Curves of limiting voltage (*V*) as a function of temperature (*T*) for nickel/cadmium cells and batteries can be computed by use of a mathematical cell model based on first principles. Such curves ("V/T curves" for short) are needed as guides to rapid full charging without overcharging. Charge-control techniques based on V/T curves are being developed for Ni/Cd cells aboard spacecraft in low orbits around the Earth. These techniques could also be used on Earth; for example, to control the charging of Ni/Cd batteries in electric vehicles during regenerative braking.

Full charging of a Ni/Cd cell is necessary for maintaining its charge capacity. Because a Ni/Cd cell exhibits a negative temperature coefficient of voltage, it can go into a thermal-runaway condition when it becomes heated during overcharging, especially if overcharging

occurs at a high current. Even when unaccompanied by thermal runaway, overcharging can degrade the cell and shorten its life. Thus, it is necessary to prevent overcharging as well as undercharging. The relevant measure of charge is the charge in ÷ charge out; it is denoted as the recharge fraction, the charge/discharge ratio, or the reciprocal of the cell throughput efficiency. The minimum value of this measure to ensure full charging is somewhat greater than 1, and the maximum allowable value to prevent damage is higher. V/T curves are chosen so that by adhering to them, one can achieve the desired recharge fraction between the minimum and maximum values at a given temperature in a relatively wide temperature range.

Heretofore, it has been necessary to construct V/T curves from experimental data. However, experiments to determine V/T curves for Ni/Cd cells and batteries are tedious and destructive. Moreover, the interpretation of experimental data involves uncertainties in that the characteristics of Ni/Cd cells depend partly on prior thermal and charge/discharge histories. Thus, there is a need for the present method of estimating V/T curves without having to perform experiments.

The mathematical model used in the present method is built around the following principles of:

- Material balance for the dissolved chemical species generated and consumed in electrochemical reactions and transported by diffusion and migration,
- Changes in electrochemical potential in the solid phase and in the electrolyte,
- Charge-transfer kinetics as represented by a modified Butler-Volmer rate equation,
- Conservation of charge in the electrochemical cell, and
- Effects of intercalation and slow diffusion of protons into the positive electrode.

This model involves a simplification from porous-electrode models in that mass-transport processes in the solid phase are recognized as predominating over those in the liquid phase and thus a uniform reaction layer on a planar electrode is assumed. The model can be used to predict the charge/discharge characteristics of a cell under any specified test conditions, including typical conditions like constant current, constant voltage, or constant power, with limits of time, voltage, current, or temperature. The model also accounts for the existence of two forms (the β and γ phases) of the positive active material (NiOOH) and the corresponding reduced forms [the β and α phases of Ni(OH)_{2}] to provide a more accurate prediction of discharging and charging behavior.

The figure presents a set of V/T curves computed by use of the model for a Ni/Cd cell under typical repetitive low-Earth-orbit charge/discharge cycling. These curves are shaped similar to experimentally determined V/T curves. In comparison with experimental curves, the curves give slightly reduced percent recharge at a given voltage or slightly higher voltage for a given percent recharge, less sensitivity to changes in inrush current, and less voltage span corresponding to the desired range of percent recharge. These differences are being addressed in continuing research.

*This work was done by Ratnakumar Bugga, Paul Timmerman, and Sal DiStefano of Caltech for *NASA's Jet Propulsion Laboratory*. For further information,**access the Technical Support Package (TSP) **free on-line at www.techbriefs.com** under the Electronic Components and Circuits category, *or circle no. 166* on the TSP Order Card in this issue to receive a copy by mail ($5 charge). NPO-20152*

##### NASA Tech Briefs Magazine

This article first appeared in the April, 1998 issue of *NASA Tech Briefs* Magazine.

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