Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 159

g k ð X Þ¼ y S ð n Þ y C ð n k Þ y C ð n Þ y S ð n k Þ ð 8 : 153 Þ

where

y C ð n Þ¼j F C j X m cos ð nX þ u þ b Þ;

y S ð n Þ¼j F S j X m sin ð nX þ u þ b Þ;

F is a filter gain, X m is an input magnitude.
It can be shown easily, which also results from ( 8.74 ), that the function ( 8.153 )
depends on: input magnitude, gain of orthogonal filters, frequency and number of
delay samples, which can be expressed in the form:

g k ð X Þ¼ X m j F C jj F S j sin ð kX Þ ð 8 : 154 Þ

If Expression ( 8.153 ) is calculated using the same as before input signals but for
different value of delay k then the argument of sine function is only changed. It
means that calculating ratio of function ( 8.154 ) for different delays one gets
expression, which depends on frequency only (signal magnitude and filter gains
are cancelled). To get simpler final expression one can choose in certain special
way relationships between delays. When for instance one delay is two times
greater than the other one gets:

g 2k ð X Þ
g k ð X Þ ¼ 2 cos ð kX Þ

ð 8 : 155 Þ

Calculating that ratio with use of orthogonal components of the signal (as in
( 8.153 )) the algorithm of frequency measurement becomes:

:

X ¼ 1

0 : 5 y S ð n Þ y C ð n 2k Þ y C ð n Þ y S ð n 2k Þ
y S ð n Þ y C ð n k Þ y C ð n Þ y S ð n k Þ

ð 8 : 156 Þ

k arccos

Simplified version of the method may be used for small frequency deviations
from its nominal value. Such result can be obtained assuming number of delay
samples giving particular delay time for instance a quarter of period of funda-
mental component (nominal value). For such value of k one gets:

¼ cos ¼ sin p

2

N 1
4

p
2

1 þ DX
X 1

p
2

DX
X 1

DX
X 1
ð 8 : 157 Þ

cos ð kX Þ¼ cos ð X 1 þ DX Þ

and then:

g N 1 = 2 ð X Þ

g N 1 = 4 ð X Þ p DX

:

X 1

Finally, simplified equation of measurement of frequency deviation is as
follows: