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

Abstract

In the paper, a comparison is made between the step responses of the classical integer-order biquad with the transfer function $((s/\omega_{0})^{2}+(s/\omega_{0})/Q+1)^{-1}$, where $\omega_{0}$ and $Q$ are the characteristic frequency and quality factor, and the commensurate fractional pseudo-biquad with the transfer function $((s/\omega_{0})^{2\alpha}+(s/\omega_{0})^{\alpha}/Q+1)^{-1},\ 0 < \alpha\leq 1$. 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 $\alpha$ parameter. For $\alpha < 1$, the response then settles faster than for the classical filter. Coupling conditions between the parameters $\alpha$ and $Q$ are found that lead to the so-called pseudo-critical damping and critical underdamping.