Counting the Number of Loop Currents Required

The number of loop currents required to describe any electric network may be determined by removing successive branches from the network until no closed paths remain anywhere in the network, but without disconnecting any of the nodes (this is called converting the network into a tree). The number of loop currents required equals the number of branches that had to be removed. This is a unique number, independent of which branches were removed.




Example: In the diagram to the right we show a network from which we have removed all the resistors and voltage sources so that we can focus attention on the topology of the network.

How many loop currents are required to describe the network?

Solution: Counting nodes and branches we see that this network has 6 branches and 4 nodes, labelled B1, ..., B6 and N1, ..., N4.

For example, cutting branch B2, then B4 and then B6 results in this tree which cannot be cut further without removing a node: whereas cutting branch B1, then B5 and then B6 results in this tree which cannot be cut further without removing a node:


In either case we conclude that this network can be described by 3 loop currents.



Alternative Method

If the network is planar (ie. can be laid out flat with no wires crossing over others) then the number of loop currents required to describe a network equals the number of meshes in the network. The meshes are the "holes" in the network.




Example: The network to the right is planar and has three meshes, labelled M1, M2 and M3 and therefore can be described by three loop currents.

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