42 ALTERNATING CURRENTS 



assuming that the origin from which x is measured remains unaltered, 

 and that the second winding is displaced backwards, or in the negative 



direction of x, relatively to the first for qx we must write q( x + 7) 



= qx + TJ. Hence, denoting by y% the value of the magnetic 



induction at any point x due to the current in the second winding, 

 we have 



2/2 



= B sin (pt 4- p sin (qx + |) 



Each of these simple alternating waves may, as already explained, 

 be replaced by two rotating waves, so that 



2ft = ^B{cos (pt qx} cos (pt 4- <?#)} 

 and 



2/2 = B{cos (pt qx) cos (pt -\-qx-\-ir)} 

 = B{cos (pt qx) 4- cos(pt 4- qx)} 



The resultant induction y at any point is given by 

 V = 2/i + 2/2 = B cos (pt - qx) 



and we see that the result of the superposition of the two alternating 

 waves is a pure rotating wave. Thus a two-phase system of 

 currents may be made to give rise to a rotating wave of magnetic 

 flux. 



Similarly, a three-phase system of currents may be used for pro- 

 ducing a rotating wave. Imagine three similar windings on the 

 stator or rotor, displaced relatively to each other by amounts JX, to 

 be traversed by three-phase alternating currents. If y\, 3/2, and y 3 

 denote the three component alternating flux waves, we may write 



yi = B sin pt sin qx 



T, . ( t , 27T\ . / . 



?/2 = B sin (pt +-g) sin (qx + 



and 



7/3 = B sin (pt 4- -j) sin (qx + 



or, using the simple transformation previously employed 

 2/i = B{cos (pt qx) cos (pt + qx)} 

 2/2 = BJcos (pt - qx) - cos (pt 4- qx 4- 



2/ 3 = LBJcos (pt - gx) - cos (pt 4- qx + -j^ 



