Вербицький Є. Spectral analysis of processes and synthesis of semiconductor converters control laws on the basis of a Fourier series of several variables

The dissertation for a Doctor of Science degree of electrical engineering of speciality 05.09.12 – semiconductor converters of the electric power – National Technical University of Ukraine " Igor Sikorsky Kyiv Polytechnic Institute", Kyiv, 2021. In the dissertation the theory of spectral analysis of converters currents and voltages in the basis of Fourier series of several variables and calculation of the electric power integral quality parameters in the closed analytical form is further developed. The received results together are the theoretical generalization and the new solution of important scientific and applied problem of increasing the efficiency of analysis and control of semiconductor converters based on the theory of spectral analysis and synthesis of control laws based on the Fourier series of several variables. The Fourier series is generalized to M variables and the its properties are analyzed. The conditions for the appearance of interharmonics in the spectrum are determined and formulas for their calculation are adapted. To eliminate the main limitations in the use of generalized Fourier series, the following methods and techniques have been developed: the method of calculating the spectrum of modulated signals with variable frequency carrier function; method of calculating currents and voltages at the output of the filter based on the joint use of Fourier series of one and several variables; methods of calculating integrated quality parameters of modulated currents and voltages based on the generalized Fourier series, which significantly expanded the capabilities of multidimensional Fourier series for it implementation in control systems and synthesis of power converters based on quality assessment of electricity parameters. Principles of effective inverter control and their implementation for modifications of existing modulation laws are proposed. Regulation of the inverter voltage modulation multiplicity to ensure the required value of the total harmonic distortion parameter, which reduces the dynamic losses in the converter and, accordingly, the dimensions of the converter and radiators weight. As a result, the principle of control of a single-phase inverter by the value of the total harmonic distortion is developed, which allows to reduce the amount of mathematical operations by at least two orders with a calculation error not more than 1%, and reduces the amount of dynamic losses by reducing the value of the modulation multiplicity parameter at maximum load. Also, it is proposed to use the method of PWM modulation with inversion of the polarity of every second pulse, described in the basis of the Fourier series of two variables, which reduces the mass and size parameters of the converter while providing the required value of total harmonic distortion value of the output voltage. It was found that in contrast to the AC-DC converter, in the electric drive of an asynchronous motor the interharmonics significantly distort the phase currents. To solve this problem, a method of eliminating interharmonic of voltage at the output of a three-phase inverter has been developed and it is proposed to control the quality parameters of the input current of the electric drive consumed from the power grid by the value of the motor phase current. Also, the generalized Fourier series is used in the design of modular power factor correctors to select the number of cells based on the required values of power quality parameters in a given range of parameter. The use of a generalized Fourier series made it possible to obtain analytical expressions of the dependence of electricity quality parameters on the modes of operation of the converter and the number of cells, which eliminates multiple simulations of modular converter model in the entire range of input parameters. The complexity of the spectrum calculation by existing methods is compared and it is shown that the complexity of the spectrum calculation based on the generalized Fourier series goes to the theoretical limit O(N) and has the same speed as the wavelet transform, while when calculating individual harmonics and integral quality parameters, the proposed method has the least complexity О(N^(1/M)), wherе М is variables number of Fourier series, and O(1), respectively.