When the balanced components are considered, it is seen that that the most frequently occurring angle is 120◦. In complex number theory, j is defined as the complex operator which is equal to and a magnitude of unity, and more importantly, √-1 when operated on any complex number rotates it anti-clockwise by an angle of 90◦. i.e. j = = 1 ∠90◦ √-1 In like manner, we define a new complex operator which has a magnitude of α unity. when operated on any complex number rotates it anti-clockwise by an angle of α 120◦ and square of rotates it by 240o . α i.e. = 1 = - 0.500 + j 0.866 ∠120◦ α and 2 = 1∠2400 = -0.500 j 0.866 α – The three phasors (1+j0), 2 and (taken in this order) form a balanced, α α symmetrical , set of phasors of positive sequence rotation since the phasors are of equal length displaced by equal angles of 1200 from each other, and cross the reference line in the order 1, 2 and (following the usual convention of counter-clockwise rotation for α α the phasor diagram). The phasors 1, and 2 (taken in order) form the balanced, α α symmetrical, set of phasors of negative phase-sequence, since the phasors do not cross the reference line in the order named, keeping the same convention of rotation, but third name following the first etc. Some Properties of α Phasor Addition Since is complex, it cannot be equal to 1, so that - 1 cannot be zero. α α + + 1 = 0 ∴ α2 α This also has the physical meaning that the three sides of an equilateral triangles must close as in figure 2.11. Also = and = α−1 α2 α−2 α Analysis of decomposition of phasors The sequence components of the unbalanced quantity are again examined, with each of the components written in terms of phase a components, and the operator as α, in figure shown below. Figure 14 Expressing components in terms of phase – All the sequence components can be expressed in terms of the quantities for phase a using the properties of rotation of 0◦, 120◦ or 240◦ .
When the balanced components are considered, it is seen that that the most frequently occurring angle is 120◦. In complex number theory, j is defined as the complex operator which is equal to √-1 and a magnitude of unity, and more importantly, when operated on any complex number rotates it anti-clockwise by an angle of 90◦. i.e. j = √-1 = 1 ∠90◦ In like manner, we define a new complex operator α which has a magnitude of unity. α when operated on any complex number rotates it anti-clockwise by an angle of 120◦ and square of α rotates it by 240o . i.e. α = 1 ∠120◦ = - 0.500 + j 0.866 and α 2 = 1∠2400 = -0.500 – j 0.866 The three phasors (1+j0), α 2 and α (taken in this order) form a balanced, symmetrical , set of phasors of positive sequence rotation since the phasors are of equal length displaced by equal angles of 1200 from each other, and cross the reference line in the order 1, α 2 and α (following the usual convention of counter-clockwise rotation for the phasor diagram). The phasors 1, α and α 2 (taken in order) form the balanced, symmetrical, set of phasors of negative phase-sequence, since the phasors do not cross the reference line in the order named, keeping the same convention of rotation, but third name following the first etc. Some Properties of α Phasor Addition Since α is complex, it cannot be equal to 1, so that α - 1 cannot be zero. ∴ α2 + α + 1 = 0 This also has the physical meaning that the three sides of an equilateral triangles must close as in figure 2.11. Also α−1 = α2 and α−2 = α Analysis of decomposition of phasors The sequence components of the unbalanced quantity are again examined, with each of the components written in terms of phase a components, and the operator α, as in figure shown below. Figure 14 – Expressing components in terms of phase All the sequence components can be expressed in terms of the quantities for phase a using the properties of rotation of 0◦, 120◦ or 240◦ .
16653818-FAULT-ANALYSIS-IN-TRANSMISSION-SYSTEM-USING-MATLAB.docx