[ §1 · tank ]

DRSSTC primary design: impedance, turns and detuning

Nobody sets the primary current. It comes out of a division, and the number you actually choose is the impedance of the tank.

CommonGnd4 August 202623 min read

Nobody sets the primary current. It comes out of a division, and the number you actually get to choose is the impedance of the tank.

The one free parameter at a fixed frequency

The secondary sets the frequency to aim at, so at that frequency the product L·C is fixed and the two can only move as a pair. Moving the primary off that frequency is the other choice, and it has its own section below. What is left here is:

Z = sqrt(L/C)
L = Z/(2·pi·f)
C = 1/(2·pi·f·Z)

On this coil that characteristic impedance is 37.3 Ω, from a 16.96 µH primary and its 12.2 nF tank capacitor [derived, √(L/C)]. Primary tank does that step and the two that follow it: the resonance, the RMS current the capacitors carry, and the voltage they stand.

At a fixed frequency the primary current goes as C, the power goes as C, and the voltage across the tank capacitor does not depend on C at all. Higher impedance, less current. That is what "high impedance primary" means, and Gao puts it as a large primary inductance with a small resonant capacitor, generally low capacitance to raise the impedance.

For a QCW the motive is direct: the ontime is tens of times longer than an ordinary DRSSTC's, and the switches have to carry that current for milliseconds rather than hundreds of microseconds.

Which is to say current is expensive and voltage is not

The two sides of that trade are not the same size, and the asymmetry is the whole reason the rule points the way it does. Things that grow with current: tank capacitor loss as I²·ESR and primary copper as I²R, both quadratic; device conduction, which goes as I on an IGBT and I² on SiC; and the peak rating of the switches together with the overcurrent threshold, which is a hard linear wall rather than a cost. Things that grow with voltage across the tank: the capacitor's voltage rating, which you fix by adding series links for pennies, and the risk of a primary to secondary flashover, which you fix with geometry.

So one column is bought with money and layout and the other is bought with margin you may not have, which is why the answer differs by machine: a QCW runs a much smaller tank than a DRSSTC for a reason that belongs to the ramp, and it is QCW primary design. Where the capacitance itself is sized is the MMC is sized by current, and where the current transformer may not go, which is between that capacitor and the primary, is two rings, not one.

Where the current comes from

I_pk = (4/pi) · V_bus / R_total

The 4/pi is the fundamental of the bridge's square wave. R_total is the copper, the switches, the bank's ESR, and the resistance reflected back from the secondary and the arc, and it is that last term that decides the answer.

The other formula for peak primary current, I_pk = V/sqrt(L/C), is the spark-gap one: charge the tank to V, let it ring down from a single discharge, and ½·C·V² becomes ½·L·I². There V is the gap's breakdown voltage, so applied to a solid-state coil it does not give the current. It gives what the drive is worth. It is ordinary energy conservation and it is carried here as that, because we have not found it stated on any solid-state design page, Steve Ward's included.

On Gao's QCW 1 a nominal 12.8 nF tank at 465 kHz gives 9.15 µH and a surge impedance of 26.7 Ω, so a 340 V bus discharged once would put 12.7 A through the primary, against the 100 to 126 A across his own captures [derived]. That ratio, eight to ten, is the gain from ringing up over many driven cycles rather than one, and it is a floor rather than a value: his 465 kHz is the running frequency at the start of the ramp, which is the upper pole, and a pole sits above the primary's own resonance, which he does not publish. A lower resonance means more inductance, a higher surge impedance and a smaller single-shot current, so the true ratio can only be larger than this. It should come out at about the primary's loaded Q, and the only builder's figure of that kind in this corpus is dr. kilovolt's Qpri~11 on his own ferrite-cored SiC coil, which is a third machine again and not a check on Gao's. Solid-state drive buys roughly a factor of Q in primary current for the same tank and the same bus, which is the quantitative form of what the topology is for.

The rate of rise is not the limit either. dI/dt = 2V/(pi·L) is around ten amps a microsecond on a typical tank, so the current reaches its working value in five to seven cycles. Whether that is nothing depends on which machine you are holding, and the same thread has both numbers: davekni puts a QCW ramp at 3000-plus cycles and the ordinary DRSSTC bang the thread is about at ten. So the rise is well under a per cent of a ramp and most of a bang.

