# Harmonic resonance calculation in power-factor correction

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).

• It might be because the s/c current is largely determined by leakage inductance and when the formula is rearranged it appears you get some way towards the standard formula for LC resonance. Sep 30 '18 at 18:26
• The estimation formula may be taken from IEEE Standard 519. My copy may be out of date, but I will take a look. Sep 30 '18 at 19:46
• @Andy: That's what I was thinking but wondering if the numbers just coincide for 60 Hz. Sep 30 '18 at 21:07
• @Thanks, Charles. Even if the standard is out of date the theory should still apply! Sep 30 '18 at 21:08
• My uncertainty about the standard is about what may have been included or not. The first thing in my file was not IEEE 519, but another document that includes that formula. I have not had time to look carefully, but I believe I there is sufficient information to show the derivation. I will see what I can do. Sep 30 '18 at 21:27

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.

• These generalizations can lead to series and parallel resonant power costs (losses) when the load is nonlinear DC or arc type when the transformer has very low Zpu or Pva/Psc % especially <<10% then active PFC is ideal. Passive PFC must be tune with a series reactor for desired parallel notch but while avoiding a series resonance. So the load spectrum and Zo must be known. Since Lpri is raised for 50 Hz to achieve the same excitation current (pu) as 60 Hz the formula in general may apply. But still my previous comments need comprehension. Oct 1 '18 at 1:11
• I added MathJax for your formula but I couldn't find a reference for it so I'm not sure if I got the subscript right.
– K H
Oct 1 '18 at 8:12
• Full marks for MathJAX, @KH, but you need to brush up on resonant circuit theory. :^) The 'Pix' was 'π times'. See the fix. Oct 1 '18 at 8:57
• Thanks @Transistor. Was I correct about the resonant frequency "fo" being "f naught" and not "f subscript O"? When I eventually found the formula they used $f_r$.
– K H
Oct 1 '18 at 9:23
• See Wikipedia's LC circuit and the section before that. They're using$f_0$ (zero) as the natural frequency of oscillation. Oct 1 '18 at 11:22