Complex Impedance Magnitude (RLC)

|Z| = √(R² + (ωL − 1/(ωC))²)

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Result
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Formula

|Z| = √(R² + (ωL − 1/(ωC))²)

Description

The magnitude of the impedance of a series RLC circuit combines resistance with the net reactance. Inductive reactance XL = ωL increases with frequency, while capacitive reactance XC = 1/(ωC) decreases. At resonance, XL = XC and the impedance equals R alone (minimum impedance). Below resonance the circuit is capacitive; above resonance it is inductive. This formula is essential for understanding filter behavior, resonant circuits, and the frequency-dependent impedance of any circuit containing R, L, and C elements.

Variables

  • |Z| — Impedance magnitude (Ω)
  • R — Resistance (Ω)
  • f — Frequency (Hz)
  • L — Inductance (H)
  • C — Capacitance (F)

Practical Notes

At resonance, f₀ = 1/(2π√(LC)), the impedance is purely resistive and equal to R. The quality factor Q = (1/R)√(L/C) determines the sharpness of the resonance. High-Q circuits (low R) have narrow bandwidth and high selectivity. For parallel RLC circuits, the impedance is maximum at resonance.

Related Formulas

Resonant Frequency Impedance (RLC Series) Quality Factor (Bandwidth) Quality Factor (RLC) Power Factor Apparent Power

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