The reading a coil leaves nearly exact

Five readings of one Q took a series resonator and read its quality factor from the bandwidth, from the slope of its phase, from the energy it stores against the energy it loses, from how fast its ringing dies, and from how high its response peaks. Three of the five agreed with the definition to every digit the arithmetic had, at any Q. The ring-down and the peak height did not, and it was exactly clear why. It then pointed at the practical case that its own measurements had avoided: a parallel tank’s peak is at its resonance when the loss is a parallel resistance, and not when the loss is a resistance in series with the inductor. Which of the five stays exact for that tank, the coil of a real tank circuit?

None of them does. That is not surprising; what is surprising is which reading comes closest and why. Most of what a series circuit has is unchanged here. The poles are the same, the same two numbers define them, and a ring-down reads what it read before. The change is one zero in the impedance, at $-R_s/L$. It is far below the resonance and it looks unimportant. It is why a coil’s Q is the hardest of these to pin to one definition, and it separates the readings by a pattern that does not look like what the series result would lead anyone to guess.

One zero, three more frequencies #

The tank is a coil, an inductance $L$ in series with a resistance $R_s$, in parallel with a capacitor $C$, driven by a current and read as a voltage:

$$Z(s) = \frac{R_s + sL}{1 + sR_sC + s^2LC}.$$

The denominator is the series circuit’s. It has the same undamped frequency, $\omega_0 = 1/\sqrt{LC}$, the same decay rate, $R_s/2L$, and therefore the same Q that a pole-based definition assigns, $\sqrt{L/C}/R_s$. That is the reference every reading here is compared with. The series circuit, read by its current under a voltage drive, has exactly that denominator and a constant numerator. A resonator read that way has one frequency that is everything at once.

The numerator changes that, and the frequency at which each thing happens now separates into three. The voltage is in phase with the current at $\omega_r = \omega_0\sqrt{1 - 1/Q^2}$, which is the definition of resonance a power engineer uses and which does not exist at all below $Q = 1$. The voltage is largest at $\omega_m = \omega_0\sqrt{\sqrt{1 + 2/Q^2} - 1/Q^2}$. The free ring is at $\omega_d = \omega_0\sqrt{1 - 1/4Q^2}$, a frequency the transient has regardless of what the circuit is connected to.

These do not fall away at the same rate, and that is the first thing to notice. The zero-phase point is the furthest out, by $1/2Q^2$, or half a per cent at $Q = 10$. The ring is a quarter of that, at $1/8Q^2$. The peak is a surprise: its distance is the fourth power, $1/4Q^4$, so at $Q = 10$ it sits 0.00248 per cent below $\omega_0$ when the zero-phase point is 200 times further away. Expanding the root makes it clear why: the terms in $1/Q^2$ cancel exactly. A tank’s peak is therefore a very good estimate of its undamped frequency, and its zero-phase point a noticeably worse one, although the second is what a phase detector would lock to. Two parasitics, and the resonance neither of them has showed a resonator’s parts producing a frequency none of them had; this one has several, and it hides them inside a single component.

Five instruments, each as it would be used #

Each reading is taken here the way an instrument takes it, with no use of the formula it will be compared with, since the point is to learn what the instrument reports. The bandwidth reading finds the peak by search, finds the two half-power frequencies by bisection, and divides the peak’s frequency by the difference. The phase reading takes the slope of the phase at the frequency where the phase is zero, and halves it after scaling by that frequency. The energy reading computes the stored energy at that same frequency against the energy lost per radian. The ring-down reads the damped cycles as the last essay did. The peak reading takes the impedance at the peak and divides by the characteristic impedance $\sqrt{L/C}$.

The series circuit’s division into three exact readings and two inexact is gone. All five are inexact. Four of them are wrong by amounts in $1/Q^2$, and at $Q = 3$ those are large: the phase slope is 16.2 per cent low, the stored energy 5.72 low and the peak height 5.42 high. The ring-down is the smallest of the four at 1.40, and that value is the one it has on any circuit with these poles.

