Understanding the Calculation of Internal Resistance of a Cell
Internal resistance calculation quantifies a cell’s opposition to current flow within itself. This article explores methods, formulas, and applications.
Discover detailed formulas, common values, and real-world examples to master internal resistance measurement in electrochemical cells.
- Calculate the internal resistance of a 1.5V AA alkaline cell with a terminal voltage of 1.3V under 0.5A load.
- Determine the internal resistance of a lithium-ion battery given open-circuit voltage 3.7V and loaded voltage 3.5V at 2A current.
- Find the internal resistance of a lead-acid cell with 12.6V open-circuit voltage and 11.8V under 10A load.
- Estimate the internal resistance of a NiMH rechargeable cell with 1.2V open-circuit voltage and 1.1V under 1A load.
Comprehensive Tables of Common Internal Resistance Values
| Cell Type | Nominal Voltage (V) | Typical Load Current (A) | Measured Terminal Voltage (V) | Open Circuit Voltage (V) | Calculated Internal Resistance (mΩ) |
|---|---|---|---|---|---|
| Alkaline AA | 1.5 | 0.5 | 1.3 | 1.5 | 400 |
| NiMH AA | 1.2 | 1.0 | 1.1 | 1.2 | 100 |
| Lead Acid 12V | 12.6 | 10 | 11.8 | 12.6 | 80 |
| Lithium-ion 18650 | 3.7 | 2 | 3.5 | 3.7 | 100 |
| LiFePO4 Cell | 3.3 | 5 | 3.1 | 3.3 | 40 |
| Button Cell (Silver Oxide) | 1.55 | 0.01 | 1.53 | 1.55 | 20 |
| Nickel-Cadmium (NiCd) | 1.2 | 0.5 | 1.15 | 1.2 | 100 |
| Zinc-Carbon Cell | 1.5 | 0.2 | 1.3 | 1.5 | 1000 |
| Lead Acid Deep Cycle | 12.8 | 20 | 12.0 | 12.8 | 40 |
| LiPo Battery Pack (3S) | 11.1 | 15 | 10.5 | 11.1 | 40 |
Fundamental Formulas for Calculating Internal Resistance of a Cell
The internal resistance (Rint) of a cell is a critical parameter that affects its performance, efficiency, and lifespan. It represents the opposition to current flow inside the cell, caused by electrolyte resistance, electrode material, and other internal factors.
The most common method to calculate internal resistance involves measuring the voltage drop under load and applying Ohm’s law. The primary formula is:
- Rint: Internal resistance of the cell (Ohms, Ω)
- Voc: Open circuit voltage (Volts, V) – voltage measured when no load is connected
- Vload: Loaded terminal voltage (Volts, V) – voltage measured when the cell is supplying current
- I: Load current (Amperes, A) – current drawn from the cell during measurement
Typical values for these variables depend on the cell chemistry and condition:
- Voc: Usually close to nominal voltage, e.g., 1.5V for alkaline, 3.7V for lithium-ion.
- Vload: Slightly less than Voc due to voltage drop caused by internal resistance.
- I: Depends on the test load; common test currents range from milliamps to tens of amps.
Alternative Formulas and Considerations
In some cases, internal resistance is expressed in milliohms (mΩ) for precision, especially in high-current cells. The formula remains the same, but the result is multiplied by 1000.
For dynamic or AC internal resistance measurement, impedance spectroscopy or pulse methods are used, but these require specialized equipment and are beyond the scope of this article.
Detailed Explanation of Variables and Their Typical Ranges
| Variable | Description | Typical Range | Units |
|---|---|---|---|
| Voc | Open circuit voltage of the cell | 1.2 – 12.8 (depending on chemistry) | Volts (V) |
| Vload | Terminal voltage under load | Varies, typically 0.9 × Voc to Voc | Volts (V) |
| I | Load current during measurement | 0.01 – 20 | Amperes (A) |
| Rint | Internal resistance of the cell | 0.02 – 1.0 (or 20 – 1000 mΩ) | Ohms (Ω) or milliohms (mΩ) |
Real-World Applications and Case Studies
Case Study 1: Measuring Internal Resistance of an Alkaline AA Cell
An engineer needs to determine the internal resistance of a standard AA alkaline cell to assess its health for a portable device. The open circuit voltage (Voc) is measured at 1.5V. When a 0.5A load is applied, the terminal voltage (Vload) drops to 1.3V.
Using the formula:
This internal resistance value indicates moderate cell degradation, as new alkaline cells typically have internal resistance around 200-300 mΩ. The engineer concludes the cell is still usable but nearing end-of-life for high-drain applications.
Case Study 2: Internal Resistance of a Lithium-ion Battery Pack
A technician tests a 3.7V lithium-ion 18650 cell used in an electric vehicle battery pack. The open circuit voltage is 3.7V. Under a 2A load, the terminal voltage drops to 3.5V.
Calculate the internal resistance:
This low internal resistance is typical for healthy lithium-ion cells, confirming the battery pack is in good condition. The technician uses this data to predict battery performance and schedule maintenance.
Additional Considerations for Accurate Internal Resistance Measurement
- Temperature Effects: Internal resistance varies with temperature; higher temperatures generally reduce resistance.
- State of Charge (SoC): Resistance changes with SoC; cells near full charge typically have lower internal resistance.
- Measurement Method: Using a stable load and precise voltmeters is essential for accuracy.
- Dynamic vs. Static Resistance: Static resistance is measured under steady load; dynamic resistance involves transient responses and requires advanced techniques.
Recommended Tools and Techniques for Measurement
- Digital Multimeter: For measuring open circuit and loaded voltages.
- Precision Load Resistor: To apply a known current load.
- Battery Analyzer: Specialized devices that automate internal resistance measurement.
- Electrochemical Impedance Spectroscopy (EIS): For advanced internal resistance and impedance profiling.
Further Reading and Authoritative Resources
- BatteryStuff.com – Internal Resistance of Batteries
- EE Times – Understanding Battery Internal Resistance
- NREL Technical Report on Battery Internal Resistance Measurement
- ScienceDirect – Internal Resistance in Electrochemical Cells
Mastering the calculation of internal resistance is essential for battery diagnostics, design optimization, and ensuring reliable power delivery in electronic systems. This article provides a comprehensive foundation for engineers and technicians to accurately measure and interpret internal resistance values across various cell chemistries and applications.