# How to calculate the internal resistance of a battery pack

• Introduction
• Impact of battery pack resistance on performance
• Battery cell power loss
• Series battery cell configurations
• Parallel battery cells configurations
• Series and parallel battery cells configurations
• Battery pack configuration
• Fault tolerance

### Introduction

Modern battery technology aims to make batteries more efficient and have a longer life. A key factor in the design of battery packs is the internal resistance Rint [Ω] . Internal resistance is a natural property of the battery cell that slows down the flow of electric current. It’s made up of the resistance found in the electrolyte, electrodes, and connections inside the cell. In single battery cells, this resistance decides how much energy is lost as heat when the battery charges and discharges. For larger battery systems, like those in electric vehicles (EVs), understanding and managing the internal resistance becomes very important, because has direct impact on the range, efficiency and thermal management of the EV.

Image: Cylindrical battery cells

The resistance of a battery pack depends on the internal resistance of each cell and also on the configuration of the battery cells (series or parallel). The overall performance of a battery pack depends on balancing the internal resistances of all its cells. High internal resistance in a pack can make it less efficient, reduce its range, and create too much heat in EVs, which can be dangerous and shorten the battery’s life. Therefore, calculating and reducing the internal resistance of battery packs is crucial in designing efficient, safe, and long-lasting battery systems.

### Impact of battery pack resistance on performance

High internal resistance in a battery pack can significantly impact its efficiency. As electric current flows through the battery during charging and discharging, energy is lost primarily as heat, a direct consequence of the internal resistance. In practical terms, this means a greater portion of the battery’s stored energy is wasted, reducing its effective capacity. For EVs, this translates to reduced mileage per charge and for energy storage systems, a lower overall energy output.

Excessive heat generation due to high internal resistance is not merely an efficiency concern but also a safety issue. Batteries operating at higher temperatures are at an increased risk of thermal runaway, a condition where the battery becomes self-heating, leading to potential fires or explosions. Maintaining an optimal internal resistance is, therefore, crucial to ensure the safe operation of the battery pack.

Image: Chevy Bolt battery pack
Credit: GM

The longevity of a battery is also closely tied to its internal resistance. As resistance increases, more heat is generated during each charge-discharge cycle, accelerating the degradation of battery components. This results in a reduced lifespan of the battery pack, necessitating more frequent replacements – a significant factor in both cost and environmental impact.

Battery packs in EVs are are being operated at different loads, especially during vehicle acceleration or during hill climbing. Higher internal resistance can lead to a drop in performance under these high-load conditions, as the voltage drops more significantly with increased current draw. This results in less power being available to the electric machine, affecting the vehicle’s dynamic response and driving experience.

Battery packs are composed of numerous cells, and the resistance of each cell can vary slightly. This imbalance can lead to uneven charging and discharging, stressing certain cells more than others and leading to premature failure. Balancing the cells in terms of resistance is crucial to ensure uniform performance and prolong the overall life of the battery pack.

### Battery cell power loss

Internal resistance of a battery cell is a parameter which is not often published by the cell manufacturer. One method of calculating the internal resistance of the battery cell, based on the discharge curves, can be found here:  How to calculate the internal resistance of a battery cell. For now, let’s take a battery cell and assume it’s internal resistance is 60 mΩ. Also, let’s assume that the current output of the battery cell is 2 A.

Image: Batter cell parameters

In this case the power loss of the battery cell is calculated as:

Ploss = Rcell · Icell2 = 0.06 · 22 = 0.24 W

If we calculate the output power of the battery cell as:

Pcell = Ucell · Icell = 3.6 · 2 = 7.2 W

Based on the power losses and power output, we can calculate the efficiency of the battery cell as:

ηcell = (1 – Ploss/Pcell) · 100 = (1 – 0.24/7.2) · 100 = 96.67 %

### Series battery cell configurations

Let’s assume that we have a battery pack made up by 3 identical battery cells connected in series. In a series connection, the positive terminal (+) of one battery cell is connected to the negative terminal (-) of the next battery cell, and so on, until the circuit is completed back to the power source.