The fifth is a different kind of thing. At $Q = 3$ the bandwidth reads 0.307 per cent low, and at $Q = 10$ it reads 0.0025 per cent low. Its error goes as the fourth power of the Q, like the peak’s frequency, so that it is, to anything an instrument could resolve, still exact. And this is the reading a designer would be least inclined to trust, because a half-power bandwidth is an affair of two frequencies some way from resonance on a tank whose loss is known to vary with frequency. It might be expected to pick up the tilt of the coil’s loss. It nearly does not.

A person holding the Q the components allow in mind, in which reciprocals add, might suppose the phase slope was the fundamental reading, since it is a local one at resonance, and that an average over a band would be the approximation. On this circuit the local one is the worst of the five and the average the best.

A loss that tilts #

The reason appears when the coil’s series loss is written in the parallel form that every tank circuit is drawn with. A resistance $R_s$ in series with $L$ is, at one frequency, exactly equivalent to some resistance in parallel with a slightly different inductance, and the parallel resistance is $R_p(\omega) = R_s(1 + (\omega L/R_s)^2)$, a formula that is exact at every frequency and gives the well-known $R_s(1 + Q^2)$ at $\omega_0$.

So a coil-loss tank is a parallel tank with a loss that varies as the square of frequency, and at $Q = 10$ it is 9.65 per cent low at the lower band edge and 10.1 per cent high at the upper. That is a big tilt, and it has to be somewhere in the readings. It is exactly what the phase slope reads in full, since a slope at one frequency sees a local derivative of everything.

The bandwidth, on the other hand, measures between the two edges, and an almost antisymmetric tilt about the middle cancels at first order. The lower edge moves in by one amount and the upper edge moves out by nearly the same, and the difference, which is what the reading is, stays where it was. The residue is the second-order difference, from 9.65 to 10.1, and it reaches the Q at order $1/Q^4$ because each edge is itself at $1/2Q$ from the centre. That is the structure of the result. A tilt is an odd function about the band’s centre, and a measurement that spans the centre symmetrically is even.

A familiar general principle is behind this. A symmetric difference cancels a linear error, and that fact underlies many a measurement that seems to be getting more than it deserves: a differential pair’s lost even harmonics and a chord through an inflection are other versions. It is worth carrying because it means the bandwidth reading can be recommended for an inductor’s Q for a reason beyond its convenience. It is the reading that is immune to the coil’s own frequency dependence, and it is usually the cheapest.

Why the series result does not simply carry over #

It may seem that the series circuit’s three exact readings should still be there in some form, since the poles are unchanged. What made them exact there was that those three readings were actually of the poles. Under a voltage drive and with the current read, the transfer function of the series circuit has nothing in it but the two poles and a zero at the origin, so anything computed from it is a function of the poles. On the coil-loss tank, the zero at $-R_s/L$ is in every reading of $Z$.

The ring-down does not read $Z$. It reads the free response, which has the poles and no zero. That is why it is the same reading on both circuits, with the same $1/8Q^2$ from counting a damped cycle in place of an undamped one. So a user who has both a sweep and a ring-down available has one reading that does not involve the coil’s zero, and comparing them is one way to detect the fact that a resonator has something in it beyond its poles. Read that way, the disagreement is more information than either number alone.

The stored-energy reading also becomes sensitive to where it is taken. At the zero-phase frequency it reads $(1 - 1/Q^2)$ of the pole Q. Taken at $\omega_0$ it would read something else, since a power engineer’s and a filter designer’s definition of Q differ exactly here. Resonance, and the bandwidth it sets exactly was on a circuit where they do not differ, so the question did not arise, and it is a sensible warning against carrying over the phrase “the Q” from one to the other.

What the source does #

All of this has assumed the tank is driven by a pure current source, and every practical measurement of a tank has to ask what that assumption costs. A series resonator is driven by a voltage source, whose own resistance appears in series with the loss being measured and can be small enough to neglect. A parallel tank has the same problem in dual form, and it is much harder to meet, because at resonance the tank’s own parallel resistance is high and rises as the square of the Q.

