Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 114

From the above two equations the following table can be obtained for the
assumed range of k:

k j G 1s ð j0 Þj j G 1s ð j2X 1 Þj

1 0.158 1.957

2 0.325 1.83

3 0.51 1.46

4 0.726 1.35

5 1 1

As one can see, lower delay values induce stronger suppression of the DC
component, but at the same time—evoke stronger amplification of the second
harmonic (effect of differentiation).
Another variant of orthogonalization by single delay applies addition and
subtraction of a signal and a signal delayed by k samples (Eqs. 8.1 and 8.2 ). It is
easy to notice that the orthogonal components are then described by equations:

¼ x ð n k Þþ x ð n Þ
2 cos ð kX 1 = 2 Þ

n k
2

x c ; ð 8 : 7a Þ

¼ x ð n k Þ x ð n Þ
2 sin ð kX 1 = 2 Þ

n k
2

x s ; ð 8 : 7b Þ

Similar to the method of orthogonalization by single delay one can obtain the
filter transfer functions and equivalent frequency responses (magnitudes and phase
shifts) resulting from ( 8.7a , b ). The two filters obtained (one of them is low pass
filter and the other high pass) have the following transfer functions:

G c ð jX Þ¼ exp ð jkX Þþ 1
2 cos ð kX 1 = 2 Þ

; ð 8 : 8a Þ

G s ð jX Þ¼ exp ð jkX Þ 1
2 sin ð kX 1 = 2 Þ

: ð 8 : 8b Þ

Their magnitudes and phase shifts versus frequency are:

j G c ð jX Þj ¼ cos ð kX = 2 Þ

cos ð kX 1 = 2 Þ ;

ð 8 : 9a Þ

j G s ð jX Þj ¼ sin ð kX = 2 Þ

sin ð kX 1 = 2 Þ ;

ð 8 : 9b Þ

tg ð w c w s Þ¼ tg ð w c Þ tg ð w s Þ

tg ð kX = 2 Þ ctg ð kX = 2 Þ
1 þ½ tg ð kX = 2 Þ ctg ð kX = 2 Þ ¼1;

1 þ tg ð w c Þ tg ð w s Þ ¼

ð 8 : 9c Þ

where w c ¼ arg ð G c Þ and w s ¼ arg ð G s Þ .