[ §1 · tuning ]

The frequency slides, and four things follow it

A DRSSTC has a frequency. A QCW has a different one at every instant of the bang, and the four things timed against it do not move in the same direction.

CommonGnd5 August 202623 min read

A DRSSTC has a frequency. A QCW has a different one at every instant of the bang, and every setting that is timed against it is only exactly right at the moment you set it.

What it is and why

A resonant circuit's frequency comes from its inductance and its capacitance, and in most electronics both of those are components. They are soldered down, they do not move, and the frequency is a property of the board.

Now let one of them change while the circuit is running. The resonance follows it, continuously, and nothing about that is dramatic on its own — the circuit is still resonant, just at a different frequency than it was a moment ago.

What is dramatic is everything else in the system that was set relative to the old frequency. A delay chosen as a fraction of a period is now a different fraction. A network that produces a fixed phase angle now produces that angle at a different number of nanoseconds. A filter's corner has moved relative to the signal going through it.

None of those parts broke. They are doing exactly what they were built to do. They were simply designed at one frequency and are now being asked to work at another, and they do not all fail in the same direction or at the same rate.

What you decide

On a coil the moving capacitance is the arc. It grows off the topload, and while it grows it adds capacitance to the secondary, which pulls the resonance down. On a DRSSTC that happens inside a few hundred microseconds and the feedback loop absorbs it. On a QCW the bang lasts twenty five milliseconds, the arc reaches a metre or more, and the frequency spends the whole time sliding.

The arc is capacitance hanging off the topload, so the secondary's own resonance falls 16.7 per cent as it grows. The pole the bridge actually switches falls 10.7, because the primary sits at 349.9 kHz and does not move, which is within 1.5 per cent of where the secondary ends a 1.5 m ramp and exactly where it would end a 1.65 m one. That is the detuning rule satisfied. The third track is the coil you get if nobody forces the upper pole: it starts below where the others finish and sinks under 300 kHz, which is where straight swords stop forming — the marker warms for that reason and not a thermal one, since a lower frequency is actually easier on the copper and on the gate supply. Derived from JavaTC's figures rather than measured, and only for the arc length the capacitance was calibrated on.

Three tracks there and not one, because "the frequency" is four numbers on this machine and only one of them is a single number. The primary is 349.9 kHz and never moves at all: nothing in it is exposed to the arc, which is exactly why it anchors the poles. The secondary's own resonance is what the arc loads, and it falls 16.7 per cent. The upper pole is what the bridge is switching — 519.1 kHz down to 463.3 — and it falls 10.7, which is 0.64 of the secondary's travel because of that anchor. The lower pole is the coil you get if nobody forces the upper one, and it starts below where the other two finish. So every setting timed against the switching is timed against the smaller movement, and every setting matched to the arc is matched against the larger one. That split is the rest of this page.

One arc length, and only one. Everything above is calibrated on a 1.5 m arc, which works out at 4.14 pF per metre of streamer. This page used to carry a 1.8 m column as well and it has been withdrawn, because the figure it rested on does not derive. Reproducing the published 289 kHz needs two wrong inputs at once: a 430 kHz base, which was retracted in favour of JavaTC's 426.49, and the toroid's 10.7 pF where the resonance actually runs on Ces of 14.031. Both err the same way, so they compounded into a number about 56 kHz low. The check that settles it: L from 426.49 kHz and 14.031 pF comes out at 9.9251 mH, and JavaTC prints Les = 9.925. The resonance is on Ces. On our own calibration a 1.8 m arc lands at 344.7 kHz, not 289. And the calibration should not be stretched anyway. The capacitance of a long thin conductor does not grow linearly with its length — it goes roughly as L/ln(L/r) — so pF per metre is a local approximation, good near the point it was fitted at and worse away from it.

Every table here is one coil, not a reference

The number under all of them is the arc's capacitance, and it is not the same on your machine. That is not measurement error, it is a property: how much capacitance a metre of streamer adds depends on the topload it grows from, on how the arc branches, on where the walls are.

Measurements on four coils put it between 3.3 and 5.0 pF per metre. The 4.14 used here is the middle of that, not a preferred value:

Weston, 2 ft      3.28  measured
Gao, QCW 1.5      3.31  measured
here              4.14  measured
Anders            5.03  measured

Anders' figure carries its own check: 16 pF at 5.03 pF/m is a 3.18 m arc, which is the length he reports. The number and the machine agree without being made to.

