 Optimal shunt capacitance for sinusoidal power factor correction

Optimal shunt capacitance for sinusoidal power factor correction minimizes the source's RMS electric current for an ideal fixed sinusoidal supply and a fixed load with active power $P>0$ and inductive reactive power $Q_L>0$. A lossless parallel capacitor supplies signed reactive power $-\omega C{V}^2$, where $V$ is the supply RMS electric potential difference and $\omega >0$ the angular frequency.

The squared source electric current isDifferentiating givesThus the unique minimum isThe fundamental quadrature electric current vanishes there. Strict convexity proves this is a global optimum in the stated model, rather than merely a stationary point. The capacitive sign and one-frequency complex electrical power relation follow from the primary MIT derivation.^[ref]

For an original example, $P=800\mathrm{W}$, $Q_L=600\mathrm{var}$, $V=100\mathrm{V}$ and $f=50\mathrm{Hz}$ give $C_{\ast }\approx 190.99\mu \mathrm{F}$. Source electric current falls from $10\mathrm{A}$ to $8\mathrm{A}$. A capacitance 20% below or above $C_{\ast }$ leaves $Q_{\mathrm{net}}=+120$ or $-120\mathrm{var}$, respectively, and both draw about $8.0895\mathrm{A}$. They have the same power factor magnitude, about $0.9889$, but opposite lead/lag direction.

The exact excess above the optimal squared electric current isSmall sizing errors therefore affect this loss objective quadratically around the optimum, while residual reactive power changes linearly and changes sign. An allowed interval $0\le C\le C_{\max }$ clips the optimum to its nearest feasible point. If the original load has $Q_L\le 0$, the constrained optimum for added nonnegative capacitance is instead $C=0$.

The result assumes a fixed load, ideal sinusoidal supply, lossless correction and no additional network effects. It does not optimize a system with waveform distortion, component heating, variable terminal electric potential difference or discrete stages. The current reduction interval for shunt reactive compensation describes a wider set of beneficial choices, while automatic power factor correction follows a changing load with available stages.