= Complex power
{disambiguate=electrical engineering}
{title2=$S=VI^*$}
= Complex electrical power
{synonym}
<Complex electrical power> represents the active and reactive parts of sinusoidal electrical energy transfer in one <complex number>. Using <phasors> $V$ and $I$ whose magnitudes are <root mean square> values, with the <passive sign convention>, define
$$
S=VI^*=P+jQ,\qquad j^2=-1.
$$
The star denotes a <complex conjugate>. The real part is <active power>; the imaginary part is <reactive power>. The magnitude $|S|=|V||I|=\sqrt{P^2+Q^2}$ is <apparent power>. A source using peak-amplitude phasors instead has the factor $1/2$: $S=V_{\rm peak}I_{\rm peak}^*/2$.<AC Power Flow in Linear Networks by James L. Kirtley Jr.>[Equation63 and §7.1, explicit amplitude convention]{ref}
To see why the conjugate matters, write $V=|V|e^{j\alpha}$ and $I=|I|e^{j\beta}$. Then $S=|V||I|e^{j(\alpha-\beta)}$ depends on the relative phase. Changing the common time reference multiplies both phasors by $e^{j\theta}$, leaving $VI^*$ unchanged. The product $VI$ would instead acquire $e^{2j\theta}$ and would not represent a time-reference-independent average-power quantity.
For an original example, choose $V=100\,\mathrm V$ and $I=(4-j3)\,\mathrm A$. Then
$$
S=100(4+j3)=400+j300\ \mathrm{VA},\qquad |S|=500\ \mathrm{VA}.
$$
Thus <active power> is400W, <reactive power> is300 reactive volt-amperes and <power factor> is0.8. The corresponding <electrical impedance> is $V/I=16+j12\,\Omega$. With RMS current magnitude5A, the resistive term absorbs $16(5^2)=400\,\mathrm W$, agreeing with the real part.
Under the stated sign convention and $e^{j\omega t}$ time convention, an inductive reactance gives positive $Q$ and a capacitive reactance negative $Q$. Negative $Q$ does not mean a negative dissipated-energy average. For example, adding an ideal parallel capacitive branch contributing $-j300\,\mathrm{VA}$ cancels this example's reactive part, leaving terminal complex power400VA and RMS source current4A at100V, while the original resistive dissipation stays400W.
This one-frequency representation must not be applied indiscriminately to a distorted waveform. Different harmonics need their own phase relationships; the definition of <active power> remains the time average of the actual <instantaneous power>.
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