Buck 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.

We know that $$ \hat i_{in} = D \hat i_L + \hat d I_L$$ but the dependency from d^ is nulled. Using the power stage functions found in previous articles, and substituting one in another we can find that $$ Z_{in,OL} = \frac{\hat v_{i}}{\hat i_{i}} = \frac{R}{D^2} \frac{s^2 + s\frac{L}{R} + 1}{sCR+1}$$ in DC (s=0) the open loop input impedance becomes $$ Z_{in,OL} = \frac{R}{D^2} $$ and this result can be obtained in this simple way, using the DC model of the buck converter $$\begin{cases} V_O = D V_I \\ I_O = \frac{V_O}{R} \\ P_O = P_I \ \rightarrow \ V_O I_O = V_I I_I \end{cases}$$ Substituting the to leaving just V_I and I_I in the formula, the same result appear.