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

Three fractional-order transfer functions are analyzed for differences in realizing ( $$1+\alpha $$ ) order lowpass filters approximating a traditional Butterworth magnitude response. These transfer functions are realized by replacing traditional capacitors with fractional-order capacitors ( $$Z=1/s^{\alpha }C$$ where $$0\le \alpha \le 1$$ ) 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 ( $$1+\alpha $$ ) order transfer function for the approximated Butterworth magnitude responses. Each fractional-order transfer function for $$(1+\alpha )=1.5$$ is realized using a Tow-Thomas biquad a verified using SPICE simulations. [ABSTRACT FROM AUTHOR]