# Effects of harmonics - Resonance

The simultaneous use of capacitive and inductive devices in distribution networks may result in parallel or series resonance.

The origin of the resonance is the very high or very low impedance values at the busbar level, at different frequencies. 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. M14) representing an installation made up of:

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

For harmonic analysis, the equivalent diagram is shown on Figure M15 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 Fig. M15 – Equivalent diagram of the installation shown in Figure M14

By neglecting R, the impedance Z is calculated by a simplified formula:

$Z={\frac {jLs\omega }{1-LsC\omega ^{2}}}$ with: ω = pulsation of harmonic currents

Resonance occurs when the denominator (1-LSCω2) 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 because of the circulation of harmonic currents. This results in major voltage distortion. The voltage distortion is accompanied, in the LS+C circuit, by the flow of harmonic currents greater than those drawn by the loads, as illustrated on Figure M16.

The distribution network and the power factor correction capacitors are subjected to high harmonic currents and the resulting risk of overloads. To avoid resonance, antihamonic reactors can be installed in series with the capacitors. Fig. M16 – Illustration of parallel resonance