BALANCED THREE-PHASE LOAD 



55 



28. Case of Balanced Load. Tangent 

 Formula 



When the circuit is balanced, the line p.d.'s and line currents 

 will, in the diagram of Fig. 37, be represented by lines of equal 

 length. Hence ABC will form an equilateral triangle. Further, 

 the vectors Vi, V 2 , and V 8 , representing the star p.d.'s, will radiate 

 towards the vertices of the triangle from its middle point, making 

 angles of 30 with its sides, i.e. with the line p.d. vectors. The star of 

 currents will consist of three equal arms, making angles of 120 with 



FIG. 39. Vector Diagram of Balanced Three-phase Circuit. 



each other, and displaced relatively to the p.d. star through an angle 9. 

 The cosine of this angle, which by supposition is the same for each 

 phase, is the power factor of the balanced three-phase load. The 

 diagram of Fig. 38 now takes the shape shown in Fig. 39, and it will 

 be seen that the reading of the wattmeter Wi in Fig. 36 is V^ cos 

 (30 - 0), while that of W a is Vi'I 3 ' cos (30 -f 9). Since V 3 ' = Vi' 

 = V say, and Ii = I 8 = I say, we have, using Wi and W 2 to denote 

 the two wattmeter readins 



Hence 



Wi = VI cos (30 - 0) 

 W 2 = VI cos (30 + 9) 



+ W a = 2VI cos 30 cos 9 

 - W a = 2VI sin 30 sin 9 



