article · The Journal of Physical Chemistry C
Electrical systems containing constant phase elements, which exhibit non-ideal capacitive behaviour, require tailored input signals to store energy efficiently. Mathematical modelling using a frequency-distributed resistor-capacitor network allows the determination of an optimal voltage profile that maximises energy storage efficiency over a set charging period. The optimal solutions belong to a family of power-law voltage profiles. When restricted to concave voltage waveforms, a linear ramp voltage provides the highest possible energy storage efficiency while producing a power-law current response. Across all power-law profiles, charging efficiency increases as the power-law exponent grows, approaching the element order parameter as an asymptotic upper limit. This demonstrates that the element order represents the theoretical efficiency ceiling, defining the maximum fraction of input energy that can be captured in capacitive modes under power-law excitation.
Understanding how to charge constant phase elements efficiently is essential for optimising systems that exhibit anomalous or distributed impedance. Establishing the theoretical limits of energy storage provides fundamental design rules for electrical excitation protocols, helping engineers and physicists understand the maximum energetic capability of non-ideal capacitive components.
This study represents early-stage theoretical research into optimal charging protocols for non-ideal electrical elements. The findings could potentially assist circuit designers, energy-harvesting engineers, and device developers in tailoring voltage waveforms to minimise energy loss in components exhibiting constant phase behaviour. However, the abstract does not describe physical prototyping, experimental testing, or immediate industrial implementation, indicating that the concept remains at an early, fundamental stage of development.
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Abstract In this study, we determine the optimal voltage profile for charging a constant phase element (CPE) of order α ∈]0, 1[ that maximizes the ratio of stored to total input energy over a fixed charging interval. Using the frequency-distributed RC network representation of the CPE, the efficiency maximization is reformulated as an isoperimetric variational problem, and solved via the fractional Euler–Lagrange equation. A Mellin transform analysis of the resulting optimality condition identifies the admissible solutions as a one-parameter family of power-law voltage profiles v(t; s0) = V0(t/T)β*, β* = α – s0 > 0. Within the physically relevant class of concave waveforms β* ∈]0, 1[, the optimal charging protocol is the linear ramp, which achieves the maximum energy storage efficiency η* = (4 – 22−α – α)/(2 – α), and produces a power-law current response. Extending to all β* > 0, the efficiency is shown to be monotonically increasing with β*, approaching the supremum α asymptotically as β* → ∞ but never reaching it, establishing α as the theoretical efficiency ceiling for power-law charging of a CPE. This result also provides an energetic interpretation of the CPE order α as the maximum fraction of input energy that can be stored in the capacitive modes of the CPE, over all power-law voltage excitations.
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DOI: 10.1021/acs.jpcc.6c04376
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