Page 68 - Power Electronic Control in Electrical Systems

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58 Power systems engineering ± fundamental concepts

average power is V m I m /2, this represents a peak±peak fluctuation 200% of the mean
power, at double frequency. The oscillation of power in single-phase circuits con-
tributes to lamp flicker and causes vibration in motors and transformers, producing
undesirable acoustic noise.

2.10.2 Two-phase

Suppose we have a two-phase load with phases a and b, with u a ˆ V m cos
ot, i a ˆ I m cos (ot f) and u b ˆ V m sin ot, i b ˆ I m sin (ot f). This system is said
to be balanced, because the voltages and currents have the same RMS (and peak)
values in both phases, and their phase angles are orthogonal. The total instantaneous
power is now given by

p ˆ u a i a ‡ u b i b
ˆ V m I m [ cos(ot)cos(ot f) ‡ sin(ot) sin(ot f)]
(2:32)
ˆ V m I m cos f
ˆ 2VI cos f

The oscillatory term has vanished altogether, which means that the power flow is
constant, with no fluctuation, and the average power P is therefore equal to the
instantaneous power p. Note that if the phases become unbalanced, an oscillatory
term reappears.

2.10.3 Three-phase
Suppose we have a three-phase load as in Figures 2.20 and 2.22, with phases a, b and
c, with
u a ˆ V m cos ot i a ˆ I m cos(ot f)
u b ˆ V m cos(ot 2p=3) i b ˆ I m cos(ot 2p=3 f) (2:33)
u c ˆ V m cos(ot ‡ 2p=3) i c ˆ I m cos(ot ‡ 2p=3 f)
This system is said to be balanced, because the voltages and currents have the same
RMS (and peak) values in all three phases, and their phase angles are equi-spaced
(i.e. with a 120 symmetrical phase displacement). The total instantaneous power is

now given by

p ˆ u a i a ‡ u b i b ‡ u c i c
ˆ V m I m [cos(ot) cos(ot f) ‡ cos(ot 2p=3) cos(ot 2p=3 f)
‡ cos (ot ‡ 2p=3) cos (ot ‡ 2p=3 f)]
(2:34)
3
ˆ V m I m cos f
2
ˆ 3VI cos f
As in the two-phase system, the oscillatory term has vanished. The power flow is
constant, with no fluctuation, and the average power P is equal to the instantaneous
power p. If the phases become unbalanced, an oscillatory term reappears.