Spectral Analysis of a Class of DC–AC PWM Inverters by Kapteyn Series

The pulsewidth modulation (PWM) technique for single-phase bipolar switching inverters has been revisited and analyzed on the basis of the isomorphism between the implicit equation of the unknown PWM switching instants, when the input to the modulator is a sinusoidal signal of period multiple of the carrier period, and the Kepler's equation governing the trajectory of a planet around the sun. The solution of the Kepler's problem allows to obtain the switching instants of the PWM in terms of Kapteyn series or integral forms and to find an explicit expression of the harmonic content. An expression of the weighted total harmonic distortion is also obtained in terms of Bernoulli polynomials.