Ah, one quirk about the design, L1 was a planar inductor -- it's non-pop because it literally isn't a component. It was some 100s nH I think? The jumper likewise was there to bypass it and place my own external one, in case needed (or to try).
What difference is dead time for L-H as opposed to H-L?
At the high switching frequency / low inductances, 14ns was more than enough dead time; I forget what the actual waveforms showed. That would be something to adjust if going to lower frequencies of course. Drivers with various fixed amounts, or variable, are available.
Indeed, setting up an open loop test is a start; just beware of drawing excess current around resonance. A relatively small supply bypass capacitor (so it discharges quickly under dangerous conditions) plus a current limited supply would be a good idea. (Mind the output capacitance of typical bench supplies, by the way; you may want to use an external resistor for that limiting.)
Draw out the node equations and solve for current and impedance; get a feel for how the network responds, and how to choose components. Generally Lm/Lr ~ 5-10, but smaller values can be used, particularly if a wide input/output voltage range is needed. Zo varies from sqrt(Lm/Cr) to sqrt(Lr/Cr) (and similarly for Fo) depending on load resistance, so you need to choose Zo (and Q) adequate to deliver required output voltage and current (power).
Resonant works well for high voltages, though you may want to alter the network so as to load the output with capacitance rather than inductance -- this accounts for the mismatch between winding Zo (the winding is layers of wire, it can't be much more than a few hundred ohms) and load resistance (easily 10s kohms for even fairly modest HVs of low ~kV). Thus the transformer bandwidth is quite low (<10%, maybe?) and pulsed operation is infeasible, but class E (CRT flyback) and (full wave) resonant (your classic "ZVS HV driver") are perfectly doable.
Which, let's see. The direct L-C complement of the LLC, the CCL, would be...
Inverter: voltage sourcing <--> current sourcing
series inductor Ls <--> shunt capacitor Cs
series capacitor Cr <--> shunt inductor Lr
shunt magnetizing Lm <--> series Cm*
shunt load RL <--> series load RL
*But it's not 'm'agnetic at all. "Electrotizing" C_E..??!
Okay, so that doesn't work -- load is in series with capacitor, not parallel. It would kinda work for some loads -- indeed, CCFL drivers use a parallel-resonant tank with a series cap to the load. The high voltage on said tank however is generated differently...
If we just straight up complement L and C, we basically swap the role of Ls and Cr so that's trivial, but Lm becomes Cm in parallel with the load resistance -- there we go. But, we don't have any place for transformer inductance; so it better be a good transformer (in the sense of contrasting with a "coupled inductor"!).
The other usual approach is another kind of LLC,
often called series-fed parallel-resonant. This works similar to LLC, but I think, is more common for higher Q factors? Note that the C can be decomposed into series and parallel resonant halves, and thus Lm + Cseries acts as an L-match network from Vin to Vo. So we can use standard RF design equations to work with it. And, indeed we can obtain Lm from leakage inductance.
And, as mentioned, HV transformers are often narrow band i.e. highish Q, so this works out nicely.
Power control can be frequency modulation as above, just beware the step response varies with the difference between Fdriven and Fo -- that can be quite nasty for a control loop. Consider this FSK response:
The same is true for a basic series resonant network as well. I've used both series-fed and series resonant for induction heating purposes, which was the context of this particular waveform.
At high enough voltages, eventually it approaches a Tesla coil, but we might not drive it that way -- either because that assumes double tuning, or pulsed operation (both as in the spark type TC). Though to be fair, since the resonant current is necessarily quite high to develop high voltages -- we might not mind using pulsed operation, if the output doesn't need much current. That said, again because of bandwidth limitations, pulsed operation may not save us much: it takes a long time for the output to ramp up and down, and we might be better off just using better materials (litz windings where applicable, better ferrite cores, etc.) and CW operation instead. (Note that, since secondary capacitance is dominant at very high voltages, its insulation also needs to be high quality, to give high Q factor. That may mean avoiding polyester tape; probably something like, oh I don't know, oil-impregnated polypropylene paper would be best..? Oil immersion would be necessary anyway if we're talking like 100s kV for x-ray, SEM, etc. applications.)
Tim