# Complex Power **The quantity containing all information related to the [[power]] absorbed by a load in AC.** > [!Example] Complex Power > *Complex power*, denoted $\mathbf{S}$, is measured in *volt-amperes*, $\text{VA}$. It is a [[complex number]] and can be expressed in rectangular form as > $\mathbf{S}=P+jQ$ > - $P$ - [[average power]], $\text{W}$ > - $Q$ - *reactive* power, $\text{VAR}$ > > Alternatively, in [[Complex Number#Polar form|polar form]], > $\mathbf{S}=S\angle(\theta_{v}-\theta_{i})$ > - $S$ - *apparent* power, $\text{VA}$ > - $\theta_{v}-\theta_{i}$ - [[Power Angle and Power Factor#Power Angle|power angle]] Complex power can be calculated using [[Periodic Function#^c5261f|peak]] and [[Periodic Function#Root mean square amplitude|root mean square values]] of the voltage and current. $\begin{align*} \mathbf{S}&=\frac{1}{2}\mathbf{V}\mathbf{I}^{*} \\ &=\mathbf{V}_{\text{RMS}}\mathbf{I}_{\text{RMS}}^* \\ &=I_{\text{RMS}}^{2}\mathbf{Z} \\ &=\frac{V_{\text{RMS}}^2}{\mathbf{Z}^{*}} \end{align*}$ ## Apparent power > [!Example] Apparent power > *Apparent power*, $S$, is the *magnitude* of complex power. It is so called because it is the "apparent" power draw of a load, even if not all of that power does work. > > Its unit is the *volt-ampere*, $\text{VA}$. > $S=|\mathbf{S}|=|\mathbf{V_{\text{RMS}}}||\mathbf{I_{\text{RMS}}}|=\sqrt{P^{2}+Q^{2}}$ ## Reactive power > [!Example] Reactive power > *Reactive power*, $Q$, is the part of complex power that does *no work*. It corresponds to the energy that flows to the load then back to the source. > > Its unit is *volt-ampere reactive*, $\text{VAR}$. > $Q=\Im(\mathbf{S})=S\sin(\theta_{v}-\theta_{i})$ ## Power triangle It is common to represent complex power as a vector sum of the real and reactive power in the form of a *power triangle*. The angle between the apparent power and real power vectors is the [[Power Angle and Power Factor#Power angle|power angle]]. ![[PowerTriangle.svg|500]]