Research On Passivity Of Fractional Order Circuits

Many of the equipment and systems in the field of electrical engineering show the essence of fractional order.Passivity is one of the most important properties of circuit analysis and circuit synthesis,especially for the passive circuit realization and modeling.Active model may result in unstable simulation results,which can cause system oscillation.The study of passivity is of great significance for fractional order circuit theory,fractional order modeling and the inherent characteristics of electrical materials.In traditional integer order circuits,the passivity of circuits is equivalent to the positive real property of the circuit's immittance function(matrix).With the introduction of fractional elements,the positive real property does not exist in s-domain.Passivity criteria of traditional circuits can not be used in fractional order circuits.Studying the passive criterion of fractional order circuits is of great importance for the research of fractional order circuits.From the definition of positive real property,this paper explains the definition from the point of view of circuit in the frequency domain,and proposes the concept of"pseudo-positive real property".With the analysic of fractional order components by the"pseudo-positive real property",the passive criterion of the fractional order circuit in the s domain is proposed.Based on the s-W transformation,the fractional order circuit immittance function is changed into integer order,the W-domain immittance function(matrix)of the positive real and bounded real property is defined and the equivalence criteria are obtained.And we discuss the expansion of fractional order circuit immittance considering the real pole,single and double conjugate poles.The positive real property of passive fractional order circuits in W-domain is discussed.The positive real property and the satisfied conditions of two terminal fractional order high-order components in W-domain are analyzed.Meanwhile,the characteristics of the fractional order simple RL?C?circuit impedance function in W-domain are discussed.