Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 134

y 1S ð n Þ¼þ X x ð n k Þ X

9 19

x ð n k Þ:

k ¼ 0 15

Gain values for full-cycle filters for frequency 50 Hz are identical and amount
to, ( 6.28 ):

2
sin ð p = N 1 Þ ¼

2
sin ð p = 20 Þ ¼ 12 : 78

F 1C ¼ F 1S ¼ F ¼j W 1 ð jX 1 Þj ¼ j W 2 ð jX 1 Þj ¼

therefore the magnitude measurement equation becomes:

q

y 1C ð n Þþ y 1S ð n Þ

q

y 1C ð n Þþ y 1S ð n Þ X 1 ¼ 1

F

¼ 0 : 0782 :

If the sampling frequency equals 2000 Hz, then the number of samples per
cycle is 40 and the above equations obtain the form:

y 1C ð n Þ¼ X x ð n k Þþ X x ð n k Þ X

9 29 39

x ð n k Þ;

k ¼ 0 k ¼ 10 30

y 1S ð n Þ¼ X x ð n k Þ X

19 39

x ð n k Þ;

k ¼ 0 k ¼ 20

q

y 1C ð n Þþ y 1S ð n Þ X 1 ¼ 0 : 0393 :

Example 8.8 For sampling frequencies 1000 and 1600 Hz derive equations of
full-cycle magnitude measurement algorithms with application of full-cycle FIR
filters with sine, cosine impulse responses windows.

Solution Output signals of the orthogonal filters ( 6.37 ), ( 6.38 ) are:

y 1C ð n Þ¼ X

19

x ð n k Þ cos ½ð 9 : 5 k Þ p = 10 ;

k ¼ 0

y 1S ð n Þ¼ X

19

x ð n k Þ sin ½ð 9 : 5 k Þ p = 10 :

k ¼ 0

The gain coefficients for both filters for fundamental frequency are (for f S =
1000 Hz):

F 1C ¼ F 1S ¼ F ¼j A C ð jX 1 Þj ¼ j A S ð jX 1 Þj ¼ N 1 = 2 ¼ 10 ;

thus the measurement equation becomes:

q

y 1C ð n Þþ y 1S ð n Þ

q

y 1C ð n Þþ y 1 ð n Þ X 1 ¼ 1

F

¼ 0 : 1 :