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Critical coupling, and why every coil is far above it

There is a coupling at which one peak becomes two, and with no arc every coil sits far above it. An arc drops a DRSSTC below it, because which circuit sets the threshold changes hands.

CommonGnd17 August 202612 min read

Two coupled resonators have a coupling at which one response peak becomes two. With no arc out, every Tesla coil ever built sits far above it. Put an arc on and a QCW closes to about 4.5 times critical and an ordinary DRSSTC drops below 1, because which of the two circuits sets the threshold changes hands while the coil fires.

What it is and how it works

Couple two tuned circuits and the single resonance splits. Below a certain coupling it does not: there is one peak, and the pair behaves like one circuit that rings a bit longer. Above it there are two peaks with a dip between them. The value where that changes is the critical coupling, and it is the only absolute mark on the coupling axis. Everything else on this site compares coils with each other.

The matched case is textbook. A Chinese national planned textbook, Higher Frequency Electronic Circuits under Zeng Xingwen, introduces a coupling factor A = kQ as its equation (2-28), calls A = 1 the critical coupling, and names the coupling coefficient there:

k_c = 1 / Q                                    equation (2-33)

It is a 424 page PDF rather than a page of text, so the equation numbers are how to find it. Read back a few lines from (2-28) and the assumption is stated outright: L1 = L2, C1 = C2, Q1 = Q2. A Tesla coil violates all three by orders of magnitude, so the question is not whether that form applies but what replaces it.

The unmatched case is worked in a 2018 paper on frequency splitting in wireless charging by Tang Guoshen and colleagues at Shandong University, whose stated contribution is exactly that gap: the literature before it assumed matched primary and secondary. Its equation (3) defines a generalised coupling factor lambda = wM / sqrt(r1 * R), and with M = k * sqrt(L1 * L2), Q1 = wL1/r1 and Q2 = wL2/R, the inductances and resistances cancel and it becomes

lambda = k * sqrt(Q1 * Q2)          [derived from equation (3)]

Where the threshold actually falls

The paper puts critical coupling at lambda = 1, in those words. Its own equation says otherwise. Equation (7) normalises the secondary current to

alpha = 2*lambda / sqrt( (1 - x1*x2 + lambda^2)^2 + (x1 + x2)^2 )

where x1 = Q1*d and x2 = Q2*d are the two circuits' detunings on one common d. The response splits when resonance stops being the maximum, which is when that bracket starts falling as d leaves zero. Differentiating it there gives

splits when   lambda^2 > (Q1/Q2 + Q2/Q1) / 2      [derived from equation (7)]

and substituting lambda = k * sqrt(Q1 * Q2), the coupling that does it is

k_c = sqrt( (1/Q1^2 + 1/Q2^2) / 2 )               [derived]

Set Q1 = Q2 and that is 1/Q, the textbook's own result, which is the check that this is the same statement made general rather than a different one. Set the two far apart and it is not 1/sqrt(Q1 * Q2). The two forms agree only where the circuits match, which is the case the paper set out to leave behind.

k_c is the root mean square of the two circuits' 1/Q, and a root mean square is set by its larger term. So the lossier of the two circuits fixes the threshold almost on its own and the better one barely enters. Everything below is that sentence with numbers in it.

What it does in a coil, and what you decide

The coil this site works from runs k = 0.421, and the two quality factors it needs are not equally well known.

With no arc, the primary decides, and nobody publishes it unloaded

Q2 unloaded is 199. That is what JavaTC prints as "Secondary Q" for this winding, worked at the secondary is a resonator. It is a program's output for a coil somebody typed in, not an instrument reading, and this page says so rather than calling it a measurement. Q1 is published exactly once in everything we have read, and it is the wrong end of the axis for this table: dr. kilovolt lists "Ipk=100A (bus voltage dropping to approx. 650V) which translates to Qpri~11" among the measured data for his ferrite-cored SiC coil, which is a loaded figure inferred from a peak current, on a machine with a rod core through the primary. No unloaded one is published anywhere. What coupling actually buys works with a span from 5, when an arc is loading the primary, to 100 or more when it is not, and that span is the honest state of the knowledge.

no arc, Q2 = 199
Q1          k_c        k / k_c
  5       0.1415         3.0
 20       0.0355        11.8
100       0.0079          53          [derived, all three]

A factor of 18 across a column nobody has measured. At the top of a bang, "how far above critical is this coil" has no answer.

Neither quantity in that table is a measurement. Q1 is a plausible value rather than a found one, and Q2 is a calculator's print. A multiple quoted without its two quality factors is not a measurement of anything. 53 times critical and 3.0 times critical are the same coil in the same table, and the only difference between them is which Q1 somebody had in mind. If you carry one of these numbers anywhere, carry the pair with it.

With an arc on it, the secondary decides, and this coil does publish that

An arc hangs across the topload as a parallel resistance and spends the secondary's Q. This site publishes 210 kΩ for it, which Uspring worked out of Hydron's measured topload currents on a 160 mm DRSSTC (HVF 117). Against this coil's characteristic impedance of 26.6 kΩ,

1/Q2 = 1/199 + 26.6/210    ->    Q2 = 7.6

and the table changes shape.

under an arc, Q2 = 7.6
Q1          k_c        k / k_c
  5       0.1693         2.5
 20       0.0995         4.2
100       0.0933         4.5          [derived, all three]

The column that moved the answer by 18 with no arc out moves it by 1.07 between Q1 = 20 and Q1 = 100 once there is one. With the primary's Q anywhere above the secondary's, the multiple reduces to sqrt(2) * k * Q2, which is 4.52 here [derived], and the primary has dropped out of the arithmetic entirely.

