Effect of Harmonic Components on EMF of Synchronous Generator

Effect of Harmonic Components on EMF of Synchronous Generator

Before knowing Effect of Harmonic Components on EMF let us know what are harmonics and harmonic components in alternators.In synchronous machines, voltages and currents are induced, which are in sinusoidal waveforms.But in practice, these sinusoidal waveforms will deviate to form non-sinusoidal waveforms.

These non-sinusoidal waveforms are expressed in terms of Fourier transforms which are the sum of series of sinusoidal waveforms.Now let us understand the effect of harmonic components on the synchronous generator.

The flux density distribution around the air gap in all well-designed alternators symmetrical with respect to the abscissa and also to polar axes.Thus it can be expressed with the help of a Fourier series which do not contain any even harmonics.

So flux density at an angle θ from the interpolar axis is given by,

B = Bm1 sin θ + Bm3 sin 3θ +……….. + Bmx sin xθ

where     x = Order of the 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 EMF generated in a conductor on the armature of a rotating machine is given by
ec = B.l.v

Substituting value of B,

 

ec = [Bm1 sin θ + Bm3 sin 3θ +……….. + Bmx sin xθ].l.v

l = Active length of conductor in metre
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 ,

ec = [Bm1 sin θ + Bm3 sin 3θ +……….. + Bmx sin xθ].l.(2πdf/P)

Effect-Harmonic-Components-Synchronous-Generator-Alternator-induced-emf

Area of each fundamental pole, 

A1 = πdl/P

ec = [Bm1 A1 2f sin θ + Bm3 A1 2f sin 3θ +……….. + Bmx A1 2f sin xθ]

Area of xth harmonic pole, Ax =  πdl/xP =  A1/x

This is because there are xP poles for the xth order harmonic.

ec = 2f [Bm1 A1  sin θ + Bm3 3A3 sin 3θ +……….. + Bmx xAx  sin xθ]

Bm1 A1 = Φ1m = Maximum value of fundamental flux per pole

            Φ1= (2/π)Φ1m = Average value of fundamental flux per pole

Similarly, the average value of xth harmonic flux per pole can be obtained as,

Φx = (2/π)Ax Bmx

  • Effect of Harmonic Components on EMF of Synchronous Generator

Substituting the values of flux in ec we get the expression for e.m.f. induced per conductor as

             ec = πf (Φ1  sin θ + 3Φ3 sin 3θ +……….. + xΦx  sin xθ)

 

Instantaneous value of fundamental frequency EMF generated in a conductor is,
ec1 = πfΦ1 sin θ    V

Hence the R.M.S. value of fundamental frequency EMF generated in a conductor is,

                       ec1 = πfΦ1/√2 = 2.22 fΦ1
Hence RMS value of xth harmonic frequency emf generated in a conductor is,

Effect-Harmonic-Components-Synchronous-Generator-Alternator-induced-emf

It can be observed that the magnitude of harmonic e.m.f.s is directly proportional to their corresponding flux densities.

    The RMS value of resultant EMF of a conductor is,

 

Effect-Harmonic-Components-Synchronous-Generator-Alternator-induced-emf

Effect of Harmonic Components on Pitch Factor in Synchronous Generator:

We know that,

 

α = Angle of the short pitch for fundamental flux wave then it changes for the various harmonic component of flux as, 

3α = For 3rd harmonic component
5α = For 5th harmonic component

                     xα = For xth harmonic component

Hence the pitch factor is expressed as,

Effect-Harmonic-Components-Synchronous-Generator-Alternator-induced-emf

              x = Order of the harmonic component

Effect of Harmonic Components on Distribution Factor in Alternators:

Similar to the pitch factor, the distribution factor is also different for various halt components.

The general expression to obtain distribution factor is,

Effect-Harmonic-Components-Synchronous-Generator-Alternator-induced-emf

Total EMF Generated due to Harmonics in synchronous Generator:

        Consider 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 series
Φ1  = Fundamental flux component

While the phase e.m.f. of Xth order harmonic component of frequency is given by,

                         Exph = 4.44 Kcx Kdx Φx (xf) Tph   V

The total phase EMF is given by,

Effect-Harmonic-Components-Synchronous-Generator-Alternator-induced-emf

Line EMF: For star connected, the line or terminal induced EMF is √3 times the total phase EMF.But it should be noted that with star connection, the 3rd harmonic voltages do not appear across line terminals though present in phase voltage.

 

Effect-Harmonic-Components-Synchronous-Generator-Alternator-induced-emf

Key Point: In delta connection also, 3rd,9th,15th….. harmonic voltages do not appear at the line terminals.

Taking the ratio of fundamental frequency EMF and Xth order harmonic frequency EMF we can write,

 

From the ratio in above equation, we can write

Thus if the third harmonic is given to be 10% of fundamental

 

 Φ3 = 1/3*(10% of Φ1) = 0.0333Φ1

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