Characteristics of Measurement of Criterion Values and Adaptive - Digital Signal Processing in Power System Protection and Control - page 172

Transients including both filters and measurement algorithms can be analyzed
for an example of magnitude measurement. If a pair of orthogonal filters is applied
then their output signals (during transients, i.e., for n N 1) are given by the
equations:

y C ð n Þ¼ X

n

a c ð k Þ x ð n k Þ;

ð 9 : 4a Þ

k ¼ 0

y S ð n Þ¼ X

n

a s ð k Þ x ð n k Þ:

ð 9 : 4b Þ

k ¼ 0

Let input signal of the filters be given by:

x ð n Þ¼ X 1m cos ð nX 1 þ u 1 Þ; ð 9 : 5 Þ

where

X 1 ¼ x 1 T S ¼ 2pf 1 = f S ¼ 2p = N 1 ;

f 1 is a frequency of fundamental component, f S ; T S are sampling frequency and
period, and N 1 is a number of samples in the period of fundamental frequency
component.
Delayed signal samples in equations of the filters ( 9.4a , b ) can be written in the
form:

x ð n k Þ¼ X 1m cos ½ð n k Þ X 1 þ u 1 ¼ b c ð k Þ x C ð n Þþ b s ð k Þ x S ð n Þ; ð 9 : 6 Þ

where

x C ð n Þ¼ X 1m cos ½ð n þ 0 : 5 Þ X 1 þ u 1 ;

x S ð n Þ¼ X 1m sin ½ð n þ 0 : 5 Þ X 1 þ u 1 ;

b c ð k Þ¼ cos ½ð k þ 0 : 5 Þ X 1 ;

b s ð k Þ¼ sin ½ð k þ 0 : 5 Þ X 1 :

Substituting ( 9.6 ) into ( 9.4a , b ) one obtains:

y C ð n Þ¼ d CC ð n Þ x C ð n Þþ d CS ð n Þ x S ð n Þ; ð 9 : 7a Þ

y S ð n Þ¼ d SC ð n Þ x C ð n Þþ d SS ð n Þ x S ð n Þ; ð 9 : 7b Þ