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Quite interesting. Of course, the suggested supply circuit hasn't much to do with a home microwave oven, which exclusively uses unfiltered 50 Hz half-wave DC (from a voltage doubler circuit).You can find here some links to experiments on phase locking a microwave oven magnetron by controlling the cathode voltage via a high voltage valve voltage regulator.
When you are going to operate a magnetron with stabilized DC supply (without the said frequency locking), you'll notice that stable operation is much easier achieved with constant current rather than constant voltage mode. With constant voltage, the magnetron output is likely to jump between modes.