A key parameter to calculate and then measure is the battery pack internal resistance. This is the DC internal resistance (DCIR) and would be quoted against temperature, state of charge, state of health and charge/discharge time.

#### DCIR of a Cell

Symbolically we can show a cell with the internal resistance as a resistor in series.

R_{int} is the DC internal resistance, sometimes abbreviated as DCIR.

The DCIR is not just a single number for any given cell as it varies with State of Charge, State of Health, temperature and discharge time.

### Components

The internal resistance of the battery pack is made up of the cells, busbars, busbar joints, fuses, contactors, current shunt and connectors. As the cells are connected in parallel and series you need to take this into account when calculating the total resistance. The other components are normally connected to the cells in series and typically add up to around 10mΩ to 30mΩ.

### Series and Parallel

When we look at the cells arranged in parallel and series we need to take that into account when calculating the overall resistance. Consider each cell as a resistor, then we get:

If P is the number of cells in parallel and S is the number of cells in series, we can calculate the total cell resistance for this configuration (note: this is assuming that the cell resistance, R_{cell}, values are all the same.

*R _{T} = S x R_{cell} / P*

### Impact

Simple Ohms Law shows us that the voltage under load is determined by the internal resistance.

If we have an OCV of 3.7V @ 50% SOC and an internal resistance of 0.025Ω and we draw 10A from the cell the voltage will drop 0.25V This is simply Ohms Law.

V = 3.7V – 10A x 0.025Ω = 3.45V

Hence the voltage of the cell under a 10A load will be 3.45V

The heat generated by the cells is dominated by Joule heating and this is equal to the resistance multiplied by the current squared. The heat generated in the busbars is related to the resistance of the busbar. This is the same for the contactors, fuses and connectors.

Hence. high power capability is related to low internal resistance, this is true for single cells and packs.

The following plot shows the peak power capability plotted versus the estimated battery pack internal resistance.

This is a very simple overview that will get you to an estimation of the internal resistance. There are a number of factors that need to be included in a more detailed study:

- variation in cell resistance part to part
- temperature dependence
- variation with SoC
- variation with SoH

All of these need to be taken into account to fully understand how the battery will behave when new and over it’s lifetime.