[ §1 · tuning ]

DRSSTC tuning: the two poles, and which one the driver lands on

Couple a primary to a secondary and the resonance splits in two. The frequency you carefully measured is a mode that no longer exists.

CommonGnd4 August 202648 min read

Couple a primary to a secondary and the resonance splits in two. The driver sits on one of them, and that choice decides most of what the coil does. It also means the frequency you carefully measured on the secondary is a mode that no longer exists.

At loose coupling there is one hill and one frequency to find. Tighten it and the hill opens into two poles, and the bridge has to be driven at one of them rather than between them.

Where the two come from

The resonance condition for the coupled system is

(1 - w1^2/w^2)(1 - w2^2/w^2) = k^2

and solving it for w gives

w±^2 =
  [ w1^2 + w2^2
    ± sqrt( (w1^2-w2^2)^2
            + 4k^2 w1^2 w2^2 ) ]
  / [ 2(1 - k^2) ]

Tuned to each other, that collapses to the familiar form:

f± = f0 / sqrt(1 -+ k)

Tighter coupling drives them apart. Worked on the QCW this corpus keeps its figures for (primary 349.9 kHz, secondary 426.49, both from JavaTC) using the general form above rather than the tuned one, because that machine is deliberately detuned:

k      lower  upper  apart
0.30   328.9  475.7  146.8
0.39   320.0  506.5  186.5
0.421  317.0  519.1  202.1  <-
0.45   314.2  531.9  217.7
0.50   309.5  556.8  247.3

<- as built

Ours is the marked row. The trajectory those two poles then follow as the arc grows is its own page.

The familiar form is not useless on a detuned machine, but it only works if the f0 in it is read as the primary's own resonance rather than the secondary's, which is worth saying because taking the secondary looks almost right and is wrong. loneoceans publishes a primary alone at 328.4 kHz and a measured upper pole at 420.2. Inverted, those two return k = 0.389 against the 0.38 he states; run forwards they give 417.1 against the 420.2 he measured. Both directions are what coupled poles does, on a primary and a secondary that are tuned apart rather than together. Started from that coil's secondary, 339.3, the same arithmetic misses by enough to invite an explanation that is not there. Two further coils attach the same way, and on both the route through the secondary is several per cent out. How closely they agree is not quoted here, because the pair this page can check are davekni's at 0.17 per cent and MS.Lab's at 0.8, and a tighter figure than either would need naming the coil and linking it.

Tuned to each other, the distance between the two poles depends on nothing but k, and at these couplings it is not a detail: 15 per cent of f0 at a DRSSTC's 0.15, 52 per cent at a QCW's 0.45, and 109 at the 0.71 of the coil checked further down this page, where the two end up further apart than the frequency they came from. That is our arithmetic on the form above rather than anyone's measurement.

Two peaks are not a property of every coupled pair, and the sentence this page opens with will let you think they are. Below a certain coupling there is one peak, and splitting is what running above it looks like. An article from 1991 arguing that the coils of its day were overcoupled reports the same thing with the value judgement the other way round: at k = 0.25 it sees "dual resonant peaks with broad skirts", and at critical coupling one sharp peak. Its list of complaints is this corpus's list of failure modes, beating and sputtering output and a secondary destroyed by arcing along itself, and its remedy is to come back down towards critical. What changed since is the answer rather than the physics: a slow ramp lets the arc form and clamp the secondary's voltage, which makes the overcoupled regime survivable and lets k go up instead of down. That article is about spark-gap coils and the source does not name its author; setting its regime beside a QCW's, and the conclusion drawn from that, are ours.

Which one you land on

davekni and Uspring arrived at the same rule independently:

Where the driver starts, and the ball rolls downhill from there

Released at

350 kHz

Distance to lower / upper

33 / 169

Lands on

lower

The lower well is genuinely deeper. Across the useful part of the ramp the lower pole has two to nine decibels more gain than the upper, so if the machine started from nothing it would end up there. That is not a defect in the drawing; it is the thing the modification exists to beat.

But depth is not what decides. A ball rolls into whichever well it is over, and the driver settles on whichever resonance it is nearest. Released at the primary's own 350 kHz, which is where a plain zero-current driver starts, it is 33 kHz from the lower pole and 169 from the upper, so it goes down. Not close, and not only at the start: at a metre and a half of arc the distances are 54 and 113, and the answer has not changed.

Which is what the modification buys. Release at 556 kHz instead and the upper well is now between the ball and the lower one. The ball never sees the deeper well, because it would have to climb out of the shallower one first, and it does not. The gain advantage is still there and still bigger; it simply never gets a chance to act, because gain decides which mode grows out of noise, and nothing here is starting from noise.

Derived. Poles 317.0 and 519.1 kHz from f_pri 349.9, f_sec 426.49 and k 0.421, all JavaTC. Well depths are drawn in proportion to each pole's gain, the lower one 5.5 dB ahead with no arc. The landscape is a picture of the argument, not a computed potential. The frequencies and the gap are the real numbers in it.

Applied to the loaded coil, not the bare one

The rule is right and it is easy to apply to the wrong state. A published build looks at first like a flat contradiction of it: on loneoceans' first QCW an ordinary UD2.7, no PLL and no self-oscillation, runs with the primary tapped at turn 7.5, 392 kHz, and he reports that the coil oscillates itself to the upper pole. Every figure below is his, off that page: the primary alone at turn 7.5, the secondary alone with its single toroid, the secondary with a metre of wire on it standing in for a streamer, and k = 0.32.

