonance and divides the resonance by their difference. The phase reading takes the slope of the current’s phase at the resonance by a central difference and multiplies by $\omega_0/2$. The energy reading drives the resonator at its resonance, takes the energy stored — which at resonance swaps between the inductor and the capacitor without changing its total, checked on the solved voltages — and divides $2\pi$ times it by the energy the resistor dissipates in one cycle. The ring-down reading builds the step response from the network’s poles, finds two successive overshoots, and takes π over the logarithm of their ratio. The peak reading golden-sections the capacitor voltage’s magnitude for its largest value.

Three are one equation #

The bandwidth, the phase slope and the energy ratio give 1.0000 at Q = 1, and to six figures at every Q from 0.75 to 50. That is not three coincidences; it is one equation read three ways.

The series resonator’s current is the source voltage over $R + j(\omega L - 1/\omega C)$, so its phase obeys $\tan\varphi = -Q(\omega/\omega_0 - \omega_0/\omega)$ exactly, at every Q. The half-power points are where the reactance equals the resistance in magnitude — where the phase is exactly ±45° — and the figure finds them at the same frequencies as the magnitude bisection, to six figures. So the bandwidth is the frequency span between the two 45° points of one phase curve, and the phase slope is the derivative of the same curve at its centre; both are properties of $\tan\varphi$ and both give Q exactly.

The energy ratio is the same statement in another unit. At resonance the stored energy is $\tfrac12 L\hat I^2$ and the energy dissipated per cycle is $\tfrac12\hat I^2 R \cdot 2\pi/\omega_0$, so their ratio times $2\pi$ is $\omega_0 L/R$, which is how Q is defined. The measurement confirms on the solved voltages that the capacitor’s peak energy equals the inductor’s at resonance, which is the condition that makes “the stored energy” one number. Off resonance it is not, and the energy definition becomes ambiguous — but at resonance, where Q is read, it is exact.

Flat magnitude, unflat delay read a filter’s group delay from its phase slope, and here that slope is $2Q/\omega_0$ at the centre: a resonator’s group delay at resonance is its Q in units of time. The envelope that leaves early used exactly this to read a band-pass’s Q off the delay of a tone burst’s envelope, and that was a sixth reading belonging to this same family.

Two are not #

The ring-down and the peak are different. At Q = 1 the ring-down reads 0.8660 and the peak 1.1547; at Q = 5, 4.975 and 5.025. They depart in opposite directions, by factors that are reciprocals of each other.

Both departures fall as $1/8Q^2$ at large Q, so each is within one per cent of Q above 3.5 and within a tenth of a per cent above 11. For most resonators that anyone calls a resonator the five readings agree to the precision of the instrument, which is why they are used interchangeably. Below Q of about three they do not, and a resonator of Q = 1 — a lightly damped servo, a loudspeaker in its box, an LC filter section — has a “Q” that depends by fifteen per cent on how it was measured.

The ring-down counts a damped cycle #

The ring-down’s error comes from the clock it uses.

A free resonator rings at its damped frequency, $\omega_d = \omega_0\sqrt{1 - 1/4Q^2}$, not at its natural frequency, because damping slows the oscillation as well as shrinking it. Successive peaks are therefore one damped period apart — 1.0610 natural periods at Q = 1.5 — and the logarithmic decrement measures the decay per damped period. The decay rate itself is $\omega_0/2Q$ whatever the damping, so the decay per damped period is larger than the decay per natural period by $\omega_0/\omega_d$, and π over the decrement is $Q\sqrt{1-1/4Q^2}$: 1.4142 at Q = 1.5, which is $\sqrt2$.

So the ring-down is not wrong; it is measuring a different thing, the damped oscillation’s own quality, which is the right number for anyone who cares how many visible cycles a ringing waveform shows before it dies. The overshoot that the cliff before the fastest settling found deciding how fast a step settles is the same decrement read once, and a designer reading a damping ratio off an overshoot is making this reading.

