For the automatic circuit analysis purpose the nodal analysis is more convenient.
According to this method the potentials at the nodes are the independent unknowns. The use
of node potentials, rather than mesh currents, makes for greater efficiency in the analysis of
circuits that are predominantly running in parallel to other branches, since the number of
simultaneous equations that have to be solved is significantly smaller. The general procedure
to obtain the equation set for a complex circuit is the following:
1) choose a convenient reference (given potential for a set of nodes - in our case the
liquid metal of the previous cell where the internal electric current distribution is computed
in the liquid layers) and assume a potential (unknown before the solution) for all remaining
nodes;
2) in the equation for a particular node, the known coefficient at the potential of that
node is the sum of the conductances (inverse to resistances) connected to it; the coefficients
of the other potentials are the conductances joining their respective nodes to the node in
question;
3) this algebraic sum of potentials multiplied by conductances is equal to an external
current entering this node.
The following equation set arises for the total number of M nodes each of which has
N
directly connected neighbor resistances:
(4)
where U
m
is the potential at a node, U
n
- for nodes at other ends of neighbor bars, R
n
-
resistances of neighbor bars, I
m
- external current entering the node. In our case total current
`I' enters the reference nodes in the liquid metal of previous cell and `-I' current leaves the
nodes at the liquid metal of the downstream cell. For all other nodes the external current -
right side of the equation - is zero.
This is just another statement of the current law, and the voltage law is satisfied
because the sum of the potential differences, with U
m
expressed from the equation (4), over
the closed mesh is identically zero. Formally this law can be applied even for two resistance
elements connected in series, and this property is essential in order to generate automatically
the set of equations to solve.
After finding the potentials at the nodes, the potential difference between two
neighbor nodes multiplied by the connecting resistance gives the current in each resistance.
The main task of the present program is to find the current distribution in the bar network,
yet a further improvement in accuracy can be achieved if computing Joule heating:
R
I
m
m
2
for each of the resistance elements.