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Combine the two we can get the graphs for the change in inductor voltage and current over a period T, and we also know that T_on+T_off=T
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Since we have assumed 'average steady state', which implies that the change in current are the same, so we can equate the two changes in current
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Finally we get the transfer function Vout/Vin=D, the duty cycle
As for how to design a buck converter, I referred