Complex-Order Single-Dispersion Models

The well-known Cole-Cole, Cole-Davidson and Havriliak-Negami electrochemical impedance models are efficient tools used to describe electrochemical materials, including biological tissues, in impedance spectroscopy characterization. These models are capacitive and can generally be realized using resistors, capacitors and Constant Phase Elements (CPEs). In this work, we propose modified versions of these models with a complex-order rather than a real valued order only. In particular, the modified models are based on the complex-order operator <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(j\omega)^{\alpha+j\beta}$</tex> which can be approximated as a stable integer-order rational transfer function. The approximation method uses stan-dard MATLAB functions applied to the frequency response data of the complex-order impedance models. The derived integer-order transfer functions can be easily realized using a Field Programmable Analog Array (FPAA), as demonstrated here. The time-domain and frequency-domain behaviors of the proposed models are verified in simulation and experimentally.