Two figures that used to sit in this list have been taken out of it, and the reason is worth more than the figures. A textbook isolated-cylinder capacitance gives 7.94 pF/m, and Weston's own three-foot case gives 7.66 — but his two-foot and three-foot together imply C ∝ L³·¹, which no conductor does. The cylinder formula, meanwhile, is a model of a wire alone in space and ignores the toroid that has already taken part of the field. This page carried them for a day as evidence that 4.14 was a low outlier under dispute. That was wrong, and it was wrong in a specific way worth naming: ratios were compared against per-metre figures without normalising — "about twice the topload" set beside "pF per metre" as though the two were the same kind of number. Once everything is in pF/m, 4.14 is the middle of the measured range and there is no dispute to have.

What the spread actually moves

Not much, in kilohertz. The frequencies shift by single-figure percentages across the whole range:

arc     secondary   lower pole
       3.3 -> 5.0   3.3 -> 5.0
0.5 m  403 -> 393   312 -> 309
1.5 m  367 -> 344   300 -> 291
3.0 m  327 -> 297   283 -> 266

The marks move by metres, though, and that is the part to carry away. The lower pole leaving the straight-sword band:

3.3 pF/m   1.51 m
4.14       1.20 m
5.00       1.00 m

Half a metre of spread on a mark that reads like a specification. So quote it as around a metre to a metre and a half, and compute it against your own capacitance when you have one — never as 1.20 m.

And what the spread does not touch

More useful than the caveat, because this is what transfers to your coil whatever its arc turns out to be:

The lower pole is out of the straight-sword band before 1.5 m. At every capacitance in the measured range.
The upper pole never leaves it, at any arc length this coil can reach, at any capacitance in the range.
The poles with no arc out — 519.1 and 317.0 — have no arc capacitance in them at all. There is no arc. Those two are as firm as k and the two resonances that produced them.
The order and the sign of every conclusion on this page. What moves is the kilohertz, not which pole is where or which way anything goes.

So the real decision is not what to set anything to. It is which end of the ramp you set each thing at — and the reason that is a decision at all is that the ends disagree.

The phase lead gives you least time where you need most

The lead network holds an angle. The hardware — the comparator, the logic, the transformer — needs nanoseconds. That mismatch is well known and it is the limit of the technique; what the lead is made of, and how much of it is the ZVS floor rather than the delay chain, is on its own page, and winding the inductor is another. What is not usually written down is which way round it goes, so here it is worked on the trajectory above, taking L/R such that the network gives 45 degrees at the bottom of the ramp:

phi = arctan(w·L/R)
t   = phi / w

  f kHz    phi       t
    519  48.3 deg  258 ns   start
    490  46.6 deg  264 ns
    463  45.0 deg  270 ns   end

The angle grows with frequency and the time falls. The arctangent compresses faster than ω rises, so the network delivers 11.7 ns less at the top of the ramp than at the bottom — 4.3 per cent down, exactly where the period is shortest and every fixed nanosecond of hardware delay is worth the most.

Which settles the direction. Set the lead at the end of the ramp and there is not enough of it at the beginning. Set it at the beginning and there is a small surplus at the end, meaning the bridge switches slightly early. A surplus is the gentler failure: early switching costs a little efficiency, late switching commutates into a conducting diode.

And the dead time points the same way

2·t_dead·f is directly proportional to frequency, so a fixed dead time takes its largest share of the period at the top of the sweep. On the 377 ns figure that already appears in dead time is a fraction, across this trajectory:

39.1 % of the period at 519 kHz
34.9 %              at 463 kHz

Same setting. Four points of period given back purely because the period grew.

Where you are on the ramp — the upper pole falls as the arc grows

Frequency

519.1 kHz

half-period 963 ns

Dead time — fixed 377 ns

39.1 % of the period

the setting never moves

Phase lead — fixed network

258 ns

48.2 degrees

Left for switching

328 ns

half-period minus both

What to watch. The bar is one half-period, drawn to scale. Drag the ramp and it grows, because the frequency falls. The two blocks on it barely move: the dead time is a fixed number of nanoseconds by definition, and the lead network — which holds an angle, not a time — delivers only 12 ns more across the whole ramp. So both take their largest bite exactly where the bar is shortest, and that is the start.

Which is the conclusion you cannot reach one page at a time. The dead-time page is honest about dead time and the phase-lead page is honest about phase lead, and neither can know the other is looking the same way. Set both at the top of the ramp and they are safe all the way down it.

