The maximum efficiency of a half-wave rectifier is 

Explanation:

• To calculate the maximum efficiency (η) of a half-wave rectifier, we typically use the following formula: η = (DC power delivered to the load) / (AC power supplied by the source)
• The DC power delivered to the load (P_dc) for a half-wave rectifier can be calculated using the following formula: $P_{dc}=(I_{dc}{)}^2\times R_i$Where I_dc is the DC current and R_i is the load resistance.
• For a half-wave rectifier, I_dc is given by the formula: $I_{dc}=\frac{I_m}{\pi }$ Where I_m is the peak input current.
• The AC power supplied by the source or the AC input power (P_ac), is given by the $\eta =\frac{P_{dc}}{P_{ac}}$ value of the source current, Is, times the rms value of the voltage, V_s, divided by 2, as the power factor in a rectifier is 1/2. $P_{ac}=\frac{(I_{rms}\times V_{rms})}{2}.$

$Pac=(\frac{I_m}{\sqrt{2}}{)}^2\times \frac{R_i}{2}$

• Now, plugging these DC and AC power formulas into the efficiency formula:

$\eta =\frac{P_{dc}}{P_{ac}}$

$\eta =[(\frac{I_m}{\pi }{)}^2\times R_i]/[(\frac{I_m}{\sqrt{2}}{)}^2\times R_i/2]$

• Canceling out the common factors R_i and I_m² on both numerator and denominator: 

​$\eta =(\frac{1}{{\pi }^2})/(\frac{1}{2}\times \frac{1}{2})$

η ≈ 0.406 or 40.6%