Impedance Capacitor⚠ unverified

Understanding the Parameters

• Capacitance $C$ and angular frequency $\omega$ both reduce the impedance magnitude: a bigger capacitor, or a higher frequency, passes AC more easily.

• $Z_C$ is a complex quantity. Its size (in ohms) tells you the voltage‑to‑current amplitude ratio; its angle ($-90^\circ$) tells you the phase relationship.

Derivation (Approaching a Proof)

Use phasors: represent a sinusoid by the complex amplitude of $e^{j\omega t}$. The capacitor's defining law $i = C\,\dfrac{dv}{dt}$ becomes, for $v(t) = V e^{j\omega t}$,

$$i(t) = C\frac{d}{dt}\big(V e^{j\omega t}\big) = j\omega C\, V e^{j\omega t}.$$

Impedance is the phasor ratio of voltage to current (Ohm's law for AC):

$$Z_C = \frac{V}{I} = \frac{1}{j\omega C} = -\frac{j}{\omega C}.$$

The factor $\dfrac{1}{j} = -j$ is a $-90^\circ$ rotation, so the current leads the voltage by $90^\circ$; the magnitude $|Z_C| = 1/(\omega C)$ is the reactance $X_C$ (see Capacitive Reactance).

Units. $\dfrac{1}{(\text{rad/s})(\text{F})} = \dfrac{1}{(\text{1/s})(\text{C/V})} = \dfrac{\text{V}}{\text{A}} = \Omega$. (The live calculator currently mislabels the output unit; the physical unit is the ohm — flagged for verification.)

History

Complex impedance is due to Charles Proteus Steinmetz (1893), who replaced the calculus of AC transients with algebra on complex numbers. Representing $R$, $L$, and $C$ by $R$, $j\omega L$, and $1/(j\omega C)$ turned AC network analysis into the same series/parallel arithmetic as DC — one of the most consequential simplifications in electrical engineering.

Related Concepts: Capacitive Reactance, Impedance Inductor, Impedance Resistor, Impedance

Notes: Registry calculator impedance-capacitor (unverified). Ideal capacitor; real parts add series resistance (ESR) and inductance (ESL) at high frequency.