Reactive Power in Capacitive Circuit

Capacitors in electronic devices and long cables have capacitive loads. Capacitive loads require power to charge their capacity. This power is referred to as capacitive reactive power. The apparent power (S1) is defined as the vector sum of actual power (P) and capacitive reactive power (Q1). For a purely capacitive circuit (such as that in [Fig. 1]), i leads v by $90^\circ$, as shown in [Fig. 2]. Therefore, in Eq. (1), $\theta = -90^\circ$. Substituting $\theta = -90^\circ$ into Eq. (1), yields or where $-VI \sin 2wt$ is a sine wave with twice the frequency of either input quantity (v or i) and a peak value of $VI$. Again, note the absence of an average or constant term in the equation. Plotting the waveform for $p_C$ [Fig. 2], we obtain Note that the same situation exists here for the $p_C$ curve as existed for the $p_L$ curve. The power delivered by the source to the capacitor is exactly equal to that returned to the source by the capacitor over one full cycle. and no energy is lost in the transaction. The power absorbed or returned by the capacitor at any instant of time t1 can be found by substituting t1 into Eq. (2).
The reactive power associated with the capacitor is equal to the peak value of the $p_C$ curve, as follows: or, since $V = IX_C$ or $I = V/X_C$, or The apparent power associated with an inductor is $S = VI$, and the average power is $P = 0$, as noted in [Fig. 2]. The power factor is therefore The energy stored by the capacitor during the positive portion of the cycle ([Fig. 2]) is equal to that returned during the negative portion and can be determined using the equation $W = Pt$.
Proceeding in a manner similar to that used for the inductor, we can show that and or, since $T_2 = 1/f_2$, where $f_2$ is the frequency of the $p_C$ curve, we have Since the frequency $f_2$ of the power curve is twice that of the input quantity, if we substitute the frequency f1 of the input voltage or current, Equation (4) becomes However, $I = V/X_C = V w_1 C$, so that providing an equation for the energy stored or released by the capacitor in one half-cycle of the applied voltage in terms of the capacitance and rms value of the voltage squared.