Page 279 - Power Electronic Control in Electrical Systems

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Power electronic control in electrical systems 267

impedance of the supply, including the transformer, then the following relations
will apply
Z s
I fn ˆ I n ˆ r I n (7:3)
f
Z f ‡ Z s
Z f
I sn ˆ I n ˆ r I n (7:4)
s
Z f ‡ Z s
From these equations, it is not difficult to see that if the distribution factor r is large
s
at a particular harmonic frequency coincident with one of the harmonics generated
by a harmonic source, then amplification of the harmonic current will occur and the
currents in the capacitor and the supply may become excessive. This would particu-
larly be the case if Z f ‡ Z s ! 0 at some harmonic frequency. Hence, r must be kept
s
low at these frequencies if excited by coincident harmonic currents.
The function of the tuning reactor shown in series with the capacitor in Figure
7.1(a) is to form a series-resonant branch or filter, for which Z f ! 0 at the resonant
frequency. As a result, r ! 0 thus minimizing the possibility of harmonic currents
s
flowing into the utility network. The ideal outcome is when r ! 1 so that I fn ˆ I n ,
f
meaning that all the harmonic current generated enters the filter.
7.3.1 Numerical example 1

A simple numerical example may be used to illustrate the performance of a detuned
and a tuned capacitor filter. Assume that the step-down transformer impedance is
much greater than the source impedance so that for a narrow range of frequencies the
approximation X s /R s ˆ constant may be used. Assume the following parameters

Bus voltage ˆ 13:8kV
Short circuit MVA ˆ 476
Capacitor reactive power ˆ 19:04 MVAr
X s /R s ˆ 10
Knowing that inductive reactance is directly proportional to frequency and that
capacitive reactance is inversely proportional to frequency
n(13:8) 2
X s ˆ ˆ 0:4n (7:5)
476
(13:8) 2 10
X fc ˆ ˆ (7:6)
19:04n n
X s
R s ˆ ˆ 0:04n (7:7)
10
0:04 ‡ j0:4
r ˆ 2 (7:8)
f
0:04 ‡ j(0:4 10=n )
In Figure 7.2, r and r are plotted against the harmonic order n. There is a parallel
s
f
resonance between the capacitor and the supply at the 5th harmonic. It should be
noted that at that point r r .
s
f