Complex power (electrical engineering, $S=V{I}^{\ast }$)

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, defineThe 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 $1/2$: $S=V_{\mathrm{peak}}{I}_{\mathrm{peak}}^{\ast }/2$.^[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 $V{I}^{\ast }$ 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 $I=(4-j3)\mathrm{A}$. ThenThus 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.

Apparent power is the product of RMS terminal electric potential difference and electric current, conventionally expressed in volt-amperes:For sinusoidal signals at one frequency, it equals the magnitude $|S|$ of complex electrical power and satisfies $S_{\mathrm{app}}^2=P^2+Q^2$. It is not generally the average rate of net energy absorption, which is active power.^[ref]

For an original example, a100V terminal pair carrying5A RMS has500VA of apparent power. If active power is400W, the power factor is0.8. In the sinusoidal case the magnitude of reactive power is $\sqrt{500^2-400^2}=300$ reactive volt-amperes; the magnitude calculation alone does not determine whether the current leads or lags.

Magnitudes do not add like complex quantities. Consider two parallel sinusoidal loads at the same supply, with complex powers $300+j400$ and $300-j400$ in volt-amperes. Each has500VA of apparent power, but the combined terminal complex power is600VA with zero imaginary part. The source therefore sees600VA rather than the1000VA sum of individual magnitudes. Their internal branch currents and individual component requirements still exist; reactive cancellation at the source does not erase those branch conditions.

For nonsinusoidal waveforms, the product definition remains useful, but the simple equality with the magnitude of one fundamental-frequency complex power needs qualification. With $v(t)=\sqrt{2}(100)\cos \omega t$ and $i(t)=\sqrt{2}[3\cos \omega t+4\cos 3\omega t]$, RMS current is5A and apparent power is500VA. The fundamental-frequency complex electrical power is only300VA. The extra RMS current is not represented by that one complex number.

Thus record the waveform convention with an apparent power rating or measurement. Separately measuring RMS potential difference and current gives this product, while recovering active power requires the corresponding average product or adequate phase-resolved information. The Cauchy-Schwarz inequality guarantees $|P|\le S_{\mathrm{app}}$ without assuming a sinusoidal waveform.

The volt-ampere, symbolVA, is the unit commonly used to express apparent power. At a terminal pair, RMS potential difference in volts multiplied by RMS current in amperes gives the numerical value in volt-amperes. The unit product has the same coherent SI dimensions as the watt:This unit identity does not make apparent power and active power the same quantity. The SI text explicitly warns that different physical quantities can share a unit and that the quantity must be specified.^[ref]

For an original example,250V RMS at8A RMS gives2000VA, or2kVA. If the absorbing load has power factor0.75, its active power is1500W. A1500W value cannot be inferred from the2kVA alone without the phase/waveform information summarized by that factor.

Hypothetical equipment can have separate apparent- and active-power limits. A1000VA/600W limit would permit a500W, factor0.5 load only at the apparent-power boundary, since it requires1000VA. A700W, factor-one load would require only700VA but exceed the separate600W limit. These examples interpret stated numerical constraints rather than specify an actual product's behavior.

For single-frequency sinusoids, the magnitude of complex electrical power is expressed in volt-amperes, while its real part is commonly expressed in watts and its imaginary part in reactive volt-amperes. Those names help keep the quantities distinct even though their base-unit dimensions agree.^[ref] With distorted signals, retain the actual RMS-product definition of apparent power rather than assuming a single fundamental phasor accounts for all current.

Reactive power is the imaginary part of complex electrical power in sinusoidal steady operation. With RMS phasors, the passive sign convention and $e^{j\omega t}$ time convention,where $\phi$ is potential-difference phase minus current phase. Inductive current lag gives positive $Q$, while capacitive current lead gives negative $Q$. Its conventional unit is the reactive volt-ampere, written var. These definitions retain the specified phasor and sign conventions.^[ref]

The physical role is periodic energy exchange rather than an additional dissipated-energy average. For an ideal inductor, $v=Ldi/dt$ and stored energy is $w=L{i}^2/2$, so $p=dw/dt$. Its energy returns to its initial value after a steady cycle and mean absorbed power is zero. Nevertheless,For an ideal capacitor, $Q_C=-\omega C{V}_{\mathrm{rms}}^2$, with the negative sign representing the opposite exchange phase. Primary circuit derivations explicitly separate these exchanges from resistor dissipation.^[ref]

For an original example, $L=20\mathrm{mH}$, $I_{\mathrm{rms}}=5\mathrm{A}$ and $f=50\mathrm{Hz}$ give $Q_L\approx 157.08\mathrm{var}$. Peak stored energy is0.50J and mean stored energy0.25J, consistent with $2\omega ⟨w⟩$. Zero ideal average dissipation therefore does not mean zero current or zero energy storage.

For a separate ideal single-frequency compensation example, a100V load with $S=400+j300\mathrm{VA}$ can have its source reactive component canceled by a parallel capacitor satisfyingThe capacitor contributes $-j300\mathrm{VA}$, leaving400W of source active power and4A RMS source current. This illustrates power factor correction under the stated ideal assumptions, rather than selecting a real component or proving its ratings.

Reactive exchange can increase the current carried through resistive wiring and thereby affect real losses. For distorted waveforms, definitions of non-active power need additional care; the one-frequency imaginary-part definition should not silently be extended to every waveform.

The reactive volt-ampere, conventionally called the var and written with symbolvar, expresses reactive power. The official SI text notes the IEC's special name and states that its coherent unit is identical to the volt ampere. The name identifies the reactive quantity; it is not a different base-unit dimension.^[ref]

In sinusoidal steady operation with RMS magnitudes $V,I$ and relative phase $\phi$,Thus100V RMS,2A RMS and a lagging phase with $\sin \phi =0.6$ give120var. Under the same passive and phasor conventions, a leading phase of equal magnitude gives−120var. The signed value describes exchange phase rather than negative dissipated heating. The prefix relation is $1\mathrm{kvar}=1000\mathrm{var}$.

A reactive-power value is also not an amount of stored energy in joules. For an ideal inductor in sinusoidal steady operation,For an original200var example at50Hz, mean stored energy is $200/[2(2\pi 50)]\approx 0.3183\mathrm{J}$ and peak stored energy is twice that. The average absorbed active power is nevertheless zero in the ideal inductor. The primary circuit treatment derives this separation between energy storage and mean dissipation.^[ref]

In a sinusoidal100V example with $S=400+j300\mathrm{VA}$, the reactive component is300var, the active component400W and apparent power500VA. Adding400 and300 as ordinary scalar powers would not give the500VA magnitude. Complex electrical power retains the phase relation needed for this combination.