Cutset Matrix Concept of Electric Circuit

Contents

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Key learnings:

Cut-Set Matrix Definition: A cut-set matrix is defined as a matrix showing the cut-sets of a connected graph in an electric circuit.
Cut-Set Concept: A cut-set is the minimum set of branches that separates a graph into two sub-graphs when removed.
Fundamental Cut-Set: Formed using one twig and the remaining links, fundamental to understanding graph structures.
Matrix Construction: To construct the cut-set matrix, identify twigs and links, and form rows for cut-sets and columns for branches.
Orientation in Matrix: Orientation of branches in the cut-set matrix is positive if it matches the twig’s orientation, negative if opposite, and zero if not included.

When discussing the cut-set matrix in graph theory, we generally refer to the fundamental cut-set matrix. A cut-set is the smallest set of branches in a connected graph that, when removed, separates the graph into two sub-graphs. The cut-set matrix is created by taking each cut-set row-wise. It is denoted by the symbol [Q_f].

Example of Cutsets Matrix of a Circuit

Two sub-graphs are obtained from a graph by selecting cut-sets consisting of branches [1, 2, 5, 6].
In other words, the fundamental cut-set of a graph, in reference to a tree, consists of one twig and the remaining links. Twigs are the branches of the tree, and links are the branches of the co-tree.
Thus, the number of cutset is equal to the number of twigs.
[Number of twigs = N – 1]
Where, N is the number of nodes of the given graph or drawn tree.
The orientation of cut-set is the same as that of twig and that is taken positive.

Steps to Draw Cut Set Matrix

There are some steps one should follow while drawing the cut-set matrix. The steps are as follows-

Draw the graph of given network or circuit (if given).
Then draw its tree. The branches of the tree will be twig.
Then draw the remaining branches of the graph by dotted line. These branches will be links.
Each branch or twig of tree will form an independent cut-set.
Write the matrix with rows as cut-set and column as branches.

	Branchase ⇒	1	2	3	.	.	b
Cutsets
C₁
C₂
C₃
.
.
C_n

n = number of cut-set.
b = number of branches.

Orientation in Cut Set Matrix

Q_ij = 1; if branch J is in cut-set with orientation same as that of tree branch.
Q_ij = -1; if branch J is in cut-set with orientation opposite to that of branch of tree.
Q_ij = 0; if branch J is not in cut-set.
Example 1

Draw the cut-set matrix for the following graph.
Answer:
Step 1: Draw the tree for the following graph.

Step 2: Now identify the cut-set. Cut-set will be that node which will contain only one twig and any number of links.

Here C₂, C₃ and C₄ are cut-sets.
Step 3: Now draw the matrix.

	Branchase ⇒	1	2	3	4	5	6
Cutsets
C₂		+1	+1	0	0	-1	0
C₃		0	0	+1	0	+1	-1
C₄		-1	0	0	+1	0	+1

This is the required matrix.
Example 2:

Draw the cut-set of the given graph.
Answer:
Again in this question we have to repeat the same steps as done in previous question.
Step 1: Draw the tree for the following graph.

Step 2: Now identify the cut-set. Cut-set will be that node which will contain only one twig and any number of links.

Here C₁ and C₅ are cut-sets.
Step 3: Now draw the matrix.