Moving the primary up the secondary to get a longer arc is the obvious first move, and it works the wrong way round. Tighter coupling raises the ratio of arc length to the size of the coil that threw it. That is a different quantity from the arc length, even though both are measured in centimetres.
Two coupled resonators have a coupling at which one response peak becomes two. With no arc out, every Tesla coil sits far above it. With an arc on, a QCW comes down to about 4.5 times critical and an ordinary DRSSTC drops below 1, because the circuit that sets the threshold changes while the coil fires.
Critical coupling, the one absolute mark on the axis
Couple two tuned circuits and the single resonance splits, but only above a certain coupling. Below it 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. It is the only absolute mark on the coupling axis. Every other figure in these articles compares one coil with another.
The matched case is in the textbooks. 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 gives the coupling coefficient there as
k_c = 1 / Q equation (2-33)It is a 424 page PDF, so use the equation numbers to find it. A few lines before (2-28) it states its assumption: L1 = L2, C1 = C2, Q1 = Q2. A Tesla coil breaks all three by orders of magnitude, so the real question is what replaces that form.
The unmatched case is worked in a 2018 paper on frequency splitting in wireless charging by Tang Guoshen and colleagues at Shandong University. They say the earlier literature assumed matched primary and secondary, and they set out to fill that gap. Their 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, but its own equation gives something else. 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, that is, when the bracket starts falling as d moves away from 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 at which it splits is
k_c = sqrt( (1/Q1^2 + 1/Q2^2) / 2 ) [derived]Set Q1 = Q2 and this is 1/Q, the textbook's own result, which checks that it is the same statement in a general form. Set the two far apart and it is not 1/sqrt(Q1 * Q2). The two forms agree only where the circuits match, and that is the case the paper was trying to get away from.
In other words, k_c is the root mean square of the two circuits' 1/Q, and a root mean square is dominated by its larger term. The lossier circuit sets the threshold almost by itself and the better one barely enters. The numbers below all follow from that.
What that threshold does in a coil
The reference coil used across these articles 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, which is what JavaTC prints as "Secondary Q" for this winding, worked at the secondary is a resonator. That is a program's output for a coil somebody typed in, not an instrument reading. Q1 turns up exactly once in everything we have read, and at 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. That is a loaded figure inferred from a peak current, on a machine with a rod core through the primary. No unloaded figure is published anywhere. The span used below runs from 5, with an arc loading the primary, to 100 or more without one, and that is as much as is known.
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]That is a factor of 18 across a column with no measurement in it. At the top of a bang, there is no way to say how far above critical the coil is.
Neither quantity in that table is a measurement. Q1 is a plausible value, not a found one, and Q2 is a calculator's print. A multiple quoted without its two quality factors tells you nothing. 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 quote one of these numbers, quote the pair with it.
With an arc on it, the secondary decides, and this coil does publish that
An arc hanging off the topload acts as a parallel resistance and pulls the secondary's Q down. The figure used here is 210 kΩ, which Uspring worked out from 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 = 7.6and 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]With no arc, the Q1 column moved the answer by a factor of 18. Under an arc it moves it by 1.07 between Q1 = 20 and Q1 = 100. With the primary's Q anywhere above the secondary's, the multiple reduces to sqrt(2) * k * Q2, 4.52 here [derived], and the primary drops out of the arithmetic.
This 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 and sets the threshold instead, and the answer comes out at 4.5 whatever the primary is doing. A coil is least knowable at the start of a bang and most knowable at the end, the opposite of the order these numbers usually get quoted in.
It is the same mechanism as the frequency slide on its own page, seen on another axis. The arc loads the resonator, so the resonator's frequency moves and its Q falls. One shows up as the tuning walking down the ramp, 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 rather than its square root, because the root mean square is then just the secondary and no longer a compromise between the two circuits. So the last part of the fall is twice as steep as the geometric mean form would make it. Over a whole bang, at Q1 = 100, the multiple goes from 53 down to 4.5, a factor of 12 [derived].
That multiple is not the distance to the poles merging
These 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. His condition is 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 through it in full, including Lenz's own warning that a large split between the two uncoupled resonances brings the collapse on sooner than the condition alone predicts.
Side by side, the two use 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]. They are 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, and neither cites the other.