Image: Battery cells connected in series

In a series circuit, the same current flows through each battery cell, which means that the current output of the battery pack will be equal with the current output of one cell. If we assume that the current through the battery cells is Icell = 2 A, the current through the battery pack will be:

Ipack = Icell = 2 A

In series circuits, the voltages of individual cells add up to give the total voltage across the battery pack. If each cell has the same voltage Ucell = 3.6 V the battery pack voltage will be the sum of all battery cell voltages. This is also equal to the product between number of cells connected in series Ns = 3 and the voltage of the cells Ucell:

Upack = Ns · Ucell = 3 · 3.6 = 10.8 V

The total resistance in a series circuit is the sum of the individual resistances. If each cell has the same resistance of Rcell = 60 mΩ, the internal resistance of the battery pack will be the sum of battery cells resistances, which is equal with the product between the number of battery cells in series Ns and the resistance of the cells in series Rcell.

Rpack = Ns · Rcell = 3 · 0.06 = 180 mΩ

By connecting cells in series, the total resistance increases, which can affect the discharge rate of the battery pack. In practical applications, a balance must be struck between the desired voltage output and the internal resistance to ensure efficient operation of the battery pack.

### Parallel battery cells configurations

In a parallel connection, all the positive terminals (+) of the batteries are connected together, and all the negative terminals (-) are also connected together. This forms a parallel circuit with the main power leads at either end. In a battery pack with 3 identical battery cells connected in parallel, the positive terminals are connected by one conductor, and the negative terminals are connected by another, forming parallel paths for the current.

Image: Battery cells in parallel

In a parallel circuit, the total current of the battery pack is the sum of the currents through each individual branch. If the current through each battery cell is Icell = 2 A and there are 3 cells connected in parallel (Np = 3), the battery pack current is calculated as:

Ipack = Np · Icell = 3 · 2 = 6 A

In parallel circuits, the voltage across each cell is the same and equal to the voltage of the power source. If the voltage drop across all cells is Ucell = 3.6 V, the voltage of the battery pack is equal with the cell voltage:

Upack = Ucell = 3.6 V

The total resistance in a parallel circuit is less than the smallest resistance of any of the branches. Thus, if the resistance of the battery cells is Rcell = 60 mΩ, the internal resistance of the battery pack will be:

1/Rpack = 1/Rcell + 1/Rcell + 1/Rcell = 3/Rcell = Np/Rcell

Solving the equation above makes the resistance of the battery pack equal with the ratio between the resistance of the battery cells and the total number of cells connected in parallel (Np = 3):

Rpack = Rcell/Np = 0.06/3 = 0.02 = 20 mΩ

Parallel connections are typically used to increase the capacity and discharge current of a battery pack without increasing the voltage. The total capacity of the battery pack is the sum of the capacities of the individual cells. However, the voltage of the pack remains the same as the voltage of a single cell.

### Series and parallel battery cells configurations

Battery packs used for electric vehicles have a combination of battery cells connected in series and parallel. The number of battery cells in series if mainly function of the requested battery pack voltage. The number of battery cells connected in parallel is given by the maximum current output required from the battery pack. To obtain a predefined battery pack voltage and current is possible through different battery cells configurations.

 Image: Battery cells 3S2P configuration Image: Battery cells 2P3S configuration

The two configurations above have the same number of battery cells in series and parallel: 3 cells in series and 2 cells in parallel. In the 3S2P configuration, 3 battery cells are first connected in series (Ns = 3), and then the resulting battery cells strings are connected in parallel (Np = 2). In the 2P3S configuration, the battery cells are first put 2 in parallel (Np = 2) and then the 3 pairs of 2 cells are connected in series (Ns = 3). From the electric connection point of view, both configuration would have the same parameters.

Given that all battery cells are identical and have the following parameters: Icell = 2 A, Ucell = 3.6 V and Rcell = 60 mΩ, applying the equations used in series and parallel battery cells connections, the current, voltage and resistance of both battery pack configurations are calculated as:

Ipack = Np · Icell = 2 · 2 = 4 A

Upack = Ns · Ucell = 3 · 3.6 = 10.8 V

Rpack = (Ns/Np) · Rcell = (3/2) · 0.06 = 0.09 = 90 mΩ

The power output of the battery pack is equal to:

Ppack = Ipack · Upack = 43.4 W

The power loss of the battery pack is calculated as:

Ploss = Rpack · Ipack2 = 0.09 · 42 = 1.44 W

Based on the power losses and power output, we can calculate the efficiency of the battery pack as:

ηpack = (1 – Ploss/Ppack) · 100 = (1 – 1.44/43.4) · 100 = 96.682 %

The 2P3S configuration is generally more fault-tolerant compared to the 3S2P configuration.

