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Consider the following statements for a network graph, if Bf is its fundamental tie set matrix, and Bt and Bl are its sub-matrices corresponding to twigs and links, respectively: 1. Bt is a unit matrix 2. Bl is a rectangular matrix 3. Rank of Bf is (b – n + 1) where b is the number of branches and n is the number of nodes. Which of the above statements are correct?

Consider the following statements for a network graph, if Bf is its fundamental tie set matrix, and Bt and Bl are its sub-matrices corresponding to twigs and links, respectively: 1. Bt is a unit matrix 2. Bl is a rectangular matrix 3. Rank of Bf is (b – n + 1) where b is the number of branches and n is the number of nodes. Which of the above statements are correct? Correct Answer 2 and 3 only

Fundamental loop: A fundamental loop is a closed path of a given graph with only one Link and the rest of them as twigs.

The number of fundamental loops for any given graph = b – (n – 1) = number of Links

These fundamental loop currents are called Tie set currents and the orientation of the tie set currents governed by the link.

=

= +1, If jth branch current is incident at ith tie set current at oriented in the same direction.

= –1, if jth branch current is incident at ith tie set current at oriented in the opposite direction.

= 0, If jth branch current is not incident with ith tie set current.

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

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We can rearrange the matrix as given below.

[ alt="F3 U.B Madhu 24.04.20 D 26" src="//storage.googleapis.com/tb-img/production/20/04/F3_U.B_Madhu_24.04.20_D%2026.png" style="width: 275px; height: 117px;">

Tie set currents and branch currents together form an identity matrix as marked in the above Tie set matrix.

Now, from the above matrix, it is clear that,

  • The submatrix corresponding to twigs (Bt)­ is not an identity matrix
  • The submatrix corresponding to twigs (Bl)­ is an identity matrix and a rectangular matrix
  • The rank of tie set matrix is b – (n – 1) i.e. 3 for the above matrix

Related Questions

For a network graph having its fundamental loop matrix Bf and its sub-matrices Bt and Bl corresponding to twigs and links, which of the following statements are correct? 1) Bl is always an identity matrix. 2) Bt is an identity matrix. 3) Bf has rank of b – (n – 1), where b is the number of branches and n is the number of nodes of the graph.
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