Work it in both states and the contradiction dissolves.

primary tapped at 392, k = 0.32

no arc, secondary 408
  lower 44 below
  upper 94 above  -> lower

1 m streamer, secondary 310
  lower 100 below
  upper 47 above  -> upper

With no arc out the primary is nearer the lower pole, and with the arc loading the secondary it is nearer the upper. His coil settles where the rule says it will once the arc is out, which is the state that matters. The rule is not wrong; applying it to a bare coil is.

And the criterion is the midpoint, not the loaded secondary

Which sharpens something this corpus had been using in a weaker form. "Primary above the loaded secondary" is necessary and not sufficient. What decides which pole the primary is nearer is the midpoint between the two poles, and because coupling is not symmetric about the secondary those are different tests.

On the coil this corpus keeps its figures for:

arc    midpoint   primary 349.9
1.5 m    379.5    30 below
2.0 m    370.4    20 below
2.81     358.1     8 below
3.46     349.9    level

The primary only reaches the midpoint at 3.46 m, and the pole structure comes apart at or before 2.81. So on this machine the upper pole is never the nearer one at any arc length it can reach: the forcing is required for the whole ramp, not merely until the arc grows enough to take over. Anders' criterion is satisfied from about 1.65 m onward and it is not enough.

And the mechanism underneath both is gain

The proximity rule describes the outcome without giving the cause, and the cause is worth having because it is what a modification has to overcome. Anders Mikkelsen states it directly:

[a self-oscillating UD2.7 and similar driver] would need more gain at the upper pole, which means that the primary needs to be tuned above the loaded secondary frequency. This is not possible for a plain self-oscillating driver, as the lower pole would have more gain without spark detuning.

He puts the excess the upper pole needs at "maybe 6 dB?".

That is checkable on a coil with known numbers, by working out the primary current each pole would draw at the same bridge voltage:

arc    lower is stronger by
0                +5.5 dB
0.5 m            +9.4 dB
1.0 m            +5.5 dB
1.5 m            +1.9 dB
1.8 m            crossover
2.0 m            -1.0 dB

The crossover at about 1.8 m arrives at the same place as Anders' tuning criterion, by a different route. His condition, primary above the loaded secondary, is met between 1.65 and 2.0 m on this coil, since the primary sits at 349.9 against a loaded secondary of 355.1 at 1.5 m and 338.2 at 2.0. Two independent calculations, one through currents and one through tuning, landing on one point.

So across the whole useful part of a ramp the lower pole holds an advantage of roughly 2 to 9 dB, while the upper one needs a 6 dB excess to be chosen. The total deficit is 8 to 15 dB, and it is why feedback left to itself goes down rather than sometimes going down. What that deficit means for a driver that starts from noise (and why a modification can beat it anyway) belongs with the modification.

One figure from that other subject belongs here rather than there, because it is what the deficit above has to be measured against. An injection-locked driver holds the upper pole through 14 dB of excess gain at the lower one and loses it at 20, a limit stated in gain terms in HVF 3140. Whether that sets it against the 8 to 15 dB above depends on a definition nobody has stated. The 8 to 15 is a sum: the lower pole's own 2 to 9 dB advantage, plus the 6 dB the upper needs before a driver will choose it. His 14 is the excess gain at the lower pole. Read against the sum, the lock covers the ramp with nothing to spare, the deepest part of the deficit sitting at the edge of what it will hold. Read against the advantage alone, 2 to 9 against 14, it has room and the question does not arise. Two readings, opposite answers, and the source does not say which quantity his figure is measured against, so neither is asserted here. It is also why the line falls where it does between boards. An unmodified UD2.7 needs its primary at or above the loaded secondary, which is the second of the three strategies below, while a PLL-locked board can be tuned further down than that and still be held on the upper pole.

So there are three strategies, not two

Primary below the loaded secondary, upper pole forced. The detuning rule is satisfied and the driver has to be modified. davekni calls this the ideal and says plainly that it requires either a PLL or self-oscillation.
Primary above the loaded secondary. The upper pole comes free, with no starting oscillator and no PLL, and the detuning rule is given up to get it. loneoceans sits 15.0 per cent above his loaded secondary and takes about a metre of arc for it [derived from his own two figures: a primary alone at 328.4 kHz against a secondary reading 285.6 with a metre of wire to the table, which is his stand-in for arc loading].
Lower pole, unmodified. Which is where an ordinary DRSSTC lives happily and where a QCW loses the linear ramp it exists for.

The corpus had the first and the third and treated the second as a mistake. It is a trade: the pole for the tuning. Which one is right depends on whether you would rather modify a driver or give up matching under the arc's load.

And the first of the three is the common one rather than the exotic one, which is worth saying because "the ideal, requiring a PLL or self-oscillation" reads like a fringe position:

builder      primary vs secondary
Lucasww       244 < 300
CJ            395-420 < 430-450
ours          349.9 < 426.5

Lucasww draws 1.30 m on a UD2.9 in pulse-skip mode; CJ draws 2.39 m and is the only one of the three with both frequencies measured rather than computed. All three sit with the primary below the loaded secondary and work the upper pole. So the configuration this page spends its length on is the one builders actually end up in.

