Impedance

The three element impedances

Each element contributes only one axis. A resistor is purely real at any frequency, an inductor's reactance grows with frequency as +jωL, and a capacitor's falls as −j/(ωC). At one frequency an inductor and a capacitor can present identical magnitudes but opposite signs, which is the basis of every resonance and every filter.

• Resistor: Z = R, real, frequency-independent.
• Inductor: Z = +jωL, magnitude rises 6 dB per octave.
• Capacitor: Z = −j/(ωC), magnitude falls 6 dB per octave.

Series and parallel in the complex plane

The combination rules are unchanged — series impedances add, parallel impedances combine as the reciprocal of the sum of reciprocals — but the arithmetic becomes complex, so the result has both a magnitude and a phase. Two impedances in parallel can produce a magnitude smaller than either, or larger, depending on whether their reactances have the same or opposite sign.

Resonance and Q

When an inductor's +jωL exactly cancels a capacitor's −j/(ωC), the impedance of the series combination collapses to the resistance alone. That happens at ω₀ = 1/√(LC). The sharpness of the effect is the quality factor Q = ω₀L/R, which for a series circuit is also the ratio of stored energy to energy lost per radian.

Why phase matters in practice

A load with a large reactive component draws current that does not line up with the voltage, so the product V · I overestimates the useful power and the wiring must still carry the larger current. This is why power factor correction exists, and why a motor's rated current is quoted at a stated power factor rather than at a stated impedance.