1 express the following operatorsj j2 j3 a a2 a3

Question

1. Express the following operators:

j, j², j³, a, a², a³, -a², -a in rectangular form and sketch the operators in the complex plane.

2. Evaluate the following expressions in polar form

a. a²- 1

b. 1- a - a²

c. 2a² + 3 + 2a

d.1 + a + a²

3.Determine analytically the voltages to neutral V_an, V_bn. V_cn in a circuit where

Van1 =	50	∠0° V
Van2 =	10	∠900 V
Van° =	10	∠1800 V

4.Solve Problem 3 graphically

5.   Determine the symmetrical components of the three currents l_a= 10∠0A    l_b = 10∠250A,  = 10∠110 A

6.The current flowing in the fines towards a balanced load connected in delta formation are l_a=100∠0A, l_b=141.4∠225A, and I_c=100∠90A . Determine a general expression for the relation between the symmetrical components of the line currents flowing towards a delta-connected load and the phase currents in the load, that is, between l_a1 and l_ab1 and between l_a2 and l_ab2. Start by drawing phasor diagrams of the positive and negative sequence line and phase currents. Find l_ab in amperes from the symmetrical components of the given line currents

7. State Fortescue's Theorem and illustrate its principle by means of a simple phasor diagram of an unbalanced three-phase system.

(ii) Calculate in polar form the sequence components of unbalanced line­to-neutral voltages given as follows:

(iii) Calculate in polar form the phase currents corresponding to the sequence components

(ii)  V_a =180∠45^° v, V_b = 50∠-30^° V, V_c= 80∠0^o V

(iii) I_aa = 50∠15^°A , l_a1, = 200∠60^°A , 1_a2 = 160∠290°A