### 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**per phase, the source inductance

_{C}**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**so that:

_{L}**S _{sc} » 3E^{2} / X_{L}**

The reactive power rating **Q _{C}** of the power factor correction capacitors for a capacitive reactance

**X**per phase at phase voltage

_{C}**E**is:

**Q _{C} = 3E^{2} / X_{C}**

The harmonic number **f _{r} / f** of the series resonance of

**X**with

_{L}**X**is:

_{C}**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**is:

_{r}/ f**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**

NOTATION |
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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] |
QRStVWXYZfw |
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] |