Harmonic Analysis Of Pole Voltage Waveform
Harmonic Analysis of Pole Voltage Waveform:
Fig: A typical pole-voltage waveform of a PWM inverter
- The pole voltage waveform shown has half wave odd symmetry and quarter-wave mirror symmetry.
- The half wave odd symmetry of any repetitive waveform f(ωt), repeating after every 2π/ω duration, is defined by f(ωt) = - f(π ωt). Such a symmetry in the waveform amounts to absence of dc and even harmonic components from the waveform.
- All inverter output voltages maintain half wave odd symmetry to eliminate the unwanted dc voltage and the even harmonics.
- The half wave odd symmetry followed by quarter wave mirror symmetry, defined by f(ωt) = f(π-ωt), results in presence of only sine components in the Fourier series representation of the waveform. It may be verified that quarter wave symmetry may not hold good once the time origin is shifted arbitrarily. However the half-wave odd symmetry is maintained in spite of shifting of time origin.
- By shifting the time origin, new (even) harmonic frequencies will not creep up in the voltage waveform, whereas by shifting time origin the sine wave may become cosine or may have some other phase-shift. The quarter wave symmetry talked above is not necessary for improvement of the output waveform quality; it merely simplifies the Fourier analysis of the pole voltage waveform.
- The quarter wave symmetry is not achieved at the cost of compromising the inverter’s output capability (in terms of magnitude and quality of achievable output voltage).
- With the assumed quarter wave mirror symmetry and half wave odd symmetry the waveform shown in Fig. may be decomposed in terms of its Fourier components as below:-
where, VAO is the instantaneous magnitude of the pole voltage and bn is the peak magnitude of its nth harmonic component.
- Because of the half wave and quarter wave symmetry of the waveform, mentioned before, the pole voltage has only odd harmonics and has only sinusoidal components in the Fourier expansion. Thus the pole voltage will have fundamental, third, fifth, seventh, ninth, eleventh and other odd harmonics. The peak magnitude of nth harmonic voltage is given as:
where α1 , α2 ¸α3 and α4 are the four notch angles in the quarter cycle of the waveform.