The driver does the same four jobs it does on any coil: it finds the frequency, it fires early to make up for its own delay, it decides when a bang runs, and it stops when the current is too high. On a ramped coil each of those four changes, and one of them changes into something else entirely.
Finding the frequency, while the frequency runs away
Every coil's frequency falls as the arc grows, because the arc adds capacitance. On a DRSSTC that happens over a few hundred microseconds and the feedback loop follows it without anybody thinking about it.
On a QCW it happens over twenty five milliseconds and it is enormous. The secondary can move by tens of per cent across one ramp. The driver has to track that the whole way up, which is why this department talks about the two poles so much: coupling splits the resonance in two, the driver sits on one of them, and on a QCW it has to sit on the upper one and stay there. Left alone, a coil settles on the lower one.
Getting up there needs a starting oscillator or a phase locked loop. That is the first thing that has to be added to an ordinary driver.
Firing early, while the target moves
The lead is set by an L and an R in the feedback path, and that network holds a constant angle. The hardware needs a constant time: the delays through the comparator, the logic and the transformer are fixed nanoseconds.
At one frequency you can make those agree. Across a ramp that sweeps by a sixth they drift apart by construction. It is the limit of the whole technique, and a QCW walks straight into it.
Deciding when to run: this is now two jobs
On a DRSSTC the interrupter says start and stop. On a QCW the same box also tells the buck converter what voltage to hold at each instant, and that instruction is the ramp.
So the box in that position is a modulator rather than an interrupter, and it has no ontime of its own: the bridge runs for exactly as long as the ramp lasts. Bangs per second drops from hundreds to five or seven, because a QCW run at two hundred is not a QCW, it is an explosion.
Stopping: cutting the bang is now a failure
An overcurrent detector that ends the bang is right for a DRSSTC. On a ramp it is a disaster: cut in the middle and the arc never finishes growing, and what you get is an audible pop instead of a sword.
So the protection has to skip a single cycle and let the bang continue. That is the second thing to add to an ordinary driver.
Two more, and then the list is done
The burden resistor has to come down, so that the feedback strength lands in its working window, and the lead coil has to come down with it, because the lead is set by the ratio of the two and not by the inductance.
Where to go next
The full list of what to change on a UD board, with the self-oscillation recipe as the people who did it wrote it, is converting a DRSSTC driver. What the driver does in general is four jobs on one board, and the pole question has its own piece.
The driver on the QCW diagram is labelled "driver and lead" for the reason in the second section, and the modulator below it has a line running up the outside to the buck.