Computing the Capacitance of Electric Double-Layer Capacitors from Time-Domain and Frequency-Domain Measurements

Abstract

In this study we show the relationship between capacitances of electric double-layer ca- pacitors (EDLC) computed from time-domain and frequency-domain measurements. EDLC are assumed to be linear time-invariant (LTI), causal and stable systems that map an input time-domain voltage signal v = [v0, v1, . , vN ] to an output time-domain charge q = [q0, q1, ..., qN ] = [f(v0), f(v1), ..., f(vN)]. When a voltage v0 is applied on an uncharged EDLC, the corresponding measured charge is q0 = c0v0 , where c0 is the value of the system impulse response at that instant. Subsequently, by considering the superposition principle in causal LTI systems, the input voltage v1 applied after a lapse of time ∆t with the impulse response of the device c0 results in a certain amount of charge c0v1 , but the preceding value of voltage v0 contributes also with another amount of charge c1v0 that depends on the current response of the device c1 at the instant ∆t. Thus the charge q1 accumulated on the device after ∆t is the sum c0v1 + c1v0 . Repeating this procedure for all discrete times tk = k∆t until N∆t (time duration of the applied signal) corresponds to the (discrete) convolution sum of the input voltage by the systems impulse response which results in the output charge: qτ = c ∗ v =Σj cτ− j vj . In continuous-time form, one can write: q(t) = ∫ c(t − τ )v(τ )dτ. The function c(t) can be viewed as the system-level, macroscopic capacitance of the device that takes into account (in a lumped form) all microscopic events of charge transport taking place inside the device. The convolution operation is carried out here in the time domain, but in the frequency domain, by virtue of the convolution theorem, it turns out to be the point-by-point multiplication Q = CV , where Q, C and V are the Fourier transforms of q, c and v respectively [1–3]. Thus, one cannot compute the capacitance of EDLCs by dividing charge-by-voltage both in the time-domain and frequency-domain as commonly done in the literature [3]. We will show examples using linear voltage ramp and constant current excitations [1,2].REFERENCES A. Allagui and M. E. Fouda, “Inverse problem of reconstructing the capacitance of electric double-layer capacitors,” Electrochim. Acta , 138848 (2021).A. Allagui, A. S. Elwakil, and M. E. Fouda, “Revisiting the time-domain and frequency-domain definitions of capacitance,” IEEE Trans. Electron Devices 68 (2021).A. Allagui, A. S. Elwakil, and H. Eleuch, “Highlighting a common confusion in the computation of capacitance of electrochemical energy storage devices,” J. Phys. Chem. C 125, 9591–9592 (2021).