Comparison of Fractional-Order Transfer Functions to Approximate Lowpass Butterworth Magnitude Responses

Three fractional-order transfer functions are analyzed for differences in realizing () order lowpass filters approximating a traditional Butterworth magnitude response. These transfer functions are realized by replacing traditional capacitors with fractional-order capacitors ( where ) 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 () order transfer function for the approximated Butterworth magnitude responses. Each fractional-order transfer function for is realized using a Tow-Thomas biquad a verified using SPICE simulations.