s ? y
The 1/2 CV^2 produces an Imax in boost L of Vsolar x sqrt (C/L), then in flyback, current into battery jumps to Imax and linearly slopes to zero at rate of dI/dt=3D (-)Vbatt/L ( neglecting a few things like solar cell potential and state of voltage on storage C)) for a total charge of Qbatt=3D1/2 Imax x Imax/(Vbatt/L)=3D1/2 Imax ^2 x L /Vbatt=3D 1/2 x C x Vsolar^2 /Vbatt=3D 1/2 x ( C x Vsolar) x (Vsolar/Vbatt)=3D1/2 x Qsolar x (Vsolar/Vbatt). Therefore Qbatt/Qsolar=3D1/2 x (Vsolar/Vbatt) or ratio of battery charge to solar harvest charge is 1/2 ratio of solar cell to battery potential. That's the circuit theortic analysis. The energy analysis simply notes that the energy increase in battery is Vbatt x Qbatt, and this is derived from the capacitor energy store of 1/2 x C x Vsolar^2=3D 1/2 x (C x Vsolar) x Vsolar=3D 1/2 x Qsolar x Vsolar making Qbatt/Qsolar=3D 1/2 x Vsolar/Vbatt again.