E.M.F. Equation of an Alternator

E.M.F. Equation of an Alternator : Part1
Let              Φ   = Flux per pole, in Wb

                    P = Number of poles

                    N_s = Synchronous speed in r.p.m.

                    f = Frequency of induced e.m.f. in Hz

                    Z = Total number of conductors

                    Z_ph = Conductors per phase connected in series

.^..                 Z_ph  = Z/3 as number of phases = 3.

       Consider a single conductor placed in a slot.

       The average value of e.m.f. induced in a conductor

                          = dΦ/dt

       For one revolution of a conductor,

       e_avg  per conductor = (Flux cut in one revolution)/(time taken for one revolution)

       Total flux cut in one revolution is Φ x P

       Time taken for one revolution is 60/N_s seconds.

.^..     e_avg  per conductor = ΦP / (60/N_s)

                                       = Φ (PN_s/60)                ............. (1)

       But                        f = PN_s/6120
.^..                                  PN_s/60= 2f

       Substation in (1),

        e_avg    per conductor = 2 f Φ volts

       Assume full pitch winding for simplicity i.e. this conductor is connected to a conductor which is  180^o electrical apart. So there two e.m.f.s will try to set up a current in the same direction i.e. the two e.m.f. are helping each other and hence resultant e.m.f. per turn will be twice the e.m.f. induced in a conductor.

.^..      e.m.f. per turn = 2 x (e.m.f. per conductor)

 = 2 x (2 f Φ)

= 4 f Φ volts

       Let T_ph   be the total number of turn per phase connected in series. Assuming concentrated winding, we can say that all are placed in single slot per pole per phase. So induced e.m.f.s in all turns will be in phase as placed in single slot. Hence net e.m.f. per phase will be algebraic sum of the e.m.f.s per turn.

.^..         Average E_ph   = T_ph   x (Average e.m.f. per turn)
.^..         Average Eph   = T_ph   x 4 f Φ

       But in a.c. circuits R.M.S. value of an alternating quantity is used for the analysis. The form factor is 1.11 of sinusoidal e.m.f.

       K_f    = (R.M.S.)/Average = 1.11                ......... for sinusoidal
.^..    R.M.S. value of E_ph    = K x Average value
       E = 4.44 x f Φ T_ph   volts                                                     ........... (2)

Note : This is the basic e.m.f. equation for an induced e.m.f. per phase for full pitch, concentrated type of winding.

       Where T_ph   = Number of turns per phase
        T_ph   = Z_ph  /2                                              ....... as 2 conductors constitute 1 turn

       But as mentioned earlier, the winding used for the alternators is distributed and short pitch hence e.m.f. induced slightly gets affected. Let us see now the effect of distributed and short pitch type of winding on the e.m.f. equation.