Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 163

2
4

3
5 ¼ 1
3

2
4

3
5

2
4

3
5 1
3

2
4

3
5

2
4

3
5 :

x 0
x 1
x 2

x L1
x L2
x L3

x L1
x L2
x L3

11 1
1

00 0
0 a I a I
0 a I

ð 8 : 164 Þ

a R a R

1 a R a R a I

The equation can be implemented directly using orthogonal components of
three phase sinusoidal variables obtained applying any methods of

orthogonalization.

8.2.6.2 Symmetrical Component Filters Using Signal Delays

Simpler filters of symmetrical components can be realized using signal delays.
Simplicity of such a solution results from the simplest operation in a digital
system—delay by certain number of samples. To get adequate shape of

transformation matrix one can use transformation operators a and a 2 in the

form:

a ¼ exp ð j2p = 3 Þ¼ exp ð jc Þ¼ exp ð j4p = 3 Þ¼ exp ð j2c Þ; ð 8 : 165 Þ

a 2 ¼ exp ð j4p = 3 Þ¼ exp ð j2c Þ¼ exp ð j2p = 3 Þ¼ exp ð jc Þ:

ð 8 : 166 Þ

Using these expressions the filters of symmetrical components can be written in
the form similar to ( 8.159 ):

2 3

5 ¼ 1
3

2 3 2 3

x 0
x 1
x 2

1 1 1 X L1 exp j ð nX 1 þ u L1 f g

X L2 exp j ð nX 1 þ u L2 f g
X L3 exp j ð nX 1 þ u L3 Þ

4 4 5 4 5 :

1 exp ð j2c Þ exp ð jc Þ ð 8 : 167 Þ

1 exp ð jc Þ exp ð j2c Þ f g

If one multiplies matrices of right-hand side of the above equation and then
compares either real or imaginary components of both sides one can get resulting
equations of transformation, which need addition of adequately delayed three
phase system components. Comparing real components one gets:

x 0 ð n Þ¼ 1 f X L1 cos ð nX 1 þ u L1 Þþ X L2 cos ð nX 1 þ u L2 Þþ X L3 cos ð nX 1 þ u L3 Þ g;

3

ð 8 : 168a Þ

x 1 ð n Þ¼ 1
3

f X L1 cos ð nX 1 þ u L1 Þþ X L2 cos ð nX 1 þ u L2 2c Þþ X L3 cos ð nX 1 þ u L3 c Þ g;

ð 8 : 168b Þ

x 2 ð n Þ¼ 1
3

f X L1 cos ð nX 1 þ u L1 Þþ X L2 cos ð nX 1 þ u L2 c Þþ X L3 cos ð nX 1 þ u L3 2c Þ g:

ð 8 : 168c Þ

The delay angles are equal either one-third or two-thirds of period of fun-
damental frequency component. It means that signals should be delayed by