A spark propagates at two to three metres a microsecond. A DRSSTC's bang lasts about three hundred microseconds. At the spark's own speed that is seven hundred and fifty metres of arc, and what one bang gives you is ten centimetres.
The arc grows about seven and a half thousand times slower than its own limit [derived, 750 m against 0.1 m], and the thing holding it back is the supply of charge. That one fact explains the branching, the thickness, the speed, and why a DRSSTC looks like lightning while a QCW looks like a sword.
How it grows
Charge runs up the channel and collects on the leader's head. The head behaves roughly like a one centimetre ball. When the field on it passes 30 kV/cm the channel jumps to a new segment, and the current flowing through the channel keeps it hot enough to stay conductive.
A one centimetre ball, a centimetre across, so five millimetres of radius, is 0.56 pF and needs 15 kV to reach that field, so each jump costs about 8 nC [derived, C = 4·pi·e0·r, E = V/r, Q = C·V on r = 5 mm and 30 kV/cm].
Step length is just voltage over breakdown
one step = topload voltage / 30 kV per cm200 kV gives 6.7 cm, 250 kV gives 8.3, 300 kV gives 10 [derived, that division].
Where the ten centimetres at the top of this page comes from, since everything here rests on it. It is that division at a 300 kV topload, and nothing else. It is not a measurement, and no published measurement of one bang's leader length has been found for this corpus. The live what a QCW arc is works from the same figure and the same arithmetic, so the two pages agree because they share a rule rather than because two people measured it.
The one published figure of that kind is per RF voltage peak rather than per bang, and it is the right order. Uspring, simulating davekni's forty-transistor DRSSTC against its captures, has the arc reaching full length 130 microseconds into the bang and puts the average growth at about 15 centimetres for each voltage peak (HVF 798). On the rule above that is a tip sitting near 450 kV [derived, 15 cm at 30 kV/cm], which is the size of coil he was modelling.
Between bursts, if the channel has not had time to disperse, the energy of the next burst rushes to the end of the ionised channel and makes another leader there, each one slightly diminished by losses. That diminishing is why the stack does not go on for ever: each burst adds less than the last, and the length settles. And it is why the length breathes from bang to bang, because a channel that did disperse starts again from ten centimetres.
Against a QCW
Same metre and a half, different clock. There are two ways to count it and they do not say the same thing, which is worth having straight before either number is used.
Per microsecond:
- DRSSTC, one 300 µs burst reaching ten centimetres: 0.33 mm/µs [derived].
- QCW, CJ's 2.39 m off a 23 ms ramp (HVF 2434): 0.10 mm/µs [derived], which puts a metre and a half at about fourteen milliseconds of ramp.
A factor of three, and a factor of three is not what you are looking at when you watch the two machines.
Per RF cycle, which is one delivery of charge to the head:
- DRSSTC at the 50 to 150 kHz a coil of that size runs: 2.2 to 6.7 mm per cycle [derived, 0.33 mm/µs divided by the frequency].
- QCW on CJ's measured 430 to 450 kHz secondary: 0.23 to 0.24 mm per cycle [derived, the same division].
Nine to thirty times [derived, those two bands divided], and that is most of the difference in appearance. The section below on why it branches has the rest. The DRSSTC's channel takes so much charge per cycle that it can break out in several directions at once; the QCW's gains 0.23 to 0.24 mm a cycle and grows one channel.
What it is, electrically
In 2018 Uspring worked the arc's load out of Hydron's measured topload currents on a 160 mm DRSSTC: about 210 kΩ of resistance with about 29 pF across it, at the current peak roughly 180 to 200 microseconds into the burst (HVF 117).
Those two numbers close on the measurement, and it is worth showing that they do, because the topload figures further down look at first as though they contradict them. Take them at the frequency Hydron's own coil runs at, near 75 kHz, which is the figure he uses himself when he sizes the protection on his scope inputs. In parallel, 1/R is 4.8 µS and w·C is 13.7 µS, so the admittance is 14.5 µS and at 200 kV that is 2.9 A [derived], against the 3 A measured at that instant. The load comes out about 71 degrees capacitive [derived], which fits: Uspring measures 50 to 60 degrees on his own coil and says Hydron's phase shifts run a little larger.
