## Harmonic resonance

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} = 3E^{2} / |Z| = 3E^{2} / |R + jX_{L}|
If the ratio X_{L} / R of the source impedance Z is sufficiently
large, |Z|
»
X_{L} so that:
S_{sc}
» 3E^{2} / 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} = 3E^{2} / 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
**
**