Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 140

Z 2 ¼ U 2

I 2 ¼ U m

ð 8 : 91 Þ

I m

and

R ¼j Z j cos ð u Þ¼ U m I m cos ð u Þ
I m

¼ 2P
I m

; ð 8 : 92 Þ

as well as

X ¼ 2Q
I m

; ð 8 : 93 Þ

where U ; I are effective values of voltage and current, U m ; I m are their magnitudes.
These fundamental equations or similar for conductance and components can
be used to find many particular algorithms using any power and any magnitude
algorithms. It is reasonable, however, to use such equations for power and mag-
nitude that constant coefficients in numerator and denominator are cancelled.
The general form of the equations uses components from the outputs of
orthogonal filters, i.e. using ( 8.71 ), ( 8.79a , b , c , d ) and ( 8.81 ). Substituting them
into ( 8.80 ) equations of measurement of impedance are obtained:

2 þ u F1S ð n Þ
F 1S

2

u F1C ð n Þ
F 1C

Z 2 ¼

; ð 8 : 94 Þ

2 þ i F1S ð n Þ
F 1S

2

i F1C ð n Þ
F 1C

resistance

u F1C ð n Þ i F1C ð n Þ
F 1C

þ u F1S ð n Þ i F1S ð n Þ
F 1S

R ¼

; ð 8 : 95 Þ

2

i F1C ð n Þ
F 1C

þ i F1S ð n Þ
F 1S

and reactance

1

F 1C F 1S ½ u F1S ð n Þ i F1C ð n Þ u F1C ð n Þ i F1S ð n Þ
i F1C ð n Þ
F 1C

X ¼ : ð 8 : 96 Þ

þ i F1S ð n Þ
F 1S

When gains F 1C ; F 1S are identical, they are being cancelled and simple equa-
tions using orthogonal components of voltage and current are received:

Z 2 ¼ u F1C ð n Þþ u F1S ð n Þ
i F1C ð n Þþ i F1S ð n Þ

; ð 8 : 97 Þ

R ¼ u F1C ð n Þ i F1C ð n Þþ u F1S ð n Þ i F1S ð n Þ
i F1C ð n Þþ i F1S ð n Þ

; ð 8 : 98 Þ