Harmonics

 In case of alternators, the voltage and current induced are having sinusoidal waveforms. But practically we can not get sinusoidal waveform when such alternators are loaded. Due to the loading condition, the generated waveform deviates from ideal waveform. Such a non-sinusoidal waveform is called complex wave. By fourier transform this complex waveform can be shown to be built of a series of sinusoidal waves whose frequencies are integral multiples of the frequency of fundamental wave. These sinusoidal components or harmonic functions are called harmonics of the complex wave.

       The fundamental wave is defined as that component which is having same frequency as that of complex wave. The component which is having double the frequency of that of fundamental wave called second harmonic. While the component which is having the frequency three times that of fundamental is called third harmonic and so on. The complex waveform contains both the even as well as odd harmonics. Consider a complex wave which is represented by,

             e = E_1m sin(ω t + Φ₁) + E_2m sin(2ωt +Φ₂) + E_3m sin(3ωt + Φ₃) + ... + E_nm sin (nω t + Φ_n)

       where E_1m sin(ω t + Φ₁) is fundamental component of maximum value E_1m having an angle Φ₁ from instant of zero of the complex wave. Similarly E_nm sin (nω t + Φ_n) represents n^th harmonic of maximum value E_nm and having phase angle Φ_n with respect to complex wave.

       Out of the even and odd harmonics a complex wave containing fundamental component and even harmonics only is always unsymmetrical about x-axis whereas a complex wave containing fundamental component and odd harmonics only is always symmetrical about x-axis. In case of alternators the voltage generated is mostly symmetrical as the filed system and coils are all symmetrical.

       So the generated voltage or current will not have any even harmonics in most of the cases.

       The complex waveform of voltage can be analysed experimentally by using the phenomenon of resonance. If voltage waveform containing harmonic content is applied to the circuit containing resistance, inductance and capacitance, then the circuit will resonate at one of the harmonic frequencies. The voltage drop across the resistance can be analysed by using an oscillograph. The values of inductance and capacitance can be changed so that resonance can be obtained at fundamental, third harmonic, fifth harmonic etc. The voltage on the oscillograph indicates the presence of particular harmonics.

1.1 Slot Harmonics

       The voltage generated in armature winding is derived assuming that the surface of armature to be smooth. However in practice armature is not smooth but is made slotted. Due to this slotting certain harmonic e.m.f.s. of undesirable order are produced.

       The reluctance at any point in the air gap depends on whether there is a slot or teeth in the magnetic path. Since in case of alternators armature is moving, the teeth and slots alternately occupy positions at this point. This will vary the reluctance. The ripples will be formed due to variation of reluctance from point to point in the air gap which is shown in the Fig. 1. These ripples will not move with respect to conductors but glide on the distribution of flux. The ripples due to slotting of armature are always opposite to slots and teeth which are causing them. Thus the harmonics which are generated in the e.m.f. due to slotting is called slot harmonics.

       It can be seen that the main source of harmonics is the non-sinusoidal field from which can be made sinusoidal and the harmonics can be eliminated.

Fig. 1

       The air gap offers maximum reluctance to the path. This air gap if made to vary sinusoidally around the machine, the filed from would also be sinusoidal. Even the air gap is made to vary sinusoidally, the field from can not be sinusoidal due to saturation in iron parts which is unavoidable. But there should no be high degree of saturation so that approximately sinusoidal waveform will be obtained.

       Thus in general it can be seen that ideal sinusoidal field from is very difficult to obtain whether the machine is salient pole type or cylindrical rotor construction rotor construction.

1.2 Harmonics Minimization

       To eliminate or minimize the harmonics from the voltage waveform, the winding must be properly designed. The different ways to eliminate the harmonics from generated voltage are,

1) Distribution of armature windings : Instead of having concentrated type of windings, they should be distributed in different slots. The distribution factor for harmonics is comparatively less than that of the fundamental and hence magnitude of harmonic e.m.f. is small.

2) Chording : The e.m.f. generated in the winding is proportional to cos (x /2) where is angle of chording and x is order of harmonic. If proper value of angle of chording is selected then harmonic e.m.f.s can be reduced significantly.

3) Fractional slot windings : The output voltage waveform will be free of harmonics by facilitating the use of fractional slot windings as the distribution factor will be smaller compared to that with the fundametal.

4) Skewing : Skewing the pole face will help in eliminating the slot harmonics.

5) Large length of air gap : The reluctance will be increased by increasing the air gap and slot harmonics can be reduced.