Finite Impulse Response Filters - Digital Signal Processing in Power System Protection and Control - page 77

magnitudes of frequency responses are shown in Figs. 6.7 and 6.8 , respectively. It
is seen that especially advantageous are filters with full cycle windows. They are
band-pass filters that reject all harmonics of fundamental frequency. The charac-
teristics of half cycle filters are worse, especially in respect of passing DC com-
ponent, however, they are two times faster. For that filters, very popular in
protection applications, we can collect their coefficients, gains and arguments for
fundamental frequency:

• for full cycle window, i.e. for N ¼ N 0 :
a c ð k Þ¼ cos ð k þ 0 : 5 Þ X 0

½ ; ð 6 : 33a Þ

a s ð k Þ¼ sin ð k þ 0 : 5 Þ X 0

½ ; ð 6 : 33b Þ

j A c ð jX 0 Þ j ¼ A s ð jX 0 Þ j j ¼ N 0 = 2 ; ð 6 : 33c Þ

arg A c ð jX 0 ½ ¼ p þ p = N 0 ; ð 6 : 33d Þ

arg A s ð jX 0 ½ ¼ p = 2 þ p = N 0 ; ð 6 : 33e Þ

• and for half cycle window, i.e. for N ¼ N 0 = 2:

a c ð k Þ¼ sin ð k þ 0 : 5 Þ X 0 ½ ; ð 6 : 34a Þ

a s ð k Þ¼ cos ð k þ 0 : 5 Þ X 0

½ ; ð 6 : 34b Þ

j A c ð jX 0 Þ j ¼ A s ð jX 0 Þ j j ¼ N 0 = 4 ; ð 6 : 34c Þ

arg A s ð jX 0 Þ ½ ¼ p = 2 þ p = N 0 ; ð 6 : 34d Þ

arg A s ð jX 0 Þ ½ ¼ p = N 0 ; ð 6 : 34e Þ

where X 0 ¼ 2p = N 0 :
Next figures show frequency responses of the filters modified in a different way,
i.e. having different natural relative angular frequency X 0 and different window
length N. The case when the filter natural frequency is the same as before but the
filter window is two times longer ð N ¼ 2N 0 Þ is shown in Fig. 6.9 . This pass band
filter has even better ability to reject or dump signal noise. When such window
length is allowed taking into account protection dynamics such filters might be
recommended. The case with filters of half cycle window ð N ¼ N 0 = 2 Þ but two
times greater natural frequency ð X 0 ¼ 4p = N 0 Þ is presented in Fig. 6.10 . Such
frequency was chosen to get filter frequency response rejecting DC for both
orthogonal filters, which may be important in many applications. The effect of
such natural frequency matched filter window is improving the filter frequency
response in region of small frequencies and worsening in the region of high
frequencies. Such filters may be applied in case of low-level of expected high
frequency noise and for high content of DC or decaying DC component, with
additional requirements of fast response. One can see that increasing this filter