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Suppose Vs = 0V. You are right into the NPN and PNP transistor. What basically happens is that if the MOSFET gate capacitance uncharged, it is sitting at zero V (relative to the source of the MOSFET). WHen you flick on the NPN BJT it connects connects this capacitance to a supply Vgatedrive to which it will charge up. BUt as it charges up the capacitor, the NPN transistor slowly turns off as the voltage difference between it's collector and emitter gets reduced since the gate capacitance voltage is rising up towards Vgatedrive but is the same node as the NPN's emitter. THis means the NPN transistor will not saturate and thus be able to switch very very fast.
It's a similar thing for the PNP.
This "quasi-saturation" that allows for incomplete turn and this much faster switching at the expense of efficiency is one of the reasons I think it is preferable to use BJTs for driving MOSFET gates (rather than using smaller MOSFETs to drive the gates of larger MOSFETs) since you cannot get this effect with a MOSFET. Also something about peak-current capability or something and how the BJT lacks the same capacitances a MOSFET does.
If you flipped the NPN and PNP transistors around (so that the PNP was on the high-side and NPN was on the low-side), it would still charge up the gate capacitance but it would not slowly turn off as the gate capacitance got more charged, it would stay on and saturate reducing it's switching time. THis is the method where the BJTs saturating has is the equivelant used for MOSFETs in bridges (where the N-channel is on the low-side and and the P-channel is on the high side), and is also used for BJT motor drivers where you want more efficiency rather than faster switching. You can make BJT motor drivers that quasi-saturate however for faster switching, but I do not know if the NPN/PNP configuration is the same as the gate driver we are talking about.
If you really want to understand how it works. BTW the configuration is apparently called a complimentary emitter follower in case you want to find out even more:
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Remember, the voltage required to turn on a MOSFET is the voltage between it's GATE AND SOURCE. Not the gate and ground. SO if the source is not at ground, you need some more complex circuitry gate drive circuitry, like something that floats and is referenced to the source of the MOSFET like an isolated floating supply or a capacitor that charges when connected between V and ground, but then gets floated up to the source via a charge-pump or something.