This paper deals with the analysis of electrical circuits with classical one-port elements including two novel defined one-port fractional order elements: fractional-order resistive-capacitive RC-α and fractional-order inductive RL-α element. The definitions and analytical relations between current, voltage and power of introduced fractional elements are provided. An example of fractional element realization via ladder electrical circuit composed of classical resistors, capacitors and/or inductors is presented. Several examples are analyzed to illustrate the behavior of electrical circuit with fractional order elements for different values of fractional order α including differentiator/integrator circuits as well as complex circuits without accumulated energy.

This paper presents simple, flexible and effective approximation method for fractional order lead-lag compensators. The proposed method relies on a Padé approximation of linear fractional order transfer functions, giving rational approximations of order N accurate enough for control applications as soon as N is greater than 2 or 3. An example of feedback loop incorporating this approximation is adopted from the well-known car suspension problem, wherein an iso-damping property of the closed loop response is achieved, with regard to a variation of the vehicle mass.

The paper introduces a novel the second order band-pass and notch filter with dynamic damping factor β _d of fractional order. Indeed, the factor β _d has a form of fractional differentiator of order α, i.e. β _d =β/s ^α , where β and α are adjustable parameters. Shaping of the frequency response enables achieving better phase response compared to the integer-order counterparts which is of great concern in many applications. The aim of the paper is to exploit an extra degree of freedoms of presented filters to achieve the desired filter specifications and obtain desired response in the frequency and the time domain.