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BRIDGELESS AC TO DC FLYBACK CONVERTER SIMPLIFIED

hi,

we are working on something which should convert (300-450) AC voltage to dc voltage of 12V without using a rectifier circuit (no diodes connected to primary) (flyback with ac input with no rectifier circuit and no auxiliary coil only operated in buck mode with primary and secondary coil and the primary circuit must function only in the positive half cycle)

given  components are a capacitor, a switch(n channel MOSFET or transistor), a flyback transformer, load ,

can you get me the design equations corresponding to this configuration of flyback transformer and capacitor and the input and the output equations