A spark propagates at two to three metres a microsecond, and a DRSSTC's bang lasts about three hundred microseconds. At the spark's own speed that would be seven hundred and fifty metres of arc. What one bang actually gives you is ten centimetres.
So the arc grows about seven and a half thousand times slower than its own limit [derived, 750 m against 0.1 m], and what holds it back is the supply of charge. That one fact explains the branching, the thickness and 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 ball one centimetre across, so five millimetres in 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].
This works at all only because the field is concentrated at the head. The mean field along a metre and a half of arc from a 300 kV topload is only 2 kV/cm, fifteen times below the breakdown field of air. The channel behind the head is already conducting and barely needs any.
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].
The ten centimetres in the first paragraph comes from that division at a 300 kV topload and from nothing else. A lot rests on it, and it is not a measurement. No published measurement of one bang's leader length has been found. What a QCW arc is works from the same figure and the same arithmetic, so the two articles agree because they share a rule, not because two people measured it.
The only 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], and that matches the size of coil he was modelling.
A metre and a half is therefore about fifteen ten-centimetre steps stuck end to end, each step its own burst laid on the channel left by the one before. A single 300 µs burst reaches only that first ten centimetres. The length comes from repeating the burst, not from one long flash.
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 smaller because of losses. That is why the stack does not go on for ever: each burst adds less than the last, and the length levels off. It is also 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 give different answers, so get them straight before using either number.
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 see 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 accounts for most of the difference in appearance. The section below on branching has the rest. The DRSSTC's channel takes so much charge per cycle that it can break out in several directions at once, while the QCW's gains 0.23 to 0.24 mm a cycle and grows one channel.
What a QCW arc is runs the same division and prints 2.2 to 6.7 against 0.13, because its QCW input is a metre and a half over a 25 ms ramp, where CJ's measured 2.39 m over 23 ms is used here. The DRSSTC half is the same in both. Across the two builds on file the QCW rate is 0.13 to 0.24 mm a cycle, so the ratio runs from about nine at one end to about fifty at the other [derived], and the comparison holds either way.
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 agree with the measurement. The check is shown here because the topload figures further down look at first as if they contradict them. Take them at the frequency Hydron's own coil runs at, near 75 kHz, 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.
Do not run that arithmetic at the reference coil's 426 kHz. The same 29 pF then passes 15 A rather than 2.7, five times the measured current, and the pair stops agreeing for no reason except the substitution. An R and a C belong to the coil they were fitted on, and these were fitted on Hydron's.
The earlier point in the same burst does not agree the same way, and it 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 is comparable with the topload's, so it pulls the secondary's resonance down. That is the detuning. The resistance loads the Q, and by how much depends on the coil rather than 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 the reference coil used in these articles, 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.
The factor here is 26. The number to watch out for is fifty. Fifty is 199 divided by 4, this coil's unloaded Q over another coil's loaded one, on machines whose impedances differ by a factor of two. Tuning is more forgiving under load either way. How much more is something to work out for your own coil, not to borrow.
The secondary article also says the 210 kΩ agrees with Terry Fritz's SPICE fitting of about twenty years earlier, at around 220 kΩ. That half carries no link there and we have not found one either, so treat the agreement as an unsourced claim, not as two sources you can go and read.
Inside one bang
The bridge turns on and the current rises for five to seven cycles. The secondary voltage climbs until the breakout fires, and the channel goes. It grows, branches and adds capacitance, so the secondary slides down in frequency and loses Q. The reflected resistance changes and takes the primary current with it. Then the bridge turns off and most of the primary's energy goes back into the bus capacitors.
Current is a better indicator of length than voltage. That is Uspring's reading of Hydron's captures, not a rule of thumb. The topload voltage clamps: 300 kV at 0.5 A about 40 µs in, 200 kV at 3 A about 200 µs in, on the same burst (HVF 117). More current is more charge, and more charge needs the larger capacitance of a longer arc. The voltage does not go up. It redistributes into length.
A bang cut short is energy that never reached the arc. A DRSSTC is not a spark gap coil, and its 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, and in captures that shows up 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 number you want, and it 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 is 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. That 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, 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).
Nothing more specific than that is established. What the question really wants is 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. No source gives one, and a single time constant cannot produce one. Take the constant and your own interrupter period and decide which side of it you are on.
Most "continuous" five minute videos are interrupted. Fast bangs look like CW. Real CW gives short thick arcs.
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, and 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. A toroid that is too small gives many weak streamers instead of one strong one.
All the energy in one channel gives maximum length, and split between three it gives three short ones, so single-channel operation is a design goal rather than a side effect. The drawing above puts the exponents on it: length goes as the cube root of the energy in a channel, so n channels each reach 1/n^(1/3) of the single-channel length while the total channel length grows as n^(2/3). More arc, less reach.
Two things often get added to that list, and neither belongs on it. Both are covered in why the sword bends, which works through the evidence for each.
The first is frequency. Raising it is often said to run the channel hotter and give fewer branches, and that article withdraws exactly that half. The causes of branching it can name do not include frequency, and the band table on the QCW side 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.
The second is the zigzag, which is not a branch. The mechanism on 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). That makes a heavily branched tree, denser than a flat pattern and nowhere near filling space.
A second reconstruction, Ichiki and colleagues in 2011, photographs the discharge from three directions rather than through a narrow stereo pair and reports a different mean and a tighter spread. We have not been able to read that paper, so its figures are not given here and the difference between the two studies is left open.
Those measurements are centimetre gaps and streamers, not metre leaders off a Tesla coil. They are fine as a guide for drawing, but they are not the physics of a coil.
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 a falling resistance, not a fixed one. As the strike channel establishes, the load goes from capacitive to resistive and bottoms out around 25 kΩ. That 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.
Two other articles, overcurrent is not a setting and what RF ground is, put the topload voltage collapse at 100 to 250 nanoseconds. Neither gives a source and we have not found one, so the microsecond figures above are the measured ones and the nanosecond figure is unsourced.
The arc takes the load off the secondary and the primary current goes up, by about forty per cent at that one measured resistance. 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 in the ground strike article 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, faster route to the same threshold, and it 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 article puts 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, so no figure is given for it. The two cases are not two ends of one range with the same answer at both. The mild strike is the one the machine runs straight through, and it is the one to design 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, the shape everything above assumes. The length came out at barely half of reality.
His own explanation for the miss is calibration, not physics. 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.
Uspring's own conclusion is that the arc's load, its R and its C, is the great unsolved problem of Tesla coils.
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 seen from the QCW side.
- the secondary is a resonator, where the loaded
Qand the 210 kΩ used here are worked out. - why the sword bends, for wander, which is not branching.
- what a ground strike does to a DRSSTC, for the detector question in full.
The discharge on the DRSSTC department page is drawn to match: 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 come from none of them. The step length is our own arithmetic, as its section says, 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 carry weight, and neither is anybody's measurement.