Response of Capacitor to a Sinusoidal Voltage

Our investigation of the inductor revealed that the inductive voltage across a coil opposes the instantaneous change in current through the coil. For capacitive networks, the voltage across the capacitor is limited by the rate at which charge can be deposited on, or released by, the plates of the capacitor during the charging and discharging phases, respectively. In other words, an instantaneous change in voltage across a capacitor is opposed by the fact that there is an element of time required to deposit charge on (or release charge from) the plates of a capacitor, and $V = Q/C$. In addition, the fundamental equation relating the voltage across a capacitor to the current of a capacitor [$i = C(dv/dt)$] indicates that for a particular capacitance, the greater the rate of change of voltage across the capacitor, the greater the capacitive current. Certainly, an increase in frequency corresponds to an increase in the rate of change of voltage across the capacitor and to an increase in the current of the capacitor. The current of a capacitor is therefore directly related to the frequency (or, again more specifically, the angular velocity) and the capacitance of the capacitor. An increase in either quantity will result in an increase in the current of the capacitor. For the basic configuration of [Fig. 1], however, we are interested in determining the opposition of the capacitor as related to the resistance of a resistor and $wL$ for the inductor. Since an increase in current corresponds to a decrease in opposition, and $i_C$ is proportional to w and C, the opposition of a capacitor is inversely related to $w (= 2\pi f )$ and C. We will now verify, as we did for the inductor, some of the above conclusions using a more mathematical approach. For the capacitor of [Fig. 2], applying differentiation, where Note that the peak value of iC is directly related to $w (= 2 \pi f )$ and C, as predicted in the discussion above. A plot of vC and iC in [Fig. 3] reveals that If a phase angle is included in the sinusoidal expression for $v_C$, such as then Applying and substituting values, we obtain which agrees with the results obtained above. The quantity $1/wC$, called the reactance of a capacitor, is symbolically represented by $X_C$ and is measured in ohms; that is, In an Ohm's law format, its magnitude can be determined from Capacitive reactance is the opposition to the flow of charge, which results in the continual interchange of energy between the source and the electric field of the capacitor. Like the inductor, the capacitor does not dissipate energy in any form (ignoring the effects of the leakage resistance). It is possible now to determine whether a network with one or more elements is predominantly capacitive or inductive by noting the phase relationship between the input voltage and current. If the source current leads the applied voltage, the network is predominantly capacitive, and if the applied voltage leads the source current, it is predominantly inductive. Since we now have an equation for the reactance of an inductor or capacitor, we do not need to use derivatives or integration in the examples to be considered. Simply applying Ohm's law, $I_m = E_m/X_L$ (or $X_C$), and keeping in mind the phase relationship between the voltage and current for each element, will be sufficient to complete the examples.