Deriving capacitor impedance

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For an ideal capacitor, the capacitor current is proportional to the time rate of change of the voltage across the capacitor where the constant of proportionality is the capacitance, C:

[i(t) = C \frac]

In the frequency domain, this equation becomes:

[I(j \omega) = j \omega C V(j \omega) \,]

The ratio of the voltage and current in the frequency domain is called the impedance, and is then:

[Z(j \omega) = \frac = \frac = \frac = - j X_C ]

It should be noted that while this relation (between the frequency domain voltage and current associated with a capacitor) is always true, the ratio of the time domain voltage and current amplitudes is equal to [X_C] only for sinusoidal (AC) circuits in steady state.

Deriving capacitor impedance

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