THEORY OF SINGLE-PHASE MOTOR 229 



induction at this instant is B sin pt. At a distance z from the origin, 

 tin- induction has the value B sin pt . sin - :, so that if / = length of 

 coil (in cms.) measured parallel to the shaft, the flux through a 

 1 1; i now strip of width dz is B? sin pt . sin ^-z.dz, and hence the total 

 tlux through the coil in the position shown is 



X+ r 



f = I B/ sin pt . sin --. . <l: 



X 



X + T 



= BZ sin pt \ sin -2 . dz 

 J T 



2rl . * 



= B sin pt . cos -x 



Assuming the rotor to be at rest, and hence x constant, we may 

 write for the e.m.f. induced in the coil shown in Fi. 144 



df Spr^T) 7T 7T 



e = ) = r B cos -x . cos pt = E cos -x . cos 

 at TT r r 



where E = -* R Hence if r, X stand for the resistance and leakage 



Tf 



self-inductance respectively of the coil, the current is 



Hi TT / * n\ j. 



I = /To o r COS "# COS 0^ ~ ") T 



where tan = 



r 



This shows (a result which is also otherwise obvious from an 

 inspection of Fig. 144) that, so long as the rotor is at rest (i.e. so long 

 as x is not a function of the time), there are different currents circu- 

 lating in different portions of the rotor winding ; the greatest current 

 circulating in the coil which faces a pole-piece, and for which x = 0, 



and cos - x = 1, and zero current in the coil whose sides are in line 



T 



with the centre lines of the pole-pieces (x = AT, cos -x = 0). 



* The negative sign being taken in accordance with Lenz's Law ( 4). It is to be 

 particularly noted that/ here stands for the resultant actually existing oscillating flux 

 common to the stator and rotor, and due to the combined action of their windings ; it 

 is not the hypothetical flux due to the stator winding alone. 



t The current being expressed, for the sake of simplicity, in C.G.S. unite. 



