# Power Angle and Power Factor **The [[phase]] of [[complex power]] and the ratio of the [[Average Power|real power]] dissipated in the load to the [[Complex Power#Apparent power|apparent power]] of the load.** > [!NOTE] Power Angle > The *power angle* is the phase of complex power. It is also the phase difference between the *voltage* and the *current* of the load and the phase of the load [[impedance]]. > $\theta_{v}-\theta_i$ > [!NOTE] Power Factor > The *power factor* of complex power is defined as the cosine of the power angle. Power factor is *unitless* and is expressed with whether it is *leading* or *lagging*. This corresponds with whether the *current* is leading or lagging. > $pf=\cos(\theta_{v}-\theta_{i})$ > > - If $\theta_{v}-\theta_{i}=0$, the load is *purely resistive*, $pf=1$, and all of apparent power delivered is [[Average Power|real power]]. > - If $\theta_{v}-\theta_{i}=\pm 90^{\circ}$, then the load is *purely reactive*, $pf=0$, and no real power is delivered. > - If $0^{\circ}<\theta_{v}-\theta_{i}<90^{\circ}$, then the load is *net inductive* and the $pf$ is *lagging*. > - If $-90^{\circ}<\theta_{v}-\theta_{i}<0^{\circ}$, then the load is *net capacitive* and the $pf$ is *leading*. ### Power factor correction The closer the power factor is to unity, the less total [[Complex Power#Reactive power|reactive power]] within the system and thus the less total [[Complex Power#Apparent power|apparent power]] required to be sourced. *Power factor correction* is the manipulation of the power factor towards unity. To achieve power factor correction, a purely reactive load is added *in parallel* to an existing load. Using a purely reactive load theoretically keeps the magnitude of the [[Average Power|real power]] the same. - If the power factor is *lagging*, implying a *net inductive* load, then a *shunt* [[capacitor]] is required with a capacitance of $C=\frac{Q_{C}}{\omega V^{2}_{\text{RMS}}}$ $Q_{C}$ refers to the *leading reactive power* that the capacitor needs to *eliminate* and $V$ is the source voltage. - If the power factor is *leading*, implying a *net capacitive* load, then a *shunt* [[inductor]] is required with an inductance of $L=\frac{V_{\text{RMS}}^{2}}{\omega Q_{L}}$ $Q_L$ is the *leading reactive power* that the inductor needs to *provide*. #### Examples These power triangles show before and after correction using a *shunt capacitor*. ![[CapacitativePowerFactorCorrection.svg]] *** These power triangles show correction using a *shunt inductor*. ![[InductivePowerFactorCorrection.svg]]