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140 J. Iwaszkiewicz 147. Tolbert L. M., Chiasson J., McKenzie K., Zhong Du: Elimination of harmonics in a multilevel converter with nonequal DC sources, Applied Power Electronics Conference and Exposition, 2003. APEC '03. Eighteenth Annual IEEE, vol. 9–13 Feb., 2003, pp. 589 – 595. 148. Tolbert L.M., Peng F.Z., Habetler T.G.: Multilevel PWM Methods at Low Modulation Indices, IEEE Transactions on Power Electronics, vol. 15, No. 4, July 2000, pp. 719-725. 149. Ueda F., Asao M., Tsuboi K.: Parallel–Connections of Pulsewidth Modulated Inverters Using Current Sharing Reactors, IEEE Transactions on Power Electronics, vol. 10, no 6, pp. 673–679, November 1995. 150. Venkataramanan G., Bendre A.: Reciprocity–Transposition–Based Sinusoidal Pulsewidth Modulation for Diode–Clamped Multilevel Converters, IEEE Transactions on Industrial Electronics, vol. 49, no. 5, October 2002, pp. 1035–1047. 151. Wajs K.: Funkcje Walsha i ich zastosowanie w elektrotechnice. Przegląd Elektrotechniczny, 1976, nr 11, ss. 413–418. 152. Walsh J. L.: A closed set of orthonormal functions. Amer. Journal of Mathematics, 1923, no. 1 pp. 5–24. 153. Wang F.: Sine–Triangle Versus Space–Vector Modulation for Three–Level PWM Voltage– Source Inverters, IEEE Transactions on Industry Applications, vol. 38, no. 2, pp. 500–506, March/April 2002. 154. Yamanaka K., Yamada K., Kumagae A., Terada T.: Three–Level Neutral Point Clamping Type Inverter Circuit, U.S. patent number 06,226,192, assigned to Kabushiki Kaisha Yasukawa Denki, May 2001. 155. Yin Y., Wu A.Y.: A Low-Harmonic Electric Drive System Based on Current-Source Inverter, IEEE Transactions on Industry Applications, vol. 34, No. 1, January/February 1998, pp. 227-235. 156. Yuan X., Barbi I.: Fundamentals of a New Diode Clamping Multilevel Inverter, IEEE Transactions on Power Electronics, vol. 15, no. 4, pp. 711–718, July 2000. 157. Zare F., Ledwich G.: Hysteresis Current Control for Single–Phase Multilevel Voltage Source Inverters: PLD Implementation, IEEE Transactions on Power Electronics, vol. 17, no 5, pp. 731– 738, Sept. 2002. Rękopis dostarczono, dnia 08.09.2005 r. Opiniowali: prof. dr hab. inż. Roman Barlik, prof. dr hab. inż. Krystyn Pawluk MATHEMATICAL MODELS OF POWER ELECTRONICS MULTILEVEL CONVERTERS ANALYSIS AND APPLICATIONS Jan IWASZKIEWICZ ABSTRACT The monograph is dedicated to mathematical models which can be useful in analysis and designing of structures and control strategies of multilevel converters. The selected models are based on synthesis of the step waveforms which comply to the best approximation criterion. The following models have been discussed:
Modele <strong>matematyczne</strong> <strong>energoelektronicznych</strong> przekształtników wielopoziomowych. Analiza ... 141 The analytical model of the two-level three-phase converter for discrete waveforms, based on a set of expressions describing in time domain voltage and current waveforms in the converter-load circuitry. The model is useful in analysis of stationary and transitory states of digitally controlled voltage and current source converters and also for research in the field of real-time control algorithms. The idea of this model has been used for elaborating the models of multilevel converters. The model of three-level converter with the discussion concerning output voltage waveforms formed by use the transformation of polar voltages to complex stationary coordinates system (α , β ). The expressions describing the three-level and five-level converter models in time domain have been formulated. The further development of the presented ideas enabled the construction of universal n-level converter model. The proposed unified denoting system for output voltage space vectors of multilevel converters makes possible the fast analysis of space vectors positions on complex plane and finding the number of multiplied vectors. The Fourier-style model is based on the approximation of the function f (x) = sin(x) using series of the square-wave pulses described by series of g n (x) functions. The parameters of this series have been calculated with Fourier factors of the orthogonal series composed from the g n (x) functions. The harmonic spectra of the converter’s output voltages have been analyzed. As a result of this analysis the proposal of a new THD factor has been presented. The new THD factor facilitates the estimation of the step waveforms from the filtration point of view. The examples of the converter structures using Fourier-style model for synthesis of alternating voltage waveforms have been presented. The wavelet-style model is using the new mathematical tool – a wavelet transform for the step waveforms synthesis. The defined wavelet-style model of the converter is based on transform similar to the Haar transform. By properties comparison of Fourier-style and wavelet-style models it was proved, that approximation method, based on wavelet theory is a useful mathematical tool in developing the structures and control algorithms of the multilevel converters. The orthogonal model is based on adding the orthogonal vectors and further development of this idea resulted in the recurrent model. The structures and control methods of these converters are promising alternatives for other well known multilevel converters’ solutions. The defined mathematical models are describing the output voltages of multilevel converters as results of the combination of the orthogonal functions (Fourier-style and wavelet-style models) or orthogonal vectors (recurrent model). Thanks to this approach the presented models show cleanly the relations between the component voltage waveforms and output voltage waveform of the whole converter. It facilitates designing the structures and control algorithms and allows to treat the multilevel converter as one unit. The examples of circuitry solutions based on defined models as well as simulation and experimental results of the chosen converters are also presented in the monograph.
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