Comparison of $$(1+\alpha )$$ ( 1 + α ) Fractional-Order Transfer Functions to Approximate Lowpass Butterworth Magnitude Responses

• DOI: 10.1007/s00034-015-0226-y
• Publication Date: 2015-12-28T10:01:59Z
• Authors: Todd J. Freeborn
• Keywords: Digital biquad filter; Butterworth filter; Alpha (finance); Spice

Three fractional-order transfer functions are analyzed for differences in realizing ($$1+\alpha $$1+?) order lowpass filters approximating a traditional Butterworth magnitude response. These transfer functions are realized by replacing traditional capacitors with fractional-order capacitors ($$Z=1/s^{\alpha }C$$Z=1/s?C where $$0\le \alpha \le 1$$0≤?≤1) in biquadratic filter topologies. This analysis examines the differences in least squares error, stability, $$-$$-3 dB frequency, higher-order implementations, and parameter sensitivity to determine the most suitable ($$1+\alpha $$1+?) order transfer function for the approximated Butterworth magnitude responses. Each fractional-order transfer function for $$(1+\alpha )=1.5$$(1+?)=1.5 is realized using a Tow---Thomas biquad a verified using SPICE simulations.