Simple circuit equivalents for the constant phase element

The constant phase element (CPE) is a capacitive element with a frequency-independent negative phase between current and voltage which interpolates between a capacitor and a resistor. It is used extensively to model the complexity of the physics in e.g. the bioimpedance and electrochemistry fields. There is also a similar element with a positive phase angle, and both the capacitive and inductive CPEs are members of the family of fractional circuit elements or fractance. The physical meaning of the CPE is only partially understood and many consider it an idealized circuit element. The goal here is to provide alternative equivalent circuits, which may give rise to better interpretations of the fractance. Both the capacitive and the inductive CPEs can be interpreted in the time-domain, where the impulse and step responses are temporal power laws. Here we show that the current impulse responses of the capacitive CPE is the same as that of a simple time-varying series RL-circuit where the inductor's value increases linearly with time. Similarly, the voltage response of the inductive CPE corresponds to that of a simple parallel RC circuit where the capacitor's value increases linearly with time. We use the Micro-Cap circuit simulation program, which can handle time-varying circuits, for independent verification. The simulation corresponds exactly to the expected response from the proposed equivalents within 0.1% error. The realization with time-varying components correlates with known time-varying properties in applications, and may lead to a better understanding of the link between CPE and applications.