Lucasww does not state his primary frequency. It is implied: the triplet he does give (300 kHz, k = 0.44, an upper pole of 370) only closes with a primary at 244. Treat it as inferred rather than reported.

Two external checks of the pole equation

The first is the sharper of them, because nothing in it comes from a description: davekni's oversized QCW in HVF 2397 is specified by component values, and he states the pole he measured.

primary   5.6 uH · 64.8 nF
            -> 264.2 kHz
secondary 11.8 mH · 32 pF
            -> 259.0 kHz
k = 0.71

full coupled pair, both
resonances in
         -> upper 485.8
         -> lower 200.0
he measured    485
             +0.17 %

Which equation matters here, and it is not the familiar one. His two circuits are not at the same frequency, so f/√(1−k) does not apply: it assumes they are, and fed his secondary alone it returns 481.0 while his primary alone returns 490.6 [derived]. The form used above is the full coupled pair, f²± = [f₁²+f₂² ± √((f₁²−f₂²)² + 4k²f₁²f₂²)] / [2(1−k²)], which takes both resonances and returns one answer. It reproduces both of his poles, 485.8 against the 485 he measured and 200.0 below, and the question of which circuit is the natural frequency never arises because the equation is given both.

From parts to a measured frequency with nothing adjusted in between, and at k = 0.71: twice as tight as this corpus's coil, so the agreement is not the equation being fitted to one case. It also puts his lower pole at 200.0 kHz: at that coupling the two are a factor of 2.43 apart, which is what the QCW habit of tight coupling looks like taken further than anything else on this site takes it.

It runs backwards too, and backwards it produces a number nobody published. Take his k = 0.71 and the 485 he measured, invert the familiar form, and the natural frequency underneath the pair comes out at 261.2 kHz, against the 259.0 his inductance and capacitance give in the block above: 0.84 per cent, from two routes that share no input. The limit that arrives with it is worth as much as the agreement. He holds primary and secondary at about the same 260 kHz, so his coil cannot separate the two circuits as inputs. That limits what his case tests, not what the equation needs: the full form takes both and does not ask which is which.

And one against a coil described rather than specified

MS.Lab publishes enough to do it: primary 313 kHz, k = 0.5, and a working frequency reported as 480 to 490 kHz. Those three are not linked, here or anywhere else on this site, because the page they were read on has not been found again. Take the check below as reported rather than as citable, and weigh it accordingly: it is the only one of the three on this page whose inputs a reader cannot go and look at.

secondary measured in place
  356 -> upper 476.0   -0.8 %

secondary measured on a bench
  411 -> upper 525.2   +9.4 %

This one agrees to within a per cent, and only with the secondary measured where the coil actually stands. That is a second lesson at no extra cost: the number that goes into this equation is the resonance in situ, not the one from the bench, and the two differ here by 55 kHz.

Which one a ramped coil can actually use

Everything above is true of any coupled pair. This is the part that is only true of a machine with a ramp, and it is the reason the choice matters at all: the frequency slides the whole way down, so the question is not which pole is better but which one is still doing the job at the end.

Coupling splits the resonance in two and the driver holds one of them for the whole ramp. It is usually put as a preference, and put that way the lower pole wins most of the comparison. That is the wrong question. The question is which one still does the job, and on this machine there is only one answer. And worse, it is not the one the coil goes to by itself.

Start with where the driver lands if nobody stops it. With plain feedback off the primary's current zero, it settles on the pole nearest the primary's own frequency, and the primary is at 349.9:

distance from the primary's 349.9
           to lower  to upper  lands
no arc         32.9     169.2  lower
1.5 m arc      54.2     113.4  lower

Both ends of the ramp, and not close. So forcing the upper pole is not a refinement: without it the machine runs in the mode it was not built for, and that is what the starting oscillator or PLL is for. Which changes to make to an ordinary board, in the words of the people who made them, is converting a DRSSTC driver.

Then the reason it matters. The first one is not frequency at all, it is that the lower pole breaks the ramp, which is the entire machine:

Anders Mikkelsen: Lower pole operation leads to the tank drawing more power as the secondary gets additional capacitive loading from the arc, making it hard to slowly ramp the power up. Most people prefer to run QCWs on the upper pole as it gives a more linear response of power vs bridge voltage.

A QCW exists to raise power smoothly and predictably. A pole on which power rises by itself as the arc grows does not degrade that job, it cancels it. An ordinary DRSSTC does not care, because it has no ramp. Which is exactly why the lower pole is ordinary there and wrong here.

And the same answer arrives from the band. Straight swords need 300 kHz and up, which is Anders' threshold, and the lower pole leaves that band almost immediately:

arc     sec   lower   upper
0     426.5   317.0   519.1
1.0   374.8   302.9   477.2
1.2   366.1   299.9   471.0  band
1.5   355.1   295.7   463.3
2.0   338.2   288.4   452.4

band = lower leaves 300 kHz

The lower pole is out of the band at about 1.2 metres. The upper one does not leave it at all, and the reason is structural rather than a matter of how far the arc gets. Grow the arc without limit and the loaded secondary goes to zero, at which point the upper pole is f_pri/√(1−k²), 385.8 kHz here [derived, 349.9 / √(1 − 0.421²)]. That is a floor, and it sits above the threshold. There is no arc length at which the upper pole leaves the band, so no length is quoted and the arc capacitance never has to be stretched to find one.

