Question

Proved that in AC circuit average AC power consumed is zero.

Answer 1

Answer on Question # 82736, Physics / Electromagnetism

Question 1. Proved that in AC circuit average AC power consumed is zero.

Solution. A circuit element dissipates or produces power according to $P=IV$, where $I$ is the current through the element and $V$ is the voltage across it. Let $p(t)=i(t)v(t)$.

$P_{ave}=\frac{1}{T}\int\limits_{0}^{T}p(t)\,dt$, where $T=\frac{2\pi}{\omega}$ is the period of the oscillations. With the substitutions $v(t)=V_{0}\sin\,\omega t$ and $i(t)=I_{0}\sin\,(\omega t-\phi)$, this integral becomes

$P_{ave}=\frac{I_{0}V_{0}}{T}\int\limits_{0}^{T}\sin\,(\omega t-\phi)\,\sin\,\omega t\,dt,$

$P_{ave}=\frac{1}{2}I_{0}V_{0}\,\cos\,\phi.$

For the resistor $\phi=0$ and $P_{ave}=\frac{1}{2}I_{0}V_{0}$.

For a capacitor $\phi=\frac{\pi}{2}$ and for an inductor $\phi=-\frac{\pi}{2}$, so $P_{ave}=0$ for both.

The phase angle for an AC generator may have any value. If $cos\,\phi>0$, the generator produces power; if $cos\,\phi<0$, it absorbs power. $\Box$