By Petar J. Grbovic
Use of extremely capacitors in strength electronics
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5 An approximated model of the electrochemical double layer capacitor taking into account the porosity of the electrodes The voltage dependent capacitances CP1 . . CP . . N and CN1 . . CN . . N ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ 0 C0N 1 KCN 1 0 0 uCN 1 CN 1 ⎢ ⎥ ⎢ CN 2 ⎥ ⎢ C0N 2 ⎥ ⎢ 0 KCN 2 0 0 ⎥ ⎥=⎢ ⎥+⎢ ⎥ ⎢ uCN 2 ⎥ . N The coefficients KCN1 . . KCP . . N model the voltage dependency of the capacitance due to the diffused layer. Voltages uCN1 . . uCP . . N are the voltages across each elementary capacitor cell.
6 is a nonlinear circuit because the capacitances depend on the voltage. To develop a model in the frequency domain we have to linearize the nonlinear circuit. 8, yields ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ 1 ⎤ ⎡ 1 1 1 C1 ⎢ 1 ⎥ ⎢ ⎢C ⎥ ⎢ ⎢ 2⎥∼ ⎢ ⎢ .. ⎥ = ⎢ ⎣ . N C0P 1 +KCP 1 ·UC01 1 C0P 2 +KCP 2 ·UC02 ⎥ ⎢ ⎥ ⎢ ⎥+⎢ ⎥ ⎢ ⎦ ⎣ .. N C0N1 +KCN 1 ·UC01 1 C0N2 +KCN 2 ·UC02 ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎦ ⎣ .. N C01 +KC1 ·UC01 1 C02 +KC2 ·UC02 .. ⎥ ⎥ ⎥ . N Now, having a linearized model of the ultra-capacitor ladder network one can develop the ultra-capacitor input impedance ZC0 (ω).
The first is a constant shunt resistor RP , while the second is a voltage-dependent shunt current iP . A brief discussion of both methods follows below. Constant Shunt Resistance Model The ultra-capacitor leakage current is usually modeled with a shunt resistor RP . There are two ways to compute the resistance for given capacitance as a parameter. 1. The first way to determine the resistance RP is from the criterion that the ultra-capacitor voltage decays at 4% of UC0max after T = 72 hours. 35 · 106 ∼ = .
Ultra capacitors in power conversion systems by Petar J. Grbovic