The two do not sink the same way, and the difference is not the one this page used to claim. Differentiate the pole equation along the ramp and the lower track runs at a near-constant rate the whole way: 13.4 kHz per metre at the start, 14.6 at its steepest around 1.5 m, 13.7 by three metres. It never accelerates. What changes is the upper one, which starts at 54.7 kHz per metre and is down to 12.4 by three metres: a factor of four of braking. The two rates only meet at 2.73 m.

This page previously said the lower pole "starts falling fast" past 1.65 m, where the secondary crosses the primary. That was wrong, and it is worth saying why rather than quietly deleting it. 1.65 m is where the two natural frequencies cross, and it is a real point: it is where the detuning rule is exactly satisfied. It is not where the modes trade character. At k = 0.421 the coupling is strong enough to smear that handover across metres instead of snapping it at the crossing, and the arithmetic puts the crossover at 2.73 m. Which is almost certainly further than the machine goes. The pole structure comes apart at or before 2.81 m, sooner given a large split, so the handover probably never happens inside a real bang at all.

Be honest about what the lower pole does give, because it is not nothing and it is not only efficiency. It reaches its own pole unaided, it drifts less with a spread in coupling, and it is easier on the hardware: copper loss follows skin depth and goes as the square root of frequency, and the gate supply's average current goes as Q_g·f. With no arc that is 1.28 times less copper loss and 1.64 times less gate current; under a 1.5 m arc, 1.25 and 1.57. If heat ever appears as a mark against the lower pole, the sign has been flipped.

And the upper pole has one numerical advantage of its own, which belongs here for the same reason the lower one's do: 12 dB more power into the arc in davekni's simulation. Twelve decibels of power is a factor of 16, not the factor of four this corpus used to print. Four is what you get by applying the amplitude form, 10^(dB/20), to a quantity that was stated as power.

Count them. For the lower pole: it reaches itself unaided, it drifts less with a spread in coupling, it moves a smaller fraction across the ramp, and it costs less in copper and less in gate current. Five. For the upper: the ramp stays linear, the band stays above the sword threshold, and the arc gets more power. Three. And still the upper, because two of those three are not about quality at all; they are about whether the machine does its job. Efficiency in a mode that will not ramp and makes a branching discharge instead of a sword is efficiency in the wrong machine.

But "the upper pole is better" is the wrong sentence too, and this is the part that does not travel. Which poles exist within reach of a bridge is decided by where your secondary lies, not by preference. On a loaded f0 above about 380 kHz the upper pole climbs out of what a bridge can switch and only the lower is available; below about 358 the lower is under the sword threshold and only the upper is. The window where you genuinely get to choose is those twenty-two kilohertz. Ours sits at 352 under the arc, so there is no choice here, and Anders' sits far below it: he reports arc loading pulling his secondary "all the way down to 180 kHz" (HVF 3140). The 332.65 this line used to carry is withdrawn. He publishes no such figure anywhere, it was doing nothing but supplying a percentage, and what does come out of his own numbers is worked through on the frequency page. The pole is not selected. It is computed from the secondary you wound.

And all of it has a scope: the start and the middle of the ramp. Past a certain loading the pole structure itself comes apart: the centre frequency and one pole go away and a single working frequency is left, near the primary's resonance, which on this coil is 349.9 kHz and therefore still above the sword threshold. The condition and the source are on this page, above. At that point there is no pole to choose and nothing above this line applies. And read "a certain loading" as possibly nearer than it sounds. Lenz adds that a large split between the two resonances brings the collapse on earlier, and this coil's split is 21.9 per cent measured against the primary, which is large. The scope of everything above may be narrower than an arc length computed from the condition alone would suggest.

The trap

The secondary frequency you measured is the resonance with the primary taken away. In the assembled system there is no such mode.

The same machine makes the point without a hypothetical: 426.49 kHz is what the secondary measures on its own, and with the primary beside it the real working points with no arc out are 317.0 and 519.1. Neither of them is 426.5, and nothing in the assembled coil is.

One asymmetry that is not about power

The lower pole is where the system lands unaided and is easier on everything, and it carries a higher flashover risk when the windings come together. On the upper pole the secondary's field is lower inside the winding, which is why the usual advice about tight coupling being survivable holds there and not on the lower.

What a room full of builders does with this, on a different forum

The argument on this page is not this site's invention, and the cleanest evidence is a thread nobody involved appears to have thought was about theory. On hvdiy in 2016 a builder posted a method for measuring the secondary's frequency: hold a scope probe in the air near the running coil, not close enough to be struck, and if the trace is a sine then "然后示波器显示的频率就是TC次级的频率了", the frequency the scope shows is the TC's secondary frequency. No current transformer, no ring, just a probe and the coil (hvdiy 34298).

Two people objected within ten minutes, and between them they took the whole page apart.

The first asked what happens when the two are far apart: "如果初级的频率和次级的频率相差较大，不在次级的通频带之内，这种方法是否可行？", if the primary's frequency and the secondary's differ a lot, outside the secondary's passband, is the method still viable? The author's answer is the interesting part, because it is a measurement rather than an opinion: "我初级给的频率是200多k，次级输出460k", his primary is driven at two hundred and something kilohertz and the secondary reads

A factor of more than two is not a pole pair. Whatever that probe was hearing, it was not the mode his primary was driving, and the second mode is where that goes.

