Page 85 - Power Electronic Control in Electrical Systems

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74 Power systems engineering ± fundamental concepts

Similar considerations apply to the voltage, such that
s 
1
X
2
V rms ˆ V m (2:64)
mˆ0
2.12.3 Phase sequence of harmonics in balanced three-phase
systems
The three phase voltages can be expanded in terms of their harmonic components

u an ˆ V 1 cos(ot) ‡ V 3 cos(3ot) ‡ V 5 cos(5ot) ‡ ...
u bn ˆ V 1 cos(ot 2p=3) ‡ V 3 cos 3(ot 2p=3) ‡ V 5 cos 5(ot 2p=3) ‡ ... (2:65)
u cn ˆ V 1 cos(ot ‡ 2p=3) ‡ V 3 cos 3(ot ‡ 2p=3) ‡ V 5 cos 5(ot ‡ 2p=3) ‡ ...
that is

u an ˆV 1 cos (ot) ‡ V 3 cos (3ot) ‡ V 5 cos (5ot) ‡ ...
u bn ˆV 1 cos (ot 2p=3) ‡ V 3 cos (3ot) ‡ V 5 cos (5ot ‡ 2p=3) ‡ ... (2:66)
u cn ˆV 1 cos (ot ‡ 2p=3) ‡ V 3 cos (3ot) ‡ V 5 cos (5ot 2p=3) ‡ ...

The three fundamental components form a balanced three-phase set of phasors
rotating at the fundamental electrical angular velocity o rad/s with positive sequence
abc. Likewise the fifth harmonic phasors form a balanced set rotating at 5 o rad/s,
but with negative sequence acb. The third harmonic components rotate at 3 o
radians/s but they are all in phase with one another and are said to have zero
phase sequence. They do not form a balanced set. The phasors are illustrated in
Figure 2.46.
Positive sequence harmonics include those of orders 1, 7, 13, 19, 25, 31, 37, ...;
negative sequence those of orders 5, 11, 17, 23, 29, 35, ...; and zero-sequence
harmonics all the triplen harmonic orders 3, 9, 15, 21, 27. Note that V a1 ‡ V b1 ‡
V c1 ˆ 0, V a3 ‡ V b3 ‡ V c3 ˆ 3V a3 6ˆ 0, and V a5 ‡ V b5 ‡ V c5 ˆ 0.

Fig. 2.46 Harmonic phasors.