Theoretical Verification of the Formula of Charge Function in Time of Capacitor (q = c*v) for Few Cases of Excitation Voltage

Abstract

We have a developed and derived a formula for capacitor i.e. charge as a function of time, which is convolution operation of time varying capacity function and time-varying voltage function. This is different to the usual and conventional way of writing capacitance multiplied by voltage to get charge stored in a capacitor. This new deliberation with convolution operation works well for classical capacitors (i.e. ideal loss less capacitors), that is of a constant capacity at all frequencies, and also for a time varying capacity function given by decaying power-law: that gives the formation of a fractional capacitor. In this paper, we use this developed new charge storage expression and apply to various types of inputs excitation voltage-sinusoidal, step, ramp voltage and then analyze and interpret the results for charge stored, the current expressions, the loss-tangent and the memory effects. With this new formulation, we also evaluate impedance function of a classical capacitor as well as a fractional capacitor, and also elaborate on the Nyquist’s diagram, that is employed to study various dielectric materials via impedance spectroscopy. This new approach of charge storage concept is yet to be practically as well as theoretically applied-though some initial work has started. This paper gives a theoretical validity test i.e. analytically obtained in several applications for this new formulation, of charge storage formula. This paper will be useful in various super-capacitor studies, dielectric relaxation experiments, and impedance spectroscopy for various material developments for electrical energy storage missions; however, this concept is yet to be used to its full potential.