The gain and phase responses of the delay element \(e^{-\sqrt{\tau s}}\) are related through a non-linear relation, unlike the conventional \(e^{-{\tau s}}\) delay element where the gain is constant over the entire frequency range. This makes this fractional-order element attractive for describing the dynamics of systems with dead-time zone. In this work, the transfer function of this element is approximated through a rational integer-order function resulting from a curve-fitting based method. The performed comparison with the literature shows that the curve-fitting based approximation method achieves errors in gain and phase less than 0.3dB and 2o, respectively, while the available method in the literature achieves 3.7dB and 8o, respectively. The behavior of some filters involving this delay element is evaluated and it is demonstrated that the implementation of these filters can be performed by the same circuit core, simply by adjusting the coefficients of the approximation transfer function. The findings of this work are supported by MATLAB simulation results and, also, by experimental results obtained through the utilization of a Field Programmable Analog Array device.
