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I'm trying to design a basic bootstrap circuit as shown in the imagem below. It can be controlled by connecting the MOSFET source to ground.

enter image description here

Since it is my first time this type of circuit I have some doubts about the voltage in the MOSFET gate and about leakage voltage in the capacitor and diode.

About the voltage in the mosfet gate: the circuit is designed to work with 11 to 16 V, so because of the capacitor there will be a voltage between 22 to 32 V in it's gate. Since the MOSFET can withstand only 20 V, should I add a Zener diode to protect its gate? My MOSFET VGSth is only 3.6 V, because of that I was thinking of using a 18 V zener.

About leakage voltage from the capacitor and diode: this circuit is designed to operate with both low frequency PWM (maximum of 1 kHz) and also work as an on/off switch (like a relay). Because of the leakage voltage I'm afraid that this circuit might not work as I intended, is there a way to make it work without sending a 99.9% PWM to control it as a switch? Is there also a better way of doing it?

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Sorry, this will not work at all, but I'm not sure how I can explain it to you. Assume left FET is conducting the cap will be charged to 13.5V (14V - diode drop) and the right FET is off. Now you turn off the left FET and the cap will be discharged via 1k resistor. The cap pin connected to the diode will stay at 13.5V because the resistor also charges the gate of the second FET until there is 13.5V as well. During this the output load voltage will also rise with an offset of Vgsth below the gate voltage. You end there with about 10V output and a partial conducting (hot) right FET. Nothing else will happen.

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