Electrochemical capacitor performance

The performance characteristics of electrochemical capacitors differ somewhat from those of onventional capacitors. In Fig. 4 the impedance plane representation (Nyquist plot) of an ideal capacitor and a simplified electrochemical capacitor, both having the same ESR (equivalent series resistance at 1 kHz), are compared.

While the ideal capacitor exhibits a vertical line, the electrochemical capacitor starts with a 45° impedance line and approaching an almost vertical line

only at low frequencies. The non-vertical slope of the low frequency impedance of any real electrochemical capacitor can be easily reproduced in any model equation by replacing the capacitance expression with a constant phase element (CPE) expression. This amounts to replacing every jv expression with ( jv)p, where 0BpB1, and where p‑1 represents an ideal capacitor with no frequency dependence. This non-ideality is a typical feature of electrochemical charging processes, and may be interpreted as resulting from a distribution in macroscopic path lengths (non-uniform active layer thickness) [25] or a distribution in microscopic charge transfer rates [26], adsorption processes, or surface roughness. The 45° region (Warburg region) is a consequence of the distributed resistance:capacitance in a porous electrode. At higher frequencies the resistance as well as the capacitance of a porous electrode decreases, because only part of the active porous layer is accessible at high frequencies. The electrochemical capacitor may thus be represented by an ideal capacitor with an ESR increased by the ‘equivalent distributed resistance’ EDR. 4.1. Porous electrode

The porous electrode is often described by a truncated RC-transmission line according to Fig. 5. The

equivalent circuit of the pore of a porous electrode is approximated by a line of R and C elements representing the elemental double layer capacitance and the

respective electrolyte resistance at a particular depth of the pore. The resistance of the bulk material is assumed to be much smaller than the electrolyte resistance. At high frequencies the capacitors behave like small impedance elements and the current flows predominantly along R1 and C1 into the bulk material and almost no current flows deep down the pore. Consequently, resistance and double layer capacitance are reduced at high frequencies . A more complete description of the porous electrode behavior was given by De Levie . Assuming straight cylindrical pores with a radius r and length l, a double layer capacitance and an electrolyte conductivity k one

can calculate the impedance according to