Why the lead moves at all, and why so little. The network holds φ = arctan(ω·L/R) and the hardware needs nanoseconds: t = φ/ω. The arctangent compresses faster than ω rises, so the angle grows while the time shrinks — the wrong way round, and worst where every fixed nanosecond of comparator and transformer delay is worth the most.

Ours, and derived. Trajectory 519.1 → 463.3 kHz on a 1.5 m arc, off JavaTC figures. Dead time 377 ns is the setting the corpus already carries. The lead network is anchored to 45 degrees at the bottom of the ramp — move that anchor and the spread moves with it, but not the direction. Neither line is a measurement.

So the two worst cases share an end, and it is the start. That is the one conclusion in this subject that cannot be reached by reading the four pages separately: each of them is honest about its own parameter and none can know that the other is looking the same way. Set both at the top of the ramp and they are safe all the way down it. Both calculations here are ours, off phi = arctan(w·L/R) and 2·t_dead·f on a published 519-to-463 trajectory. Neither is a measurement, and the 11.7 ns depends on where L/R is anchored — 45 degrees at the bottom of the ramp, here. Move the anchor and the spread moves; the direction does not.

But the things you can see are at the other end

Detuning is the opposite case. Anders' rule is to detune by as much as the arc detunes you, and you cannot check that against no arc: the evidence is the current envelope, and the envelope only says anything once the coil is loaded. Turns and tap follow the same logic. So does the tank current, which is not a point on the ramp at all — an MMC integrates the whole envelope.

Which is the actual subject of this page. There is no single operating point to tune a QCW at. "Tuned" means tuned along a trajectory, and the settings do not all read out at the same place on it.

Where on the ramp to set each thing

set at the START
  dead time     2·t_dead·f worst
  phase lead    least time up top

set at the END
  detune, turns  match the arc

set over the WHOLE
  MMC current    envelope total

The top two are arithmetic and the bottom two are evidence. That is why they land at opposite ends: you compute what you cannot see, and you look at what you cannot compute.
Do the start-of-ramp pair first. They are the ones that end bangs, and neither of them needs an arc to be set, so they can be settled before the coil is making anything worth watching.
Then tune against a full arc without going back and touching the first two, because nothing you see down there is evidence about the top.
The tank current is checked last and over the whole bang, because it is the only one that is not a point at all.

And no measurement transfers across the ramp

Which generalises. A capture is taken at an instant, so it is a statement about one frequency and not about the bang. A gate waveform photographed at the start of the ramp is not evidence about the end. A dead time judged at full arc says nothing about the first millisecond. An overcurrent threshold set where the current happens to be visible is being applied across the whole sweep.

The cure is not more captures. It is knowing, for each number, whether it gets better or worse as the frequency falls.

Two more that move, and are owned elsewhere

The envelope and the bank under it: read the envelope for the tuning instrument, and chosen by RMS current for what the shape costs the capacitors.

And one of them is which pole you are on

The frequency moving is also what decides which of the two coupled modes a ramped coil can use, and that argument got long enough to leave. It is with the poles now, in full: where the driver lands if nobody stops it, why forcing the upper pole is required rather than preferred, the three strategies, and the point past which there is no pole to choose at all.

What belongs here is only the part that is about the ramp. The detuning rule and the gain criterion contradict each other by construction, so following the first commits you to modifying the driver:

detuning rule
  primary ~= loaded secondary
  at the arc you designed for

Anders' gain criterion
  primary ABOVE loaded secondary
  for the upper pole to win

Put the primary at the loaded secondary and it sits below it everywhere short of the design arc, which is the whole useful part of a ramp. davekni says the same thing from the other end: the ideal is the upper pole with the primary tuned below the secondary, and that ideal requires either a PLL or self-oscillation. Detuning and the driver modification are not two decisions. The first makes the second compulsory.

The numbers

The trajectory used throughout: 519.1 kHz to 463.3, a fall of 10.7 per cent, on a 1.5 m arc. That is the upper pole — see the section below on why that is the one that belongs here, and why it is not the number you will find quoted for the same coil elsewhere.
Phase lead across it: 258 ns at the top against 270 at the bottom. 11.7 ns of spread, 4.3 per cent, with the network anchored at 45 degrees at the bottom.
Dead time across it: 39.1 per cent of the period against 34.9, on a 377 ns setting.
Both worst at the top. Set both there.

Which frequency, though — and it is not the secondary's

There are two different numbers in circulation for how far a QCW's frequency moves and they differ by a factor of nearly two, and neither of them says which quantity it is measuring.

