If the A.C. main supply is given to be 220 V in a circuit. The average emf during a positive half cycle is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( 198\,\mathrm{V} \)
The quoted mains voltage of 220 V is the rms value. The peak value follows from \( V_{\mathrm{rms}} = V_0/\sqrt{2} \).
\( \displaystyle V_0 = \sqrt{2}\,V_{\mathrm{rms}} = 1.414 \times 220 \approx 311\,\mathrm{V} \)
Over a full cycle the average of a sine wave is zero, but over a positive half cycle it is not. For \( V = V_0\sin\omega t \):
\( \displaystyle V_{\mathrm{av}} = \frac{1}{T/2}\int_0^{T/2} V_0\sin\omega t\,dt = \frac{2V_0}{\pi} \)
\( \displaystyle V_{\mathrm{av}} = \frac{2 \times 311}{\pi} \approx 198\,\mathrm{V} \)
Shortcut: The half-cycle average is \( 2\sqrt{2}/\pi \approx 0.9 \) times the rms value, so \( V_{\mathrm{av}} \approx 0.9 \times 220 = 198\,\mathrm{V} \).