34 



ALTERNATING CURRENTS 



and if the current density in all the wires be the same, the cross- 

 section of the common wire will have to be y^2 times the cross-section 

 of one of the outer wires. 



Three-phase circuits are invariably linked together, as this effects 



a saving in the amount of copper required in the mains. There are 



two methods of coupling such circuits. One of 



these, shown in Fig. 25, is known as the star or 



Y coupling. In this, it will be noticed, the line 

 current is the same as the phase current, and 

 there is a point known as the neutral point, 



marked N in Fig. 25 (a) common to the three 



phases. The possibility of so coupling the cir- 

 cuits is due to the fact, at once evident from 

 Fig. 21 (&), or from an inspection of the equa- 



tions of 14, that the algebraical sum of the 



FIG. 24. interconnected three currents vanishes at every instant. With 

 Two-phase System. the star method of connection, the line currents 

 are equal to the phase currents, but the line p.d. 

 is much higher than the phase p.d. If we assume for the positive direc- 

 tions of the three p.d.'s the directions away from the neutral point N, 



FIG. 25. Star-connected Three-phase System. 



as indicated by the arrows in the figure, then since in proceeding from 

 B to N we move in the negative direction, and in proceeding from 

 N to A in the positive direction, it is evident that the line p.d., i.e. the 

 p.d. across BA, will at every instant be equal to the difference 'of the 

 instantaneous p.d.'s across NA and NB. Now, if the instantaneous 

 p.d. across NA be represented by the projection of OA in the vector 

 diagram of Fig. 25 (&), and that across NB by the projection of OB, 

 then the difference of these two projections, which gives the instan- 

 taneous value of the p.d. across BA, is equal to the projection of OA, 

 plus the projection of OB reversed. In Fig. 25 (&), OB' is OB reversed, 

 so that the p.d. across BA is equal to the sum of the projections of 