They also agree on the machine where it matters. An ordinary DRSSTC runs k somewhere around 0.14 to 0.19, which is where the three DRSSTCs further down ended up (BrOdin's, davekni's and Ward's DRSSTC-1), 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]That is below critical coupling for every primary Q in the band. Lenz's margin on the same two numbers is 0.6, also below. By both conditions, from two unrelated literatures, an ordinary DRSSTC under a full arc has no pole pair left to choose from. On a QCW, what keeps one alive is the 0.30 to 0.50 band. That is where the wall puts the two air-cored 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. The two-pole page makes the same point 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.
The loaded Q2 used here for this coil is 7.6. 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 warns 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 a QCW then sits further from the collapse than the 4 would suggest.
The threshold used here is our own derivation. The textbook states k_c = 1/Q along with the assumptions behind it. The 2018 paper states lambda = 1 for the unmatched case, yet 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. The form used here comes out of that paper's own equation (7) and reduces to the textbook's result when the circuits match, which is as far as arithmetic can check itself. We have not found it stated in any source.
Both models also tune the two circuits to one common resonance, and this coil's are not. The primary sits at 349.9 kHz and the secondary at 426.49, 21.9 per cent apart measured against the primary. Every figure above therefore puts the boundary in roughly the right place and is not something to design against. Lenz says the same from his side.
What raising k does to the tank
The secondary's losses show up across the primary as a resistance, and the coupling sets how big that resistance is:
R_ref = w^2 · k^2 · L_p · L_s / R_sThe square is what matters. Double the coupling and the secondary looks four times as lossy from where the bridge is standing. The bridge is a voltage source into a series circuit, so what falls is the current it can push through that circuit:
I_pk = (4/pi) · V_bus / R_totalRaise k and R_total goes up, I_pk comes down, and the power going into the arc comes down with it. None of this is subtle or disputed. Uspring describes it as the primary looking lossy: the transfer of power to the secondary is itself what takes the primary current down.
That does not argue for loose coupling. It only means k and length are separate knobs. Turning k changes what the bridge sees, and if you want the same current afterwards you have to get it from somewhere else.
What the record holders actually run
If coupling gave length, the longest arcs would come off the most tightly coupled coils. It is the other way round.
- BrOdin measures his ground strikes with a tape: 14 feet from breakout point to ladder, and later a 15 foot ground strike, which is 427 to 457 cm [derived, 30.48 cm to the foot], off a 12.75 by 48 inch secondary, so 324 mm in diameter [derived]. The 17 foot figure is only a mark on the wall: the ladder is at 17 feet, and he says he did not get an arc to it that time. His coupling is JavaTC's throughout and it moves: 0.18 on the secondary that flashed over at long ontime, then 0.185 with the first turn of the next one at the bottom edge of the primary, then 0.164 to 0.146 by raising the secondary an inch to stop racing sparks, and 0.141 on the secondary after that. These are the longest arcs on the list, and the coupling is ordinary, the same range davekni's 160 mm coil runs at further down, 0.14 to 0.157. The coil is simply a large one.
- Anders Mikkelsen gets 318 cm off a 16.4 cm secondary at
k= 0.497. That is 19.4 times the coil, and about seven tenths of BrOdin's longest [derived]. - ZakW gets 122 cm off a 5.0 cm secondary: 24.4 times the coil, and just over a quarter of BrOdin's length [derived].
- Daniel Eindhoven's RAIKIRI staff gets 350 cm off a 14 cm secondary, at 25 times the coil the highest ratio here, on a 900 V bus into a full bridge of silicon carbide.
The metres sit at the bottom of the coupling range and the ratios at the top. The two highest ratios on the wall come off fourteen and five centimetre secondaries, throwing three and a half metres and a metre respectively. BrOdin's coil throws more than either from the bottom of the band. Which of his couplings was on the coil for the 15 foot strike is not established. The strike and the figures are in different posts, and the coil was rewound between several of them.
Before any of those numbers are used
None of the eight published coupling figures used here is stated by its author as a measurement of k. That is not sloppiness. No meter reads coupling. It comes out of geometry, or out of inductances, or out of a pair of frequencies, and the route changes the answer.
