Analog Realization of Fractional Order Hybrid Differentiators via Carlson’s Approach Saptarshi Das 1,3 , Suman Saha 2,3 , Amitava Gupta 1,3 , and Shantanu Das 4 1 School of Nuclear Studies & Applications, Jadavpur University, Salt Lake Campus, LB-8, Sector 3, Kolkata-700089, India, Email of Author 1 : saptarshi@pe.jusl.ac.in 2 Drives and Control System Technology Group, Central Mechanical Engineering Research Institute (CMERI), CSIR, Mahatma Gandhi Avenue, Durgapur-713209, India, Email of Author 2 : s_saha@cmeri.res.in 3 Department of Power Engineering, Jadavpur University, Salt Lake Campus, LB-8, Sector 3, Kolkata-700089, India, Email of Author 3 : amitg@pe.jusl.ac.in 4 Reactor Control Division, Bhabha Atomic Research Centre (BARC), Mumbai-400085, India, Email of Author 4 : shantanu@magnum.barc.gov.in Abstract—Rational approximation of fractional order (FO) elements are now of prime importance for the need of its hardware implementation in control design and signal processing. Among the well known analog realizations methods, the Carlson’s approach has been used in this paper due to its simplicity of calculation for designing a certain class of FO differentiators as hybrid filters. Impact of the independent parameters of the hybrid differentiator on the phase response has been depicted along with the achievable accuracies for increasing order of realization for filter design. I. INTRODUCTION Fractional order elements are the building blocks for the fractional order system theory, control and signal processing [1]. Fractional order differ-integrators show constant-phase frequency response, thus are also known as Constant Phase Elements (CPE) [2]. The only problem with FO elements is its hardware realization due to its infinite dimensional nature. In practice, FO elements can be approximated as higher order rational transfer functions which have a constant phase curve within a certain frequency band. The fractional order elements can be rationalized as analog filters by various iterative techniques like Carlson’s, Oustaloup’s, Charef’s method etc. [3]-[5]. In this paper, the Carlson’s method has been used to realize a certain class of FO hybrid differentiators, commonly used in control and signal processing applications. Accuracy of the Carlson’s approach for approximating FO elements with increasing level of deviation from semi-integro-differential operators has been studied extensively in the present paper. Rest of the paper is organized as follows. Section II discusses about the theoretical framework of Carlson’s approach. Section III shows the simulation studies with the FO differentiators and hybrid filters. The accuracy of this technique has been illustrated in Section IV. The paper ends with the conclusion as section V, followed by the references. II. RATIONAL APPROXIMATION OF FRACTIONAL ORDER ELEMENTS BY CARLSON’S METHOD Analog circuit realization of fractance or FO immitance has been first studied in [6]-[9]. The recursive approximation of FO elements using the Carlson’s method is discussed next. If () Gs be a rational transfer function and () Hs be a fractional order transfer function such that [ ] () () , q Hs Gs q = ∈ \ (1) where, q m p = is fractional order of the transfer function model, then () Hs can be recursively approximated as [3]-[6]: [ ] [ ] 2 1 1 2 1 ( ) () ( ) () () () ( ) () ( ) () i i i i p m H s p mGs H s H s p m H s p mGs - - - - + + = + + - (2) with an initial guess of 0 () 1 H s = . Equation (2) can be rewritten in a more compact form as: [ ] [ ] 2 1 1 2 1 () () () () () () i i i i Gs H s H s H s Gs H s α α - - - + = + (3) where, ( ) ( ) 1 1 : 1 1 mp p m q p m mp q α ⎡ ⎤ - - - = = = ⎢ ⎥ + + + ⎣ ⎦ and 1 : 1 q α α - = + (4) For the simplest case, () q Hs s = , (1) will behave like a FO differentiator for 0 q > implying 1 α < and like a FO integrator for 0 q < implying 1 α > . The Carlson’s formula recursively places interlacing negative real poles and zeros in complex s- plane so as to meet the constant phase criteria [3]-[5], corresponding to the FO differ-integration. Also, from (3), it is evident that all the intermediate poles and zeros lie in the negative s-plane that enables direct inversion of the filter transfer functions to obtain a FO integrator, while also preserving the stability of the model. 60 2011 International Conference on Multimedia, Signal Processing and Communication Technologies 9781457711077/11/$26.00 ©2011 IEEE