The peak is not at resonance #

The peak’s error has a simpler cause: the largest response is not at the resonant frequency.

The capacitor voltage at the resonance is exactly Q times the source, which is the Q meter’s definition. But the capacitor’s voltage magnitude does not peak at the resonance: the current peaks there, and the capacitor’s impedance falls with frequency, so their product peaks a little lower, at $f_0\sqrt{1 - 1/2Q^2}$. A Q meter that tunes for the largest reading rather than for the resonance therefore reads the higher peak, $Q/\sqrt{1-1/4Q^2}$, overstating Q by 15.5 per cent at Q = 1.

At Q below $1/\sqrt2$ there is no peak at all: the capacitor voltage falls monotonically from its DC value, and the peak reading at Q = 0.75 is a barely visible 1.006 at a third of the resonance. A resonator with a Q of 0.7 has no resonant peak to measure, though its bandwidth, phase and energy readings still give its Q exactly. Had the voltage been read across the inductor instead, the peak would have been above the resonance by the reciprocal factor and equally high, which is the cleanest demonstration that the peak’s position belongs to where the voltage is read and not to the resonator.

What each instrument is for #

The split decides which reading suits which purpose. A filter designer specifying a bandwidth needs the bandwidth reading, and it is exact. A designer of an oscillator or a narrow-band amplifier who needs the phase slope at resonance — which sets frequency stability and group delay — needs the phase reading, and it is exact too. A designer of a resonant power converter who needs to know how much energy circulates for a given loss needs the energy reading, and at resonance it is exact. Those three are one quantity, and a resonator’s Q in the strict sense is it.

The ring-down is the right reading for a time-domain question — how many cycles does the ringing last, how fast does a struck resonator decay — and its departure from Q is real information about how the decay looks. The peak is the right reading for the largest voltage a component must withstand, and for a capacitor in a series resonator driven near resonance that peak is higher than Q times the source by the factor the figure shows, which a component rating should include.

The trap is reading one of these and quoting it as another. A ring-down measurement of a Q = 1 resonator, quoted as its bandwidth Q, predicts a bandwidth 15 per cent too wide; a peak-tuned Q meter reading, used to rate a capacitor’s voltage, is the right number for that purpose but the wrong one for the filter’s bandwidth. The Q the components allow found the resonator’s Q limited by its worst component’s; that sum of reciprocals is a statement about the energy reading, and holds for the others only where they agree with it.

The instruments’ own errors, which are larger #

The five readings above are taken from an exact solution, so the only differences between them are the ones the definitions make. A bench measurement adds the instrument’s own error, and for each of the five it is a different error with a different size, which is a second reason the readings disagree in practice even at high Q.

A bandwidth read by a swept analyser is exact only if the sweep waits long enough at each frequency for the resonator to settle; how long a sweep waits at each step found a resonance measured with a wait of Q cycles a step coming out 19.6 per cent too wide, and within one per cent only after about 1.6Q cycles. So the one reading that is exact by definition is, on a bench, the one most easily made too wide by impatience, and the error grows with Q — the opposite of the definitional errors measured here, which shrink with Q.

A ring-down read on an oscilloscope is limited by noise on the later, smaller peaks, so it is usually fitted over a few cycles near the start; at high Q the decay per cycle is small and a fit over few cycles is imprecise, while at low Q the definitional error of the damped period dominates. A peak read by a Q meter is affected by the meter’s own loading, a small capacitance and resistance across the capacitor being read, which lowers both the peak and the frequency at which it occurs. A phase slope read by a network analyser is limited by the analyser’s phase noise and the step between points. An energy ratio is rarely measured directly at all except in calorimetric measurements of very high-Q cavities.

So the choice of reading on a bench is also a choice of which error to accept. For a resonator of Q above ten the definitional differences are below a tenth of a per cent and the instrument errors decide; for one of Q below three the definitional differences are several per cent and decide before the instrument has had any say. A measurement report that gives a Q should give the method with it, and for a low-Q resonator the method changes the number by more than most instruments’ own uncertainty.

