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
.^{.}. 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)
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 phaseT_{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.
Related Articles :- Pitch Factor or Coil Span Factor (Kc)
- Distribution Factor (Kd)
- Generalized Expression for E.M.F. Equation of an
- Line Value of Induced E.M.F
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Your website is very helpful for electrical engineers.Here i found easy derivation of alternator emf equation
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