Page 284 - Power Electronic Control in Electrical Systems

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272 Harmonic studies of power compensating plant

Applying a Fourier transform to both sides of equation (7.14), and taking the
Integration and Frequency Convolution theorems into account, it is possible to write
an expression in the frequency domain for the TCR current, I R
1
I R ˆ S R C (7:15)
L R
where S R and are the switching function and the excitation flux vectors, respectively.

7.4.2.1 Harmonic switching vectors
One way to obtain the harmonic coefficients of the switching vector, S R ,is by
drawing a fundamental frequency cycle of the derivative of the TCR current with
respect to the flux with respect to time and then using an FFT routine (Acha, 1991).
The harmonic content of S R would then be obtained by multiplying this result by L R ,
the inductance of the linear reactor. However, this is an inefficient and error prone
option. A more elegant and efficient alternative to determine the harmonic coeffi-
cients of the switching function was put forward by Rico et al. (1996).
The more efficient formulation uses a periodic train of pulses to model the switch-
ing function s R (t), which in the frequency domain is well represented by a complex
harmonic vector, S R , with the following structure
2 3
a h j b h
2 2
6 . 7
.
6 . 7
6 7
6 a 1 b 1 7
2
6 2 j 7
6 7 (7:16)
a 0
S R ˆ 6 7
a 1 b 1
6 7
2 2
6 ‡ j 7
.
6 7
6 . 7
4 . 5
a h ‡ j b h
2 2
where the vector accommodates the complex harmonic coefficients from h to ‡h.
Also, the generic coefficient S h is
a h b h 1 Z p jhot
S h ˆ ‡ j ˆ s(t)e dot (7:17)
2 2 p p
The Fourier coefficients a h and b h are calculated from equations (7.18) to (7.20),
which take into account the fact that the switching function has a value of one in the
following intervals [ p/2 s 1 /2, p/2 ‡ s 1 /2] and [p/2 s 2 /2, p/2 ‡ s 2 /2]
" p s 1 Z p s 2 #
1 Z ‡ 2 2 ‡ 2 s 1 ‡ s 2
2
a 0 ˆ dot ‡ dot ˆ (7:18)
p p s 1 p s 2 2p
2 2 2
2
" p s 1 Z p s 2 #
1 Z ‡ 2 2 ‡ 2
2
a h ˆ cos (hot)dot ‡ cos (hot)dot
p p s 1 p s 2

2 2 2 2 (7:19)
2 hs 2 hs 1 hp
ˆ sin ‡ sin cos
hp 2 2 2