W Transform And Its Application In Fractional Circuits With Rational Orders

Compared to integer calculus,fractional calculus can better describe some endemic phenomena in the circuit.Fractional linear systems have attracted widespread attention from scholars and researchers for their excellent performance and potential application prospects.In the analysis and design of fractional linear systems,the solution of fractional linear systems is an important part.So far,the powers of s in the complex-frequencydomain equations obtained by the existing fractional Laplace transform are fractions,which makes it very difficult to solve algebraic equations formed by multiple fractional powers.To solve this issue,based on the traditional Laplace transform,this thesis proposes a new fractional Laplace transform—W transform.When the transformation is applied to fractional linear systems with rational powers,the powers of algebraic equation in the Wdomain are integers,which makes the transform domain analysis of the fractional system with rational powers not only feasible,but also simple.When the order of w higher,the image function is directly expanded by the traditional decomposition method,then the expansion formula will have a large number of terms,which makes the time-domain form more complex.Therefore,this paper uses the W transform to propose a fraction expansion method for the rational image function in the W-domain.Any rational function Y(w)can be obtained by combining six basic types.By finding the inverse W transform of these six basic types of image functions,we can get the expression of their original functions y(t).At the same time,this fraction expansion method for the rational image function in the Wdomain is used to obtain the analytical solutions of linear constant coefficient fractional differential equations and the fractional state equations,which avoids the complexity of other analytical solutions.On this basis,the W transform is applied to the circuit solution,the form of the fractional-order elements and Kirchhoff’s law in W-domain are derived,and the general steps of circuit analysis in the W-domain are given,then the correctness and feasibility of the application are verified by an example.Finally,a new method to prove the stability of fractional circuit in W domain is given.