Do not trust the calculation. Etesla, building a small DRSSTC, simulated "primary currents of like 300A over maybe 10 or 12 cycles without the secondary" and then watched a primary that "sort of levels out at 57A and does not get any higher no matter what" (EEVblog, December 2021): a factor of 5.3 [derived, 300 over 57]. Put a current transformer on it, because the transformer is the only reason he knew. The cause was not in the arithmetic at all. Wolfram asked in reply whether the primary was copper coated steel, and it was, worth 2.08 Ω of AC resistance where ten milliohms of copper had been assumed. Feed that back into the division on this page and it comes out: a half bridge's fundamental is (2/pi)·V_bus, so 170 V across 2.08 Ω is 52 A against the 57 A he read [derived]. He replaced the tube the next day and the coil worked.

Two regimes, and almost everything else depends on which you are in

Etesla's coil had a resistance in it that had no business being there, and that is the easy version: something was wrong and it was found. A coil can also stop short of the calculated current with nothing wrong with it at all, and then the question is which of two things ended the bang:

OCD-limited
  current reached the
  threshold; protection
  is what cuts it

impedance-limited
  current stopped below
  the threshold; the
  bridge gave all it had

The boundary comes out of the same division as everything else on this page. To reach the threshold at all:

R_refl < (4/pi)·V_bus/I_OCD
           - R_copper

At 325 V and a 160 A detector that is 2.59 Ω [derived] before subtracting the losses that are not the arc: the copper, the switches, the bank's ESR, which on this machine's own published figures come to roughly 0.3 Ω. So the reflected resistance has to stay under about 2.3 Ω, and past that the detector is decoration.

R_refl    current            [derived, all four, on 0.3 Ω of copper]
1.0 Ω      318 A   OCD-limited
2.3 Ω      160 A   the boundary
4.0 Ω       96 A   impedance-limited
11.9 Ω      34 A   impedance-limited

And the sign of the impedance question flips between them

Which is the reason to care, because two competent answers to "should the primary be high impedance" point in opposite directions and both are right.

Uspring, on HVF 1928, argues against:

power transfer from the primary to the secondary tank will reduce primary current… in a high impedance primary, the current might be reduced to levels far below the bridge's capability. That limits input power and consequently arc length.

And the arithmetic argues for it, though only under a detector that is actually reached. Power goes as I²·R_refl, and R_refl goes as the primary's inductance at the same coupling, so at a fixed threshold the power rises in proportion to the inductance:

L     I same arc   P    arc
x1        1.00    1.00  1.50 m
x1.5      0.82    1.50   1.72
x2        0.71    2.00   1.89

Arc length goes as the cube root of energy, so 50 per cent more power buys about 14 per cent more arc. That column is derived, not measured, and it is worth saying that it inherits every assumption in L ∝ E^⅓.

Both statements are true and they describe different machines. A high impedance primary gives you more arc if you are limited by current and less if you are limited by impedance. No amount of reasoning settles which one you have.

Coupling

0.2500.421, this coil0.750

Arc you are aiming for, where the ramp ends

0.5 m1.5 m3.0 m

What is plotted. The resistance the bridge sees looking into the primary, as the arc loads the secondary. It comes from a published closed form for the primary's Q:

Q_pri = (Q_sec/k²)·(1 − f²/f_sec²)² + 1/(k²·Q_sec)

with Q_sec = R_arc/(2πf·L_sec) and f_sec the loaded secondary frequency. Both modes fall straight out of it: Q_sec sits in the numerator of the first term and the denominator of the second, so as loading grows the primary Q falls and then rises again. Its minimum is the mode boundary, and since resistance goes as 2πf·L_pri/Q_pri, that minimum is the peak on this graph.

Why the peak position is the number that matters. Left of it, a growing arc raises the load and the ramp gets help it did not ask for. Right of it, the load falls away again. A ramp that crosses the peak changes character halfway through, and that is what makes a coil unpredictable. The goal is not a flat curve. It is a peak that sits past the end of your ramp.

