Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 151

x ð n Þ¼ X 1m sin ð nX 1 þ u Þ¼ X 1m cos ð u Þ sin ð nX 1 Þþ X 1m sin ð u Þ cos ð nX 1 Þð 8 : 132 Þ

or

x ð n Þ¼ X 1C sin ð nX 1 Þþ X 1S cos ð nX 1 Þ

ð 8 : 133 Þ

Components X 1C ; X 1S can be calculated in many ways, using correlation or
DFT, for instance. Knowing these variables one can directly calculate phase shift
of the signal:

X 1S
X 1C

u ¼ arctg ð 8 : 134 Þ

Example 8.13 Applying the method of full-cycle correlation calculate the phase
shift of given input signal with respect to the correlating functions. Sampling
frequency is equal to 1000 Hz.

Solution In considered case one can apply both non-recursive and recursive
correlation procedures. In the latter version, the following values hold:

2 = N 1 ¼ 2 = 20 ¼ 0 : 1; X 1 ¼ 2p = N 1 ¼ 0 : 1p

and the correlation components are calculated according to

X 1C ð n Þ¼ X 1C ð n 1 Þþ 2 = N 1 ½ x ð n Þ x ð n N 1 Þ cos ð nX 1 Þ
¼ X 1C ð n 1 Þþ 0 : 1 ½ x ð n Þ x ð n 20 Þ cos ð 0 : 1np Þ;

X 1S ð n Þ¼ X 1S ð n 1 Þþ 0 : 1 ½ x ð n Þ x ð n 20 Þ sin ð 0 : 1np Þ:

The correlation components can now be directly used for calculation of the
sought phase shift:

:

X 1S
X 1C

u ¼ arctg

8.2.4.2 Measurement of Phase Shift Between Two Signals

Phase shift of the signal calculated above was referenced to phase of correlation
functions. One can measure the phase shift between two signals in the same way.
Useful practical approach could be application of algorithm of either power or
impedance components. This is unimportant whether the signals are current and
voltage—one simply needs any two signals. Applying the algorithm of calculation
active and reactive power one reaches:

:

Q
P

u ¼ arctg ð 8 : 135 Þ