Page 86 - Power Electronic Control in Electrical Systems

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Power electronic control in electrical systems 75

2.12.4 Harmonics in balanced networks

In a wye connection we can observe that the actual instantaneous line±line voltage
obeys the equation u ab ˆ u an u bn , and so does its fundamental component: u ab1 ˆ
u an1 u bn1 . However, the third harmonic component is u ab3 ˆ u an3 u bn3 ˆ 0, which
means that no triplen harmonic voltage can appear between two lines in a balanced
system.
If there is no neutral, i a ‡ i b ‡ i c ˆ 0. Since i a3 ˆ i b3 ˆ i c3 , they must all be zero. In
a three-wire balanced system, no triplen harmonic currents can flow in the lines. This
is true for wye-connected and delta-connected loads. However, triplen harmonic
currents can circulate around a delta without appearing in the lines. This property
is used to provide the third-harmonic component of magnetizing current in saturated
transformers. If the neutral (4th wire) is connected, the neutral current is

i N ˆ i a ‡ i b ‡ i c ˆ‡3(i 3 ‡ i 9 ‡ ... ) (2:67)
The neutral connection helps to prevent oscillation of the neutral voltage.
A non-linear load can draw non-sinusoidal currents in each phase, including 3rd
harmonics. If such a load is connected in delta, the triplen harmonics can flow in the
delta without appearing in the lines. The equivalent circuit of such a load must
include a fictitious voltage source for each triplen harmonic, in series with the non-
linear load impedance. In a delta connection, the sum of the triplen source voltage
and the triplen harmonic voltage drop across the non-linear load impedance will be
zero, so that no triplen harmonic voltage component appears between the lines. This
is illustrated in Figure 2.47, with
E 3 ‡ Z 3 I 3 ˆ 0 (2:68)

Fig. 2.47 Equivalent circuit of non-linear load.