Deriving capacitor impedance
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The derivation of the complex impedance of a capacitor, or phasor impedance, is a fairly simple application of the Fourier Transform on a first-order differential equation to move to the frequency domain.
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:
Applying the Fourier Transform, this equation becomes:
- where ω is the angular frequency, and j is the imaginary unit.
The ratio of the voltage and current in the frequency domain is called the impedance, and is then:
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 XC only for sinusoidal (AC) circuits with a constant frequency as a consequence of the Fourier Transform.
