Infineon H7 is pretty recent IGBT technology, with an impressive combination of low conduction and switching losses. I didn't realize it was on the market yet in 650V. Icm is 600 A, which is pretty impressive for a TO-247 part, and the Vce vs Ic curves really support that these devices can handle a lot of current.

Congratulations on getting it working. As you've seen, not having a matching transformer leads to a lot of losses in the IGBTs, given that it restricts you to using a pretty low bus voltage in order to keep the power at a reasonable level. This will be even less favorable with more difficult loads, since the series resonant tank will draw maximum current when the loading is the lightest (highest Q), while a ferromagnetic steel tube will present the lowest Q of any practical load.

Your IGBTs are also pretty oversized in terms of rated current. I would aim for some more modern 1200 V parts of lower rated current, and operate it off rectified 3-phase mains. A full bridge running from 560 V bus would only need to push some 25 - 30 A to deliver your target power, so using 400 A parts just leads to uneccessary drive power requirements.

For the matching transformer, 0.5 mm wire is technically within a few skin depths of the operating frequency, but as long as you have more than a single wire then you also need to consider proximity effect. For this frequency, I would go with 0.1 mm or finer litz, but 0.2 mm is likely also fine, especially if you use a single layer winding on your toroid.

The ferrite should work fine. At these low frequencies, any power grade of ferrite should be fine.

Your caps are rated at 2.5 A RMS a piece for 15 degree internal heating. If we aim for the best practical external cooling, we can get away with 60 degrees heating, so double the current, that's 5 A per cap. .22 uF is 15 ohms at 50 kHz, which gives you a cap voltage of 15*5 = 75 V RMS at the maximum rated cap current, or 5 A * 75 V = 375 VA per cap, for a total of 34 kVAR for the 90 caps, independently of how they are configured. Right now it also looks like you have some stray inductance in your tank circuit which will eat up some of those VARs as well, by the ratio of work coil inductance to total tank inductance.

With a steel tube in the coil, you might end up with a Q as low as 4, which allows you to transfer 34/4 = 8.5 kW while staying within the ratings of your capacitors. For solid steel well matched to the size of the coil, Q might be 5 - 8, up to a few times that for smaller work pieces. For aluminium workpieces, a Q in the 20 - 50 range is not unrealistic, and for copper it can exceed 100. It follows that the power throughput will be very limited. I would definitely look for a larger capacitor if you're aiming for 10 kW. I would aim for at least 300 kVAR for a heater of this size, to give some leeway for processing a usable fraction of the rated power across a range of loads. Where are you located? A proper induction heating cap doesn't need to be particularly expensive.

I missed the fact that it was operating around 13 MHz, in which case the situation is a bit better. 47 pF is around 250 ohms at this frequency, so the additional loading from the series 1k and clamping diodes will have less of an impact than I expected, though maybe a bit much for comfort when using PC1. I usually use a D flip-flop together with an XOR gate to unwrap the PC1 range around zero degrees, so it covers -180 to +180, but I also know people had success using PC3 for tesla coil applications. PC2 will not work well when both inputs are frequency-locked together.

In my experience, these kind of supplies lend themselves well to power control by changing the input voltage. My experience is based on Hongba HB- series SSNSTs. A variac would work, and a dimmer might work in a pinch but this is not guaranteed. I've used variable DC supplies to power them, but you need one that goes up to some 200+ V.

Exactly what are you trying to achieve with this? Some more context makes it easier to give useful advice. How much power are you aiming for and what is the nature of the load?

You can't connect them in parallel, the rectifier would short the AC output, and the AC transformer secondary would short the DC output. The former can be addressed by a DC blocking cap (at the cost of some reactive current flow), while latter issue would require an impractically large AC blocking inductor in series with the rectifier. Relays would be a lot more compact and cheaper compared to a kilohenry 30 kV inductor

For the current transducers, the Tamura would probably be fine, but the dynamic behavior is so badly specified (only a vague di/dt rise time specified) that I would prefer something with better documented behavior. The ICE part is meant for busbar mounting, and I would be a bit worried about stray field rejection and sensitivity deviation depending on your busbar geometry vs. the reference case. I would consider something like this https://www.digikey.com/en/products/detail/mornsun-america-llc/TL300-D1C/18735421 or https://www.digikey.com/en/products/detail/lem-usa-inc/HASS-200-S/1680530 , with preference to the Mornsun part.