A source resistance across the tank simply adds in parallel, so the reading becomes $Q/(1 + R_p/R_{source})$, and that sum agrees with the solve within three per cent wherever the source is not itself the main loss. A source ten times $R_p$ reads 9.10 for a true 10; a source equal to $R_p$ reads 5.05, half the coil’s Q. Getting within a per cent means a source a hundred times $R_p$. On a good 1 mH coil with a Q of 100 at this resonance, $R_p$ is about ten kilohms, and a source a hundred times that is a megohm, which is not a generator’s output impedance. So a parallel tank is measured through a very small coupling capacitor or a loosely coupled loop, which is the arrangement a classic Q meter uses, so as not to read the pair.

That sets up a trade the series circuit does not have. The coupling that keeps the source’s loading out of the reading also reduces the signal, by the same ratio. And the better the coil, the stiffer the requirement, because $R_p$ grows as $Q^2$. The instrument that reads the highest Q is therefore the one with the least signal to read, and the probe is part of the circuit is the same argument with a voltmeter on the other side of the tank.

What the numbers change for a design #

For a filter designer, the zero-phase frequency and the band’s centre are where trouble starts. A tank tuned by nulling its phase sits $1/2Q^2$ below where its centre is: half a per cent at $Q = 10$ and 5.7 per cent at $Q = 3$. A low-Q matching network tuned that way comes out noticeably off its intended centre, in the direction that a design from the undamped formula never predicted. Which section goes first concerns filters where sections of various Q are stacked, and the frequency of each is an input to that calculation.

For an oscillator, the question is which of these frequencies the circuit actually runs at. An oscillator’s amplifier closes its loop where the phase around the loop is zero, and for a tank in the loop that is the zero-phase point, not the peak, unless the amplifier’s own phase changes this. So a crystal-less LC oscillator with a modest coil runs a little low, by very nearly $1/2Q^2$, and any reasoning from the peak’s frequency misses it. That is a much bigger error than the 200 times smaller distance of the peak from $\omega_0$ would suggest.

What the model leaves out #

The coil here has one resistance, at every frequency, and a real coil’s rises. Skin effect makes $R_s$ grow as $\sqrt{f}$, and the resistance that grows with frequency measures how much. A second tilt is thus added to the loss, and with it the bandwidth’s cancellation is less complete, although it is still a symmetric reading of a smooth function. The capacitor also has a loss of its own, which enters as a true parallel resistance, and the fair statement of a real tank has both. And the coil’s own winding capacitance is in parallel with $C$, moving the frequency and not the Q, until it is a significant part of $C$. On a low-frequency tank with a large capacitor that does not matter; on a tank near a coil’s self-resonance, as the inductor that is a capacitor found, it is the main thing happening.

The number to keep #

A coil-loss tank’s three resonances are $1/2Q^2$, $1/8Q^2$ and $1/4Q^4$ below $1/\sqrt{LC}$, for zero phase, for the ring and for the peak. Of the five readings of its Q, four are out by amounts in $1/Q^2$: at $Q = 3$, phase slope −16.2 per cent, stored energy −5.72, ring-down −1.40 and peak height +5.42. The bandwidth is out by −0.307 per cent there and by −0.0025 at $Q = 10$, because the coil’s loss tilts across the band in a way nearly antisymmetric about its centre, 9.65 per cent low at one edge and 10.1 high at the other, and the reading that spans the band cancels it. So the bandwidth is the reading to take, provided that the source driving the tank is far larger than its parallel resistance, which grows as $Q^2$.

Still open: the skin effect in the coil, the loop that is a resonator, and the coupling that keeps the signal #

A resistance that rises with frequency. With $R_s$ rising as $\sqrt{f}$ the tilt has a second term, and the bandwidth’s cancellation is no longer of a purely quadratic function. How far the near-exact bandwidth then departs from the reading that a phase slope gives, and which of them is closer to what the coil does at its centre, 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 coil’s pattern, with a zero in the response, or the series circuit’s, without one, depends on whether the loop has a zero, and most compensated ones do.

A coupling that keeps a signal. A Q meter has to couple loosely and read a small signal. The noise floor then sets the highest Q that can be read at a stated accuracy, and there is an optimum coupling at which the loading error and the noise error are equal. Where it lies for a coil of a stated Q is a measurement that would finish the question of what the source costs.