In a three-phase power system, the harmonics of one phase have a rotational and phase angle relationship with the harmonics of the other phases.

In power system studies involving harmonics, this relationship is important. In a balanced three-phase electrical system, the voltages and currents have a positional relationship as shown in Fig. 4.6. The three voltages are 120° apart and so are the three currents.

The normal phase rotation or sequence is a-b-c, which is counterclockwise and designated as the positive-phase sequence. For harmonic analyses, these relationships are still applicable, but the fundamental components of voltages and currents are used as reference. All other harmonics use the fundamental frequency as the reference. The fundamental frequencies have a positive-phase sequence. The angle between the fundamental voltage and the fundamental current is the displacement power factor angle.

So how do the harmonics fit into this space-time picture? For a clearer understanding, let us look only at the current harmonic phasors. We can further simplify the picture by limiting the discussion to odd harmonics only, which under normal and balanced conditions are the most prevalent. The following relationships are true for the fundamental frequency current components in a three-phase power system-

ia1 = Ia1 sin ωt… (4.7)

ib1 = Ib1 sin (ωt – 120°)… (4.8)

ic1 = Ic1 sin (ωt – 240°) … (4.9)

The negative displacement angles indicate that the fundamental phasors ib1 and ic1 trail the ia1 phasor by the indicated angle. Figure 4.7(a) shows the fundamental current phasors.

The expressions for the third harmonic currents are:

ia3 = Ia3 sin 3ωt…(4.10)

ib3 = Ib3 sin 3 (ωt – 120°) = Ib3 sin (3ωt – 360°)

= Ib3 sin 3ωt …(4.11)

ic3 = Ic3 sin 3(ωt – 240°) = Ic3 sin (3ωt – 720°)

= Ic3 sin 3ωt …(4.12)

The expressions for the third harmonics show that they are in phase and have zero displacement angle between them. Figure 4.7(b) shows the third harmonic phasors. The third harmonic currents are known as zero sequence harmonics due to the zero displacement angle between the three phasors.

The expressions for the fifth harmonic currents are:

ia5 = Ia5 sin 5ωt…(4.13)

ib5 = Ib5 sin 5(ωt – 120°) = Ib5 sin (5ωt – 600°)

= Ib5 sin (5ωt – 240°)…(4.14)

ic5 = Ic5 sin 5(ωt – 240°)

ic5 = ic5 sin 5(ωt – 240°) = Ic5 sin (5ωt – 1200°)

= Ic5 sin (5ωt – 120°) …(4.15)

Figure 4.7(c) shows the fifth harmonic phasors. Note that the phase sequence of the fifth harmonic currents is clockwise and opposite to that of the fundamental. The fifth harmonics are negative sequence harmonics.

Similarly the expressions for the seventh harmonic currents are:

ia7 = Ia7 sin 7ωt…(4.16)

ib7 = Ib7 sin 7(ωt – 120°) = Ib7 sin (7ωt – 840°)

= Ib7 sin (7ωt – 120°)…(4.17)

ic7 = Ic7 sin 7(ωt – 240°) = Ic7 sin (7ωt – 1680°)

= Ic7 sin (7ωt – 240°) …(4.18)

Figure 4.7(d) shows the seventh harmonic current phasors. The seventh harmonics have the same phase sequence as the fundamental and are positive sequence harmonics.

The expressions shown so far for harmonics have zero phase shifts with respect to the fundamental. It is not uncommon for the harmonics to have a phase-angle shift with respect to the fundamental. Figure 4.8 depicts a fifth harmonic current waveform with and without phase shift from the fundamental.

Expressions for the fifth harmonics with a phase-shift angle of 0 degrees are:

ia5 = Ia5 sin 5 (ωt – θ) …(4.19)

ib5 = Ib5 sin 5 (ωt – 120°- θ) …(4.20)

ic5 = Ic5 sin 5 (ωt – 240°- θ) …(4.21)

While the phase-shift angle has the effect of altering the shape of the composite waveform, the phase sequence order of the harmonics is not affected.