## Theoretical Part

### Calculation method

#### Positive Plate

The positive grid is regarded as a network of interconnected linear ohmic resistors and an external current distribution. These currents are being collected finally in a single point (the lug).

#### Negative Plate

The picture of an equipotential negative plate is an oversimplification of the real picture and leads to a false channel current distribution. A more realistic picture which is easily incorporated in the existing model is described in Figure 1. The negative plate is considered as a bunch of wires leading from the lug to the different nodes. All wires have the same cross section area. By this way the resistance of the negative plate is reduced into a set of linear ohmic resistors which can be easily calculated and added to the channel resistance.

###### Figure 1

#### Elementary Cell

The two dimensional networks, both the positive grid and the negative grid, will be linked with the electrolyte resistance and the polarization to a lumped network (Figure 1).

The current flows through the different elements of the positive grid and will be collected in the positive lug. From there it is distributed through the negative plate and the electrolyte into the different channels. The theory of such a network is based on the Kirchhoff’s laws and the Tellegen theorem. The result is a matrix version of the ohmic law:

I = U * G

- I is the distribution of the current in the branches,
- U is the distribution of the potential of the nodes,
- and G is the conductance matrix.

Known parameters are the total current, a set of start values for the distribution of the current into the different cannels and the conductance matrix G.

To solve the above equation an iterative approach is used.

- A set of starting values for the node potentials is selected. Normally the same potential is assigned to all nodes with exception of the lug node.
- With these starting values a new potential can be calculated for each node because the sum of all currents has to be zero.
- A new current distribution results from these new potentials.
- Step two and three are repeated until the change of the potential for each node falls below a specified value.

This method generally will converge in very good, if some certain mathematical constraints are satisfied. Finally the internal resistance of the elementary cell results from the given total current and the difference of the potentials of the positive and negative lug.

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