Approximation of Fractional-Order Butterworth Filter Using Pole-Placement in W-Plane

This work presents a new technique to approximate the fractional-order Butterworth filter using the pole-placement in the W-plane. Initially, poles of the fractional-order Butterworth transfer function are obtained by replacing the complex frequency <italic>s with a new variable</italic> <inline-formula> <tex-math notation="LaTeX">$(\boldsymbol {s}')^{2N/\alpha }$ </tex-math></inline-formula>. These poles and their complex-conjugate counterparts are plotted in <inline-formula> <tex-math notation="LaTeX">${W}$ </tex-math></inline-formula>-plane. Half of these poles, lying in the stable regions, are chosen to derive the characteristic equation of the fractional Butterworth filter. Therefore, the proposed fractional-order filters are inherently stable. Further, a 1.5<sup>th</sup> order Butterworth filter is designed using the proposed technique, and realized using the fractional-order <italic>Sallen-Key</italic> circuit. Additionally, several other higher order fractional Butterworth filters are realized using the proposed pole-placement technique. Finally, PSpice and MATLAB based simulations have been done to see the accuracy and effectiveness of the proposed design technique.