Page 84 - Power Electronic Control in Electrical Systems

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Power electronic control in electrical systems 73

fundamental frequency in Hz and o ˆ 2pf . According to Fourier the coefficients a m
and b m can be determined from the original waveform by the integrations
2 Z 2p
a m ˆ u(ot)cos (mot) d(ot)
2p 0 (2:59)
2 Z 2p
b m ˆ u(ot) sin (mot) d(ot)
2p 0
and the DC value from the integral
1 Z 2p
u 0 ˆ u(ot) d(ot) (2:60)
2p 0

2.12.1 Harmonic power
In general p ˆ ui so

1 Z 2p
P avg ˆ p(ot) d(ot)
2p 0
1 Z 2p X p p 
1
ˆ 2V m cos (mot) 2I n cos (not ‡ f ) d(ot)
n
2p 0 mˆ0
nˆ0
1 Z 2p
X 1 (2:61)
ˆ V m I n fcos [(m ‡ n)ot ‡ f ] ‡ cos [(m n)ot f ]g
n
n
2p
mˆ0 0
nˆ0
1
X
ˆ V m I m cos f m
mˆ0
ˆ V 0 I 0 ‡ V 1 I 1 cos f ‡ V 2 I 2 cos f ‡ ...
1
2
Products of the mth voltage harmonic and the nth current harmonic integrate to zero
over one period, if m 6ˆ n, leaving only the products of harmonics of the same order.
The power associated with each harmonic can be determined individually with an
equation of the form VI cos f, where V and I are the rms voltage and current of that
harmonic and f is the phase angle between them.
2.12.2 RMS values in the presence of harmonics
If the current flows through a resistor R, V m ˆ RI m and the average power dissipa-
tion is

s 
1 1
X 2 2 X 2
P avg ˆ I R ˆ I R where I ˆ I m (2:62)
m
mˆ0 mˆ0
I is the rms current and equation (2.62) is consistent with the definition of rms current
s 
1 Z T
2
I rms ˆ i (t)dt (2:63)
T 0