Approximation of Fractional-Order Butterworth Filter Using Pole-Placement in<i>W</i>-Plane

Abstract

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 xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s with a new variable</i> <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(\boldsymbol {s}')^{2N/\alpha }$ </tex-math></inline-formula> . These poles and their complex-conjugate counterparts are plotted in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <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 xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> order Butterworth filter is designed using the proposed technique, and realized using the fractional-order <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Sallen-Key</i> 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.