The second objected to the noun: "这样是测打火频率吧，并非固有频率吧", this measures the sparking frequency, not the natural frequency. Which is this page's whole thesis in nine characters, arrived at by somebody looking at a photograph of a scope.

And a third, eight days later, put the correction in the form this page spends several thousand words on: the measurement gives the coil's own resonant frequency, and 实际频率要大于这个测试频率才真的谐振, the actual frequency has to be higher than the measured one for the system to be truly resonant. That is the upper pole, described by somebody who does not use the word.

The frequency falls, the pole does not change

The arc adds capacitance and pulls the secondary's frequency down, and both poles come down with it. Same mode. You stay on whichever pole you landed on at the start, for the whole bang.

Until they stop existing

Which is the other half of the same subject rather than a separate one: a pair of resonances that can be moved can also be destroyed, and the condition for it is exact rather than a vague "eventually". Udo Lenz, on the Tesla Coil Mailing List in February 2013:

system loading makes the center frequency and one pole go away, leaving only one frequency near the primary resonance… That will happen when Qsec drops below 1/k.

Near the primary's resonance, not the secondary's. Which this page had backwards until the source turned up. It also has to be that way round: the condition is written on the secondary's Q, so the secondary is the resonator being damped out, and a frequency cannot be set by the resonator that was just killed.

Secondary Q, drag right as the arc grows

Coupling

Secondary Q

199

Rule of thumb 1/k

2.38

Lower peak

317.0 kHz

Upper peak

519.0 kHz

response of the coupled pair primary's own resonance, 349.9 kHz, fixed below 300 kHz, straight swords stop forming

What to watch. At the top of the ramp the arc is small, the secondary's Q is high and there are two clean peaks, the poles. As the arc grows it loads the secondary, Q falls, and the peaks broaden and slide towards each other. Past a certain point they are no longer two: a single working frequency is left, and it sits near the primary, not near the secondary. That is the resonator the arc did not damp.

Where the threshold really is. The condition quoted for this is Q_sec < 1/k, which is 2.38 on this coil. In the model drawn here the two peaks actually stop being distinguishable at Q = 2.12, about 11 per cent lower. 1/k is a rule of thumb rather than an exact boundary, and it is quoted here as one. Lenz adds that a large split between the two resonances brings the collapse on earlier still, and this coil's two uncoupled resonances are 21.9 per cent apart measured against the primary. That is not the gap between the poles drawn above, which is 39 per cent of the upper pole on the same coil.

Why coupling is on the panel. Switch to k 0.15 and the same damping wipes the structure out far sooner. An ordinary DRSSTC under a heavy arc has no pole structure left to choose from. High coupling is what keeps it standing, which is one more reason a QCW is wound for k 0.4–0.5.

Derived, not measured. Primary 16.959 µH with 12.2 nF, secondary 9.925 mH with 14.031 pF, both from JavaTC; the curve is the steady-state response of the coupled pair, normalised to its own peak at each setting so the shape stays visible. Damping is applied as a series resistance in the secondary, which is a stand-in for the arc, not a model of one.

Sweep the coil with no arc out and there are two sharp peaks, one at 317 kHz and one at 519, with a dip between them. Now load it. The secondary's Q is what the load spends: 199 with nothing drawn, and 7.6 with a metre and a half out, which is 1/199 + Z/R_arc on this coil's 26.6 kΩ against the arc's 210 [derived]. The 4 often quoted beside that figure belongs to Hydron's coil, whose impedance is nearer 51 kΩ, and it is the arc's resistance divided by whichever impedance you hang it on. As it falls, both peaks get broader and lower, and they slide towards each other. The dip fills in. Past Q = 2.375 there is no dip left and no two peaks: one broad hump remains, near 349.9 kHz, which is the primary.

The secondary stops being a resonator, the primary never stopped being one, and what is left is the primary's own resonance with a lossy load hung on it.

Which explains a piece of QCW practice that otherwise looks like superstition. A DRSSTC runs k around 0.15, and 1/0.15 is 6.67. At a loaded Q of 4, which is what a DRSSTC of that impedance runs, that coil is already past the condition with a margin of 0.6, and at k = 0.25 it is exactly 1.00, on the line. Our own coil is the one to read at 7.6 rather than 4, so at 0.421 the margin is 3.20 [derived] and not the 1.68 this page used to print, which paired our coupling with somebody else's loaded Q. Tight coupling is what keeps the pole structure alive under load. So the QCW habit of running k = 0.4 to 0.5 is not only about the arc-to-secondary ratio it buys. It is also what stops the two poles from merging the moment the arc gets long. That last step is ours; the condition and the numbers are Lenz's.

And the same threshold turns up in Lenz's other well-known result, which is worth putting beside it because nobody usually does. On optimal secondary impedance, quoted in Kaizer's DRSSTC guide, he writes that power transfer into the arc peaks, for a given primary current and with the two tuned together, "approximately when Qsec = 1/k": the same Qsec, and the same number. So the point of best power transfer and the point where the pole structure comes apart are the same point. A coil loaded as hard as it usefully can be is sitting on that boundary rather than safely inside it, which is a different picture from "the poles collapse eventually if you overdo it". Both statements are his; setting them side by side is ours.

