Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 120

(a)

1

X 1 m

0.8

0.6

0.4

0.2

0

-0.2

-0.4

x

-0.6

-0.8

-1

0 2 4 6 8 10 12

n

(b)

1

X 1 m

0.8

0.6

0.4

0.2

0

-0.2

x

-0.4

-0.6

-0.8

-1

0 2 4 6 8 10 12

n

Fig. 8.5 Signal magnitude estimation results with LSE approach for: a pure sine wave and
b signal containing 10% of the 2nd harmonic

;

N 1 1

X

N 1 1

X

X kC ¼ 2
N 1

x ð n Þ cos ð nkX 1 Þ¼ 2
N 1

nk 2p
N 1

x ð n Þ cos ð 8 : 22 Þ

n ¼ 0 n ¼ 0

N 1 1

X

N 1 1

X

X kS ¼ 2
N 1

x ð n Þ sin ð nkX 1 Þ¼ 2
N 1

nk 2p
N 1

x ð n Þ sin ð 8 : 23 Þ

n ¼ 0 n ¼ 0

and

;

X

N 1 1

X ð k Þ¼ X C ð k Þ jX S ð k Þ¼ 2
N 1

x ð n Þ exp jnk 2p
N 1

ð 8 : 24 Þ

n ¼ 0