For comparing devices, the datasheet current rating is pretty much meaningless, it's basically the current the device can run at with the package immersed in boiling freon without exceeding the maximum junction temperature. It does not consider realistic cooling systems, and it does not consider switching losses, so it's useless for comparing devices in *SSTC applications. The main issue is that IGBTs have much higher switching losses than IGBTs, and soft switching only helps to a limited extent due to effects like channel forward recovery. The best practical comparison I have found is from the paper "The practical use of SiC devices in high power, high frequency inverters for industrial induction heating applications", because it considers both resonant operation and frequencies in the range where QCW coils operate, I've attached a figure here:



This is comparing 300 A class SiC MOSFET bricks with 300 A class fast IGBT bricks, and even with three times as many IGBTs, you get less than half the output power at 400 kHz. There will not be a factor six improvement for pulsed coils, as MOSFET conduction losses rise quicker than those of IGBTs, favoring CW operation, but I would not be surprised if you get a factor 3 improvement, i.e. three times the amount of power for a given device current rating.

SiC device performance peaks at higher voltages, so you get better value from using 1200 V devices and much higher bus voltage. Both Steve Ward and Jan Martis' SiC QCW designs run at 800 V bus. Looking at existing designs gives a good idea of how much you can get out of a given device size. Steve Ward's 3-phase QCW only used two 21 mohm 1200 V MOSFETs to drive each coil, and that's using frequency detuning which is even more lossy than phase shift control.

Sil-pads are easy to use, but the performance is not amazing. For a TO-247 package, they add 2 - 3 k/W thermal resistance for a TO-247 for typical materials, down to around 1.5 k/W for high performance versions. That's 70+ degrees Celcius of additional junction temperature rise at a dissipation of 50W per device. Micas can be around twice as good, at the cost of more drain-heatsink capacitance on account of their lower thickness. Kapton film can perform better, but add even more capacitance for a given thermal resistance, and I've had failures from less than perfect heatsink surface condition and high dV/dt stress. That's 8 kV rated kapton thermal foil that failed after around half an hour at 800 V, 50 V/ns, 75 kHz.

Ceramics are by far the best. Alumina (Al2O3) at 1 mm thickness gives around 0.25 k/w, which can be reduced by going down to 0.63 or 0.50 mm at the cost of some capacitance (which will still be a lot lower than with mica or kapton). Electrical robustness is also extremely good. For even better performance, AlN can be used, which is almost as thermally conductive as aluminium or BeO, at some cost. Note that ceramics are a bit more finicky when it comes to flatness, both of the insulator sheet and the heatsink surface. Some extruded heatsinks are far from flat, others are pretty good, but ones with machined surfaces will always be best.

Spring clip clamping is also superior to using a screw through the package hole, by providing more pressure, more even distribution, and better creepage distances.

Some Aliexpress ceramic insulators, checked or sorted for flatness within say 50 µm, paired with a flat heatsink, a good paste in a not too thick layer, and a good spring clip like the MAX08NG, gives very good performance without being too expensive or difiicult to work with. Beware that some pastes are somewhat conductive, but there are good ones without metallic fillers. Even metallic fillers pastes can be used, as long as it's not smeared near the package pins or across the edge of the insulator.

Edit: as mentioned, direct contact reduces the interface thermal resistance even more, with the same concerns about using a good paste, not applying too much, having a flat heatsink and applying even pressure. Small heatsinks with a lot of air flow can perform better than large heatsinks under any air flow conditions. A bottleneck is how far the heat can spread from the transistor, due to the bulk thermal resistance of the heatsink. Having the fins close to the device and good forced airflow is hard to beat.

MCP6561/MCP6562 are also nice options. Not as fast as the TLV3501 or TL3116, but very good value if your application can live with some 50 ns propagation delay.

Each multiplier stage gives a voltage equal to Vpp, which is 2.8 * Vrms for a sinewave. So 60 kV out (neglecting voltage drop from loadin) would correspond to 60 / (4*2. = 5.3 kV RMS, or 7.5 kV peak.

If I understand correctly, the problem is that the circuit is not drawing much power when you load the output of the multiplier? How are you loading the multiplier and what happens when you do so? Does the multiplier output voltage collapse? Does the transformer secondary voltage disappear?

I've played a bit with inductance fed parallel resonant halfbridge circuits, of which the classic "ZVS" circuit is the simplest possible form. My experiences were:

1: It's critical to have some conduction overlap in the two transistors, in order to not get overvoltage spikes from interrupting the current in the DC link inductor(s). This conduction overlap is only guaranteed by the balance of turn-on and turn-off times in the transistor, which is not a well controlled parameter, especially when people use whichever transistors they can find.
2: Conduction overlap is problematic though, since it essentially shorts out the tank circuit. Depending on the tank impedance and VARs, and the actual overlap time and frequency, this can represent significant power loss in the devices.

To deal with (1), it's good to enforce some overlap using RC delays, but this makes (2) worse. The common solution for larger CFPR inverters is to have diodes in series with the switching devices to block this current, this makes it possible to have an arbitrary amount of overlap. Then a bit of phase lead is required to ensure ZVS. As long as all these things are in place, the circuit can be scaled up to the low MHz range and hundreds of kilowatts without any major challenges, but it does add enough complexity to make voltage fed half bridge inverters look attractive again.