Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 150

x ð n 2k Þ x ð n Þ
2x ð n k Þ sin ð kX 1 Þ :

c ð n k Þ¼ arctg ð 8 : 130 Þ

To measure phase shift of the signal one can also use a pair of orthogonal filters.
Assuming that input signal of orthogonal filters is equal to x ð n Þ¼ X 1m sin ð nX 1 þ u Þ
the output signal of one filter equals to: y 1C ¼ F 1C X 1m sin ð nX 1 þ u þ b Þ , while the
second equals y 1S ¼ F 1S X 1m cos ð nX 1 þ u þ b Þ . Taking ratio of these signals yields:

¼ c ð n Þþ b ¼ nX 1 þ u þ b

F 1S
F 1C

y 1C ð n Þ
y 1S ð n Þ

arctg ð 8 : 131 Þ

From ( 8.131 ) one can calculate phase shift, since filters parameters are known.

Example 8.12 Provide an algorithm for signal phase estimation with orthogonal
components obtained from signal filtration with use of FIR sin, cos filters. Assume
sampling with 1000 Hz.
Solution For given sampling rate the number of samples per cycle of 50 Hz is
N 1 ¼ 20 thus the filter gains are F 1C ¼ F 1C ¼ N 1 = 2 ¼ 10 and their arguments
amount to b ¼ p þ p = N 1 ¼ 0 : 95p : Taking that into account one obtains:

b ¼ arctg

F 1S
F 1C

y 1C ð n Þ
y 1S ð n Þ

y 1C ð n Þ
y 1S ð n Þ

u þ nX 1 ¼ arctg þ 0 : 95p ;

where (according to ( 4.30 ), ( 4.31 )) the output signals of orthogonal filters are given
by

y 1C ð n Þ¼ X

19

x ð n k Þ cos ½ 0 : 1p ð 9 : 5 k Þ;

k ¼ 0

y 1S ð n Þ¼ X

19

x ð n k Þ sin ½ 0 : 1p ð 9 : 5 k Þ;

k ¼ 0

for the input signal of the standard form ( 8.125 ):

x ð n Þ¼ X 1m sin ð nX 1 þ u Þ¼ X 1m sin ð 0 : 1np þ u Þ:

For realizing of the signal phase measurement the input signal is to be delivered
to both filters, which—besides orthogonalization function—are helpful in elimi-
nation of possible signal distortions, if any, that are not included in assumed signal
model ( 8.127 ). It is good to recall that full-cycle sin, cos filters are able to com-
pletely eliminate all harmonic components, while other frequencies are only
reduced, according to the filter frequency characteristics described in previous
chapters.
The simplest possibility of phase shift calculation results from correlation
method or DFT. Let the signal ( 8.125 ) be given in the form: