Question

The DC and RMS components of currents in a half-wave rectifier are given by the relations?

• A. \( \frac{I_m}{2} \) and \( \frac{I_m}{\pi} \)
• B. \( \frac{I_m}{\pi} \) and \( \frac{I_m}{2} \)
• C. \( \frac{2I_m}{\pi} \) and \( \frac{2I_m}{2} \)
• D. \( \frac{I_m}{2\pi} \) and \( \frac{2I_m}{2} \)

Hint:

For a half-wave rectified sine wave: \( I_{DC} = \frac{I_m}{\pi} \), \( I_{rms} = \frac{I_m}{2} \)

Answer 1

The Correct Option is B

Step 1: Understand the wave shape
Half-wave rectifier only allows positive half of a sine wave. So, the waveform is non-symmetric.
Step 2: DC component calculation
Average value over one full cycle: \[ I_{DC} = \frac{1}{\pi} \int_0^\pi I_m \sin \theta d\theta = \frac{I_m}{\pi} \] Step 3: RMS component calculation
\[ I_{rms} = \sqrt{ \frac{1}{2\pi} \int_0^\pi I_m^2 \sin^2 \theta d\theta } = \frac{I_m}{2} \] Hence, Option (2) is correct.