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 t (t)=c t (t) ×v t (t)], and in others as a multiplicative function in the frequency domain [i.e., Q f (s)=C f (s) × V f (s) derived from the definition of admittance I f (s)/V f (s) = s C f (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 t (t) = C f (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.

中文翻译：

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重新审视电容的时域和频域定义


电容是电能存储设备的特征函数，它将设备上施加的电压与累积的电荷联系起来。在一些研究中，它被不一致地视为时域中的乘法函数[即 qt (t)=ct (t) ×vt (t)]，而在其他研究中，它被视为频域中的乘法函数[即 Q f (s)=C f (s) × V f (s) 由导纳 I f (s)/V f (s) = s C f (s) 的定义导出，尽管事实上电容是时间-和频率相关。然而，卷积定理指出，时域中的函数乘法相当于频域中的卷积运算，反之亦然。在这项工作中，我们重新审视并比较了理想电容器和有损分数阶电容器的电容的两个概述定义。尽管对于理想的恒定电容器来说 ct (t) = C f (s) = C，但我们表明分数阶电容器的情况并非如此，它表现出频率分散阻抗、记忆效应和非指数弛豫函数。这一事实对于超级电容器和电池的精确建模和表征至关重要。对于这些设备，为了与使用传统阻抗分析仪进行的测量保持一致，建议在时域中应用积分卷积定义，这将恢复为频域中的乘法定义。