Two things follow that are easy to miss. It arrives sooner than the arithmetic says: Lenz adds that a large split between the primary and secondary resonances brings it on earlier, and this coil's split is 21.9 per cent measured against the primary, so any arc length computed from Qsec < 1/k alone is a ceiling rather than an estimate. And past that point there is no pole to choose: "for large system loads, you'll have just one ZCS frequency, so there is no choice between poles at this point." Everything on this page belongs to the start and the middle of a bang.

An arc reaching an earthed object does the same thing, but only across a band of strike resistances, and it does three other things besides. That is its own event rather than a property of coupled resonators.

The lock points go before the poles do

A sweep sees the pair of peaks. The driver sees something narrower: the frequencies where the primary current's phase crosses zero, the ones it can lock to. A coupled pair has three of those, the same three the sweep shows as two dips and the peak between them, but what the driver does at each is not symmetric.

Every figure in this section is on the kechuang model whose component values are given further down, both circuits near 205 kHz, and not on the 349.9 against 426.5 the rest of this page runs. It is the machine the widget below is built from, and the argument is about the shape rather than the frequencies.

k = 0.3, no arc, both circuits near 205 kHz
180.5 kHz    62.7 A    phase slope -21.97 °/kHz    holds
204.9 kHz     2.1 A    phase slope +11.31 °/kHz    repels
245.1 kHz   114.0 A    phase slope -21.81 °/kHz    holds

The crossing happens at all three, but a lock is only stable where the phase crosses going down, from above. Where it crosses going up the smallest drift is amplified and the driver is thrown off. So the middle frequency is not a lock point, it is the boundary between the two basins, and it is the secondary's own resonance: tuned from 0.80 to 1.10 of nominal the two outer points move but the watershed stays at 204 to 206 kHz.

And the middle does more than repel: the ground between the poles is not neutral territory to pass through. On the build in HVF 3057, with a lower pole around 75 to 81 kHz and an upper around 115 to 121, the space between them holds an antiresonance, a zero near 100 kHz, and the transitions across it drew a surge of about 135 A. So a driver that hunts, or a tuning that lets the operating point wander over the middle, is not merely inefficient in there; it is briefly exposed to a current excursion. That machine is continuous-wave, which is already outside what the rest of this page describes, so whether the surge scales down is not established. The 0.8 on that page is not the coupling those poles belong to, and the arithmetic said so before the schematic did. The pair sits 1.42 to 1.61 apart, which the equation this page runs on everything else puts at k = 0.34 to 0.45, where 0.8 would want them a factor of three apart [derived]. davekni's own line resolves it: "L1 and L2 with 0.8 coupling are the 5-turn DRSSTC primary coil", and L3 is the secondary. The 0.8 is between the two halves of a centre-tapped primary, not between primary and secondary, so nothing is misread and the pole pair stands.

How long the arc is

no arc yetgrowinglong arc

Where the primary is tuned, against the secondary's own frequency

0.80 belowtuned together1.15 above

Coupling

k 0.050.300.60

A zero crossing driver does not choose a pole. It chooses a phase zero, and there are three of them. The two outer ones are the poles. The middle one sits between them, where the primary current is at its smallest, and a driver's capture condition is satisfied there just as well as at the other two.

What separates them is the slope, not the height. A lock is stable only where the phase falls through zero: a small drift then pushes the loop back. At the middle crossing the phase rises through zero, so the same drift pushes the loop away. It is not a place to sit. It is the watershed between the two poles, and it stands at the secondary's own frequency: move the primary anywhere you like and the watershed does not follow it.

Which is why neither pole is the strong one by nature. Tune the primary below the secondary and the lower pole carries the larger current; tune it above and the upper one does. The crossover is around nine tenths of the secondary's frequency. A self oscillating driver lives on primary current, so it settles wherever that current is larger, and the tuning is what decides that, not the coil. That is the whole of the old rule that a QCW is tuned above its secondary: it puts the starting point on the upper pole's side of the watershed.

So the capture is bistable, and only while the arc is short. Grow the arc and the lower pole loses its current and walks up towards the watershed until the two meet and cancel each other. After that there is one phase zero left and it is the upper pole. The driver is not holding the upper pole because it is the better one. It is holding it because by then there is nowhere else to be.

Which is a third job for the wick nobody lists. A shelf at the start loads the coil before the ramp begins, and a loaded coil has already lost the lower pole and the watershed. The shelf does not only light the arc. It removes the driver's choice.

Below a coupling of about 0.06 the third crossing does not exist at all, and capture is unambiguous from the start. Every practical coil is far above that, which is why the ambiguity is the normal case rather than the exception.

The model is rb_sama's, published with its netlist, and this corpus reproduced his own measurement table from it to within 0.025 dB before using it for anything. Primary 60 µH and 10 nF, secondary 35.4 mH and 17 to 26 pF as the arc grows, 310 V of drive. It is a simulation and not a measurement, and the phase slopes are the part of it that matters rather than the exact frequencies.

Then grow the arc:

Rload     lower           boundary        upper
1000k    180.5 / 62.7 A   204.9 / 2.1 A   245.1 / 114.0 A
 500k    170.3 / 12.3     176.7 / 5.2     232.9 / 155.7
 300k    one lock point                   229.9 / 134.5

The lower point weakens and slides up into the boundary, and at a few hundred kΩ of load the two meet and annihilate. Below that a single lock point is left, the upper one, and there is no choice to make.