The one check available, and it passes. A builder who has swept this published a verbal scale rather than a plot: discontinuities at 30 to 60 cm around k 0.2–0.3; at 0.4 it "rises to a metre, then falls"; at 0.5 it "starts to flatten out but is far from flat"; flat from 0.6 up. The turnover computed here lands at 0.41 m, 0.88 m, 1.25 m, 1.71 m and 2.58 m for those same couplings. "Flatter" turns out not to mean a smaller spread. The spread is not monotonic in k at a fixed primary tap. It means the turnover leaves the ramp. The fraction of a 1.5 m ramp spent past the peak runs 73, 41, 17, 0 and 0 per cent.

Every curve here is computed, and this coil is not built. Primary 16.959 µH, secondary 9.925 mH on Ces 14.031 pF, unloaded Q 199, arc taken as 75 kΩ·m in parallel and 4.14 pF per metre. The conversion from Q to ohms is ours. Nothing on this page has been measured on this machine, because there is no machine yet.

And the closed form has never been made to fit a measured coil. Uspring put a published dataset into it, a secondary Q of 4 and a coupling of 0.15, running at 70 kHz against a loaded secondary at 55, and got a primary Q near 80, which he called "seriously too large". He blamed either heavy detuning or his own figure for the loaded secondary frequency. The equation is very sensitive to that frequency, and the loaded secondary frequency is not observable while a coil is running. The question was asked in that thread and never answered. So take the shape of this curve as the argument; the ohms are not a claim about anyone's machine.

Sweeping coupling here holds the primary tap fixed at 349.9 kHz. A builder changing coupling would retune. That makes this "what happens if you change only the coupling", which is a real question but not the same experiment as theirs.

The measurement that settles it

There is a cheaper test of the same split and it needs only a variac. Two things can end a bang and they leave opposite fingerprints on the bus. If the current trip ends it, the energy per bang is L·I²/2, which contains no bus voltage term at all, so raising the bus does not lengthen the arc; it only gets to the same threshold sooner. If a fixed on-time ends it, the current climbs as V·t/L and the energy goes as V², so tripling the bus is nine times the energy. Worked on an illustrative 40 µH and a 300 A trip: the ramp to threshold takes 240 µs at 50 V and 80 µs at 150 V, and the energy is the same 1800 mJ both times, while the same inductance under a fixed 200 µs on-time gives 1250 mJ at 50 V against 11250 mJ at 150 [derived]. The result does not depend on the inductance, which scales both cases alike, so the numbers here are an illustration and the test is not. An arc that will not move when the bus is raised threefold is current clamped, and the autotransformer is the wrong knob: what is left is a higher threshold, less primary inductance or better coupling, and none of those is voltage.

Peak primary current on the first bangs

read peak I_pri, first flashes

reached the OCD threshold
  -> current-limited
  -> more turns help

stopped well short of it
  -> impedance-limited
  -> more turns make it worse

One number decides all of these at once:

Whether to add primary turns at all, which is the question above.
Cone or cylinder. A cylinder runs at lower current, which is easier on the bridge, on the copper and on the bank's RMS rating; a cone puts its top turn further down the secondary and lowers the flashover risk. The electrical argument and the geometric one genuinely disagree, and the current is what says which one you can afford to lose.
Whether the bank's RMS overload is real, since it scales with the same current.
Whether the detector threshold is reachable, which is the boundary above.
Whether P ∝ I²·R_refl describes this coil at all.

And the rule of thumb "better too high an impedance than too low" only holds in the first regime. It belongs to Jan (dr. kilovolt), who wrote it about a primary he had potted and could no longer rewind, and who called a lower impedance tank and a lower primary tuning the better design in the same paragraph (HVF 1073). In the second regime the rule buys nothing: the current is already short of the threshold, and raising the impedance moves it further short. And the other side of it is not electrical at all. Mads Barnkob, on HVF 1563: "Designing for low impedance could make you able to use fewer primary turns and by that make a smaller base and then still use a 160mm diameter secondary coil form." Everything above prices a low impedance in current and in devices. It also buys floor area, and that is the constraint that decides real builds more often than the arithmetic admits.

