Wednesday, February 22, 2012

Effect of Harmonics Components on an Induced E.M.F.

      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 = Bm1 sinθ + Bm3 sin 3θ + ... + Bmx sin xθ + ...
       where   x = Order of harmonic component which is odd
       Bm1 = Amplitude of fundamental component of flux density
       Bm3 = Amplitude of 3rd harmonic component of flux density
       Bmx = Amplitude of xth (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 = B l  v
       Substituting value of B,
       ec   =  (Bm1 sinθ + Bm3   sin3θ+ .... + Bmx sinxθ + ...... ) l  v
        l   = Active length of conductor in meter
       d = Diameter of the armature at the air gap
       v = Linear velocity = π d ns  
       where ns   = Synchronous speed in r.p.s.
Now   Ns    = 120f/P
...         ns   = 120f/60P = 2f/P
...         v = π d (2f/P)
       Substituting in the expression for ec,

       Area of each fundamental pole, A1   = (π d l )/P
...     ec   = (Bm1 A1   2f sinθ + Bm3  A2f   sin3θ+ .... + Bmx A2f sinxθ )
       Area of xth harmonic pole, Ax   = (π d l )/(xP) = A1/x
       This is because, there are xP poles for the xth  order harmonic
...     ec   = 2f (Bm1 A1 sinθ + Bm3  3A sin3θ+ .... + Bmx Asinxθ )
Now Bm1 A1 = Φ1m = maximum value of fundamental flux per pole
...      Φ1 = (2/π) Φ1m = Average value of fundamental flux per pole 
       Similarly average value of xth harmonic flux per pole can be obtained as,
       Φ= (2/π) Ax Bmx
       Substituting the values of flux in ewe get the expression for e.m.f. induced per conductor as,
       ec   = π f (Φ1 sinθ +   sin3θ+ .... + x Φsinxθ )
       Instantaneous value of fundamental frequency e.m.f. generated in a conductor is,
       ec1   = π f Φ1 sinθ     V
       Hence the R.M.S. value of fundamental frequency e.m.f. generated in a conductor is,
       Ec1   = (π f Φ1 )/√2  = 2.22 fΦ1
       Hence R.M.S. value of xth harmonic frequency e.m.f. generated in a conductor is,
       Ecx  = 2.22Φx . xf
But  Φx = (2/π )Ax Bmx = (2/π) . (A1 /x) Bmx
...     Ecx  = 2.22  . (2/π) (A1/x) . x f Bmx
              = 1.4132 A1 f Bmx
Now  Ec1 = 2.22 f Φ1 = 2.22 f (2/π) Bm1 A1 = 1.4132 f Bm1 A1
...        Ecx/Ec1 = (1.4132 A1 f Bmx)/(1.4132 A1 f Bm1)= Bmx/Bm1
Ecx =Ec1 . (Bmx/Bm1)
       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 3rd harmonic component 
       5 α = For 5th harmonic component
       x α = For xth 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,
       E1ph = 4.44 Kc1 Kd1 Φ1 f Tph    V
       where Tph  = Turns per phase in series
        Φ1 = Fundamental flux component
       While the phase e.m.f. of order harmonic component of frequency is given by,
       Exph = 4.44 Kcx Kdx    x Φx f Tph      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 3rd harmonic voltages do not appear across line terminal though present in phase voltage.

Note : In data connection also, 3rd, 9th, 15th   ... harmonic voltages do not appear at the line terminals.
Taking ratio of fundamental frequency e.m.f. and xth order harmonic frequency e.m.f. we can write,

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