The positive-reality and regularity of fractional- order biquadratic impedance functions

With the wide application of fractional calculus, fractional- order circuits and systems find a widespread in different engineering applications. Researchers began to study and analyze the driving- point function synthesis with fractional order elements. In this paper, necessary and sufficient conditions for the fractional biaquadratic functions to be positive-real are given by aid of variables substitution method and the concept of multivariable positive-real functions. In addition, by using the concept of regular function in s domain, we get some properties for a fractional biquadratic impedance can realize a regular function. Finally, based on the research on integer order five- element with two reactive elements, we find that in fractional-order five-element with two reactive elements could also achieve regular positive-real fractional biquadratic impedance function.