Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 131

8.2.1 Measurement of Magnitude of Voltage or Current

Magnitude measurement can be realized using the methods described in general in
last paragraph. The most important are given by Eqs. 8.50 , 8.53 and 8.54 . It can be
easily recognized that all of these algorithms can be applied when orthogonal
components from rotating phasors are used, as examples in the case of FIR digital
filters. In such cases it is also possible to obtain delayed signals. In contrary, for
constant phasor, as in the correlation method, Eq. 8.50 can only be applied.
Considering the first case, the output signals of a pair of orthogonal FIR filters
can be written in the form:

y 1C ð n Þ¼ F 1C X 1m cos ð nX 1 þ u 1 þ b Þ; ð 8 : 69a Þ

y 1S ð n Þ¼ F 1S X 1m sin ð nX 1 þ u 1 þ b Þ;

ð 8 : 69b Þ

where y 1 ð n Þ¼ P

N 1

a ð k Þ x 1 ð n k Þ is an output signal of digital FIR filter for the

k ¼ 0

input signal,

x 1 ð n Þ¼ X 1m cos ð nX 1 þ u 1 Þ;

X 1 ¼ x 1 T S ;

x 1 is an angular frequency of fundamental component (50 Hz), F are gains of
filters of odd (S) and even (C) symmetry of impulse response for fundamental
frequency components, b is a phase shift of the filter of even symmetry.
The orthogonal components ( 8.47a , b ) can be obtained from ( 8.69a , b ) just by
dividing the filter outputs by filter gains:

x 1C ð n Þ¼ y 1C ð n Þ
F 1C

; ð 8 : 70a Þ

x 1S ð n Þ¼ y 1S ð n Þ
F 1S

: ð 8 : 70b Þ

Applying them in ( 8.50 ) one gets:

s

y 1C ð n Þ
F 1C

þ y 1S ð n Þ
F 1S

X 1m ¼ : ð 8 : 71 Þ

This is the fundamental equation of current or voltage magnitude measurement
using a pair of orthogonal filters. In practice, in most cases the orthogonal filter
gains are the same and then Eq. 8.71 is simplified to:

q

y 1C ð n Þþ y 1S ð n Þ

X 1m ¼ 1
F 1

: ð 8 : 72 Þ