A New Look at Formulation of Charge Storage in Capacitors and Application to Classical Capacitor and Fractional Capacitor Theory

A New Look at Formulation of Charge Storage in Capacitors and Application to Classical Capacitor and Fractional Capacitor Theory

In this study, we revisit the concept of classical capacitor theory-and derive a possible new explanation of the definition charge stored in a capacitor. We introduce a ‘capacity function’ with respect to time to describe the charge storage in a classical capacitor and for a fractional capacitor. The observation regarding capacitor breakdown is very interesting. This study practically described for DC link capacitors (in power supply circuits), that the capacitor breaks down even it were never exceeded its maximum voltage limit. Here we will describe that the charge stored at any time in a capacitor as a ‘convolution integral’ of defined capacity function of a capacitor and voltage stress across it which comes from the causality principle. This approach, however, is different from the conventional method, where we multiply the capacity and the voltage functions to obtain charge stored. This new concept is in line with the observation of that charge stored as a step function and the relaxation current in form of impulse function for ‘ideal geometrical capacitor’ of constant capacity; when an uncharged capacitor is impressed with a constant voltage stress. Also this new formulation is valid for a power-law decay current that is given by ‘universal dielectric relaxation law’ called as ‘Curie von-Schweidler law’, when an uncharged capacitor is impressed with a constant voltage stress. This universal dielectric relaxation law gives rise to fractional derivative relating voltage stress and relaxation current that is formulation of ‘fractional capacitor’. A ‘fractional capacitor’ we will discuss with this new concept of redefining the charge store definition i.e. via this ‘convolution integral’ approach, and obtain the loss tangent value. We will also show how for a ‘fractional capacitor’ by use of ‘fractional integration’ we can convert the fractional capacity a constant that is in terms of fractional units (Farads per sec to the power of fractional number); to normal units of Farads. From the defined capacity function, we will also derive integrated capacity of capacitor. We will also give a possible physical explanation by taking example of porous and non-porous pitchers of constant volume holding water and thus, explaining the various interesting aspects of a classical capacitor and a fractional capacitor that we arrive with this new formulation; and also relates to a capacitor breakdown theory-due to electrostatic forces. Study investigates the charge stored in a capacitor, as a function of time is not the usual multiplication operation of capacity and voltage; instead, the charge is convolution integral of these two functions, derived from causality principles. With this formulation, we showed for a fractional capacitor, the charge goes to infinity for large times, when the fractional capacitor is placed on a constant voltage; and is in line with earlier fractional order models and observations.  With this formulation of convolution integral, this study also showed that the relaxation current is in the form of impulse function for ideal geometrical capacitor of constant capacity, when stressed by a constant voltage and for fractional capacitor with power-law decay current that is given by universal dielectric relaxation law called as Curie von-Schweidler law. Practically, this new ‘generalized- formulation’ has use while getting the charge stored in a capacitor which is a function of time with time-varying voltage stress across it, and to convert the fractional capacity units to usual capacity units in Farads.

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