Batteries

Advanced: electrochemistry and engineering

A practical battery joins electrochemistry, materials science, heat transfer, circuit theory, controls, and manufacturing. The equations below are useful models, but real cells are nonlinear and time-dependent. Accurate design should use the battery manufacturer's data over the required temperature, current, and state-of-charge range.

Cell voltage and chemical free energy

A cell reaction can perform electrical work because the reactants and products have different Gibbs free energy. For a reversible electrochemical reaction:

ΔG = −n F E

ΔG is the Gibbs free-energy change in joules per mole.
n is the number of electrons transferred per reaction.
F is Faraday's constant, approximately 96,485 coulombs per mole.
E is the reversible cell voltage.

A reaction with a more negative free-energy change can, in principle, provide a larger positive cell voltage. The usable voltage is lower under load because of resistive, kinetic, diffusion, and concentration losses.

The Nernst equation

The equilibrium voltage depends on temperature and chemical activities. For a reaction quotient Q:

E = E^° − (R T / n F) ln Q

Here E^° is the standard electrode potential, R is the gas constant, and T is absolute temperature. The Nernst equation helps explain why cell voltage changes with composition and state of charge. It does not, by itself, describe voltage drop caused by current.

Charge, capacity, and state of charge

Electric charge is the time integral of current:

q = ∫ I(t) dt

One ampere-hour equals 3,600 coulombs. A battery-management system may estimate state of charge by coulomb counting:

SOC(t) = SOC(t₀) − [1 / C_nom] ∫ η I(t) dt

The sign convention varies by system. η represents charge or discharge efficiency and C_nom is nominal capacity. Coulomb counting accumulates measurement error, so practical systems often correct it with open-circuit voltage, model-based observers, or periodic reference points.

Energy and power

P(t) = V(t) I(t)
Energy = ∫ V(t) I(t) dt

Ampere-hours measure charge, not energy. Two batteries with the same ampere-hour rating can store different energy if their voltages differ. Watt-hours are usually the more useful quantity for comparing the work available from complete batteries.

C-rate

C-rate expresses current relative to rated capacity. For a 4 Ah cell:

1C means 4 A.
0.5C means 2 A.
2C means 8 A.

The notation does not guarantee that the cell is safe or capable of that rate. The datasheet's current, temperature, voltage, and duty-cycle limits still control.

Equivalent-circuit models

Simple Thevenin model

V_terminal = V_OC − I R₀

The open-circuit source V_OC changes with state of charge and temperature. The series resistance R₀ represents the immediate voltage drop and ohmic heating.

Dynamic model

One or more resistor-capacitor branches may be added to represent polarization and diffusion. A common model uses a voltage source, a series resistance, and one or two parallel RC networks. Its parameters are identified from pulse tests and often stored in tables against temperature and state of charge.

Maximum-power transfer occurs in the simplest source model when load resistance equals internal resistance. That condition wastes half the generated power inside the source and can cause severe battery heating, so it is usually a mathematical limit rather than a desirable battery operating point.

Rate, temperature, and usable capacity

Rated capacity is measured under specified conditions. High current can reduce usable capacity because the terminal voltage reaches the equipment cutoff earlier. Low temperature slows reaction and diffusion rates and raises effective impedance. High temperature can improve short-term performance while accelerating aging and side reactions.

For lead-acid batteries, Peukert-type relationships are often used to describe how available capacity falls as discharge current rises. The exponent and reference conditions are empirical and battery-specific. Applying Peukert's law unchanged to lithium-ion cells can give misleading results.

Charging methods

Chemistry	Typical engineering approach	Important concerns
Lead-acid	Current-limited charging followed by controlled voltage stages.	Temperature compensation, gas generation, sulfation, float service.
Nickel-based	Controlled current with voltage, temperature, time, or pressure-related termination methods.	Overcharge heat, cell matching, charge efficiency.
Lithium-ion	Constant-current then constant-voltage charging within strict limits.	Overvoltage, undervoltage, temperature, current, cell balancing, protection.

The table describes broad engineering ideas, not construction instructions. Charge limits differ among chemistries and even among cells that share a common name. Use an approved charger or a properly engineered battery-management system matched to the exact cell manufacturer's requirements.

Series packs and balancing

Cells connected in series carry the same current, but they do not remain at identical state of charge. Small differences in capacity, leakage, temperature, and aging accumulate. A battery-management system therefore measures individual cell-group voltages and may remove charge from stronger groups or redistribute energy so that no group exceeds its limits.

Parallel groups require careful current sharing. Differences in resistance, connection length, cooling, and cell condition can make one branch carry more current than another. Pack design therefore includes bus-bar resistance, fusing, contact resistance, thermal paths, and fault containment.

State of health

State of health is not one directly measurable quantity. It may refer to remaining capacity, power capability, impedance, self-discharge, or ability to meet a particular duty cycle. Useful methods include controlled capacity tests, pulse-resistance measurements, electrochemical impedance techniques, model identification, and long-term trend analysis.

Applications and design choices

Application	Dominant requirements
Clock or memory backup	Very low self-discharge, long shelf life, modest current.
Vehicle starting	Very high short-duration current, low cost, cold-temperature capability.
Portable electronics	High specific energy, low mass and volume, accurate fuel gauging.
Cordless tools	High power, repeated cycling, robust thermal design.
Electric vehicles	Energy, power, cycle life, crash protection, thermal management, diagnostics.
Stationary storage	Cost per delivered kilowatt-hour, calendar life, maintainability, fire protection.

Sources and further reading

U.S. Department of Energy — DOE Explains: Batteries
Royal Swedish Academy of Sciences — scientific background on lithium-ion batteries
For practical design values, use the exact cell manufacturer's current datasheet, charging specification, safety information, and application notes.

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