The flux density distribution around the air gap in all well designed alternators is symmetrical with respect to abscissa and also to polar axis. Thus it can be expressed with the help of a Fourier series which do not contain any even harmonics. So flux density at any angle from the interpolar axis is given by, B = B m1 sinθ + B m3 sin 3θ + … + B mx sin xθ + … where x = Order of harmonic component which is odd B m1 = Amplitude of fundamental component of flux density B m3 = Amplitude of 3 rd harmonic component of flux density B mx = Amplitude of x th (odd) harmonic component of flux density The e.m.f. generated in a conductor on the armature of a rotating machine is given by, e c = B l v Substituting value of B, e c = (B m1 sinθ + B m3 sin3θ+ …. + B mx sinxθ + …… ) l v l = Active length of conductor in meter d = Diameter of the armature at the air gap v = Linear velocity = π d n s where n s = Synchronous speed in r.p.s. Now N s = 120f/P . . . n s = 120f/60P = 2f/P . . . v = π d (2f/P) Substituting in the expression for e c , Area of each fundamental pole, A 1 = ( π d l )/P . . . e c = (B m1 A 1 2f sinθ + B m3 A 1 2f sin3θ+ …. + B mx A 1 2f sinxθ ) Area of x th harmonic pole, A x = ( π d l )/(xP) = A 1 /x This is because, there are xP poles for the x th order harmonic . . . e c = 2f (B m1 A 1 sinθ + B m3 3A 3 sin3θ+ …. + B mx A x sinxθ ) Now B m1 A 1 = Φ 1m = maximum value of fundamental flux per pole . . . Φ 1 = (2/ π ) Φ 1m = Average value of fundamental flux per pole Similarly average value of x th harmonic flux per pole can be obtained as, Φ x = (2/ π ) A x B mx Substituting the values of flux in e c we get the expression for e.m.f. induced per conductor as, e c = π f ( Φ 1 sinθ + 3Φ 3 sin3θ+ …. + x Φ x sinxθ ) Instantaneous value of fundamental frequency e.m.f. generated in a conductor is, e c1 = π f Φ 1 sinθ V Hence the R.M.S. value of fundamental frequency e.m.f. generated in a conductor is, E c1 = ( π f Φ 1 )/√2 = 2.22 fΦ 1 Hence R.M.S. value of x th harmonic frequency e.m.f. generated in a conductor is, E cx = 2.22Φ x . xf But Φ x = (2/ π )A x B mx = (2/ π ) . (A 1 /x) B mx . . . E cx = 2.22 . (2/ π ) (A 1 /x) . x f B mx = 1.4132 A 1 f B mx Now E c1 = 2.22 f Φ 1 = 2.22 f (2/ π ) B m1 A 1 = 1.4132 f B m1 A 1 . . . E cx / E c1 = (1.4132 A 1 f B mx )/(1.4132 A 1 f B m1 )= B mx /B m1 E cx = E c1 . (B mx /B m1 ) It can be observed that the magnitude of harmonic e.m.f.s are directly proportional to their corresponding flux densities. The R.M.S. value of resultant e.m.f. of a conductor is, 1.1 Effect of Harmonic Components on Pitch Factor We know that, α = Angle of short pitch for fundamental flux wave The it changes for various harmonic component of flux as, 3 α = For 3 rd harmonic component 5 α = For 5 th harmonic component . . . x α = For x th harmonic component Hence the pitch factor is expressed as, where x = Order of harmonic component 1.2 Effect of Harmonic Components on Distribution Factor Similar to the pitch factor, the distribution factor is also different for various harmonic components. The general expression to obtain distribution factor is, where x = order of harmonic component 1.3 Total E.M.F. Generated due to Harmonic Components Considering the windings to short pitch and distributed, the e.m.f. of a fundamental frequency is given by, E 1ph = 4.44 K c1 K d1 Φ 1 f T ph V where T ph = Turns per phase in series Φ 1 = Fundamental flux component While the phase e.m.f. of order harmonic component of frequency is given by, E xph = 4.44 K cx K dx x Φ x f T ph V The total phase e.m.f. is given by, Line e.m.f . : For star connected, the line or terminal induced e.m.f. is √3 times the total phase e.m.f. but it should be noted that with star connection, the 3 rd harmonic voltages do not appear across line terminal though present in phase voltage. Note : In data connection also, 3 rd , 9 th , 15 th … harmonic voltages do not appear at the line terminals. Taking ratio of fundamental frequency e.m.f. and x th order harmonic frequency e.m.f. we can write,
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