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Chapter 3. Characterisation Methods 3.6.1.2 Capacitor Capacitors are used to describe the capacitance of electrical double-layers, which exist at the interface between an electronically conducting surface and an electrolyte. As opposed to the resistor, it only has an imaginary part. The impedance response is shown in Figure 3.12 (black line) and defined in Equation 3.33. ZC(ω) = 1 (3.33) jωC 3.6.1.3 Constant-Phase In real electrochemical systems, the capacitor is not able to fully describe the surface im- pedance due to the inherent imperfections of real surfaces, such as roughness, distribution of reaction rates and solution resistances, and adsorption of impurities [67]. A constant- phase element (CPE) is the generalised version of a capacitor that, to some degree, takes into account these imperfections. The impedance is defined as follows: ZQ(ω) = 1 (3.34) (jω)qQ0 where Q0 is a parameter related to the electrode capacitance and q is the dimensionless constant phase exponent, which can take on values between 0 and 1. The resulting impedance takes on the shape of a straight line with a phase of (−90q)◦, as shown in Figure 3.12 for various values of q (red lines). When q = 1, the impedance of a CPE corresponds to that of a capacitor. The parameter Q0 is not a real capacitance, but it can be used to calculate an equivalent double-layer capacitance, Ceq. For a blocking electrode represented by the equivalent circuit RsQ (a series connection of a resistor and CPE), it is given by [68]: 1/q 1 (1−1/q) Ceq = Q0 R (3.35) s where Rs is the series resistance. For a non-blocking electrode represented by the equiva- lent circuit Rs(RctQ) (resistor in series with a parallel connection of a resistor and CPE), the equivalent double-layer capacitance is given by [68]: 1/q 1 1 (1−1/q) Ceq = Q0 R + R (3.36) s ct where Rct is the charge transfer resistance. 3.6.1.4 Inductor An inductor is represented as a coil in which a current induces a magnetic field. This field in turn induces an electromotive force, or voltage, that opposes any change in current, which means a constant current flows through an inductor with 0 resistance. In a typical 34PDF Image | Organic Redox Flow Batteries 2023
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