Extending the concept of analog Butterworth filter for fractional order systems

•Concept of analog Butterworth filter extended using fractional order linear systems.•Proposed fractional filter has poles on a circle in the transformed complex w-plane.•The unstable poles of fractional filter are discarded using Matignon′s theorem.•Number of poles in w-plane and commensurate order are taken as the tuning knobs.•Practical design example is added to show the advantage of this fractional filter.

This paper proposes the design of fractional order (FO) Butterworth filter in complex w-plane (w=sq; q being any real number) considering the presence of under-damped, hyper-damped, ultra-damped poles. This is the first attempt to design such fractional Butterworth filters in complex w-plane instead of complex s-plane, as conventionally done for integer order filters. First, the concept of fractional derivatives and w-plane stability of linear fractional order systems are discussed. Detailed mathematical formulation for the design of fractional Butterworth-like filter (FBWF) in w-plane is then presented. Simulation examples are given along with a practical example to design the FO Butterworth filter with given specifications in frequency domain to show the practicability of the proposed formulation.