Optimal Time-Domain Pulse Width Modulation in Power Electronics

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## Abstract

The technique of pulse width modulation is used in the design of dc-to-ac power converters and controlling the peak value and frequency of ac output voltages. These dc-to-ac converters are called inverters. They are of interest in their own right, but they are also used as the main buildings blocks of ac-to-ac converters. Inverters are widely used for the frequency control of the speed of various ac motors (for instance, in hard disk drives for magnetic data storage, or in electric cars), for integrating renewable sources of energy with existing power grids and for the design of uninterruptible power supplies (UPS) to maintain the availability of power in the case of power blackouts.

The central idea of the pulse width modulation is to use the available input dc voltage to generate a periodic train of rectangular pulses. The widths of these pulses are properly modulated to generate the desired output ac voltages. The latter is achieved by suppressing low order harmonics at the expense of high-order harmonics in the pulse width modulated voltages. The high-order harmonics are then suppressed in output voltages by energy storage elements of load circuits.

The study of pulse width modulation for voltage source inverters is the main focus of this dissertation. Single-phase inverters are first considered. The train of rectangular pulses generated by pulse width modulation are characterized by the corresponding switching time-instants. Then, analytical formulas for the output voltages of inverters in terms of these switching time-instants are derived for first-, second- and third-order linear circuits. The obtained formulas for output voltages are then used to derive the explicit analytical expressions for the $L_2$-norm of error between the output voltages and their desired sinusoidal values. This $L_2$-norm is shown to be a measure of the total harmonic distortion (THD) in the desired output voltages of the inverter. Next, the derived explicit expressions are used to obtain the optimal switching time-instants which minimize the THD in output voltages, by numerically solving the THD minimization problem by using the interior point or sequential quadratic programming techniques. Remarkably, the elimination of specific harmonics in the output voltages within the framework of THD minimization can be achieved by the use of constrained optimization. As presented in the dissertation, significant improvement (around 40%) in the THD can be obtained, especially for small numbers of rectangular pulses (i.e., low switching frequencies).

Next, pulse width modulation for three-phase inverters is discussed. The most important difference from the case of single-phase inverters is the role of symmetries (such as half-wave symmetry, time-translational symmetry and odd-symmetry) in the structure of three-phase pulse width modulated voltages. These symmetries must be preserved under constraints imposed by Kirchhoff Voltage Law as well as the single-leg switching utilized in modern three-phase inverters. To account for these symmetries and constraints, a novel $pqr$-technique for analysis of three-phase inverters is developed. Then, using the $pqr$-technique the approximate Fourier spectra of sinusoidally modulated three-phase line-voltages are analytically derived by using the midpoint approximation, and the appearance of intermediate bands of high-order harmonics is revealed. Next, the $pqr$-technique is used to develop the per-phase time-domain analysis, which reveals that the outputs of three-phase inverters are driven by five-level phase voltages. Explicit analytical formulas for output voltages of three-phase inverters in terms of the inverter switching time-instants are derived. By using the above analytical formulas, the expressions for the THD of output voltages are obtained, which are then used for THD minimization. Furthermore, the $pqr$-technique is used to obtain algebraic constraints on the switching time-instants consistent with the desired symmetries of three-phase PWM voltages, which reduce the number of variables in the minimization problem.

At the end of the dissertation, a brief discussion is presented on how the latest developments in spintronics may be promising for power conversion at nanoscale and at very low values of voltages and power. It is suggested that spintronics based power converters may be designed to operate without repetitive switching by using the unique physical phenomena occurring in nanomagnetic devices. The possible designs of dc-to-dc, dc-to-ac, ac-to-dc and ac-to-ac converters using spintronics is discussed.