Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 127

x 1S ð n Þ¼ X 1m sin ð nX 1 þ u 1 Þ: ð 8 : 47b Þ

Rotating, time-dependent phasor, is a complex function of these components:

x 1 ð n Þ¼ x 1C ð n Þþ jx 1S ð n Þ¼ X 1m exp ½ j ð nX 1 þ u 1 Þ ð 8 : 48 Þ

One can easily notice that the product of this phasor and its conjugate is equal
to magnitude squared, being also equal to sum of squared orthogonal components:

x 1 ð n Þ x 1 ð n Þ¼ X 1m ¼ x 1C ð n Þþ x 1S ð n Þ: ð 8 : 49 Þ

Then the simplest and the most typical algorithm of magnitude measurement is
as follows:

q

x 1C ð n Þþ x 1S ð n Þ X 1m ¼ : ð 8 : 50 Þ

More general version can be obtained using the product of the phasor and its
delayed conjugate. Then one obtains:

x 1 ð n Þ x 1 ð n k Þ¼ X 1m exp ð jkX 1 Þ

ð 8 : 51 Þ

Calculating the same, but applying orthogonal components, one gets:

x 1 ð n Þ x 1 ð n k Þ¼ x 1C ð n Þ x 1C ð n k Þþ x 1S ð n Þ x 1S ð n k Þþ j ½ x 1S ð n Þ x 1C ð n k Þ
x 1S ð n k Þ x 1C ð n Þ

ð 8 : 52 Þ

Comparing two above equations, i.e. their real and imaginary components,
yields:

s

x 1C ð n Þ x 1C ð n k Þþ x 1S ð n Þ x 1S ð n k Þ
cos ð kX 1 Þ

X 1m ¼

; ð 8 : 53 Þ

s

x 1S ð n Þ x 1C ð n k Þ x 1S ð n k Þ x 1C ð n Þ
sin ð kX 1 Þ

X 1m ¼ : ð 8 : 54 Þ

The first of these equations is more general version of ( 8.50 ) for delay value
equal to zero. The second is a new measurement possibility, which requires delays
different from zero.
The three presented algorithms required orthogonal components. There is also a
possibility to design the algorithm for signal samples only, given in the form:

s

x 1 ð n k Þ x 1 ð n m Þ x 1 ð n Þ x 1 ð n k m Þ
sin ð kX 1 Þ sin ð mX 1 Þ

X 1m ¼ : ð 8 : 55 Þ