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GATE EE 2021 Official Paper

Option 4 : 1 ∠-120° + 0.1 ∠0°

CT 1: Ratio and Proportion

3536

10 Questions
16 Marks
30 Mins

**Concept:**

**Fortescue’s Theorem:**

A unbalance set of ‘n’ phasors may be resolved into (n - 1) balance n-phase system of different phase sequence and one zero phase sequence system.

A zero-phase sequence system is one in which all phasors are of equal magnitude and angle.

Considered three phasors are represented by a, b, c in such a way that their phase sequence is (a b c).

The positive phase sequence will be (a b c) and the negative phase sequence will be (a c b).

Assumed that subscript 0, 1, 2 refer to zero sequences, positive sequence, negative sequence respectively.

Current I_{a}, I_{b}, I_{c} represented an unbalance set of current phasor as shown,

Each of the original unbalance phasor is the sum of its component and it can be written as,

I_{a} = I_{a0} + I_{a1} + I_{a2}

I_{b} = I_{b0} + I_{b1} + I_{b2}

I_{c} = I_{c0} + I_{c1} + I_{c2}

For a balance position phase sequence (a b c) we can write the following relation,

I_{a0} = I_{b0} = I_{c0}

I_{b1} = α^{2} I_{a1}

I_{c1} = α I_{a1}

I_{b2} = α I_{a2}

I_{c2} =α^{2} I_{a2}

From the above equation I_{a}, I_{b}, I_{c} can be written in terms of phase sequence component,

I_{a} = I_{a0} + I_{a1} + I_{a2}

I_{b} = I_{b0} + α^{2} I_{a1} + α I_{a2}

I_{c} = I_{a0} + α I_{a1} + α^{2} I_{a2}

The above equation can be written in form of Matrix as shown,

\(\left[ {\begin{array}{*{20}{c}} {{I_a}}\\ {{I_b}}\\ {{I_c}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&1&1\\ 1&{{\alpha ^2}}&\alpha \\ 1&\alpha &{{\alpha ^2}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{I_{a0}}}\\ {{I_{a1}}}\\ {{I_{a2}}} \end{array}} \right]\)

**Calculation:**

Given,

IB0 = 0.1 ∠0° p.u

IA0 = IB0 = IC0 = 0.1 ∠0° p.u

IA = 1.1 ∠0° p.u.

IC = (1 ∠120° + 0.1) p.u.

From the above concept,

I_{A} = I_{A0} + I_{A1} + I_{A2}

I_{A} - I_{A0} = I_{A1} + I_{A2} = 1.1 ∠0° + 0.1 ∠0° p.u = 1 ∠0° p.u

I_{A1} + I_{A2} = 1 ∠0° p.u .... (1)

I_{C} = I_{C0} + I_{C1} + I_{C2} = I_{A0} + α I_{A1} +α^{2} I_{A2}

1 ∠120° + 0.1 = 0.1 ∠0° + α I_{A1} +α^{2} I_{A2}

α I_{A1} +α^{2} I_{A2} = 1 ∠120° .... (2)

**Adding equation (1) and (2),**

(I_{A1} + I_{A2}) + (α I_{A1} +α^{2} I_{A2}) = 1 ∠0° + 1 ∠120°

I_{A1}(α + 1) + I_{A2}(α^{2} + 1) = 1 ∠0° + 1 ∠120°

α2 I_{A1} + α I_{A2} = 1 ∠-120° .... (3)

From above concept,

I_{B} = I_{A0} + α^{2} I_{A1} + α I_{A2}

**I _{C} = 0.1 ∠0° + 1 ∠-120°**