Effects of harmonics - Resonance

The simultaneous use of capacitive and inductive devices in distribution networks results in parallel or series resonance manifested by very high or very low impedance values respectively. The variations in impedance modify the current and voltage in the distribution network. Here, only parallel resonance phenomena, the most common, will be discussed.
Consider the following simplified diagram (see Fig. M6) representing an installation made up of:

• A supply transformer
• Linear loads
• Non-linear loads drawing harmonic currents
• Power factor correction capacitors

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Fig. M6: Diagram of an installation

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For harmonic analysis, the equivalent diagram (see Fig. M7) is shown below.
Impedance Z is calculated by:
[math]\displaystyle{ Z=\frac{jLs \omega}{1- LsC \omega^2} }[/math]
neglecting R and where:
Ls = Supply inductance (upstream network + transformer + line)
C = Capacitance of the power factor correction capacitors
R = Resistance of the linear loads
Ih = Harmonic current
Resonance occurs when the denominator 1-LsCw2 tends toward zero. The corresponding frequency is called the resonance frequency of the circuit. At that frequency, impedance is at its maximum and high amounts of harmonic voltages appear with the resulting major distortion in the voltage. The voltage distortion is accompanied, in the Ls+C circuit, by the flow of harmonic currents greater than those drawn by the loads.
The distribution network and the power factor correction capacitors are subjected to high harmonic currents and the resulting risk of overloads. To avoid resonance, anti-harmonic coils can be installed in series with the capacitors.

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Fig. M7: Equivalent diagram of the installation shown in Figure M6

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