# Harmonics in Three-Phase Systems

The sine waves of the currents and voltages in a power system may not be perfect and so may contain harmonics. Harmonics are created by domestic non-linear loads, particularly computer and TV power supplies but also by large industrial rectifiers (e.g. for aluminium smelters). Consider the instantaneous values of phase voltages in a balanced system containing harmonics up to the third:

Here V_{1}, *V _{2}* and

*V*are the magnitudes of the harmonic voltages.

_{3}From equations (2.20)-(2.22) the phasor diagram for fundamental, second and third harmonic components of v_{a}, *Vb* and v_{c} shown in Figure 2.26 can be obtained. As shown in Figure 2.26, the fundamental terms have the normal phase rotation (as do the fourth, seventh and tenth, etc., harmonics). However, the second harmonic terms possess a reversed phase rotation (also the fifth, eighth, eleventh, etc.), and the third harmonic terms are all in phase (also all harmonics of multiples three).

**Figure 2.26 **Phase rotation of harmonic voltage (fundamental, 2nd and 3rd harmonics)

When substantial harmonics are present the л/3 relation between line and phase quantities no longer holds. As can be seen from Figure 2.26, harmonic voltages other than that of triple harmonics (n = 3, 6, 9, etc.) in successive phases are 2p/3 out of phase with each other. In the resulting line voltages (as V_{a}b is given by (V_{a} — Vb)) no triple harmonics exist. The mesh connection forms a complete path for the triple harmonic currents which flow in phase around the loop.

When analysing the penetration of harmonics into the power network an initial approximation is to assume that the effective reactance for the nth harmonic is n times the fundamental value.