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Cut-Set Matrix and Node-Pair Potential

Cut-Set A cut-set is a minimum set of elements that when cut, or removed, separates the graph into two groups of nodes. A cut-set is a minimum set of branches of a connected graph, such that the removal of these branches from the graph reduces the rank of the graph by one.

In other words, for a given connected graph (G), a set of branches (C) is defined as a cut-set if and only if:
  1. the removal of all the branches of C results in an unconnected graph.
  2. the removal of all but one of the branches of C leaves the graph still connected.

Fundamental Cut-Set

A fundamental cut-set (FCS) is a cut-set that cuts or contains one and only one tree branch. Therefore, for a given tree, the number of fundamental cut-sets will be equal to the number of twigs.

Properties of Cut-Set

  1. A cut-set divides the set of nodes into two subsets.
  2. Each fundamental cut-set contains one tree-branch, the remaining elements being links.
  3. Each branch of the cut-set has one of its terminals incident at a node in one subset and its other terminal at a node in the other subset.
  4. A cut-set is oriented by selecting an orientation from one of the two parts to the other. Generally, the direction of cut-set is chosen same as the direction of the tree branch.

Cut-Set Matrix (QC)

For a given graph, a cut-set matrix (QC) is defined as a rectangular matrix whose rows correspond to cut-sets and columns correspond to the branches of the graph. Its elements have the following values:

Qij = 1, if branch j is in the cut-set i and the orientations coincide.
= –1, if branch j is in the cut-set i and the orientations do not coincide.
= 0, if branch j is not in the cut-set i.

Cut-Set Matrix and KVL

By cut-set schedule, the branch voltages can be expressed in terms of the tree-branch voltages.

A cut-set consists of one and only one branch of the tree together with any links which must be cut to divide the network into two parts. A set of fundamental cut-sets includes those cut-sets which are obtained by applying cut-set division for each of the branches of the network tree.

(a) Graph

(b) Tree
To summarize, KVL and KCL equations in three matrix forms are given below.
Matrix KCL KVL
Incidence Matrix (Aa) Aa × Ib = 0 Vb = AaT × Vn
Tie-Set Matrix (Ba) Ib = BaT × IL Ba × Vb = 0
Cut-Set Matrix (QC) QC × Ib = 0 Vb = QCT × Vt

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