Why does the axle of a VVC needs to be isolated and that of a Butterfly-C not?
I think it would be going too far to describe the electrical states during operation using mathematical formulas at this point. I'll try to do that with a few drawings and a more simple explanation.
If tuned, a mag-loop is a parallel resonant circuit. Let's start with the capacitor. The states on the capacitor change from charged positive to zero and then to charged negative. These states alternate in the form of a sine wave in time with the resonant frequency.
The charge on the capacitor generates a current in the parallel inductance (loop) until the capacitor is completely discharged. When the capacitor is discharged, all of the energy is stored in the inductor's magnetic field.
The magnetic field now generates a current in the opposite direction and thus charges the capacitor again, but with reversed polarity. Losses that occur are compensated by supplying energy. In this way, the oscillation is maintained. A part of the magnetic field is also radiated. The loop "sends".
Now to the starting question:
The difference is caused by the design of the capacitors.
The VVC is a two-electrode unit. The resonant voltage of the oscillating circuit is always present at the electrodes, alternating between positive and negative. The drive axle is electrically connected to one of the capacitor plates. So there is the full high voltage during operation. The electrical zero point of the loop is the mechanical middle of the loop.
The Butterfly-C consists of two capacitors connected in series. The inner plates are like a third electrode in the middle of the capacitor. The drive axle is connected to this. The voltage at this point is almost zero, as result of the addition of the outer voltages. The outer electrodes oscillate in terms of voltage around this midpoint. This is why you can attach the axle here without insulation. However, some differences in the two capacitors (synchronism) can also cause small voltages (a few volts) to occur. An isolation of the axle is recommanded but not essential.