That is a second threshold, and it is not the pole collapse. The lock points go well before the peaks merge:

k      lock points gone   gain peaks merged   ratio
0.3        458 kΩ              126 kΩ            3.6×
0.6        213                  62               3.4×

The driver loses the lower option about three times earlier in arc resistance, and so noticeably earlier in arc length, than a sweep loses the lower peak. A coil can be firmly on the upper pole with the lower one still plainly there on a sweep.

In ohms, the condition is on resistance, not capacitance

Qsec < 1/k is written on the secondary's Q, but Q is the arc's loading in disguise, and putting it in ohms says which term actually sets the collapse. The arc hangs on the topload as a parallel resistance; referred into the tank it damps the secondary, so Qsec ≈ R_arc / Z, where Z = √(L/C) is the secondary's characteristic impedance, 26.6 kΩ here. The condition becomes

Qsec < 1/k   ⇔   R_arc < Z/k = 63.2 kΩ

The poles come apart when the arc becomes more conductive than that, and there is no capacitance in it at all. Everything on this page about the collapse, the Q under arc and the ringdown rests on the arc's resistance, the worse-known half of it.

Worse-known, and worse than it needs to be until it is put in one form. **Half the published figures are quoted as a series R–C where the criterion wants the parallel one**; compared as written they spread by 28× and flip the verdict. Reduced to the same form and to a 1.5 m arc on this coil (R ∝ 1/L):

source                      as quoted   R∥ kΩ   margin
Anders, 8 pF/m ∥ 75 kΩ·m    parallel       50   0.79×
Uspring, 47°                parallel       77   1.22×
Weston, 210 kΩ·ft           series         79   1.26×
Uspring, 55°                parallel      103   1.63×
Anders, 16 pF + 8 kΩ        series        225   3.56×
textbook, 220 kΩ + 1 pF/ft  series        258   4.08×

Five of the six put the poles alive at 1.5 m. Their margins run 1.22 to 4.08×, and this corpus's own row sits inside them rather than outside: the 210 kΩ it publishes for a metre and a half of arc is 3.32× on the same form, R∥ over Z/k [derived, 210 / 63.2]. Across those five the conversion leaves the resistances spanning 3.3×, not the 28× the raw quotes threaten.

The top row is the exception and it is the newest. Anders Mikkelsen revised his own series figure, the other row here carrying his name, on the grounds that "a parallel one is more representative", and published 8 pF per meter in parallel with 75 kohm*meter fitted to his measured 50 degrees near 340 kHz and 60 near 460 (HVF 3140). The constant reproduces both angles: 1.5 m of it is 12 pF and 50 kΩ, whose reactance is 39.0 and 28.8 kΩ at those two frequencies, so arctan(R/X_C) is 52 and 60 degrees [derived]. That is the only row of the six fitted to measured phase at two frequencies in the parallel form the criterion actually wants, and it lands below Z/k: on it, an ordinary metre-and-a-half arc takes this coil's poles apart by itself. Including it the six span 5.2× [derived, 258 over 50] and the verdict on the collapse is five to one rather than unanimous. The same builder is in the table twice, on either side of the answer, which is the honest state of the question and not an argument for dropping either row.

That 3.32 and the 3.20 at the top of this page are one margin counted two ways, not two estimates. The rows above use the arc alone, R∥ over Z/k; the 3.20 puts the winding's own 199 in beside it, so Qsec is 7.59 rather than 7.90 and the product with k comes down four per cent. Neither is the 1.68 this page used to print. That one paired our coupling with the loaded Q of 4 Uspring measured on Hydron's coil, whose impedance is near twice ours, and it is withdrawn here as it is on the coupling page.

Two of these rows earn a footnote for disagreeing with themselves rather than a share of the spread:

The textbook "220 kΩ + 1 pF/ft", in parallel form, implies 0.48 pF/m of arc capacitance, a tenth of every measurement. It was written for a spark gap and does not carry to a QCW streamer.
Anders' "16 pF + 8 kΩ" implies a voltage-to-current angle of 74°, well outside every other row here. His capacitance is corroborated by the detuning; the resistance is the suspect term: 52° at 16 pF wants 22 kΩ in series, not 8.

One word in that comparison used to be wrong. This page said the band Anders sits outside of was the measured band is 45 to 60, and neither end is a measurement: they are the 47° and 55° rows of the table above, both Uspring's, one man's two figures rounded outward. The measurement exists and it belongs to somebody else. davekni reads the angle off an instrument and states it from the capacitive end rather than the resistive one, which is the complement of the convention used here: 40 to 45 degrees is the common case on his coil, and one arc came in under 30. Converted, that is 45 to 50 in these terms with the outlier past 60 [derived, the conventions are complements], so the band barely moves and now has an instrument inside it instead of only a name. At 45 the two conventions meet, which is where the arc's resistance equals its reactance, and that is Steve Conner's hungry streamer arriving at the same place from a model rather than a probe.

And the threshold itself checks out on someone else's coil

Every row above is a published arc resistance asked whether it clears the threshold. None of them tests the threshold. R_arc < Z/k is our own reduction, and the one check it had not had was against a coil solved independently of this corpus.

