You don't set a DRSSTC's primary current directly. It comes out of a division, bus voltage over the resistance in the tank, and the figure you actually choose at design time is the tank's impedance.
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. That leaves:
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.
Some search results define the primary's impedance as the time its current takes to rise from zero to maximum. Impedance is not a time. You can catch that kind of error for free, without knowing who wrote it, by checking that the definition makes sense in units.
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 means 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 reason 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 that is why 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 limit 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.
The voltage side costs money and layout, while the current side costs margin you may not have. The answer differs by machine. A QCW runs a much smaller tank than a DRSSTC for a reason that comes from the ramp, covered in QCW primary design. How the capacitance itself is sized is in the MMC is sized by current. Where the current transformer must not go (between that capacitor and the primary) is in two rings, not one.
Where the current comes from
I_pk = (4/pi) · V_bus / R_totalThe 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. The last term is the one 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 on a solid-state coil it does not give the current. It gives a baseline to compare the drive against. It is plain energy conservation and is used 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 lower bound. 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. It should come out at about the primary's loaded Q. The only builder's figure of that kind we have is dr. kilovolt's Qpri~11 on his own ferrite-cored SiC coil, which is a third machine and not a check on Gao's. Roughly, solid-state drive gets you a factor of Q more primary current for the same tank and the same bus.
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 matters depends on the machine, and one 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.
Two regimes, and almost everything else depends on which you are in
Etesla's coil had a resistance in it that should not have been there, and once found it was fixed. A coil can also stop short of the calculated current with nothing wrong with it, and then you need to know 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 hadThe boundary comes out of the same division. To reach the threshold at all:
R_refl < (4/pi)·V_bus/I_OCD
- R_copperAt 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 and the bank's ESR, which on this machine's own published figures come to roughly 0.3 Ω. The reflected resistance has to stay under about 2.3 Ω. Above that the detector never fires.
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-limitedAnd the sign of the impedance question flips between them
This matters 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.
The arithmetic argues for it, but only when the detector 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 1.89Arc length goes as the cube root of energy, so 50 per cent more power gives about 14 per cent more arc. That column is derived, not measured, and it carries every assumption in L ∝ E^⅓.
The two statements 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. You can't tell which one you have by reasoning about it.
The curve is a closed form for the primary Q against arc loading, with the arc's capacitance taken as 4.14 pF/m. That is one coil's calibration, not a measurement. The two published measurements that hold up, 5.03 and 7.66 pF/m, both sit above it, so treat where the impedance turns over as a shape and not a length to the metre. The direction of every conclusion here is the same across that range. The capacitance only moves the turnover length.
The measurement that settles it
A cheaper test of the same split needs only a variac. The two things that can end a bang leave opposite signatures on the bus. If the current trip ends it, the energy per bang is L·I²/2, which has no bus voltage term, so raising the bus does not lengthen the arc. It only reaches 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 gives 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. The same inductance under a fixed 200 µs on-time gives 1250 mJ at 50 V against 11250 mJ at 150 [derived]. The inductance scales both cases alike, so the numbers are only an illustration but the test holds for any coil. An arc that does not grow when the bus is raised threefold is current clamped, and the variac is the wrong knob. What is left is a higher threshold, less primary inductance or better coupling, and none of those is voltage.
On cone or cylinder, the published record leans one way regardless of the argument. 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, with a single exception wound as a cone. That is one more place where QCW practice runs the other way from DRSSTC practice. The others are secondaries built squatter, and the coupling itself, which runs at two to two and a half times an ordinary DRSSTC's, band against band, as the coupling section below sets out. No source puts that list together, so it is ours, but all of it points the same way. A QCW is a different machine from a DRSSTC, beyond the supply, even though the output has the same name.
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. Adding turns does both jobs at once: you detune downwards and unload the switches and the bank in the same step.
This coil's own tap table shows it with numbers. 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 from the same inductance. The inductance 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 really one measurement, not four. Wheeler's flat spiral formula is that square in both directions: the inductance a winding's shape gives, and the turns a stated inductance implies. The second use is the handy 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. For the arithmetic itself use JavaTC, which does the whole geometry at once. The range Wheeler's forms hold over, and what a tap does to a published set, are on coil geometry.
The tap table also shows 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 needs a different primary rather than a different tap. Cut taps at half turns where the geometry allows it.
How far to detune is not a fixed percentage. The detuning should equal the amount your arc detunes you, and the difference is large. Primary turns go as 1/(1-d), so detuning 25 per cent rather than 15 takes about thirteen per cent more of them, which on a ten-turn primary is 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 are spread as widely 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 page gives. At least one published coil tunes its primary above its secondary, so even the sign varies. None of this lets you carry one coil's ratio onto another. The coils differ in coupling, the poles move with coupling, and the primary has to meet a pole rather than a natural frequency, so two raw ratios side by side are not the same quantity.
The detuning figure is also the secondary's movement rather than the pole's, and it means nothing without an arc length beside it. One worked machine detunes 16.7 per cent under a 1.5 m arc, and a percentage taken from somebody else's coil comes with their arc. Which movement is which, and the trajectory the percentages come from, are kept on one page so they can be updated in one place once someone measures them. One figure previously quoted for this has already been withdrawn for resting on the wrong capacitance.
The old reason for detuning low, that the arc pulls the primary up and you harvest more power, is wrong, and Anders Mikkelsen retracted their own version of it in public. The mechanism is matching under the arc's load, not extraction.
Coupling
An ordinary DRSSTC runs at k = 0.15 to 0.2, which is where Steve Ward brought his DRSSTC-3 down to (on pupman). A QCW runs 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, and the builders with the highest arc-to-secondary ratios sit near 0.5 on tiny ones. Coupling raises the ratio, power raises the length, and there is a whole article about that.
What coupling does give you 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.
To check the model describes 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 0.15 to 0.2 an ordinary DRSSTC calls ordinary. The limit falls inside the normal range, at a place your own coil decides.
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], so on 18 AWG at 1.02 mm across the centre of the wire is about five skin depths in from the surface and carries almost nothing [derived]. A tube puts its copper where the current is. Its advantage is where the copper sits, not how deep the current goes, and the same number is behind the parallel litz strands other builders wind instead.
- It melts at high bangs per second. Long bursts heat the copper seriously. Give it air and cross-section.
- 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 we have found, so here it is in full.
- 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". That headroom is the same advice the detuning section above reaches 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".
The tap is the highest voltage point on the primary, which is easy to miss when you mount one. He designs it to hit "about 3kV in operation", while carrying "several hundred amps" peak and up to "maybe 50A" RMS. A tap has two hard requirements in one fitting, a low resistance joint and three kilovolts of clearance, and it is often built as though only the first mattered.
His resistance figures are an outside check on the skin effect argument above, and they agree with it. The method belongs with the measurement rather than the construction: he fits the tank's real resistance against a decay envelope and reads the gap as skin effect. That check, and its limits, are with the dead time.
And it does not have to be air
Put ferrite inside the winding and the primary becomes a transformer core rather than a coil you tap. None of the above carries over to that: the impedance argument, the two regimes and the detuning rule all assume a primary you can tap and move. The worked cases, both of them QCW machines, are in QCW primary design.
The primary stands beside the secondary on the department diagrams rather than inside the bridge, because sharing one axis is the only way the two are coupled at all.