A buck can produce any curve you like. Which curve is the main setting on a QCW, and the most underrated one: most branching problems are fixed here rather than in the hardware.
The reference implementation
Nearly every ramp generator in the community grew out of Finn Hammer's Arduino code, and its parameters became the standard set of knobs: amplitude, duration, a shape control, bangs per second, and a wick, a starting shelf of about two milliseconds before the ramp proper.
There are usually two shape knobs rather than one, in time and in amplitude, called tscale and uscale in most implementations. Those two are what bend the curve.
What shape is right
Two independent statements, and they are the same statement:
Jan says the ramp should be close to linear in the bridge's voltage, not in power. Steve Ward says the power into the arc should rise as the square of time.
The load is resistive, so P = V²/R. Linear voltage gives quadratic power automatically:
V ~ t => P ~ t^2 => L ~ E^(1/3)What has actually been compared
One direct comparison exists, from MS.Lab, best to worst:
- Normal, close to linear. Best overall.
- Exponential. Fewer flashovers on the secondary, better on long ramps.
- Two-state. Branches.
- Half sine. Worst of those tried.
Read that as: "shallow first, then steeper" is the exponential shape and it makes sense. Not as a replacement for linear, but as a defence against flashover and as the option for long ramps. The logic is direct: a shallow start does not drive the voltage up while the arc is still short and all the field is sitting on the secondary.
It is absurdly sensitive at the end
This is the nastiest property of the whole thing.
Jan: microscopically different ramp rates right at the end either give branching or take it away. He calls it magic, and does not mean it kindly.
Which means: tune the shape finely and last, after the pole, the detuning and the phase lead are all set. And change one parameter at a time, or you will not know what worked.
Why smoothness matters at all
The ideal growth is at constant speed, and constant speed means a constant voltage at the tip relative to ground. Any jerk in the ramp is a jerk in the tip voltage, which is an invitation to a new branch.
Everything else about the modulator connects here. The buck sets the smoothness of the ramp, which is why pulse skipping never produced a sword, and why the ripple and the discontinuous threshold matter as shape rather than as efficiency.
The upper limit on ramp time is physics, not hardware
You cannot stretch the ramp indefinitely, and the wall is not in the switches:
Uspring: the space charge repulsion in the channel is of order rho^2/(4·pi·e0), about 0.01 newtons per cubic centimetre, while the buoyant force from heating is about 10^-5. Three orders of magnitude apart.
So the channel is pulled apart by its own charge rather than floating away on heat. That sets the upper bound on a sensible ramp time, past which the arc simply stops holding straight.
The energy side agrees. Going from 30 ms to 35 buys 12 to 25 centimetres. Returns fall away on both counts at once.
The ramp on the QCW department page is drawn as the modulator makes it: a wick, an even climb, a fall, with the primary current standing up inside the envelope. The slider under it moves the length, and the arc answers at the cube root of the energy.