The DCIR of a cell is the Direct Current Internal Resistance. This is the resistance in charge and discharge to a direct current demand applied across the terminals.

### Cells are not a Perfect Current Source

The electrical symbol for a cell. Used in any electrical circuit schematic and in it’s simplest form.

A battery cell is not a perfect current source as it also has an **internal resistance**.

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.

The DCIR of a cell is normally measured using a defined current against time pulse. Typically the pulse duration is from 1s to 30s and most quoted values are for a 10s pulse. The resistance is the maximum voltage drop divided by the current demand.

There are a number of phenomena contributing to the voltage drop, governed by their respective timescales:

- the instantaneous voltage drop is due to the pure Ohmic resistance R
_{0}which comprises all electronic resistances and the bulk electrolyte ionic resistance of the battery - the voltage drop within the first few seconds is due to the battery’s double layer capacitance and charge transfer resistance RCT which is attributed to the charge transfer reaction at the electrode/electrolyte interface
- the shallow, linear (or close to linear) voltage drop is due to polarisation resistance R
_{p}which accounts for ionic difusion in the solid phase and is usually considered to be the rate determining step for Li ion batteries.

Barai, A., Uddin, K., Widanage, W.D. et al. A study of the influence of measurement timescale on internal resistance characterisation methodologies for lithium-ion cells. Sci Rep 8, 21 (2018)

### DCIR vs Discharge Time

In the 1 to 100 second timescale the resistance will be dominated by electron transfer, ion transfer and ion diffusion. Ion transfer and diffusion dominating the longer 30 to 100s timescale and hence resulting in a larger internal resistance.

The very short timeframes will be related to the complex impedance of the cell and the 0.001s timescale shown in this graph is what we also cal the 1kHz ACIR.

This graph is from [1] where the paper compares cell resistance measurement techniques.

### DCIR vs SoC

The graph shows the DCIR for two cells versus their capacity. The P42B is a cell with a total capacity of 4200mAh and the P45B has a total capacity of 4500mAh.

Note: this graph does appear to have a reversed x-axis. Here capacity is the amount of charge removed from the cell before the 10A and 10s pulse test is applied. Therefore, 1000mAh for the P42B is 76% SoC and 1000mAh for the P45B is 77.8% SoC.

At low SoC the internal resistance of the cell increases significantly. At high SoC the cell is also has a slightly higher DCIR.

### DCIR vs Temperature

The internal resistance of a cell decreases as the temperature increases. This is very dependent on cell chemistry and on cell design. However, it generally holds that as you cool the cell down the internal resistance increases.

In simple terms this is because batteries generate current using a chemical reaction and the reaction generally goes slower at lower temperatures.

Turning this around it has been shown that the internal resistance of the cell can be used to estimate the average temperature of the cell jelly roll [3].

### DCIR vs Cycling

As the cell is cycled the capacity fades and internal resistance of the cell increases [4].

The reduced capacity and increased ageing are related to the loss of available ions to side reactions and the loss or damage of the anode and cathode structure.

### Calculated Example

For most simple peak power calculations we will be interested in the DCIR value for a new cell at 50% SOC, 25°C and for a 10s pulse.

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

We can also calculate the maximum current we can draw taking the cell down to the minimum voltage:

2.5V = 3.7V – I x 0.025Ω

I = (3.7V – 2.5V) / 0.025Ω = 48A

These numbers are quite typical of a 5Ah NMC cell. Peak discharge is around 10C.

If we want more power then we need more voltage or more current. We could:

- use a large battery cell
- put more cells together in series / parallel

#### References

- Barai, A., Uddin, K., Widanage, W.D. et al. A study of the influence of measurement timescale on internal resistance characterisation methodologies for lithium-ion cells. Sci Rep 8, 21 (2018).
- Molicel P45B Specifications
- Sebastian Ludwig, Marco Steinhardt and Andreas Jossen, Determination of Internal Temperature Differences for Various Cylindrical Lithium-Ion Batteries Using a Pulse Resistance Approach, Batteries
- Calum Strange, Gonçalo dos Reis, Prediction of future capacity and internal resistance of Li-ion cells from one cycle of input data, Energy and AI, Volume 5, 2021

#### Power versus Energy Cell

Comparing power versus energy cells we see there are some fundamental differences. A high energy cell will have better volumetric and gravimetric energy density at the expense of the ability to deliver a high current. The power cell will have a low internal resistance and will be optimised to deliver current over energy density.

#### Heat Generation in a Cell

Heat generation in a cell can be defined quite simple for the case where the cell is operating within it’s normal limits.