Harmonic resonance calculation in power-factor correction

Question

Eaton's Power factor correction: a guide for the plant engineer states,

Capacitors and transformers can create dangerous resonance conditions when capacitor banks are installed at the service entrance. Under these conditions, harmonics produced by nonlinear devices can be amplified manyfold. Problematic amplification of harmonics becomes more likely as more kVAR is added to a system that contains a significant amount of nonlinear load. You can estimate the resonant harmonic by using this formula:

\$ h = \sqrt {\frac {kVA_{sys}}{kVAR}} \$

kVAsys = short-circuit capacity of the system
kVAR = amount of capacitor kVAR on the line
h = the harmonic number referred to a 60 Hz base

If h is near the values of the major harmonics generated by a nonlinear device—for example, 3, 5, 7, 11—then the resonance circuit will greatly increase harmonic distortion.

Two questions:

1. Why would resonance depend on the short-circuit capacity of the system? (Can you explain the formula?)

2. I can't find a similar equation for 50 Hz. Any suggestions?

For reference, this is for a 1 MVA transformer (short-circuit capacity not known at this time) and 350 kVAr PF bank).

Answer 1

The formula is derived from the basic formula for a simple resonant circuit;

\$f_0 = \frac{1}{2 \pi\sqrt{LC}}\$

L and C are calculated from the short circuit impedance and the capacitor bank impedance. The short circuit impedance is assumed to be purely inductive. That assumption is justified by the assumption that the transformer X/R ratio is 5 for three-phase distribution transformers smaller than 150 kVA and progressively higher for larger transformers. X/R is assumed to be 8 for a typical 1 MVA transformer. Since the capacitor and transformer impedances are based on the relevant power frequency, that frequency is the fundamental for the harmonic order calculation. On other words, the formula is valid for either 50 or 60 Hz as long as the capacitor bank VAR rating and the short circuit VA are stated or calculated based on the frequency of interest.

The formula is given in IEEE Std 519-1992. I found the assumptions and derivation in seminar notes printed for a 1990 seminar.