The loop matrix

2.2.2 The loop matrix

Topological matrixes can be compiled for circuit loops too.

= (2.4)

where, are elements of the matrix . =1 if the branch is incident to the loop and coincides with the direction of loop path-tracing; =-1 if the branch is incident to the loop and opposite to the direction of loop path tracing; =0 if the branch is not incident to the loop . The loop matrix corresponding to the directed graph with “ ” loops and “ ” branches, is the name for the matrix . For instance, for the directed graph in Fig. 2.3,c we will get the matrix (2.5) when loop path-tracing is clockwise. Here, the branch of the current source is not used in the matrix because such a branch does not form a separate loop as mentioned above.

It’s obvious that the parts of loops not used in the matrix (2.5) are linearly dependent. A loop, that includes at least one branch not being part of any other loops, is linearly independent.

branches

loops

For the planar circuits it’s easy to define the number of independent loops if we take these loops as meshes of the graph grid or “windows”. So for the graph in Fig.2.3,c the loops will be the following: Then we get the submatrix from the matrix .

= (2.6)

The matrix is called the matrix of main loops.

For nonplanar circuits, it is difficult to define the number of dependent loops from the number of “windows” in the graph. In this case, the T graph tree is used. A loop is formed by the connection of any chord to a tree. This loop is called the main loop. The direction of main loop path-tracing is taken as the chord direction. Thus, the number of main loops is equal to the number of chords of the selected graph tree. It is evident that it is equal to . So, main loops for the tree in Fig. 2.3,b are formed by the chords: chord loop chord loop chord loop Having arranged the edges in the lower columns, we get the matrix of main loops for the graph with the tree in Fig. 2.3,b