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TN TRB EC 2012 Official Paper

Option 2 : branch voltage and tree branch voltages

Network Theory

737

30 Questions
40 Marks
40 Mins

We can express the branch voltages in terms of tree branch voltages or twig voltages, using the cut-set matrix Q.

[Vb] = [Q]T [vt]

Where, Vb is the column matrix of branch voltages

Vt is the column of matrix of twig voltages.

Cut-set matrix:

- It gives the relation between cut-set voltages and branch voltages.
- The rows of a matrix represent the cut-set voltages.
- The columns of a matrix represent the branches of the graph.
- The order of the cut set matrix is (n – 1) × b.
- The rank of a cut-set matrix is (n – 1)

The elements of a cut set matrix, [Q] = [aij]n−1×b

[aij] = +1, if jth branch is incident to ith cut set and oriented in same direction

= –1, if jth branch is incident to ith cut set oriented in opposite direction

= 0, if jth branch is not incident with ith cut set voltage.

Example:

Let the graph is

The cut-set voltages are as shown below.

For the above graph, the tie set matrix is given by

We can rearrange the matrix as given below.

Using cut-set matrix we can obtain the KCL equations at nodes.

[Q] [Ib] = 0

Where [Ib] is the column matrix of branch currents.