A model published on the Chinese forum kechuang supplies one, and it is neither our circuit nor our arithmetic on the loading. Its arc capacitance grows with the load, so Z moves as the arc does and the collapse point has to be solved self-consistently. Our R_arc < Z/k against that solution, at their component values, across coupling:

k      our Z/k      on their model     discrepancy
0.3    123.6 kΩ     125.8 kΩ           -1.8 %
0.4    92.1         95.9               -4.0 %
0.6    61.0         61.6               -1.0 %

One to four per cent from k of 0.3 up. This is the first check of the collapse criterion itself, as distinct from the pole frequencies it acts on, which the measured coils earlier on this page already confirm. They are two different claims: that the poles sit where the equation puts them, and that they come apart where R_arc crosses Z/k. Only the second is settled here.

The values, so it can be repeated: secondary 35.4 mH and 17 to 26 pF, primary 60 µH and 10 nF, k from 0.2 to 0.6.

It also dissolves an apparent contradiction. Their collapse resistance is about 126 kΩ and ours is the 63.2 kΩ above, and the two do not disagree: the difference is entirely in the machine. Their Z is around 37 to 46 kΩ against 26.6 here, and Z/k carries the rest, 123.6 against 63.2, a factor of 1.96 against the 1.99 the two solved collapse resistances stand apart [derived]. The 11.2 to 39.6 kΩ band that a ground strike carries is a different quantity and does not belong in this comparison. That one is the range of strike resistances over which a lumped model makes the structure come apart, and it has a bottom edge because a low enough channel stops damping and starts shorting. Z/k has no bottom edge.

And past it is a place you can work, not a place you have failed

Which this page had implied without ever saying, and it is wrong to imply. Anders Mikkelsen took 3.18 m operating beyond the collapse, and describes the regime rather than apologising for it: only one ZVS frequency, and arc loading with minimal effect on the operating frequency.

That last clause is the diagnostic, and it is a better one than any threshold calculation, because it needs no capacitance model:

poles alive
  frequency tracks the arc

past collapse
  frequency stops caring

A machine whose frequency stops responding to arc length has not drifted. It has left the two-pole regime, and every argument on this page about choosing between poles stopped applying at that moment.

The argument nobody should flatten

Uspring's position is that on the lower pole the arc pulls the secondary towards the working frequency and the transfer improves, while on the upper it pulls away and the coil detunes. davekni's is that the upper pole is worth 12 dB of arc power at one simulated load. Both observations are real, and the temptation to declare a winner should be resisted.

What the community does in practice is a compromise: tune the primary below the secondary, and force the upper pole from the driver. Then the arc brings the frequencies together as the ramp goes on instead of separating them. davekni puts it as being able to force upper pole operation even with the primary tuned lower, so that arc loading lowers the secondary resonance and keeps the frequencies close over the whole of the arc's growth.

Getting the inputs without trusting a model

Every number on this page comes out of k and the primary's resonance, and both are usually taken from a simulator. One sweep of the cold coil measures them instead, and it needs nothing running.

The other bench route to k, reading the primary's inductance with the secondary open and then with it shorted, asks the shorted winding to behave as a short circuit at the frequency you are working at, and that assumption is what gives way at QCW frequencies. The sweep does not rest on it.

One sweep, three frequencies

sweep |Z| across the tank,
secondary in place, no power

two dips     = the two poles
peak between = the secondary's
               own resonance

Three readings, two unknowns, and the pole equations invert:

f+ · f-  = f_pri·f_sec/sqrt(1-k^2)

w+^2 + w-^2
  = (w_pri^2 + w_sec^2)/(1-k^2)

Solve the pair and out come k and the primary's frequency, both measured on the coil as it stands rather than computed from geometry. On the figures this page carries the inversion returns k = 0.4210 and 349.97 kHz, against the 0.421 and 349.9 they were built from, which checks the algebra, not the coil.

Three readings and not two, and that is not caution. The symmetric form that recovers k from the pole pair alone is quoted in the literature with its condition attached: it holds between identical resonators. A coil's two are not identical, which is why it fails here. On loneoceans' turn-10 pole pair, 255 and 362 kHz measured with the secondary grounded, it returns 0.337 where he states 0.306, and this page threw the formula out for disagreeing with the numbers before noticing it had been used outside the domain it declares. The two figures are not the same kind of thing either: his 0.306 is, in his own word, calculated, from the geometry of the system, while the 0.337 comes out of frequencies he measured. Ten per cent apart, one coil, one tap. His ungrounded pair at the same tap, 282 and 394, gives a third answer, 0.323 [derived].

It also comes with a free consistency test, and that is worth more than the convenience. Three readings determine two unknowns, so the third is a check: measure several configurations and the recovered k has to move the way you moved the geometry, not wander. Applied to Anders' published triplets it does wander: three of his pairs return k of 0.263, 0.421 and 0.623, which is not one coil with one coupling. Something in that set is mislabelled. Said here because the same test will catch it on your own bench, and because a number that fails its own closure check should not be quoted as a measurement by anybody, including us.

Checking which one you are on

Look at the frequency at the very start of the arc, at minimum load. It tells you immediately.

A sharp jump of about a factor of two in the middle of a ramp might be a real transition. A smooth slide downwards is normal.

The two poles are why a QCW's timing has to be tracked the whole way up the climb, and why the department diagram draws a driver with a lead in it rather than a box marked driver.

polestuningPLL