How much it changes is on the record for one coil at one tap. loneoceans publishes both his figure and the frequencies behind it, and he calls the figure calculated, from the geometry of the system:
0.306 his own, from geometry his word: calculated
0.323 from his poles, secondary ungrounded 282 / 394 kHz [derived]
0.337 from his poles, secondary grounded 255 / 362 kHz [derived]Ten per cent between the first and the last, with nothing changed but the route. His final configuration is the 0.38 quoted everywhere, and that number is an inversion of frequencies he measured. It is a measurement of frequencies rather than of coupling.
The clearest case is the one where somebody asked. dr. kilovolt lists k = 0.55 under "Measured data", and davekni asks him directly, "Did you ever measure coupling factor?" The reply is only that the coupling coefficient is 0.55, as written in the first post. It gives the value again, not the method. A heading that says measured does not make everything under it a measurement.
The rule followed here, wherever k comes up, is therefore to record a coupling coefficient as measured only when the source says how it was obtained. The figures may well be right. The trouble is that a tenth of the value hangs on a choice nobody writes down.
An ordinary DRSSTC lives at about 0.14 to 0.19 on the builds linked here. A QCW lives at 0.30 to 0.50. It is the same trade made on purpose, because a QCW is trying to get a long arc off a short coil and gives up current for ratio. Against the middle of the ordinary band, that is about 1.8 to 3 times the coupling [derived, 0.30 to 0.50 over 0.165].
Two published figures sit above that band, and neither belongs in it. A k of 0.55 and one of 0.71 are both reached with ferrite in the primary's magnetic path: a rod core in dr. kilovolt's, and a ferrite floor and centre post in davekni's oversized QCW, the top of the post NiZn. Every coil in the band above is air cored, and putting a partial magnetic circuit on the same ladder compares two different machines. Those two are a separate line, not the top of this one.
Where all of this sits on an absolute scale
Everything above compares coils with each other. There is also an absolute scale, and the comparisons above have been using it all along.
Two coupled resonators have a critical coupling, the value at which the single response peak splits into two. Below it there is one peak, and above it there are two, with a dip between them. A Chinese national planned textbook, Higher Frequency Electronic Circuits under Zeng Xingwen, gives it for the matched case as k_c = 1/Q in its equation (2-33), from the coupling factor A = kQ at A = 1. A 2018 paper on frequency splitting in wireless charging does the unmatched case, the one a Tesla coil actually is, and says the earlier work assumed matched primary and secondary. Its equation (3) gives a generalised coupling factor, and substituting M = k·√(L₁L₂) turns it into λ = k·√(Q₁Q₂). The paper then puts critical coupling at λ = 1, which would make it 1/√(Q₁Q₂), a geometric mean. Its own equation (7) disagrees. Differentiating it at zero detuning gives a root mean square instead.
k_c = sqrt( (1/Q1^2 + 1/Q2^2) / 2 )The derivation above shows it reducing to the textbook's 1/Q when the two circuits match, which checks that it is the same statement in general form. The difference matters in practice. A root mean square is set by its larger term, so the lossier of the two circuits fixes the threshold almost alone, while the geometric mean lets a healthy primary cover for a spent secondary.
Put numbers in it. The unloaded secondary Q used here is 199, and a primary's runs from a handful when the arc is loading it to a hundred or more when it is not:
Q1 Q2 k_c
5 199 0.142
20 199 0.036
100 199 0.008
100 100 0.010 [derived, all four]Most of these machines sit far above it, and the exception is the interesting part. Against the three rows with a primary Q of 20 or more, an ordinary DRSSTC at 0.14 to 0.19 runs 4 to 24 times critical coupling and a QCW at 0.30 to 0.50 runs 8 to 63 [derived]. Against the top row, where an arc is already loading the primary, the same two bands land at 1.0 to 1.3 and 2.1 to 3.5. The ordinary machine is then on the boundary rather than far above it. This is the loaded case showing up early, in the one row where the primary is already loaded. The spread in all those multiples comes from Q, not from k. The four rows span a primary Q from five to a hundred, and the corpus has one builder's published figure for that column, dr. kilovolt's Qpri~11 on his own coil. On these machines the coupling is the better known of the two.
That changes how to think about the poles. They do not only turn up when you couple too hard. Every coil in the field has them before the arc is out, and a coil with a single response peak would be one nobody wants, because critical coupling is where energy transfer stops being fast. The two-pole page works out which of them a driver lands on, and this is why it can never skip that question.
