Construction Of Fractional Order Component And Its Application In DC-DC Converter

Fractional calculus has the history of more than three hundred years, it is widely applied in more and more fields recently. Integer-order elements of the real world are actually similar while the actual external characteristics present fractional-order relationships, namely fractional elements. The fabrication process of a fractional-order element involves the calculation of fractional calculus. Hence in this paper, three common definitions of fractional calculus and two related special functions are introduced as a start. The fractional-order Laplace transform is applied to avoid the calculation in time domain. Numerical method and analytic method are compared when solving fractional-order state equations. Then, different construction methods of fractional-order elements are studied in order to analyze its new external characteristics. By replacing capacitors and inductors with fractional-order ones, some fractional-order DC-DC converters are built to analyze and verify the effects of fractional-order elements. The main research results are as follows:Taking use of the self-similar fractal topology, fractional elements of 1/2 order are constructed with domino ladder circuit and tree circuit respectively. The iteration expressions and differential limits are calculated. As a result, the former circuit would exist some theoretical errors, while the latter one is completely a half-order differential. Amplitude-frequency curves and phase-frequency curves are obtained by iteration expressions based on MATLAB simulations. Compared to ideal curves, the larger the approximation number is, the smaller the error will be. The characteristics of the domino ladder network and the tree network are compared as well. The approximation of variable fractance in a certain frequency band is expressed as a polynomial of S operator, and then the specific analog circuits are obtained. In spite of a high efficiency, the approximate band cannot be confirmed in a regular Newton process, and the original 1/n-order approximation is expanded to(n-1)/n-order by integer-order differential. Several specific fractional-order capacitors and inductors are constructed. The waveforms of the error functions are analyzed from the order-frequency responses.Fractional-order DC-DC converters are built with constructed fractional-order elements using resistors, inductors and capacitors. The impacts on the parameters of the converters caused by orders of inductors and capacitors. When the order is less than 1, the ripples of the state variables increase, while the ripples will decrease if the order is greater than 1. Ignoring high order infinitesimal, the order makes no effect on the DC components of the state variables. To simplify the calculation, the DC-DC converters are modeled in S domain, separating DC components from AC components. The waveforms of the output voltages and currents are obtained by solving state equations in Simulink. The circuit simulation models are built in PSIM. Both the numerical simulation results and the circuit simulation results are in agree with the theoretical calculations. Finally, experiments of fractance approximations and fractional-order Buck converter are conducted to verify the above simulation results.