On the cone or cylinder question the published record is one-sided, whatever the argument says. The guidance for an ordinary DRSSTC is a flat or a conical primary, on the reasoning that the coupling cannot be made too tight; the QCWs with published builds are mostly solenoids, or primaries interleaved with the secondary, or in one case the "half-donut" of Gao's QCW 2, against a single exception wound as a cone. That is the fourth place where QCW practice runs the other way from DRSSTC practice, after secondaries three to four times squatter than the rule, toploads two to two and a half times larger than the sizing rule, and the coupling itself at two to four times the baseline. Band against band that factor is exactly 2.5 at both ends, 0.12 to 0.30 and 0.2 to 0.50, but the baseline moves with the diameter of the secondary, and once each coil is compared with the starting figure for its own size the factor spreads to roughly 2.2 to 4.2. The coupling page does that arithmetic and says why the spread is the answer. No source draws that list together and it is ours, but all four point one way. A QCW is not a DRSSTC with a different supply, it is a different machine that shares a name for its output.

Turns, detuning, and one happy coincidence

More turns means more inductance, which means higher impedance and less current, and at the same time a lower primary frequency. Those are two separate jobs and adding turns does both: you detune downwards and unload the switches and the bank in the same movement.

This coil's own tap table is the movement with numbers on it. Four positions, a turn apart, against a 12 nF bank and a secondary at 430 kHz:

turns    L         primary f    detune from fs    Z0 = sqrt(L/C)
 9.43    12.4 µH    412.6 kHz      4.0 %            32.1 Ω
10.43    15.2       372.6         13.3              35.6
11.43    18.2       340.6         20.8              38.9
12.43    21.5       313.3         27.1              42.3

[derived, every column, from the turns and the bank]. One turn moves the frequency by 27 to 40 kHz and the impedance by about 3.4 Ω, and the two move together because both come out of the same inductance. The inductance itself goes as the square of the turns, which those four rows confirm to better than half a per cent, so a tap table on a wound primary is not four measurements but one. Wheeler's flat spiral is that square in both directions: the inductance a winding's shape gives, and the turns a stated inductance implies. The second is the useful one on somebody else's coil, because a published set of diameter, length, turns and inductance has only three free figures and the fourth is a check rather than a fact. For the arithmetic itself use JavaTC, which does the whole geometry at once; what it does not tell you, the range Wheeler's forms hold over and what a tap does to a published set, is on coil geometry.

Wheeler's flat spiral was checked against three published machines and it reproduces them. Barnkob's featured coils gives each one an inside diameter, a conductor size, a turn spacing, a turns count, a tap position, an outside diameter, a tank capacitance and a resonant frequency, which is two figures more than the arithmetic needs, so it can fail in public. ``

 machine                       L at tap     computed     stated Barnkob, Kaizer DRSSTC II      3.15 µH     327.4 kHz     327      +0.1 % Franzoli, DRSSTC 3            26.77        37.3           36      +3.6 Slawinski and Prince, no. 6   15.69        35.9           36      -0.2

`` Three builders, three sizes from 0.37 to 5 metres of spark, and a factor of nine in frequency [derived, Wheeler's flat spiral on their figures]. Two things fall out of it. Their stated outside diameters reproduce to under half a millimetre, which settles that "turn spacing" is the gap and the pitch is conductor plus gap. And the tap is not decoration: run the first machine on its whole six turns instead of the 4.8 it is tapped at and the answer lands twenty per cent low.

What it also shows is how coarse a tap is. There is no position between 13.3 and 20.8 per cent of detuning, so a design that wants 15 wants a different primary rather than a different tap, and that is the reason to cut taps at half turns where the geometry allows it.

How far to detune is not a fixed percentage. The rule is that the amount of detuning should equal the amount your arc detunes you, and the difference matters. Primary turns go as 1/(1-d), so detuning 25 per cent rather than 15 is about thirteen per cent more of them: on a ten-turn primary, more than a whole turn. Make your taps generous in both directions, because until you have measured the detuning you do not know the number.

The published machines sit as far apart as that suggests. Two of Gao's, on the QCW 2 and QCW 1.5 pages, work out at 11.4 per cent low and at 3.2 per cent low, the second of them 328.4 kHz against 339.3. Neither page states a ratio; both figures are ours, computed from the frequencies each of them gives. And at least one published coil tunes its primary above its secondary, so the sign is not settled either. What none of this licenses is carrying one coil's ratio onto another. The coils differ in coupling, the poles move with coupling, and it is a pole the primary has to meet rather than a natural frequency, so two raw ratios put side by side are not the same quantity twice.

