A crystal diode having   internal resistance  $r_f$  is used  for full wave rectifier .  If the  applied voltage is $V=V_0\sin wt$  and load resistance is $R_L$ , find  $i_{avg}and{i}_{rms}$.

$Average current in full wave rectifier, i_{avg}=\frac{{\displaystyle \underset{0}{\overset{\frac{T}{2}}{\int }}}{I}_0\sin \omega tdt}{\frac{T}{2}} \left[here, T=\frac{2\pi }{\omega }\right]$

$$\begin{gathered} \Rightarrow i_{avg}=\frac{I_0}{\pi }{\left(- \mathrm{cos\omega t}\right)}_0^{\frac{\pi }{\omega }} \\ \Rightarrow i_{avg}=\frac{2i_0}{\pi } \end{gathered}$$

$RMS current in full wave rectifier , i_{rms}=\sqrt{\frac{1}{{\displaystyle \raisebox{1ex}{$T$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\underset{0}{\overset{\frac{T}{2}}{\int }}{I_0}^2{\sin }^2\omega tdt}$

$\Rightarrow i_{rms}=\sqrt{\frac{{I_0}^2}{T}{\left(t-\frac{2\sin \omega t}{\omega }\right)}_0^{\frac{T}{2}}} \left[here, T=\frac{2\pi }{\omega }\right]$

$\Rightarrow i_{rms}=\frac{i_0}{\sqrt{2}}$