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# Harmonic Resonance

### Formulas

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

 NOTATION The symbol font is used for some notation and formulae. If the Greek symbols for alpha beta delta do not appear here [ a b d ] the symbol font needs to be installed for correct display of notation and formulae. B C E f G h I j L P Q susceptance capacitance voltage source frequency conductance h-operator current j-operator inductance active power reactive power [siemens, S] [farads, F] [volts, V] [hertz, Hz] [siemens, S] [1Ð120°] [amps, A] [1Ð90°] [henrys, H] [watts, W] [VAreactive, VArs] Q R S t V W X Y Z f w quality factor resistance apparent power time voltage drop energy reactance admittance impedance phase angle angular frequency [number] [ohms, W] [volt-amps, VA] [seconds, s] [volts, V] [joules, J] [ohms, W] [siemens, S] [ohms, W] [degrees, °] [rad/sec]

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