And the figure is the secondary's movement rather than the pole's, and it is not defined without an arc length beside it: one worked machine detunes 16.7 per cent under a 1.5 m arc, and a percentage carried over from somebody else's coil is carrying their arc with it. Which movement is which, and the trajectory the percentages come off, are on one page so that they can be replaced in one place when somebody finally measures them. And one figure that used to be quoted around here has already been withdrawn for resting on the wrong capacitance.

Coupling

An ordinary DRSSTC lives at k = 0.12 to 0.2. A QCW lives at 0.30 to 0.50.

Raising k does not lengthen the arc. The reflected resistance goes as k squared, so tightening the coupling into an unchanged tank cuts the current and the power. The builders with the longest arcs sit at 0.15 on enormous coils; the builders with the highest arc-to-secondary ratios sit near 0.5 on tiny ones. Coupling buys the ratio. Power buys the length, and there is a whole article about that.

What coupling does buy is tolerance to detuning: the poles spread further apart as k rises, and each is stiffer against what the arc does to it. The cost is that a higher k eats the detuning budget at the lower pole and pushes the upper one further from what the bridge can do.

The test that the model is describing your coil: the current should fall as one over k squared. If it does not, stop and find out why before going further.

The limit on coupling is set by racing sparks, not by any calculation, and where it sits depends on the coil. They have been caught as low as k = 0.185, under the 0.30 to 0.50 a QCW runs at and inside the 0.12 to 0.2 an ordinary DRSSTC calls ordinary. The limit is not a fence above the normal range. It runs through the middle of it, at a place your own coil picks.

Building it

Copper tube or wide strip. At the working frequency the skin effect keeps the current on the surface, so most of a round conductor's cross-section is not doing anything. The skin depth in copper is 0.103 mm at 400 kHz and 0.094 mm at 485 kHz [derived; the guide's own table has no 485 kHz row, and gives 0.11 mm at 400 kHz and 0.095 mm at 500]. The working rule, from Mads Barnkob's DRSSTC design guide, is four skin depths, and it is four for tube as well as for solid wire. A tube looks like it should get twice that for having two surfaces, and the guide says why it does not: "it can only be considered a single surface as there is only a magnetic field around the entire conductor". So a conductor of either shape is doing its work in the outer 0.4 mm, and 18 AWG at 1.02 mm across is about two and a half times thicker than it can use. The guide's own comparison makes the shape argument better than a thickness rule does: a 1 mm wall tube uses at least 98 per cent of its copper up to 300 kHz, while a solid wire of the same cross section has a dead core already at 30 kHz. Tube wins on where the copper sits, not on how deep the current goes. That is the number behind the parallel litz strands other builders wind instead.
It melts at high bangs per second. Long bursts heat the copper seriously. Air and cross-section, both taken seriously.
Water cooling works with ordinary water, but arrange a pressure relief. There is a known case of a hose exploding with steam directly into the bridge.
A metal frame under the primary is a shorted turn. Ground each rail at one end only.

One built primary, with the part numbers

Shane Colton's, on DRSSTC Δt2, is the most completely written up primary this corpus has found, and it is worth having as a whole rather than as a rule.

Conductor: 8 AWG grounding wire, McMaster 7512K641, and 25 feet of it "just barely enough to make the complete spiral".
Former: a 20 inch circle of quarter inch polycarbonate, with the spiral marked by unwrapping a string from a spool at the centre.
Turns: "I calculated six turn for the primary, but added an extra turn and a half for tuning". The headroom is the point, and it is the same advice the detuning section above arrives at from the arithmetic.
Mounted upside down, so the winding hangs under the former rather than sitting on it, which he puts as a "keep-the-high-voltage-covered perspective". He moved the secondary to keep the designed spacing.
The tap is a screw terminal he made, "a small aluminum block with a copper strip bent around one side".

His resistance figures are an outside check on the skin effect argument above, and they pass it, by a route that belongs with the measurement rather than with the construction: he fits the tank's real resistance against a decay envelope and reads the gap as skin effect. That check, and what it does and does not settle, is with the dead time.

And it does not have to be air

Put ferrite inside the winding and the primary stops being a coil you tap and becomes a transformer core. Nothing on this page transfers to that: the impedance argument, the two regimes and the detuning rule are all written for a primary you can tap and move. The worked cases, both of them QCW machines, are QCW primary design.

The primary stands beside the secondary on the department diagrams rather than lying inside the bridge, because the two being on one axis is the only way they are coupled at all.

primarycouplingtuning