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.