144 THE MAGNETIC CIRCUIT [ART. 47 



<1 all the time. Thus, in a polyphase generator, the armature 

 rn.rn.f. lags behind the field m.m.f. by 90 electrical degrees in 

 space, when the currents arc 1 in phase with the voltages induced at 

 no-load. This statement is in accord with that in problem 4 in the 

 preceding article, because, if each phase shifts the flux against the 

 direction of rotation, all the phases together simply increase the 

 result. 



Let now the currents in the armature windings be lagging 

 90 electrical degrees behind the corresponding e.m.fs. induced 

 at no-load. This simply means that the armature m.m.f. is shifted 

 further back by 90 degrees as compared to the case considered 

 before; therefore, the angle between the field m.m.f. and the 

 armature m.m.f. is 180 electrical degrees, and the two m.m.f s. are 

 simply in phase opposition. This is in accord with the statement 

 in prob. 5. 



From the two preceding cases it follows that, when in a syn- 

 chronous machine with non-salient poles the currents lag by an 

 angle (p electrical degrees (Figs. 37 and 38) with respect to the 

 induced voltage at no-load, the armature m.m.f. wave lags by an 

 angle of 90 + ^ electrical degrees behind the field m.m.f. wave. In 

 the case of a generator with leading currents the angle <p is negative ; 

 in a synchronous motor <p is larger than 90 degrees. 



Let, in Fig. 37, i be the vector of the current in one of the phases, 

 and let e be the corresponding terminal voltage, the phase angle 

 between the two being <p. Adding to e in the usual way the ohmic 

 drop ir in the armature, in phase with i, and the reactive drop ix 

 in leading quadrature with i, the induced voltage E in the same 

 phase is obtained. 1 The resultant useful flux, <P, which induces 

 this e.m.f. leads E by 90 degrees in time; is in phase with the 

 net or resultant m.m.f. M n which produces it. The m.m.f. M n is a 

 sum of the field m.m.f. M f and of the armature reaction M a 



1 On account of skin effect and eddy currents in the armature conductors, 

 the effective resistance r to alternating currents is considerably higher than that 

 calculated or measured with direct current. The actual amount of increase 

 depends upon the character of the winding, the size of the conductors, the 

 shape of the slots, the frequency, etc., so that no definite rule can be given. 

 Fortunately, the ohmic drop constitutes but a small percentage of the voltage 

 of a machine, so that a considerable error committed in estimating the value 

 of the ir drop affects the voltage relations but very little. See A. B. Field, 

 "Eddy Currents in Large Slot-wound Conductors," Trans. Amer. Inst. 

 Elect. Engrs., Vol. 24 (1905), p. 761. 