Arc length — drag to run the ramp

Primary — tapped, fixed

349.9 kHz

0.0 % — nothing in it sees the arc

Secondary's own

426.5 kHz

0.0 %

Upper pole — what the bridge switches

519.1 kHz

0.0 %

Lower pole — where it lands unforced

317.0 kHz

0.0 %

What to watch. One thing moves — the arc's capacitance — and four numbers respond to it differently. The primary never moves at all, because nothing in it is exposed to the arc, and that is exactly why it anchors both poles. The secondary is what the arc loads and it travels furthest. The poles are made out of both resonators, so they travel somewhere in between: the upper one covers about 0.64 of the secondary's fall on this coil.

The two marks on the axis. At 1.20 m the lower pole drops under 300 kHz, which is Anders' threshold for straight swords — from there on that mode is out of the band the machine was built for. At 1.65 m the secondary passes the primary on the way down. That is a crossing of the two uncoupled frequencies, and it is worth marking — but it is not where the modes trade roles. Differentiating the pole equations gives the lower pole a flat rate of about 14 kHz per metre for the entire ramp, peaking at 1.65 m and slowing after it; the upper pole is the one that changes, decelerating from 54 to 12 kHz/m. The two rates only cross at about 2.7 m, and so does the exchange of modal character — at k 0.421 the coupling is strong enough to smear that exchange over metres rather than snapping it at the crossing.

And 2.7 m is probably not reachable. Pole collapse sets in at 2.81 m by the undamped condition, Lenz adds that a large split brings it earlier, and this coil's split is 21.9 per cent. So the rate crossing is a property of the arithmetic more than of any bang you will fire.

Which number belongs where. Dead time and phase lead are timed against the frequency the bridge is commutating, which is the pole — 10.7 per cent. Detuning and turns are set against the loaded secondary — 16.7. They are not roundings of each other, and quoting either without saying which one it is, and at what arc length, is the mistake this whole page exists to prevent.

Derived, not measured. Secondary 426.49 kHz on Ces 14.031 pF and Les 9.925 mH, primary 349.9 kHz, k 0.421 — all from JavaTC. Arc capacitance 4.14 pF per metre, calibrated at 1.5 m. Past about 2 m that calibration is being stretched: the capacitance of a long thin conductor goes roughly as L/ln(L/r), not linearly, so the trend is right and the kilohertz are not. One measurement of the frequency at the end of a real ramp replaces this entire chart.

The pole moves less than the secondary because the primary does not move at all. It sits where it was tapped, at 349.9 kHz, whatever the arc does, and a pole is made out of both resonators — so the primary anchors it and the secondary's 16.7 per cent arrives at the pole as 10.7. A factor of about 0.64 on this coil.

Which decides which number belongs on this page. Dead time and phase lead live at the frequency the bridge is commutating, and the bridge commutates the pole. The 10.7 per cent is the right one here, and it is not a rounding of the other.

The secondary's own movement is the right number somewhere else: detuning and turns are set against the loaded secondary, which is what the primary's page is choosing turns for and what a QCW adds capacitance is working in.

And one of these is a measurement rather than arithmetic. Anders Mikkelsen's machine ran 409 kHz down to 335 across a ramp, 18.1 per cent, measured. Everything else on this page is computed. Compare it with the right row. What you measure on a running coil is the frequency the bridge is switching, which is the pole — the secondary's own resonance is not observable while the machine is running. So his 18.1 goes against our 10.7, pole against pole, and that is a factor of 1.7 apart. It is not agreement, and calling it agreement means doing the exact substitution the section above exists to stop. The gap is explainable: his k is 0.497 against our 0.421, his primary and secondary sit at a different spacing, and his arc length at the end of that ramp is not recorded. Explainable is not the same as agreeing, and only the first one is honest. What his numbers do give away is better than a coincidence of percentages. His secondary's own frequency is 332.65, and his ramp ends at 335 — the working frequency arrives within 0.7 per cent of the secondary's unloaded resonance. That is a measured machine landing where the detuning argument says a tuned one should, and it comes from a coil that is not ours.

What will get you

Both of those are the same mistake in different clothes: treating the frequency as a setting rather than as a variable that the machine is sweeping on purpose.