In a 2P3S configuration, if one cell fails or is removed, the parallel connection ensures that the series string it’s a part of can still operate at a reduced capacity. The remaining cell in the parallel pair can continue to provide current, and thus the series connection remains intact. This means the pack maintains its voltage output but at a reduced capacity and current output. The system can continue to operate, albeit at a lower performance level.

In contrast, in a 3S2P configuration, if one cell fails or is removed from one of the series strings, that entire string will cease to function because a series circuit requires all components to be connected for current to flow. This results in a complete loss of half of the battery pack’s capacity. The overall battery pack can no longer function correctly unless there’s a bypass mechanism in place, which would still result in a significant reduction of the total voltage and energy capacity of the pack.

Therefore, while both configurations have the same voltage, current, resistance, and power under normal conditions, the 2P3S configuration offers better fault tolerance. It allows the battery pack to continue operating with one cell down, making it more robust against single-point failures. This attribute is particularly valuable in applications where reliability is critical, and it’s essential to maintain operation even in the event of a partial failure.

### Battery pack configuration

Let’s assume that we have a battery pack with a total of 3978 battery cells. To obtain the required voltage we need to have a string of 153 cells in series and 26 strings in parallel. The 153 cells should not be arranged into continuous strings since one fault on a single cell will disable the entire string and the battery capacity would be halved. Also, it would be more difficult to balance the entire string plus, in case of a malfunction the whole battery would have to be replaced.

One way to configure this battery pack is to build a battery module with 13 cells in parallel to form a group and then link 9 groups in series. The configuration of the battery module would be 13P9S (Np = 13, Ns = 9).

Image: Battery module cell arrangement 13P9S.

If a battery cell fails in one of the parallel paths, the module’s capacity and current output may slightly decrease, but the remaining cells will continue to maintain the module’s voltage.

The second step of the battery pack configuration is to create a string of 17 modules and connect 2 strings of modules in parallel. This will make the configuration of the battery pack as 17S2P (Np = 2, Ns = 17).

Image: Battery pack module arrangement 17S2P.

If one module of the battery pack will fail, the pack will still still have the same operating voltage but with reduced capacity and current output.

To verify that the number of cells add up for the entire battery pack, we are going to calculate:

• total number of cells in series Ns = 9 · 17 = 153
• total number of battery cells in parallel Np = 13 · 2 = 26
• total number of cells in the battery pack Npack = 153 · 26 = 3978

Assuming that all battery cells are identical and have the following parameters: Icell = 2 A, Ucell = 3.6 V and Rcell = 60 mΩ, calculate the following parameters of the battery pack: current, voltage, internal resistance, power, power losses and efficiency.

Ipack = Np · Icell = 26 · 2 = 52 A

Upack = Ns · Ucell = 153 · 3.6 = 550.8 V

Rpack = (Ns/Np) · Rcell = (153/26) · 0.06 = 0.353 = 353 mΩ

Ppack = Ipack · Upack = 52 · 550.8 = 28641.6 W = 28.6 kW

Ploss = Rpack · Ipack2 = 0.353 · 522 = 954.512 W

ηpack = (1 – Ploss/Ppack) · 100 = (1 – 954.512/28641.6) · 100 = 96.6674 %

### Fault tolerance

From the fault tolerance point of view, the configuration of the battery modules and pack have the following implications:

• battery module (13P9S): individual cells are grouped into modules, which means the Battery Management System (BMS) can potentially isolate issues at the module level without affecting the entire pack. The BMS must balance cells within each module, and redundancy at the module level allows for some fault tolerance.
• battery pack (17S2P): Modules are less redundant when configured in series at the pack level. A failure in one module affects the entire series string. Parallel connections provide some redundancy, but the series connection’s vulnerability highlights the importance of reliable module-level protection and fault detection.

Overall, the series connections set the voltage, and parallel connections determine the capacity and current capability of the battery system. The BMS plays a critical role in monitoring and managing the health and safety of the system, addressing issues such as cell balancing, temperature monitoring, and fault detection to prevent failures that could affect the entire system. The design must also consider ease of maintenance and the possibility of replacing individual modules or cells, as the accessibility and replaceability can greatly influence the practicality and service life of the battery pack.