Question

The instantaneous emf and current equations of an RLC series circuit are
$e=200\sin (\omega t-\frac{\pi }{6})$
$i=20\sin (\omega t+\frac{\pi }{6})$
The average power consumed per cycle is

• A. Zero
• B. $2000W$
• C. $1000W$
• D. $500W$

Answer 1

Answer: (C)

$1000W$

Instantaneous power of the circuit, $P=ei$

Hence the average power of the circuit $=\frac{{\int }_0^{2\pi /\omega }Pdt}{2\pi /\omega }$ $=\frac{{\int }_0^{2\pi /\omega }4000sin(\omega t-\frac{\pi }{6})sin(\omega t+\frac{\pi }{6})}{2\pi /\omega }W$

Using: $cosA-cosB=-2sin\left(\frac{A+B}{2}\right)sin\left(\frac{A-B}{2}\right)$

Average power $=\frac{{\int }_0^{2\pi /\omega }4000\frac{1}{2}(cos\frac{\pi }{3}-cos2\omega t)dt}{2\pi /\omega }W$ $=1000W$