c4 looks like it has a joint purpose of being the resonating cap for the buck inductor to resonate with after d1 has turned off...but also it cunningly acts as a charge pump kind of thing to keep the IC supplied.
The cct relies on the resonance between the buck inductor and c4 to resonate down when d1 turns off, and then the low voltage there is detected (by the drain pin), and that way the IC knows that the inductor has discharged, and so it knows that it must turn the fet on again.
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incidentally, d3 is there to stop charge getting sucked out of the vcc capacitor when the source of q1 flys upwards...as you know, when the drain of q1 flys upwards, the source goes with it because they are connected by a capacitor.
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d2 blocks the discharge of c3, so that the fet q1 is kept on...you dont want c3 to supply the ic , because then c3 would discharge....also, you dont really want chip power to come through R1A r2a and r2c, (as they woudl overpower and heat up) so you have the diode d2 to block current going thru those three resistors to the chip.
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actually some of the supply current for the chip will come through q1 just after the internal fet turns off.