Fundamental Cut Set Matrix - Electrical Network Graphs

This is my fourth article on an electrical network graph. The article is about the cut-set matrix and how it is useful in determining branch voltages and node voltages. As the name implies cut set is a set of branches that when removed from the graph results in an unconnected graph.

Key Questions:

• What is a cut set and a fundamental cut set?
• Procedure for obtaining cut sets of a graph and writing matrix elements
• Fundamental cut set matrix and KVL
• Fundamental cut set and KCL

Cut Set and Fundamental Cut Set:

Cut set: It is a set of branches that when cut or removed from a graph, separates the graph into two groups of nodes or the graph splits into two isolated graphs. It is a minimum set of branches of a connected graph, such that if this set of branches is removed, it will reduce the rank of the graph. And also results in an unconnected graph.

• Cut set defines the minimum set of branches C, if this set of branches removed from the graph results in an unconnected graph.

• If anyone branch from a set of branches C is removed, we still get a connected graph.

Fundamental cut set: It contains only one twig and one or more links.

Procedure For Obtaining Fundamental Cut Set:

As discussed above fundamental cut set contains only one twig and one or more links. To obtain a fundamental cut set follow these steps.

• Select an oriented graph
• Select a tree and mark the links as well with the dotted line
• The number of twigs is equal to the number of cut sets
• Remove one twig and necessary links to get the fundamental cut set
• Repeat the above procedure for all twigs

Fig 1 graph and its tree

Fig 2 Fundamental Cut Sets

Have a closer look at fig 2.

• F-cut set C1 is obtained by removing a twig 5 and a link 3. The direction or orientation of C1 is in the same direction of twig 5
• F-cut set C2 is obtained by removing twig 6 and links 4 and 3. The orientation of C2 is the same as the direction of twig 6
• F-cut set 3 C3 is obtained by removing twig 1 and links 2 & 4. The orientation of C3 is the same as the direction of twig 1

Fundamental Cut Set Matrix (Q):

It describes the branches contained in the cut set and their orientation.

In the cut-set matrix, each row represents a cut set and columns represent branch voltages.

Order of Fundamental Cut Set Matrix:

The number of fundamental cut sets is equal to the number of twigs that is N-1.

B is the total number of branches.

Order of fundamental cut matrix becomes (N-1)xB

Elements of Fundamental Cut Set:

Suppose the graph has 'b’ branches, N nodes, (N-1) twigs in a tree and (N,-1) fundamental cut sets. Let 'i’ be the rows and 'j’ be the columns of the cut-set matrix.

• aij = +1 for the twig of selected fundamental cut set
• aij = +1 if the links have the same current direction as that of the selected twig
• aij = -1 if the links have opposite current direction as that of selected twig
• aij = 0 if the twigs and  are not a part of the selected fundamental cut set

Procedure For Writing Fundamental Cut Set Matrix:

Consider first f-cut set C1

• Branch 5 (twig) = +1
• Branch 3 (link) = -1

Consider second f-cut set C2

• Branch 6 (twig) = +1
• Branch 4 (link) = -1
• Branch 3 (link) = +1

Consider third f-cut set C3

• Branch 1 (twig) = +1
• Branch 2 (link) = +1
• Branch 4 (link) = +1

Fig 3  Fundamental Cut Set Matrix

Fundamental Cut Set Matrix and KCL

Q.Ib = 0

Where Q = cut set matrix

           Ib = branch current vector

Fig 4 KCL and Cut Set Matrix

          -i3 + i5 = 0

          +i3 - i4 + i6 = 0

          i1 + i2 + i4 = 0  

Fundamental Cut Set Matrix and KVL

Vb = QT.Vt

Where Vb = branch voltage vector

          QT = transpose of the cut-set matrix

          Vt = tree branch voltage vector

Fig 5 KVL and Cut Set Matrix

v1 = vt6

v2 = vt6

v3 = vt5 - vt1

v4 = vt6 - vt5

v5 = vt1

v6 = vt5