AC circuit

[큰나무]

Impedance, the vector sum of reactance and resistance, describes the phase difference and the ratio of amplitudes between sinusoidally varying voltage and sinusoidally varying current at a given frequency. Fourier analysis allows any signal to be constructed from a spectrum of frequencies, whence the circuit's reaction to the various frequencies may be found. The reactance and impedance of a capacitor are respectively

${\displaystyle {\begin{aligned}X&=-{\frac {1}{\omega C}}=-{\frac {1}{2\pi fC}}\\Z&={\frac {1}{j\omega C}}=-{\frac {j}{\omega C}}=-{\frac {j}{2\pi fC}}\end{aligned}}}$

where j is the imaginary unit and ω is the angular frequency of the sinusoidal signal. The −j phase indicates that the AC voltage V = ZI lags the AC current by 90°: the positive current phase corresponds to increasing voltage as the capacitor charges; zero current corresponds to instantaneous constant voltage, etc.

Impedance decreases with increasing capacitance and increasing frequency. This implies that a higher-frequency signal or a larger capacitor results in a lower voltage amplitude per current amplitude—an AC "short circuit" or AC coupling. Conversely, for very low frequencies, the reactance is high, so that a capacitor is nearly an open circuit in AC analysis—those frequencies have been "filtered out".

Capacitors are different from resistors and inductors in that the impedance is inversely proportional to the defining characteristic; i.e., capacitance.

A capacitor connected to a sinusoidal voltage source causes a displacement current to flow through it. In the case that the voltage source is V₀cos(ωt), the displacement current can be expressed as:

${\displaystyle I=C{\frac {dV}{dt}}=-\omega {C}{V_{\text{0}}}\sin(\omega t)}$

At sin(ωt) = -1, the capacitor has a maximum (or peak) current whereby I₀ = ωCV₀. The ratio of peak voltage to peak current is due to capacitive reactance (denoted X_C).

${\displaystyle X_{C}={\frac {V_{\text{0}}}{I_{\text{0}}}}={\frac {V_{\text{0}}}{\omega CV_{\text{0}}}}={\frac {1}{\omega C}}}$

X_C approaches zero as ω approaches infinity. If X_C approaches 0, the capacitor resembles a short wire that strongly passes current at high frequencies. X_C approaches infinity as ω approaches zero. If X_C approaches infinity, the capacitor resembles an open circuit that poorly passes low frequencies.

The current of the capacitor may be expressed in the form of cosines to better compare with the voltage of the source:

${\displaystyle I=-{I_{\text{0}}}{\sin({\omega t}})={I_{\text{0}}}{\cos({\omega t}+{90^{\circ }})}}$

In this situation, the current is out of phase with the voltage by +π/2 radians or +90 degrees, i.e. the current leads the voltage by 90°.