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Harmonic resonance

 

If a node in a power system operating at frequency f has a inductive source reactance XL per phase and has power factor correction with a capacitive reactance XC per phase, the source inductance L and the correction capacitance C are:
L = XL /
w
C = 1 /
wXC
where
w = 2pf

The series resonance angular frequency wr of an inductance L with a capacitance C is:
wr = (1 / LC)½ = w(XC / XL)½

The three phase fault level Ssc at the node for no-load phase voltage E and source impedance Z per-phase star is:
Ssc = 3E2 / |Z| = 3E2 / |R + jXL|
If the ratio XL / R of the source impedance Z is sufficiently large, |Z|
» XL so that:
Ssc
» 3E2 / XL

The reactive power rating QC of the power factor correction capacitors for a capacitive reactance XC per phase at phase voltage E is:
QC = 3E2 / XC

The harmonic number fr / f of the series resonance of XL with XC is:
fr / f =
wr / w = (XC / XL)½ » (Ssc / QC)½

Note that the ratio XL / XC which results in a harmonic number fr / f is:
XL / XC = 1 / ( fr / f )2
so for fr / f to be equal to the geometric mean of the third and fifth harmonics:
fr / f =
Ö15 = 3.873
XL / XC = 1 / 15 = 0.067