Comprehensive Guide to Basic Formulas for Lithium Batteries: From Electrical Models to Life Prediction​ Lithium - Ion Battery Equipment

Lithium batteries rely on mathematical models to evaluate and optimize their performance. This article explains the three core models—​​electrical model​​, ​​thermal model​​, and ​​life model​​—in simple terms, helping you understand the science behind battery operation. Lithium - Ion Battery Equipment

​​1. Electrical Model: The Battery's "Circuit Diagram"​​

The electrical model uses circuit components (resistors, capacitors, etc.) to simulate battery behavior during charging and discharging, like drawing a "circuit diagram" of the battery.

​​1. Equivalent Circuit Models​​

• ​​Rint Model​​: The simplest model, treating the battery as an ideal voltage source with a series resistor (internal resistance). Good for rough voltage estimates.
• ​​Thevenin Model (1st-order RC)​​: Adds a capacitor to simulate short-term dynamic responses (e.g., voltage stabilization after charging).
• ​​Dual-Polarization Model (2nd-order RC)​​: Separates electrochemical and concentration polarization for higher accuracy, especially in fast charging.

​​2. Terminal Voltage Calculation​​

The battery's terminal voltage Vbat​ is calculated as:

Vbat​=VOCV​(SOC)−I⋅R0​−Vp1​−Vp2​

• ​​VOCV​(SOC)​​: Open-circuit voltage, related to the state of charge (SOC).
• ​​I⋅R0​​​: Ohmic voltage drop due to internal resistance.
• ​​Vp1​,Vp2​​​: Polarization voltages representing dynamic responses.

​​3. SOC Calculation (Coulomb Counting)​​

SOC (State of Charge) indicates remaining battery capacity:

SOC(t)=SOC0​+Cbat​1​∫0t​η⋅I(τ)dτ

• ​​Cbat​​​: Battery capacity (Ah), affected by temperature and health (SOH).
• ​​$\eta$​​: Coulombic efficiency (typically 1 for discharge, <1 for charge).

​​2. Thermal Model: The Battery's "Thermometer"​​

Lithium batteries generate heat during operation, and excessive temperatures can reduce lifespan or cause safety hazards. The thermal model predicts temperature changes like a "thermometer."

​​1. Heat Generation (Bernardi Model)​​

Pheat​=I⋅(Vbat​−VOCV​)+I⋅T⋅dTdVOCV​​

• ​​Irreversible heat​​: Energy loss from internal resistance (Joule heating).
• ​​Reversible heat​​: Entropy-related heat from electrode reactions.

​​2. Heat Transfer Equations​​

• ​​Conduction (Fourier’s Law)​​:qcond​=−k⋅∇T
• ​​Convection (Newton’s Cooling Law)​​:qconv​=h⋅A⋅(Tbat​−Tamb​)
• ​​Radiation (Stefan-Boltzmann Law)​​:qrad​=ϵ⋅σ⋅A⋅(Tbat4​−Tamb4​)

​​3. Temperature Change​​

Based on energy conservation:

dtdT​=m⋅Cp​Pheat​−qcond​−qconv​−qrad​​

• ​​High temperatures​​ accelerate aging or thermal runaway.
• ​​Low temperatures​​ reduce charging efficiency and may cause lithium plating.

​​3. Life Model: The Battery's "Aging Clock"​​

Battery degradation occurs over time and usage cycles. The life model predicts remaining lifespan using empirical formulas.

​​1. Calendar Aging (Time-Based Degradation)​​

Qloss,cal​=A⋅e−RTEa​​⋅tn

• Higher temperatures ($T$) accelerate capacity loss.

​​2. Cycle Aging (Charge-Discharge Degradation)​​

Qloss,cyc​=B⋅e−RTEa​​⋅(Ahthroughput​)z

• Fast charging (high current) shortens battery life more than slow charging.

​​3. Total Capacity Loss​​

Qloss,total​=Qloss,cal​+Qloss,cyc​

• Typically, a battery reaches end-of-life (EOL) at 80% capacity retention.

​​4. Practical Applications​​

1. ​​Battery Management Systems (BMS)​​: Use electrical models for real-time SOC estimation.
2. ​​Fast-Charging Optimization​​: Combine thermal models to prevent overheating and lithium plating.
3. ​​Lifespan Prediction​​: Helps plan battery replacements in EVs and energy storage systems.

​​Key Takeaways​​

• ​​Electrical Model​​: Describes voltage and SOC using circuit theory.
• ​​Thermal Model​​: Predicts temperature to prevent overheating or overcooling.
• ​​Life Model​​: Estimates how long a battery will last.

For everyday users: ​​Avoid extreme temperatures, don’t always charge to 100%, and prefer slow charging for longevity!​