Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 113

Fig. 8.3 Frequency response
of orthogonalization by single
time delay

G ( j Ω )

1 S

G ( j Ω )

1 S 1

5

k = 1

4

k = 2

3

k = 3

2

1

k = 5

0

0 2 4 6 8

Ω Ω

1

sin ð kX Þ
cos ð kX Þ cos ð kX 1 Þ :

arg G 1s ð jX Þ ½ ¼ arctg ð 8 : 6b Þ

It is easy to notice that the filter gain equals unity for X equal to X 1 while the
argument is equal to p = 2 independent of parameter k. However, when X is
different from X 1 the argument is not equal to p = 2. It means that during fre-
quency deviations from nominal value the product of the method is not orthogonal
to the signal.
Frequency response of the method depends on assumed value of parameter
k. The minimum value of this parameter equals one and its maximum value results
from equation kX 1 ¼ p = 2, which depends on sampling frequency. The frequency
responses for sampling frequency equal to 1000 Hz and k varying from 1 to 5 are
shown in Fig. 8.3 . They are a very good illustration of the well-known dilemma:
speed-accuracy.

Example 8.1 Calculate magnitudes of the frequency response of orthogonalization
by time delay for two selected frequencies: zero (DC component) and second
harmonic. Assume sampling at 1000 Hz and fundamental frequency 50 Hz.
Solution From the assumed values it results:

N 1 ¼ 20; X 1 ¼ 0 : 1p :

Use of ( 8.6a ) leads to (for the frequency equal zero):

j G 1s ð j0 Þj ¼ tg ð 0 : 5kX 1 Þ¼ tg ð 0 : 05kp Þ

and for the second harmonic:

p

1 þ cos 2 ð 0 : 1kp Þ 2 cos ð 0 : 1kp Þ cos ð 0 : 2kp Þ

j G 1s ð j2X 1 Þj ¼

sin ð 0 : 1kp Þ