Impedance Inductor⚠ unverified

Understanding the Parameters

• Inductance $L$ and angular frequency $\omega$ both increase the impedance magnitude: a bigger inductor, or a higher frequency, opposes AC more strongly (an inductor is a short at DC, an open at very high frequency).

• $Z_L$ is complex: magnitude $\omega L$ (ohms) sets the amplitude ratio, phase $+90^\circ$ sets the timing.

Derivation (Approaching a Proof)

In the phasor domain a sinusoid is the complex amplitude of $e^{j\omega t}$. The inductor's defining law $v = L\,\dfrac{di}{dt}$ becomes, for $i(t) = I e^{j\omega t}$,

$$v(t) = L\frac{d}{dt}\big(I e^{j\omega t}\big) = j\omega L\, I e^{j\omega t}.$$

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

$$Z_L = \frac{V}{I} = j\omega L.$$

The factor $j$ is a $+90^\circ$ rotation, so the voltage leads the current by $90^\circ$; the magnitude $|Z_L| = \omega L$ is the reactance $X_L$ (see Inductive Reactance).

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

History

As with the capacitor, the compact form $Z_L = j\omega L$ comes from Steinmetz's complex‑impedance method (1893), which rests physically on Faraday's law of induction. It made inductors, capacitors, and resistors combine by ordinary complex arithmetic, unifying AC circuit analysis.

Related Concepts: Inductive Reactance, Impedance Capacitor, Impedance Resistor, Impedance

Notes: Registry calculator impedance-inductor (unverified). Ideal inductor; real inductors add winding resistance and inter‑winding capacitance.