Resonance

Formulas

Series Resonance

A series circuit comprising an inductance L, a resistance R and a capacitance C has an impedance Z_S of:

Z_S = R + j(X_L – X_C)

where X_L = wL and X_C = 1 / wC

At resonance, the imaginary part of Z_S is zero:

X_C = X_L

Z_Sr = R

w_r = (1 / LC)^½ = 2pf_r

The quality factor at resonance Q_r is:

Q_r = w_rL / R = (L / CR²)^½ = (1 / R )(L / C)^½ = 1 / w_rCR

Parallel resonance

A parallel circuit comprising an inductance L with a series resistance R, connected in parallel with a capacitance C, has an admittance Y_P of:

Y_P = 1 / (R + jX_L) + 1 / (- jX_C) = (R / (R² + X_L²)) – j(X_L / (R² + X_L²) – 1 / X_C)

where X_L = wL and X_C = 1 / wC

At resonance, the imaginary part of Y_P is zero:

X_C = (R² + X_L²) / X_L = X_L + R² / X_L = X_L(1 + R² / X_L²)

Z_Pr = Y_Pr^-1 = (R² + X_L²) / R = X_LX_C / R = L / CR

w_r = (1 / LC – R² / L²)^½ = 2pf_r

The quality factor at resonance Q_r is:

Q_r = w_rL / R = (L / CR² – 1)^½ = (1 / R )(L / C – R²)^½

Note that for the same values of L, R and C, the parallel resonance frequency is lower than the series resonance frequency, but if the ratio R / L is small then the parallel resonance frequency is close to the series resonance frequency.