AC Circuit Containing Capacitance and Resistance in Series

Let an alternating voltage E = E₀ e^jωt be applied to a circuit containing a resistance R and a capacitor C in series. Let I be the current through the circuit and q, the charge on the plates of the capacitor at any instant. Then,
        I (R + 1/jωC) = E = E₀ ^ejωt                                … (1)
        I = E/R + 1/ jωC                                     … (2)

But         I = E/Z                                         … (3)
. :        Impedance of series R-C circuit,
        Z = R + 1/jωC = R – j/ωC
        I = E / R – j/ωC = E₀ e^jωt/√[(R2 + 1/ω²C²)] e^-j0                     … (4)

Here,        θ = tan^-1 (1/ωC/R)                                … (5)
. :        I = E₀ e^{j(ωt + θ)}/√{R² + (1/ωC)²}                            … (6)

The current in this circuit thus leads the applied voltage by an angle θ.


The impedance of R – C circuit, Z = √[R² + (1/ωC)²].
The actual current in the circuit is
        I = I₀ sin (ωt + θ)