What goes wrong

It behaves at low power and misbehaves as the arc grows. Something is timed as an angle and needed as a time, or the tracking is losing the pole. Both get worse with arc length because both are driven by the frequency, and the arc is what moves it.
The bang starts clean and degrades identically every time. Look for something that scales with frequency rather than with power. A supply drooping does this too, and the gate rail is the usual one.
The current envelope curls up at the end, or is flat from the start. That is the detuning against the frequency movement, and it is read off the envelope.
Everything measures correctly and the coil still misbehaves. Check where in the bang each measurement was taken. On this machine that is part of the measurement.

Where our number sits among measured ones

Everything in the trajectory above is computed, and it stays computed until this coil is switched on. What has changed is that it is no longer alone: four other builders have published the frequency at both ends of a real ramp, and the fall can be read off directly.

corn can, HVF 3522
  462 -> 440    4.8 %  measured
davekni, HVF 2397
  485 -> 440    9.3 %  measured
ours
  519 -> 463   10.7 %  COMPUTED
davekni, ground strike
  485 -> 413   14.8 %  measured
Anders Mikkelsen
  409 -> 335   18.1 %  measured

Ours lands in the middle of that spread. This is not verification. A number sitting inside a range of other people's numbers has not been checked, only found unremarkable — the four measured coils differ from each other by a factor of nearly four, so the range is wide enough to accept almost anything. But the figure previously had nothing at all behind it, and it is not in that position any more.

The equation those poles come from has since been checked properly, against a built coil rather than against a range: on davekni's machine it predicts 485.8 kHz from the component values where he measured 485, which is 0.17 per cent. That is the equation rather than our coil — the working is on the poles page — but it is the equation this whole table rests on.

The capture that replaces most of this page

Everything above is derived. The trajectory comes out of JavaTC's capacitances, a coupling factor and one constant — 4.14 pF per metre — fitted at a single arc length. It is arranged carefully and it is not measured, and those are different things.

One capture on the first real bang settles most of it. Not a campaign: a frequency at each end of one ramp.

What to catch on the first bang

probe   primary current, CT
capture whole bang, single shot

read  f at the start of the arc
read  f at the end of the ramp
watch for a step change between

The frequency at the start, at minimum loading, says which pole you are on. That is where the two are furthest apart, so the answer is unambiguous: near 519 kHz and the forcing worked, near 317 and it did not and the machine is in the mode it was not built for. Nothing else on this page matters until that reads right.
The two ends together replace the whole trajectory. Start and end give the fall directly; the 4.14 pF/m calibration is then either confirmed or gone, and with it every derived row here, on the poles' page, and on the primary's.
A step of about 1.6 in the middle is the failure mode, not a tuning fault. That is the ratio between the poles, and it is worth being exact because the ratio narrows through the ramp: 519.1/317.0 is 1.64 with no arc and 463.3/295.7 is 1.57 at 1.5 m, since both poles are falling. So the test is sharper than "about 1.6" — take where in the bang the step happened, read the expected ratio for that point, and compare. A jump well outside 1.55 to 1.65 is not a lost pole at all and wants looking at as something else. A smooth slide from one end to the other is the healthy case.
And the frequency at the start is the one to set dead time and lead from, which is the argument at the top of this page arriving as a measurement rather than as arithmetic.

Where next

Everything on this page is a consequence of one thing moving, so the rest of it is spread across the pages that own each consequence.

What moves, and why there are poles at all

There is no secondary frequency — where the two poles come from, which one survives heavy loading, and the condition that decides it.
A QCW adds capacitance — the arc as a capacitor, and what a bigger topload does to the same arithmetic.
The second mode — the other thing a coupled pair can do that gets mistaken for detuning.

The driver, which has to follow it

The driver on a QCW — the four jobs, each one changed by the frequency running away underneath it.
Converting a DRSSTC driver — the starting oscillator or PLL that holds the upper pole, as an actual list of changes.
Inside a driver — the comparator, the lead network and the protection, on any coil.
What a phase lead actually buys and choosing its inductor — the angle-against-time problem this page works across the ramp.

Set at the start of the ramp

Dead time is a fraction — the arithmetic, and the same fraction moving inside one bang.

Read at the end of it

Read the envelope — the cheapest instrument there is for the loaded end.
The primary is an impedance — turns and detuning, set against the loaded secondary.
An MMC is chosen by current — what the envelope's shape costs the bank.
The gate supply is a power supply — the other thing that degrades steadily through a bang and looks like this one.

The connecting argument here — that the four consequences move in different directions, and that this is what decides where in the bang to set each one — is ours. Each individual consequence belongs to the page linked beside it, and the measurements behind them are other people's.

tuningpolesQCW