 Current reduction interval for shunt reactive compensation

The current reduction interval for shunt reactive compensation is the range of ideal capacitive corrections that draws less source RMS electric current than the uncorrected inductive load. It is wider than the range that avoids reactive power overcompensation.

Assume fixed sinusoidal RMS electric potential difference $V>0$, active power $P>0$, and inductive reactive power $Q_L>0$. Let $q=\omega C{V}^2\ge 0$ be the magnitude of the added capacitive reactive power. Comparing squared source electric currents givesConsequentlyor $0<C<2C_{\ast }$ with $C_{\ast }=Q_L/(\omega V^2)$. The model's capacitive reactive power follows the sinusoidal complex electrical power convention.^[ref]

At $q=Q_L$, source electric current is smallest. Between $Q_L$ and $2Q_L$, the source has a leading power factor but still draws less electric current than before correction. At $q=2Q_L$, the residual reactive power is $-Q_L$: its magnitude matches the original value, restoring the original source-current magnitude. Beyond that point the corrected electric current is larger.

For an original example, choose $P=300\mathrm{W}$, $Q_L=400\mathrm{var}$ and $V=100\mathrm{V}$. Uncorrected electric current is $5\mathrm{A}$. Full cancellation $q=400\mathrm{var}$ gives $3\mathrm{A}$. An overcorrection $q=600\mathrm{var}$ leaves $-200\mathrm{var}$ and draws only $\sqrt{13}\approx 3.606\mathrm{A}$. But $q=800\mathrm{var}$ restores $5\mathrm{A}$, and $q=1000\mathrm{var}$ increases it to $\sqrt{45}\approx 6.708\mathrm{A}$.

Two choices $q$ and $2Q_L-q$ are symmetric around the optimum and give the same squared source electric current. This symmetry explains why power factor magnitude alone cannot identify the sign of the residual reactive power. If $Q_L\le 0$, no positive $q$ reduces current under the same fixed-load assumptions.

With the same upstream electrical resistance, reducing source mean-square electric current also reduces that conductor's Joule heating. It does not erase the current carried by the local capacitor branch or assess its ratings. Harmonic interactions, nonideal components and changing load conditions need their own model; this interval is an exact statement about the stated one-frequency circuit.