# 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*, $\text{VA}$ and 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#^827b3d|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. $\begin{align*} \mathbf{S}&=\frac{1}{2}\mathbf{V}\mathbf{I}^{*} \\ &=\mathbf{V}_{\text{RMS}}\mathbf{I}_{\text{RMS}}^* \\ &=I_{\text{RMS}}^{2}\mathbf{Z} \\ &=\frac{V_{\text{RMS}}^2}{\mathbf{Z}^{*}} \end{align*}$ ## Average Power *Average power* is the component of complex power that is capable of transferring energy from the source to the load. It is also common to refer to average power as *real power* since it is the real component of complex power. > [!Example] Average or Real Power > $\begin{align*} > P=\frac{1}{2}\text{Re}[\mathbf{V}\mathbf{I}^{*}]&=\frac{1}{2}V_{\text{m}}I_{\text{m}}\cos(\theta_{v}-\theta_{i}) \\ > &=V_\text{RMS}I_\text{RMS}\cos(\theta_{v}-\theta_{i}) > \end{align*}$ > - $\mathbf{V}, V_{\text{m}}, V_{\text{RMS}}$ - phasor, [[Periodic Function#^c5261f|peak]], and [[Periodic Function#Root mean square amplitude|root mean square]] voltage > - $\mathbf{I}, I_{\text{m}}, I_{\text{RMS}}$ - phasor, peak, and root mean square current > - $\cos(\theta_{v}-\theta_{i})$ - [[Power Angle and Power Factor#Power factor|power factor]] >In a *purely resistive* load, the average power is calculated with ^d7f3bd >$\begin{align*} P=\frac{1}{2}V_\text{m}I_\text{m}&=V_\text{RMS}I_\text{RMS} \\ &=\frac{1}{2}I_\text{m}^{2}R=I_\text{RMS}^{2}R \\ &=\frac{1}{2}|\mathbf{I}|^{2}R \end{align*}$ ^d8d1b7 >In a *purely reactive* load, the average power delivered is zero. ## 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]]