According to my schematic below, I would like to know the relation between the Duty Cycle (called alpha) of the NMOS gate, and the voltage Vp.

At the end, I want the relation between the current in the voltage source VS alpha, but from Vp, I will be able to find it easily.

What I understand:

If C2 is removed from the circuit, it is easy to calculate the equivalent resistor made by R1, R2 and R3 according to alpha:

Req1 = R1*(1-alpha)+alpha*R1*(R2+R3)/(R1+R2+R3)

And then calculate the current in V1 :

I(V1) = V1/Req1

But the addition of C2 creates a non linearity and the above equation is no longer correct (because C2 is charged and discharged with a different time constant due to the difference between R2 and R3)

I tried to find the charging and discharging differential equations:

Vfinal_discharge =(Vfinal_charge-V1*R3/(R3+R2))*exp(-alpha*T/(C2*Req2))+V1*R3/(R3+R2)

Vfinal_charge == (V1-Vfinal_discharge)*(1-exp(-T*(1-alpha)/(C2*Req2)))+Vfinal_discharge

with :

T=switching period

Req2 = R2+R3 / (R2*R3)

Resolving them (2 equations and 2 unknowns) does not seam to give a correct relation...

If you have any idea, I would very much appreciate.

Thank you,

Oto