RELATIONS OF LINE P.D.'S AND CURRENTS 53 



But since v\ v a is the instantaneous p.d. across the shunt circuit 

 of the wattmeter Wi,* and v a v a the instantaneous p.d. across the 

 shunt circuit of W 2 , it follows that the mean total power over a period 

 (i.e. the mean value of w) is equal to the algebraical sum of the watt- 

 meter readings. 



27. Relations of Line P.D.'s and Currents 



On account of the importance of the two-wattmeter method, we 

 shall investigate the phase relations of the line p.d.'s and line currents 

 somewhat more closely in the 

 general case of an unbalanced 

 inductive load. In Fig. 37 is 

 given a vector diagram of the line 

 currents, the star p.d.'s, and the 

 line p.d.'s.f The three vectors 

 Ii, la, and I 3 represent the line 

 currents. These lag behind their 

 respective star p.d.'s Vi, V 2 , and 

 V 3 by the angles 0i, 2 , and 3 , and 

 for the sake of generality we have FlQ . 37 ._vcctor Diagram of Three-phase 

 supposed these angles to be all Circuit, 



different. The line p.d.'s, whose 



instantaneous values are denoted by V, V, and v a ' in Fig. 36, and 

 which are reckoned positive when they have a counter-clockwise 

 direction around the mesh or triangle A'B'C' in Fig. 36, are easily 

 obtained by noticing that v\ = v% v a> v 2 ' = v d v\, and v a ' = v\ 

 - v a . Hence in the vector diagram of Fig. 37 these line p.d.'s 

 are represented by the sides of the triangle formed by joining the 

 extremities of Vi, V 2 , and V 3 ; their directions are indicated by the 

 arrow-heads. 



For the sake of clearness in the diagram, it will be convenient to 

 draw the triangle of line p.d.'s as in Fig. 38, and from the vertices of 

 this the current vectors Ii, I 2 , and I 8 . 



We next notice that the connections of the two wattmeters Wi and 

 W 2 in Fig. 36 were supposed to be so arranged that a positive couple 

 was obtained when the current and p.d. at any instant were both 

 acting away from (or towards) the junction of the current and 

 pressure coils. Now, from Fig. 36 it will be seen that the positive 

 direction of v 3 ' is the same as that of v it and since the reading of Wx 

 is equal to the mean value of + v$ii, it is represented, in Fig. 38, by 

 V 8 'Ii cos ^>i = V 3 ' x AD. On the other hand, considering the watt- 

 meter W 2) we notice that, since in Fig. 36 the positive direction of 



* Cf. 10. t This diagram should be compared with Fig. 36. 



