= AC power
{c}
{wiki}
<AC power> describes energy transfer associated with time-varying <electric potential difference> and <electric current>, including the distinction between instantaneous transfer and its average. With the <passive sign convention>, current enters the terminal marked positive, and <instantaneous power> absorbed by the element is $p(t)=v(t)i(t)$. A positive value means energy entering the element; a negative value means energy returning through that terminal pair.
For sinusoidal steady operation, choose
$$
v(t)=\sqrt2 V\cos\omega t,\qquad i(t)=\sqrt2 I\cos(\omega t-\varphi),
$$
where $V$ and $I$ are <root mean square> magnitudes. The cosine-product identity gives
$$
p(t)=VI\cos\varphi+VI\cos(2\omega t-\varphi).
$$
Averaging over a complete period removes the second term, giving <active power> $P=VI\cos\varphi$. This derivation separates the energy-transfer average from a twice-frequency oscillation.<University Physics Volume 2 by Samuel J. Ling and others>[§15.4, equations15.12–15.13, sinusoidal average power and RMS quantities]{ref}
For an original example, $V=120\,\mathrm V$, $I=5\,\mathrm A$ and $\cos\varphi=0.8$ give <active power>480W. The oscillating term has amplitude600W, so <instantaneous power> ranges from−120 to1080W. Over ten complete cycles at50Hz, the net energy absorbed is $480(10/50)=96\,\mathrm J$. Temporary negative power can accompany positive net transfer when an element stores and returns energy.
The product $VI=600$ in volt-amperes is the <apparent power>. In the single-frequency sinusoidal convention, <reactive power> is $Q=VI\sin\varphi$, giving360 in reactive volt-amperes for a lagging current with $\sin\varphi=0.6$. <Complex electrical power> packages $P$ and $Q$, while the <power factor> $P/(VI)=0.8$ distinguishes net transfer from the terminal magnitude product. The primary MIT treatment derives these quantities and makes the peak-versus-RMS convention explicit.<AC Power Flow in Linear Networks by James L. Kirtley Jr.>[Energy and Power, equations61–63; §7.1 RMS Amplitude]{ref}
A measurement must retain the relative phase of the two channels. Multiplying separately measured RMS magnitudes gives <apparent power>, not generally <active power>. For nonsinusoidal signals, the time average of $v(t)i(t)$ remains the definition of <active power>, but one phase angle need not describe every harmonic contribution.
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