Electrical Engineering ⇒ Topic : Solving Unbalanced 3-Wire Y Load by Millmans Theorem

Solving Unbalanced 3-Wire Y Load by Millman's Theorem

We have seen that problems on unbalanced 3-wire, Y load can be solved by converting the given Y load into equivalent Δ-load. However, such a procedure involves lengthy calculations. J.E. Millman proposed a new technique to obtain the solution of unbalanced Y loads. Consider an unbalanced Y load connected to a balanced 3-phase supply as shown in Fig. 15.110. Here 0 is the star point of the supply (normally at zero potential) and O' is the load star point. Due to load unbalance, the potential of O' is different from that of O. If we know V_0'0 (i.e., voltage of O' w.rt. 0), we caneasily determine the load phase voltages and the line currents of the unbalanced Y load. According to *Millman's theorem, the voltage V_0'0 is given by

where V_R0, V_Y0 and V_B0 are the phase voltages of the supply and are equal in magnitude but 120° apart in phase. The quantities Y_R, Y_Y. and Y_B are the admittances of the branches of the unbalanced Y load.

                                            (a)                                                                                                                     (b)

                                                                                                 figure         

   Fig. (b) shows the triangular phasor diagram. Here 0 is the star point of the supply and is located at the centre of the equilateral triangle RYB. The point 0' is the load star point. Due to load unbalance, 0' has some potential w.r.t. 0 and is shifted away from the centre of the triangle. Such a

diagram is very useful in analysing an unbalanced 3-wire star load because it gives at a glance the

picture of the happenings in the circuit. Thus

(i)distances OR, OY and OB are equal in magnitude but 120° apart from one another and represent the phase voltages of the supply.

(ii)distance 0'0 represents V0,0 i.e., potential of 0' w.r t. O.

(iii) distances 0' R, 0' Y and 0' B represent the load phase voltages. Note that load phase voltages are unequal in magnitude as well as differ in phase by unequal angle.

(iv) distances RY, YB and BR are equal in magnitude but *1200 apart in phase and represent the supply line voltages as well as load line voltages.