The same split in a mechanical resonator #

None of this is specific to inductors and capacitors. A mass on a spring with viscous damping has exactly the equation of the series resonator, with the mass as the inductance, the compliance as the capacitance and the damping as the resistance; its displacement under a driving force is the capacitor voltage, and its velocity is the current. So a loudspeaker cone, a vibration isolator or a cantilever has the same five readings with the same split: its velocity response peaks exactly at the resonance, and its displacement response peaks below it, at $Q/\sqrt{1-1/4Q^2}$ times its static deflection. Vibration engineers who quote a “transmissibility” peak and electrical engineers who quote a bandwidth for the same mechanical part are quoting the two ends of the spread measured here.

Where the readings agree, and the resonator that has two Qs #

Above Q of about ten the five readings agree to within a tenth of a per cent, and the distinction is a matter of definition rather than measurement. That is where most electrical resonators live: a tuned circuit, a crystal, a cavity. The distinction matters for the low-Q resonances that turn up uninvited — a decoupling capacitor’s anti-resonance with the board’s inductance, a filter section driven by a source with resistance, a feedback loop with a modest phase margin — and those are exactly the cases where an engineer is most likely to measure Q one way and use it another.

There is one more case worth naming, because it breaks the clean picture entirely. Two parasitics, and the resonance neither of them has found a loop with two resonances, whose half-power level was crossed four times. For such a circuit the bandwidth reading has no single answer, the phase crosses ±45° more than twice, and the energy at one resonance is not independent of the other. The five readings are readings of a second-order resonator, and a circuit with two resonances close together has no single Q to read.

What the measurement leaves out #

The resonator is a series RLC with a resistor that does not change with frequency. A real resonator’s resistance rises with frequency through skin effect and dielectric loss, which makes the half-power band asymmetric and gives the phase slope and the bandwidth slightly different answers; the Q that peaks before resonance measured how far a coil’s own Q moves when its resistance rises as the square root of frequency.

The energy reading is taken at resonance only. Away from resonance the energy in the inductor and in the capacitor differ, and “the stored energy” has to be defined as the peak of one, the other, or their time average; each gives a different Q, and none is standard.

And the ring-down uses the first two overshoots of a step response. A ring-down measured over many cycles of a lightly damped resonator averages the decrement and gives the same answer; one measured over the first half-cycle, where the step’s own transient is still present, does not.

The number to keep #

Of five common readings of a resonator’s Q, three — the half-power bandwidth, the phase slope at resonance and the energy ratio at resonance — are one equation and give Q exactly at every Q, measured equal to six figures from 0.75 to 50. The ring-down reads $Q\sqrt{1-1/4Q^2}$, because a damped cycle is longer than a natural one, and the peak magnification reads $Q/\sqrt{1-1/4Q^2}$, because the peak is below the resonance: 0.8660 and 1.1547 at Q = 1. Each is within one per cent of Q only above Q = 3.5.

Still open: the parallel resonator, the resonator with a rising resistance, and the loop that is a resonator #

The parallel resonator. A parallel tank read by its voltage under a current drive is the dual of the series case, and its peak is at the resonance when the loss is a parallel resistance but not when it is a resistance in series with the inductor. Which of the five readings stays exact for the series-loss tank — the coil of a real tank circuit — is the practical case.

A resistance that rises with frequency. With skin effect the three exact readings should part, because the bandwidth averages the resistance over the band while the phase slope reads it at the centre. Measuring by how much, at a stated skin depth, would bound the error of quoting a bandwidth Q for a real coil.

A feedback loop’s Q. A closed loop with a second-order dominant pair is a resonator, and its phase margin, its overshoot and its closed-loop peak are three of these readings under other names. Whether they inherit the same three-and-two split, and at what phase margin the two depart by a per cent, connects this essay to the stability measurements.