Reactance & LC Resonance

Capacitive and inductive reactance at a frequency, the resonant frequency of an LC pair, the L or C needed for a target frequency, and the impedance of series and parallel LC circuits.

Method, formulas and references

Reactance

(1)X_L = 2πfL, X_C = 1 / (2πfC)

Resonance

An LC pair resonates where X_L = X_C:

(2)f₀ = 1 / (2π √(LC)), L = 1 / ((2πf₀)² C), C = 1 / ((2πf₀)² L)

Series and parallel LC

(3)|Z_series| = √(R_s² + (X_L − X_C)²)

(4)|Z_parallel| = 1 / √(1/R_p² + (1/X_C − 1/X_L)²)

With no loss the series circuit is a short at f₀ and the parallel tank is an open; the ideal parallel |Z| reduces to X_LX_C / |X_L − X_C|. The quality factor is Q = √(L/C)/R_s for the series circuit and Q = R_p/√(L/C) for the parallel one, and the −3 dB bandwidth is f₀/Q. Real inductors and capacitors have their own self-resonance and ESR; the model here uses ideal parts plus the loss resistance you enter.

References

C. K. Alexander, M. N. O. Sadiku, Fundamentals of Electric Circuits, McGraw-Hill, ch. 9 and 14 (resonant circuits).
P. Horowitz, W. Hill, The Art of Electronics, 3rd ed., Cambridge University Press, 2015, ch. 1.7.

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