That is the finding, and it is the reverse of what the unloaded table suggests. The unknown changes hands while the coil fires. With no arc the primary is the lossier circuit, it sets the threshold, and the multiple cannot be stated. Once the arc is out the secondary is lossier by a wide margin, it sets the threshold instead, and the answer lands on 4.5 whatever the primary is doing. A coil is least knowable at the start of a bang and most knowable at the end, which is the reverse of the order these numbers usually get quoted in.

It is the same mechanism as the frequency slide on its own page, read on another axis. The arc loads the resonator; the resonator's frequency moves and its Q falls; one shows up as the tuning walking down the ramp and the other as the machine walking down this scale.

How fast it walks

Once Q2 is well below Q1 the multiple goes as Q2 itself, not as its square root, because the root mean square has stopped being a compromise between the two circuits and become the secondary alone. So the last part of the collapse costs twice what the geometric mean form would charge for it. Over a whole bang, at Q1 = 100, the walk is 53 down to 4.5, a factor of 12 [derived].

That multiple is not the distance to the poles merging

Worth separating, because the two get run together and they are conditions on different quantities. Udo Lenz, on the Tesla Coil Mailing List in February 2013, gives the Tesla case: system loading makes the centre frequency and one pole go away, leaving one frequency near the primary's resonance, and that happens when Qsec drops below 1/k. It is written on the secondary's Q alone. This coil's 1/k is 2.4, so at a loaded Q2 of 7.6 the margin is 3.2. There is no secondary frequency works the whole of it, including Lenz's own warning that a large split between the two uncoupled resonances brings the collapse on sooner than the condition alone predicts.

Put the two side by side and they are the same coordinate. With the primary's Q the higher of the two, the threshold derived above reduces to k * Q2 = 0.71, against Lenz's k * Q2 = 1 [derived]: a factor of 1.41 apart, one from a 2013 mailing list post about Tesla coils and one from a 2018 paper about charging cars, neither citing the other.

They also agree on the machine where it matters. An ordinary DRSSTC runs k = 0.12 to 0.2, the band Kaizer's design guide gives, and Uspring puts the loaded Q of Hydron's coil at about 4. At k = 0.15 with Q2 = 4:

Q1          k / k_c
  5          0.66
 20          0.83
100          0.85          [derived, all three]

Below critical coupling for every primary Q in the band. Lenz's margin on the same two numbers is 0.6, also below. Two conditions, two literatures, one verdict: an ordinary DRSSTC under a full arc has no pole pair left to choose from. A QCW's 0.30 to 0.50, which is where the wall puts the two builds that publish a coupling at all, Anders Mikkelsen's 0.497 and paulsimik's 0.40, with this coil's 0.421 between them, is what keeps one alive. That is the point the two-pole page makes from the other side, and it is a better reason to couple a QCW tightly than the arc-to-secondary ratio it is usually given for.

This page uses 7.6 for this coil's loaded Q2. The pole collapse section of there is no secondary frequency puts the same coil "down around 4 with a metre and a half out". The 4 is Uspring's figure on Hydron's coil, whose impedance is near 52 kΩ against this coil's 26.6, and loaded Q is R_arc over Z and very little else, so the two cannot both belong to this winding. An arc grows as fast as charge arrives already carries the warning against pairing this coil's 199 with that coil's 4. On 210 kΩ over 26.6 kΩ it is 7.6, the margin to Lenz's condition is 3.2 rather than 1.7, and that is a different picture of how close a QCW runs to the collapse.

What will get you

The threshold this page runs on is ours. The textbook states k_c = 1/Q and states the assumptions it made; the 2018 paper states lambda = 1 for the unmatched case, and its own worked example has a primary Q about 29 times its secondary's at the load where it reports splitting [derived from its Table 1], where the form above would want lambda past 3.8 before anything splits. What is written here comes out of that paper's own equation (7) and reduces to the textbook's result when the circuits match, which is as much checking as arithmetic can do to itself. Nobody has stated it in a source we have read, and it is carried here as this site's arithmetic.

What goes wrong

k_c is treated as a property of the coil. It is a property of the coil and its load, and the load is the arc, which changes through every bang. The same geometry and the same unchanged k sit at 53 times critical before the arc forms and at 4.5 by the time it is out.
An unloaded multiple is quoted about a loaded coil. The large figures belong to the first microseconds of a bang, before there is an arc. Quoting one of those about a machine mid ramp overstates it by about 12 at Q1 = 100.
A large multiple is read as room to spare. It is not the distance to anything happening. Whether there are still two poles is Lenz's question and it is answered on the secondary's Q alone, which is why an ordinary DRSSTC can sit at 0.83 times critical under a full arc while its coupling never moved.
The two forms of k_c are treated as interchangeable. 1/sqrt(Q1 * Q2) is a geometric mean and the form above is a root mean square. On this coil under an arc, at Q1 = 100, they differ by a factor of 2.6: 11.6 against 4.5. The geometric mean flatters the machine, because it lets a healthy primary cover for a spent secondary, and nothing in the physics allows that.

Where next

What coupling actually buys, where the bands come from and why the longest arcs are not the tightest coupled.
There is no secondary frequency, which carries Lenz's condition, the swept model behind it and what survives the collapse.
The frequency slides down the ramp, the same arc loading the same resonator, read on the other axis.

couplingpolesQ