Harmonic Resonance

Formulas

If a node in a power system operating at frequency f has a inductive source reactance X_L per phase and has power factor correction with a capacitive reactance X_C per phase, the source inductance L and the correction capacitance C are:

L = X_L / w

C = 1 / wX_C

where w = 2pf

The series resonance angular frequency w_r of an inductance L with a capacitance C is:

w_r = (1 / LC)^½ = w(X_C / X_L)^½

The three phase fault level S_sc at the node for no-load phase voltage E and source impedance Z per-phase star is:

S_sc = 3E² / |Z| = 3E² / |R + jX_L|

If the ratio X_L / R of the source impedance Z is sufficiently large, |Z| » X_L so that:

S_sc » 3E² / X_L

The reactive power rating Q_C of the power factor correction capacitors for a capacitive reactance X_C per phase at phase voltage E is:

Q_C = 3E² / X_C

The harmonic number f_r / f of the series resonance of X_L with X_C is:

f_r / f = w_r / w = (X_C / X_L)^½ » (S_sc / Q_C)^½

Note that the ratio X_L / X_C which results in a harmonic number f_r / f is:

X_L / X_C = 1 / ( f_r / f )²

so for f_r / f to be equal to the geometric mean of the third and fifth harmonics:

f_r / f = Ö15 = 3.873

X_L / X_C = 1 / 15 = 0.067