Battery electrode equivalent circuit

One possible equivalent circuit of a battery is shown in Figure 8.18, in which Csc is the capacitance of the electrical double layer, W the Warburg impedance for diffusion processes, Rt the internal resistance, and ZA and Zc the impedances of the electrode reactions [124,130],... 

The entire diagram illustrating the performance of the electrode processes, corresponding to the simple equivalent circuit diagram of a battery which is shown in Figure 8.18, is presented in Figure 8.23 [6],... 

The metal and the electroljrte also determine the DC half-cell potential, modeled by the battery B. If there is no electron transfer. Ret is very large and the battery B is decoupled, the electrode is then polarizable with a poorly defined DC potential. But if there is an electrode reaction. Ret has a lower value and connects an additional admittance in parallel with the double layer admittance. This current path is through the faradaic impedance Zf, and the current is the faradaic current if. Faradaic current is related to electrode reactions according to Faraday s law (Section 7.8). The faradaic impedance may dominate the equivalent circuit in the lower Hz and sub-Hz frequency range and at DC. The faradaic impedance is modeled by a complete Cole-like series system. It consists of the resistor Ret... 

Formulating the problem in a discretized way allows us to extend it effortlessly to more complicated cases. Let s assume, for example, that in addition to double-layer capacitance we will have an electrochemical reaction on the pore surface, as would be the case in a battery or fuel-cell electrode. The equivalent circuit for the pore surface now will involve a capacitor in parallel with a charge transfer resistance, Ra, and the surface impedance Z oss will be given as follows ... 

Figure 4.5.9. Equivalent circuit of a battery insertion electrode. Here R is the distributed resistance of the transmission line representing electronic and ionic resistance of the layer of active material the charge transfer resistance and passivation layer resistance of the particle interface and the double layer capacitance C . is the impedance of diffusion and charge storage processes inside the particles.

The anode of a Ni-Cd battery typically consists of a mix of Cd and CdO powders with the addition of a conductive additive (acetylene black). The impedance of the anode-particle surface is determined by the activated adsorption of OH anions first on the metal surface, with subsequent conversion into Cd(OH)2 and hydrated CdO layers (Duhirel et al. [1992])). Reaction products are also present in a partly dissolved Cd(OH)3" state. The activated adsorption mechanism of the anode reaction, as well as porous structure of the electrode, makes it appropriate to use for its analysis the equivalent circuit shown in Figure 4.5.14. It was shown by Xiong et al. [1996], by separate impedance measurements on the anode and cathode, that most of the impedance decrease during discharge is due to the anode, as the initial formation of a Cd(OH)Jrate limiting step of the reaction. The... 

The cathode of a modem Ni-Cd battery consists of controlled particle size spherical NiO(OH)2 particles, mixed with a conductive additive (Zn or acetylene black) and binder and pressed onto a Ni-foam current collector. Nickel hydroxide cathode kinetics is determined by a sohd state proton insertion reaction (Huggins et al. [1994]). Its impedance can therefore be treated as that of intercalation material, e.g. considering H+ diffusion toward the center of sohd-state particles and specific conductivity of the porous material itself. The porous nature of the electrode can be accounted for by using the transmission line model (D.D. Macdonald et al. [1990]). The equivalent circuit considering both diffusion within particles and layer porosity is given in Figure 4.5.9. Using the diffusion equations derived for spherical boundary conditions, as in Eq. (30), appears most appropriate. 

The Thevenin equivalent circuit is the simplest combination, since it is the association of an ideal voltage source and a resistor connected in series. This is a much more realistic way of modeling a lead-acid battery. Indeed, the resistor illustrates the voltage drop due to the current passing through the components of the battery. In the case of LABs, this instantaneous voltage drop mainly results from the low electrical conductivity of electrolyte and is proportional to the current. But, such a simple combination does not account for the polarization of the electrodes happening later on, when the battery is operated. 

Typical dimensions for the /5-alumina electrolyte tube are 380 mm long, with an outer diameter of 28 mm, and a wall thickness of 1.5 mm. A typical battery for automotive power might contain 980 of such cells (20 modules each of 49 cells) and have an open-circuit voltage of lOOV. Capacity exceeds. 50 kWh. The cells operate at an optimum temperature of 300-350°C (to ensure that the sodium polysulfides remain molten and that the /5-alumina solid electrolyte has an adequate Na" " ion conductivity). This means that the cells must be thermally insulated to reduce wasteful loss of heat atjd to maintain the electrodes molten even when not in operation. Such a system is about one-fifth of the weight of an equivalent lead-acid traction battery and has a similar life ( 1000 cycles). 

These laws (determined by Michael Faraday over a half century before the discovery of the electron) can now be shown to be simple consequences of the electrical nature of matter. In any electrolysis, an oxidation must occur at the anode to supply the electrons that leave this electrode. Also, a reduction must occur at the cathode removing electrons coming into the system from an outside source (battery or other DC source). By the principle of continuity of current, electrons must be discharged at the cathode at exactly the same rate at which they are supplied to the anode. By definition of the equivalent mass for oxidation-reduction reactions, the number of equivalents of electrode reaction must be proportional to the amount of charge transported into or out of the electrolytic cell. Further, the number of equivalents is equal to the number of moles of electrons transported in the circuit. The Faraday constant (F) is equal to the charge of one mole of electrons, as shown in this equation ... 

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