The ramp exponent: the curve is a hump

Put a shape control on a ramp and it is tempting to think of it as one axis with a good end and a bad end. It is a hump. Both ends give branching, for opposite reasons, and knowing which reason you are looking at is the difference between fixing it and chasing it.

The axis, written down

V(t) = V_max · (t/T)^p

p < 1   concave: fast start, early shelf      <- half sine lives here
p = 1   linear                                 <- "normal", the best one
p > 1   convex: slow start, sharp peak         <- "exponential" lives here

The physics is direct. A concave ramp brings the voltage to maximum before the arc has grown. Full power arrives at a short channel and it breaks out in every direction. That is not really a ramp any more, it is a long DRSSTC bang.

Convex does the opposite: the arc lights gently and grows, and the power follows it up. Which is why ZakW removed the starting shelf entirely and used a non-linear rise instead, doing the same job without a corner in the curve.

The hump

length
  |        .--.
  |      .-'    '-.            maximum around p = 1 to 1.5
  |    .-'          '--.
  |  .-'                 '---. falls to "pop" past p = 2.5
  +--+-----+-----+-----+-----+--> p
    0.4   1.0   1.5   2.5   4.0
  concave linear       sharp peak

The left side falls because full power arrives too early and the short channel branches. The right side falls because all the energy is in the tail, the overcurrent detector fires, and the bang tears: that is the pop.

The useful measure is the tail, not p

What actually trips the protection is how much of the energy arrives at the end:

• p = 0.4, strongly concave, near half sine: 33 per cent in the last fifth of the ramp.
• p = 0.7, concave: 42 per cent.
• p = 1.0, linear: 49 per cent.
• p = 1.5, slightly convex, near exponential: 59 per cent.
• p = 2.5, sharp peak: 74 per cent.
• p = 4.0: 87 per cent.

So the right curve is an S

Putting it together: convex at the start, which lights the arc gently and replaces the wick; linear through the middle and the end, in the bridge's voltage, so the power goes as t² by itself because the load is resistive; and no corners anywhere, because the two-state shape branches precisely at its step.

That is Jan's "linear in bridge voltage" and Ward's "power as the square of time" being the same statement, with a soft nose on the front.

What breaks first, in order

1. Overcurrent, and the bang tears. Immediately, as soon as the tail puts the current over the threshold. This is the pop: the bang breaks near the top, the arc never finishes growing, and you hear it.
2. Racing sparks. When the amplitude or the coupling goes up. Flashover across the secondary, visible in the dark.
3. Tank capacitors overheating. Cumulative, over minutes. Calculated from RMS current.
4. Bridge dT fatigue. Latent, thousands of cycles, and the bridge dies weeks later for no visible reason.

Three axes that get mixed up

Branching and zigzag are one defect: a dip in current raises the losses and a new branch leaves. But two other things are often folded in with them and should not be.

• Branching and zigzag are set by the ramp shape, the pole, and residual voltage on the bus. They fall as frequency rises.
• Channel thickness is set by frequency and power. Thicker at low frequency.
• The whole channel bending is set by the surroundings and the toroid's field. It rises as frequency rises.

And Jan's warning belongs here too: microscopically different rates right at the end either give branching or take it away. Which means the right-hand slope of that hump is steep and slightly unpredictable, not a gentle curve.

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The ramp scope on the QCW department page draws the shape the modulator makes, and the slider under it moves the length rather than the exponent. The exponent is the one you set once, carefully, and last.