Boost converter - Input impedance

Now, let's find the open loop input impedance of the power stage. The scheme to be used is the following one: the DC transformer is removed and the d^ source is nulled.

Actually, we already know that $$ \hat i_{in} = \hat i_L$$ We have already calculated this value when finding the AC model! It's the reciprocal of the so-called Giv $$Z_{in,OL} = \frac{\hat v_{i}}{\hat i_{i}} = G_{iv}^{-1} = D'^2\frac{s^2 \frac{LC}{D'2} + s\frac{L}{RD'2}+1}{sC+\frac{1}{R}}$$ in DC (s=0) the open loop input impedance becomes $$ Z_{in,OL} = RD'^2 $$ and this result can be obtained in this simple way, using the DC model of the buck converter $$\begin{cases} V_O = \frac{V_I}{D'} \\ I_O = V_ID' \\ P_O = P_I \ \rightarrow \ V_O I_O = V_I I_I \end{cases}$$ Substituting the to leaving just V_I and I_I in the formula, the same result appear.