The question before every upgrade: I will raise the bus, how much longer does the arc get? The answer is never proportionally, and the linear guess overshoots badly enough to be worth the arithmetic.
Four sources, all below one
- Steve Ward, from the law of feeding a branchless spark: exponent 0.33, so twice the energy gives 1.26 times the length.
- Gao's measurements, 76 J to 120 cm and 275 J to 178: 0.31, giving 1.24.
- Freau's law, from spark gap coils, length squared with power: 0.5, giving 1.41.
- Community rule of thumb, that double the length costs three or four times the energy: 0.5 to 0.63, giving 1.41 to 1.55.
- Millimetres per kilowatt, which is a linear guess: 1.0, giving 2. Wrong.
Ward's law, and where the exponent comes from
Steve Ward: branchless sparks want to be fed such that their input power goes up with the square of the time. If you want to double the length of a branchless streamer you can get there by doubling the driving period, and should expect four times the input power by the end of it.
One line of algebra:
P ~ t^2 => E = integral P dt ~ T^3
L ~ T (double the length, double the period)
=> L ~ E^(1/3) = E^0.333Which is what Gao measured independently at 0.31. One arrived from control theory and the other from a tape measure, and they differ by 8 per cent.
A worked upgrade
Taking the bus from 325 to 440 V raises the bang from 300 to 406 J, a factor of 1.35. Note that is not V², which would be 1.83: the published pair did not raise the current in proportion, which is the trap two sections down. On a coil currently making 188 cm:
- linear: 255 cm, plus 67. Wrong.
E^0.63, the optimistic ceiling: 228 cm, plus 39.E^0.5, Freau: 219 cm, plus 31.E^0.33, Ward: 208 cm, plus 20.E^0.31, Gao: 206 cm, plus 18.
The reason it is sublinear: the outer half of a growing arc costs about as much as the whole original arc did, while the inner part thickens and heats as well. Every additional centimetre is dearer than the last.
About millimetres per kilowatt
People quote figures around 58 to 74 mm/kW, and as a local indicator it is genuinely useful. Measure the power, measure the arc, divide; if it went up, your matching improved.
Use it as a thermometer on your own coil at your own ramp. Not as a scaling law.
The doubling trap
Formally E goes as V², but quadrupling the energy requires the current to double as well. If the bridge will not carry that and the current stays where it was, the energy merely doubles, and the arc then grows by E^0.31...0.63, which is tens of centimetres at best.
Does a longer ramp give a longer arc
Yes, and there is no optimum: the length rises monotonically with diminishing returns.
- 25 ms, about 500 to 620 J: 2.2 to 2.8 m.
- 30 ms, 600 to 740 J: 2.3 to 3.0 m.
- 35 ms, 700 to 870 J: 2.4 to 3.1 m.
Thirty five against thirty buys 12 to 25 centimetres. What stops people going further is not the growth physics but the consequences: the detuning carries the frequency below the band where arcs stay straight, the duty cycle and the heating rise, and the tank capacitors run out of current.
Figures here are from published builds and measurements, and the arithmetic on them is ours.