The multiple is also not a single number. With no arc it spans a wide range, because Q₁ is barely published. Under an arc it hardly depends on Q₁ and lands near 4.5 for a QCW and below 1 for an ordinary DRSSTC, because Q₂ collapses while the coil fires and the circuit that sets the threshold changes. As worked through above, the unloaded figure belongs to the first microseconds of a bang and the loaded one to its end, so they are a sequence rather than a range.
The textbook's derivation assumes the two circuits match in L, C and Q, and a Tesla coil breaks that by orders of magnitude, so the unmatched form is the one used here. Both only say roughly where the boundary is. Neither is a figure to design against.
The wall on the other side
There is a reason builders do not just wind the primary tighter and accept the lost current. Racing sparks have been seen inside the band above, not beyond it, and on the coil with the longest arcs in this article. The secondary BrOdin wound at 0.185, and quotes a week later at 0.164, gave him massive racing sparks with the current limit at 2000 A, long before any large streamers even formed, and the cure was less coupling: raising the secondary an inch, which JavaTC put at 0.164 down to 0.146, was enough to run at 2000 A without them. On that coil, lowering k fixed it. It is one coil, so the safe conclusion is a narrow one: the ceiling belongs to the winding, not to the number. One machine is built around that limit instead of against it, and gets its coupling from a third coil instead of a tighter winding: a DRSSTC magnifier is not just a longer coil. davekni had to take his own coil from 0.157 to 0.14 with more resin layers to stop secondary arcing, and says he would love 0.18 but his 160 mm secondary diameter limits getting there, while BrOdin's 324 mm one carried 0.164 to 0.185 before it flashed. The turn to turn voltage up the secondary is not distributed the way the coupling coefficient suggests, and the coil flashes over its own winding before any of the arithmetic above comes into play.
This is why the QCW's 0.30 to 0.50 is less reckless than it looks. Racing sparks are a secondary voltage failure, where the winding flashes over along itself before the energy has finished moving. A ramp slow enough for an arc to form takes that voltage away, because the arc clamps it. Anders Mikkelsen puts both halves in one sentence on his thread about poles and detuning: ramp slowly enough that an arc has time to form, in order to clamp the secondary voltage to a low value, which allows long sparks from a short resonator without the risk of flashover, and also allows high coupling to be used.
A QCW does not break the coupling rule. It removes the reason for the rule and then uses the headroom. That escape is not open to you on a banged coil, where the voltage arrives before any arc does and the limit is where it always was.
So what does buy length
Energy, delivered slowly. Across the measurements that exist, the arc goes as the cube root of the energy in the bang:
L ~ E^n, n = 0.31 to 0.33A cube root is harsh. Eight times the energy gets you twice the arc, and the eight has to come from somewhere: a higher bus, a longer ontime, more capacitance in the tank, and each of those has its own limit. This is also why millimetres per kilowatt does not work as a figure of merit. It draws a straight line through a curve, and it is not even defined until you say how long the bang lasted.
Four ways the multiple gets misused
k_cis 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 unchangedksit 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 does not measure the distance to any event. Whether there are still two poles is Lenz's question, and it is answered on the secondary's
Qalone, which is how an ordinary DRSSTC can sit at 0.83 times critical under a full arc while its coupling never moved. - The two forms of
k_care 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, atQ1= 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.
At the bench
- Set the coupling from the geometry your winding can stand, not from a length you want. Start looser than you think and creep up.
- Measure the primary current before and after every change to
k. If it fell and the arc did not, the coupling was not the limit. - When the arc stops growing, look at the bus and the ontime before you look at the coil. With a cube root, the coil is rarely the cheapest thing to change.
- Treat the first racing spark as the ceiling for that winding, permanently.
The figures above are collected from published builds, not measured here, and the ratios are arithmetic on the builders' own numbers. Four of them have been read at the source: RAIKIRI, a QCW 1.5 and ramped SSTC, and BrOdin's, whose thread is linked post by post above. RAIKIRI's page has since stopped serving any of it, so that one was read once and cannot be checked again at that address. The rest came from a compilation of forum threads, and the records page keeps them apart from the ones read at the source.