The earlier point in the same burst does not close the same way and should not. 300 kV at 0.5 A is 600 kΩ of magnitude, and no capacitance across a 210 kΩ resistance can make it look larger, only smaller. It is larger because at 40 microseconds the arc is a fraction of the length it has at the current peak the 210 kΩ was fitted to. Along the channel, a few pF and about a microhenry per metre.
The capacitance, being comparable with the topload's, pulls the secondary's resonance down. That is the detuning. The resistance loads the Q, and by how much is a property of the coil rather than of the arc, because loaded Q is R_arc over Z and very little else. On the coil the 210 kΩ was fitted to, Uspring puts the loaded Q at about four, which implies an impedance near 52 kΩ [derived, 210/4]. On this site's own coil, where Z is 26.6 kΩ, the same 210 kΩ gives 7.6 against 199 unloaded, a factor of 26, worked at the secondary is a resonator.
Inside one bang
The bridge turns on, the current rises for five to seven cycles, the secondary voltage climbs until the breakout fires, the channel goes, it grows and branches and adds capacitance, the secondary slides down in frequency and loses Q, the reflected resistance changes and takes the primary current with it, and then the bridge turns off and most of the primary's energy goes back into the bus capacitors.
A bang cut short is energy that never reached the arc, because a DRSSTC is not a spark gap coil: the energy does not leave in one transfer time, it is pumped in for the whole bang. So a longer ontime gives a longer spark, most visibly at modest peak currents.
Between bangs, the channel remembers
Successive bangs go into air that is already hot, which is thinner and breaks down at a lower voltage, so the current starts rising earlier and the voltage peak comes out slightly lower. Hydron's own run shows it: the second burst peaks lower than the first, and the bursts after that stop dropping (HVF 117). The first bang after a pause does not behave like the tenth, which shows up in captures as a drift that is not really there.
This only works if the next discharge takes the same channel. A wandering root kills it, which is a practical argument for a fixed breakout point.
How long the memory lasts is the part worth having a number for, and the number is a time constant rather than a burst rate. Uspring fits the channel a warm-up time constant of about 300 microseconds and argues the cooling constant sits in the same region, hundreds of microseconds, because the channel is thin and the hot air in it has expanded away most of its mass (HVF 798). davekni, on the same thread, reports each burst starting with little arc loading even at a 2.5 kHz burst rate and then growing straight back to where the previous one left off, which is a 400 µs repeat interval against that constant [derived, 1/2.5 kHz].
So the effect is strongest when the gap between bangs is comparable with the cooling constant, which means kilohertz burst rates. At the few hundred bangs a second an ordinary interrupter runs, the gap is milliseconds and many time constants, and what carries over is the retraced path rather than a hot channel: sequential bursts follow the same route because of the thin air the last one left, and higher burst rates leave less time to cool (HVF 1239).
Anything more specific than that is not established here. A scale in bangs per second, with a floor below which every bang starts from nothing and a ceiling above which the channel never cools, is the shape the question wants, and this corpus has no source for one and cannot get one out of a single time constant. Take the constant and your own interrupter period and decide which side of it you are on.
Why it branches
A dip in current raises the losses in the channel, the channel loses conductivity, and it breaks out in a new direction. Power delivered abruptly branches; delivered smoothly it makes one channel. Residual voltage left on the bus branches the start of the bang. A smooth breakout surface gives one large channel and a rough one gives several short ones. And a toroid that is too small gives many weak streamers instead of one strong one.
Two things are commonly added to that list and neither belongs on it, both on the authority of why the sword bends, which is live and works the evidence for both.
Frequency is not a branching cause. Raising it is often said to run the channel hotter and give fewer branches, and that page withdraws exactly that half: the causes of branching it can name do not include frequency, and this site's own band table puts 1 to 4 MHz down as a thin straight flame, which is the wrong way round for a simple hotter-is-straighter story. What frequency does move is wander, in the other direction, and wander is a different defect.
A zigzag is not a branch. The mechanism on the record for a crooked channel is the arc repelling itself, adjacent sections of one channel pushing apart, reported by davekni whose 100 kHz QCW went from straight to zigzagged (HVF 3140). That predicts a zigzag with no branching at all, so the two are not automatically one defect, and treating them as one sends you after the wrong cause.
The shape, for anyone drawing it
A tree with a definite trunk, not a bush. Branches leave from where the channel lost conductivity, which is behind the head, not from the base: a fan out of one point is wrong.
