Characteristic times in the infinite charge landscape of the universal capacitor

Abstract

Anyone who ever took a circuit theory class will be familiar with the classical capacitor theory under constant voltage operation: the current dynamics obeys a first-order differential equation and therefore follows the exponential Debye law. However, more than a century ago Curie and von Schweidler modeled, using mathematical arguments, the real behavior of the current in experimental capacitive devices using an inverse power law. Although this universal behavior has a valuable range of applications in nature and science, until now no one has presented a useful model to obtain characteristic values of time in the complex landscape of infinite transient charge found from this theory. Here we show, using the scenario of charge/discharge experiments, that the long-term charge-voltage relationship comes from an integration-based operation of the time-dependent capacitance function derived by fractional calculus. This result serves as the foundation for understanding a wide range of anomalous current responses observed in many capacitive devices, and for obtaining relevant charge-time relations corresponding to when the current reaches a pseudo-steady-state value, typically given by experimentalists, or the real energy storage devices approach the ideal charge value. Our work contributes to completing the universal capacitor theory governing the operation of real-world devices in the time-domain.