Revisiting the Time-Domain and Frequency-Domain Definitions of Capacitance

Abstract

The capacitance is a characteristic function of an electrical energy storage device that relates the applied voltage on the device to the accumulated electric charge. It is inconsistently taken in some studies as a multiplicative function in the time domain [i.e., q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> (t)=c <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> (t) ×v <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> (t)], and in others as a multiplicative function in the frequency domain [i.e., Q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> (s)=C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> (s) × V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> (s) derived from the definition of admittance I <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> (s)/V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> (s) = s C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> (s)], despite the fact that the capacitance is time- and frequency-dependent. However, the convolution theorem states that multiplication of functions in the time domain is equivalent to a convolution operation in the frequency domain, and vice versa. In this work, we revisit and compare the two outlined definitions of capacitance for an ideal capacitor and for a lossy fractional-order capacitor. Although c <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> (t) = C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> (s) = C for an ideal constant capacitor, we show that this is not the case for fractional-order capacitors which exhibit frequency-dispersed impedance, memory effects, and nonexponential relaxation functions. This fact is crucial in the accurate modeling and characterization of supercapacitors and batteries. For these devices, and for being consistent with measurements using conventional impedance analyzers, it is recommended to apply the integral convolution definition in the time domain which reverts to the multiplicative definition in the frequency domain.