Branching angle, from stereo photography of positive streamers in a point to plane gap in air, reconstructed in three dimensions rather than measured off the projection: 43° ± 12°, and Nijdam and colleagues find no significant dependence on pressure or on distance from the needle (Stereo-photography of streamers in air). Fractal dimension of the branching structure in a needle to plane corona, 2.16 ± 0.05, from Popov (Plasma Physics Reports 28(7), 2002). So a heavily branched tree, denser than flat and nowhere near filled.
Thickness is a cone, not a cylinder. The section nearest the topload carries the current of every branch above it and runs hottest. Jan, who built the SiC phase-shift QCW the modelling below is about, puts the growth side of it well: doubling the length costs the same energy again in the new outer half, and on top of that the already grown inner half has to be fattened and kept hot, which he guesses at three to four times the energy for twice the arc (HVF 1073).
Strike to ground
The ground arc is not a fixed resistance but a falling one: as the strike channel establishes, the load goes from capacitive to resistive and bottoms out around 25 kΩ, which is Hydron's measurement and the only strike resistance anybody has published (HVF 117). On the same thread the event is fast but not instant: Uspring reads the strike as lasting one to two microseconds and davekni puts the current spike at about a microsecond wide at half amplitude.
The arc takes the load off the secondary and the primary current goes up, and at that one measured resistance it goes up by about forty per cent. Whether that trips anything is decided in the detector, not in the strike. A threshold set half again over the normal ramp is not reached until the channel falls to 22.1 kΩ, below anything published; a threshold a fifth over is reached at 37.3 kΩ, well above the measured strike [both derived on the live page from the same expression]. The one measurement sits between the two settings, so the answer flips inside the range a builder might reasonably choose, and what a ground strike does to a DRSSTC works the whole curve with a calculator on it.
There is also a second and faster route to the same threshold that has nothing to do with the current the strike draws through the resonator: the switching loses sync, the primary stops being part of a tuned tank, and the current climbs at V = L·dI/dt with nothing resonant holding it. That page has it at a few microseconds, and has a real 600 A machine reported tripping on a wall strike.
A strike heavy enough to be a near short is a different case and behaves like one: the primary current hits its limit immediately, and a driver set to skip pulses rather than to latch spends much of the rest of the burst inhibited. What fraction of its pulses it loses is not established here, and no figure for it is printed on this page for that reason. The two cases are not a spectrum with the same answer at both ends. The mild strike is the one the machine runs straight through, and it is the one worth designing for.
What the models still cannot do
Uspring built an LTspice arc model and ran it against Jan's SiC phase-shift QCW using the tank figures from that coil's own build post (HVF 1073). The starting current matched, 35 A at the one to two kilowatt level. The peak came out a little high, 115 A modelled against the 100 A peak Jan measured on the machine. The growth speed came out almost constant, which is the shape everything above assumes. The length came out at barely half of reality.
His own reason for the miss is a calibration one rather than a physics one. The model's length was calibrated on ordinary DRSSTCs, breakout point to tip, and those arcs wander and branch, so a given amount of channel does not reach very far. A QCW's arc is ironed straight, and the same channel then reaches roughly twice as far. The model has no wander and no branching in it, so it carries the wiggly calibration onto a straight arc and comes up short.
For anybody simulating one, the practical consequence is small and annoying: if you draw a straight channel, take a longer length than the calculation gives. If you draw a meandering one, take the calculated length. Otherwise the picture and the number disagree by a factor of two.
Where next
- what a QCW arc is, for the same growth rate read from the other department's end.
- the secondary is a resonator, which owns the loaded
Qand the 210 kΩ this page borrows. - why the sword bends, for wander, which is not branching.
- what a ground strike does to a DRSSTC, for the detector question this page only summarises.
The discharge on the DRSSTC department page is drawn to this: segments of a fixed step, branches at forty odd degrees behind the head, thickness as a cone. The sources are Hydron's 2018 topload dataset and Uspring's analysis of it, Uspring's arc model run against Jan's and davekni's coils, Nijdam and colleagues on branching angles, and Popov on the fractal dimension. Two figures here are not from any of them: the step length is this site's own arithmetic, which the section on it says in as many words, and the two to three metres a microsecond in the first line is a textbook propagation speed for which no citable source was found. Both are load-bearing and both are the corpus's rather than somebody's measurement.