Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 156

y ð n Þ¼ CU m cos ð nX þ u þ b Þ: ð 8 : 143 Þ

A ratio of output and input magnitudes is equal to the zero-order Walsh FIR
filter gain for given frequency:

Y m
U m

¼ C ¼ W 0 ð jX Þ j j: ð 8 : 144 Þ

The filter gain is given by:

j W 0 ð jX Þj ¼ sin N 1 2

2

; ð 8 : 145 Þ

X

sin

while the frequency characteristic is shown in Fig. 6.2 .
The gain is equal to zero for nominal frequency (X ¼ X 1 ¼ 2p = N 1 ) and the
same is for magnitude of the output signal. As it is seen from Eq. 8.144 the gain
value can be measured using ratio of output Y m and input U m magnitudes. Having it
we can rearrange complex function of frequency to find it or its deviation. The last
possibility leads to simpler equation. One can notice first that:

N 1 X
2

¼ N 1 ð X 1 þ DX Þ
2

¼ p þ N 1 DX
2

ð 8 : 146 Þ

and substituting it to ( 8.145 ) and then to ( 8.144 ) yields:

N 1 DX
2

C ¼ sin : ð 8 : 147 Þ

X 1 þ DX
2

sin

If DX and N 1 DX are very small then one obtains:

:

2
N 1 sin

X 1
2

DX ¼ C ð 8 : 148 Þ

The measurement error is a result of simplifications made. Its value is a fraction
of percent for small frequency deviations.
Simpler formula is obtained when X 1 þ DX is very small (by high sampling
frequency) and both sine functions in ( 8.147 ) are substituted by their arguments.
Simple rearrangements give:

DX
X 1

C
N 1 C :

¼ ð 8 : 149 Þ

This is the simplest equation allowing to calculate small frequency deviations.
To apply it is necessary to know the ratio of output and input magnitudes of zero-
order Walsh filter. Having it one can calculate frequency deviation from ( 8.149 ).
For greater frequency deviations the measurement error increases significantly.
One can avoid it using finite difference of output signal ( 8.143 ) at discrete time
n and n 1. Then: