Step Response of Commensurate Fractional Lowpass Pseudo-Biquad: Critical Damping

In the paper, a comparison is made between the step responses of the classical integer-order biquad with the transfer function ((s/ω0)^2+(s/ω0)/Q+1)^(-1), where ω0 and Q are the characteristic frequency and quality factor, and the commensurate fractional pseudo-biquad with the transfer function ((s/ω0)^(2α)+(s/ω0)^α/Q+1)^(-1), 0< α ≤1. While the classical biquad experiences the fastest response for critical damping when Q = 0.5, a similar response can be observed for the fractional circuit, but for larger values of Q depending on the α parameter. For α < 1, the response then settles faster than for the classical filter. Coupling conditions between the parameters α and Q are found that lead to the so-called pseudo-critical damping and critical underdamping.