146 TREATISE ON ALTERNATING CURRENTS. 



If these three currents pass simultaneously along the same 

 wire, the resulting current is 



i sinpt + i sin (pt + -yj + i sin (pt -f *\ = 



that is, the resultant current is zero when three currents, mutually 

 out of phase by one-third of a period and of the same intensity and 

 frequency, act simultaneously in the same circuit. 



ROTARY MAGNETIC FIELDS. 



97. Suppose that an anchor-ring of laminated soft iron is 

 wound as in Fig. 50, and that the coils a, a 1 are traversed by an 



alternating current, and the coils &, V 

 traversed by a second alternating cur- 

 rent of the same intensity and fre- 

 quency, but lagging behind that in a, a' 

 by a quarter of a period. When the 

 current in the coils a, a' is a maximum, 

 that in b, I' is zero, so that if the direc- 

 tion of the current in a, a at that 

 instant is represented by the arrow 

 heads, the ring will be magnetized in 

 the direction b, b', the south-seeking (S) 

 pole being at b', and the north-seeking 

 (N) at 5. A quarter of a period later there will be no current in the 

 coils a, a, and that in the coils b, b' will have attained its maximum 

 value, so that the S pole of the ring will now be at a, and the N 

 pole at a'. In another quarter of a period the S pole will be at b, 

 and the N pole at b', and so on. 



We see, therefore, that the anchor- ring, magnetized by two 

 alternating currents differing in phase by a quarter of a period, 

 will have its direction of magnetization rotated at such a speed 

 that it will have turned through 180 in half the periodic time 

 of either current. 



A similar rotation of the magnetic axis would occur if the 

 anchor-ring was wound with three circuits fed by three similar 

 alternating currents, differing mutually in phase by one third of a 

 period. In Figs. 51, 52, and 53, A, B, C represent the three wind- 

 ings on an anchor-ring. If the currents in A, B, C are respectively 



