A Unified Analytical Solution for Bipolar, Unipolar, and Multistep SHE Converters

In this paper, a unified analytical approach is presented for solving a set of transcendental equations commonly encountered in Selective Harmonic Elimination (SHE). Traditionally numerical solutions are used that require an initial value and a valid value of modulation index m. Not only the initial value but also an invalid m will affect convergence. Such difficulties are not associated with analytical solution. The solution set will determine the range of m valid for that modulation. The analytical approach is also suitable for real-time applications where a microcontroller/microprocessor can be programmed for the application making it attractive for low cost application. In the proposed approach, the transcendental equations are converted to power-sum non-linear polynomials using Chebyshev expansion. These non-linear polynomials have been simplified using a successive polynomial extraction method and lead to a solution set for elementary functions. These have been formulated for the three types of modulation for a 4th order system with contiguous harmonic elimination that eliminates up to ninth harmonic. The solution trajectories as a function of m have been presented and valid range of m values has been determined. Simulation results verify the harmonic cancellations.

[1]  Toshiji Kato,et al.  Sequential homotopy-based computation of multiple solutions for selected harmonic elimination in PWM inverters , 1999 .

[2]  H. Gould,et al.  THE GIRARD-WARING POWER SUM FORMULAS FOR SYMMETRIC FUNCTIONS AND FIBONACCI SEQUENCES , 1999 .

[3]  Sraddhanjoli Bhadra,et al.  A closed-form approach for contiguous and non-contiguous harmonic elimination with application to a three-level switching waveform , 2017, 2017 IEEE 60th International Midwest Symposium on Circuits and Systems (MWSCAS).

[4]  Zhong Du,et al.  A complete solution to the harmonic elimination problem , 2003, IEEE Transactions on Power Electronics.

[5]  Sraddhanjoli Bhadra,et al.  An analytical solution of switching angles for Selective Harmonic Elimination (SHE) in a cascaded seven level inverter , 2016, 2016 IEEE 2nd Annual Southern Power Electronics Conference (SPEC).

[6]  L.M. Tolbert,et al.  Novel multilevel inverter carrier-based PWM methods , 1998, Conference Record of 1998 IEEE Industry Applications Conference. Thirty-Third IAS Annual Meeting (Cat. No.98CH36242).

[7]  V. Agelidis,et al.  Multiple sets of solutions for harmonic elimination PWM bipolar waveforms: analysis and experimental verification , 2006, IEEE Transactions on Power Electronics.

[8]  J. Sun,et al.  Pulsewidth modulation based on real-time solution of algebraic harmonic elimination equations , 1994, Proceedings of IECON'94 - 20th Annual Conference of IEEE Industrial Electronics.

[9]  V. Agelidis,et al.  On applying a minimization technique to the harmonic elimination PWM control: the bipolar waveform , 2004, IEEE Power Electronics Letters.

[10]  Sraddhanjoli Bhadra,et al.  Design equations for selective harmonic elimination and microcontroller implementation , 2014, IECON 2014 - 40th Annual Conference of the IEEE Industrial Electronics Society.

[11]  Zhong Du,et al.  A unified approach to solving the harmonic elimination equations in multilevel converters , 2004, IEEE Transactions on Power Electronics.

[12]  D. V. Chudnovsky,et al.  Solving the optimal PWM problem for single-phase inverters , 2002 .

[13]  Zhong Du,et al.  Elimination of harmonics in a multilevel converter using the theory of symmetric polynomials and resultants , 2005, IEEE Transactions on Control Systems Technology.

[14]  R. Hoft,et al.  Generalized Techniques of Harmonic Elimination and Voltage Control in Thyristor Inverters: Part II --- Voltage Control Techniques , 1974 .

[15]  W. Burnside,et al.  Theory of equations , 1886 .

[16]  L.M. Tolbert,et al.  Fundamental Frequency Switching Strategies of a Seven-Level Hybrid Cascaded H-Bridge Multilevel Inverter , 2009, IEEE Transactions on Power Electronics.