Abstract
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.
Highlights
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 constant phase element (CPE) and applications
The constant phase element (CPE) is a capacitive impedance with a phase angle in the range h−π/2, 0i which is independent of frequency
He connected it to the temporal power law step response of the Curie-von-Schweidler law [3] which was first observed for real-life capacitors more than a century ago
Summary
The constant phase element (CPE) is a capacitive impedance with a phase angle in the range h−π/2, 0i which is independent of frequency. The common capacitive CPE with a negative phase angle will have the same current response to an input voltage impulse as a resistor in series with an inductor that increases linearly with time. The results are confirmed by simulation of the responses of the time-varying circuits in Micro-Cap 12. Sometimes a simpler relationship which only includes the first term is assumed instead: diðtÞ uðtÞ 1⁄4 LðtÞ : ð9Þ dt This ambiguity is reflected in how time-varying inductors and capacitors are implemented in circuit simulators. Assuming that t τ so that the second term in the parenthesis dominates, this is iðtÞ t À This result the same functional form as (3) and this time-varying circuit approximates a CPE. This is as expected from the theory as the approximation of (17) is valid when t > >τ
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