Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 112

are presented below. One can state in general that the methods are similar to the
use of the signal and its derivative, though in digital version the finite difference is
calculated.
Let the signal,in which orthogonal components are to be calculated, be given by
the equation:

x ð n Þ¼ X 1m cos ð nX 1 þ u Þ ð 8 : 1 Þ

The signal delayed by k samples is given by:

x ð n k Þ¼ X 1m cos ð n k Þ X 1 þ u ½
¼ X 1m cos ð kX 1 Þ cos ð nX 1 þ u Þþ X 1m sin ð kX 1 Þ sin ð nX 1 þ u Þ
¼ x c ð n Þ cos ð kX 1 Þþ x s ð n Þ sin ð kX 1 Þ

ð 8 : 2 Þ

From both equations one gets:

x c ð n Þ¼ x ð n Þ; ð 8 : 3a Þ

x s ð n Þ¼ x ð n k Þ x ð n Þ cos ð kX 1 Þ
sin ð kX 1 Þ

ð 8 : 3b Þ

The values of number of delay samples can be different: from one sample of
delay giving the fastest algorithm up to a number of samples equivalent to a
quarter of fundamental frequency component. The simplest digital orthogonali-
zation method is thus given by the equations:

x c ð n Þ¼ x ð n Þ; ð 8 : 4a Þ

x s ð n Þ¼ x ð n N 1 = 4 Þ: ð 8 : 4b Þ

The latter case, though ''time consuming'', is very convenient since orthogo-
nalization being in this case simple signal delaying is not a digital filter and does
not change the signal spectrum. In general, however, in all remaining cases for
k between one and N 1 = 4 samples the orthogonalization means realizing certain
digital filter, whose settling time and frequency response depend on k. To get the
frequency response of such a filter one has to calculate its discrete transfer
function, then substitute exp ð jX Þ for operator z, and calculate magnitude and phase
shift. One obtains then:

X s ð jX Þ

X ð jX Þ ¼ G 1s ð jX Þ¼ exp ð jkX Þ cos ð kX 1 Þ

ð 8 : 5 Þ

sin ð kX 1 Þ

and from that the magnitude and phase shift are following:

p

1 2 cos ð kX Þ cos ð kX 1 Þþ cos 2 ð kX 1 Þ

j G 1s ð jX 1 Þj ¼ ; ð 8 : 6a Þ

sin ð kX 1 Þ