Subject : Power Electronics
Unit : DC to AC Converters
Trade Off Between Low Order And High Order Harmonics
Trade Off between lower order and high order harmonics:
- The root mean square (rms) of the pole voltage equals 0.5 E_{dc}. Now a periodic function ‘()ftω’ when expressed in terms of its Fourier components satisfies the following mathematical identity.
Where, f(ωt)_{rms} is the rms magnitude of the given periodic waveform whereas f(ωt)_{1},_{rms} and f(ωt)_{n,rms} are the rms magnitudes of the fundamental component and n^{th }harmonic component of the waveform respectively.
- Also, if the waveform ‘f(ωt)’ has half wave odd symmetry and quarter wave mirror symmetry, its fundamental voltage can be expressed as
- Now let f(ωt) be replaced by the two-level pole voltage waveform of the PWM inverter. The term on the left hand side of equation defining f(ωt)_{rms} equals (0.5E_{dc})^{2}. The first term on the right hand side of the equation is the square-of-rms (i.e., mean of square) magnitude of the fundamental component of pole voltage whereas the second term on the right hand side denotes the mean-of-square magnitude of the unwanted ripple in the pole voltage.
- The rms magnitude of the fundamental pole voltage is always going to be less than 0.5E_{dc}. Further, as given by the equation of f(ωt)_{1},_{rms} , the fundamental magnitude (rms) of PWM inverter’s output pole-voltage will be less than 0.45E_{dc}, which is the rms magnitude of fundamental pole voltage of a 3-phase square wave inverter. In case of square wave output, both f(ωt) and sin(ωt) are positive during 0 ≤ wt ≤π but the sign of f(ωt) in PMW waveforms alternates between positive and negative values.
- In case of PWM inverter the magnitude of fundamental output voltage is fixed by suitable pulse width modulation (by selection of suitable notch angles for the waveform. However, the reduction in fundamental magnitude leads to increase in the rms magnitude of the unwanted ripple voltage.
- Also, after fixing the fundamental voltage magnitude if it is desired to eliminate some of the low order harmonics, it will be at the cost of increasing the magnitudes of higher order harmonics. Thus, as far as the quality of inverter pole voltage alone is concerned the PWM technique is not helping.
- However considering the fact that most of the loads are inductive in nature with low pass filter type characteristics the load current quality effectively improves by eliminating lower order harmonics from the pole voltage waveform (even if the higher order harmonic magnitudes increase).
- In case the load, on its own, is not able to filter out the harmonic voltages satisfactorily the inverter output may be passed through some external filter before being applied to load. The required size of the external filter will be small if the inverter output is free from low frequency harmonics.