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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.
- a)1 and 2 only
- b)2 and 3 only
- c)1 and 3 only
- d)1, 2 and 3
Correct answer is option 'C'. Can you explain this answer?
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For a network graph having its fundamental loop matrix Bfand its sub-m...
Explanation:
Bl is always an identity matrix:
- The sub-matrix Bl corresponds to links, which represent the branches of the network graph.
- Each link in the network graph is independent of other links, hence the sub-matrix Bl will always be an identity matrix.
Bt is not an identity matrix:
- The sub-matrix Bt corresponds to twigs, which are sets of branches that form loops.
- Unlike links, twigs are not independent of each other, so the sub-matrix Bt will not be an identity matrix.
Bf has rank of b - (n - 1):
- The fundamental loop matrix Bf is a square matrix of size (b x b), where b is the number of branches in the network.
- The rank of Bf is equal to the number of linearly independent rows or columns in the matrix.
- For a connected network with n nodes, the rank of Bf is given by b - (n - 1), where n is the number of nodes.
Therefore, the correct statements are:
- 1 and 3 only.
Bl is always an identity matrix:
- The sub-matrix Bl corresponds to links, which represent the branches of the network graph.
- Each link in the network graph is independent of other links, hence the sub-matrix Bl will always be an identity matrix.
Bt is not an identity matrix:
- The sub-matrix Bt corresponds to twigs, which are sets of branches that form loops.
- Unlike links, twigs are not independent of each other, so the sub-matrix Bt will not be an identity matrix.
Bf has rank of b - (n - 1):
- The fundamental loop matrix Bf is a square matrix of size (b x b), where b is the number of branches in the network.
- The rank of Bf is equal to the number of linearly independent rows or columns in the matrix.
- For a connected network with n nodes, the rank of Bf is given by b - (n - 1), where n is the number of nodes.
Therefore, the correct statements are:
- 1 and 3 only.
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For a network graph having its fundamental loop matrix Bfand its sub-m...
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.
Number of fundamental loops = 6 – 4 + 1 = 3
Fundamental loop 1 is a, b, e with b and e as twigs and a as Link. i1 is Tie set current and the direction as same as link ‘a’.
Similarly, loop2 → b, c, d → i2
loop3 → a, e, f → i3
Tie set matrix:
- It gives the relation between tie set currents and branch currents.
- The rows of a matrix represent the tie – set currents.
- The columns of a matrix represent branches of the graph.
- The order of the tie set matrix is (b – n + 1) × b
- A tie-set matrix can be assumed to be comprised of two submatrices.
Where, Bt and Bl are the sub-matrices of tie-set matrix (Bf) corresponding to twigs and links of a connected graph, respectively.
The elements of tie set matrix [M] = [aij]
[aij] = +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
We can rearrange the matrix as given below.
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
- The rank of tie set matrix is b – (n – 1) i.e. 3 for the above matrix
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For a network graph having its fundamental loop matrix Bfand its sub-matrices Btand Blcorresponding to twigs and links, which of the following statements are correct?1) Blis always an identity matrix.2) Btis an identity matrix.3) Bfhas rank of b – (n – 1), where b is the number of branches and n is the number of nodes of the graph.a)1 and 2 onlyb)2 and 3 onlyc)1 and 3 onlyd)1, 2 and 3Correct answer is option 'C'. Can you explain this answer? for Electrical Engineering (EE) 2025 is part of Electrical Engineering (EE) preparation. The Question and answers have been prepared according to the Electrical Engineering (EE) exam syllabus. Information about For a network graph having its fundamental loop matrix Bfand its sub-matrices Btand Blcorresponding to twigs and links, which of the following statements are correct?1) Blis always an identity matrix.2) Btis an identity matrix.3) Bfhas rank of b – (n – 1), where b is the number of branches and n is the number of nodes of the graph.a)1 and 2 onlyb)2 and 3 onlyc)1 and 3 onlyd)1, 2 and 3Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Electrical Engineering (EE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For a network graph having its fundamental loop matrix Bfand its sub-matrices Btand Blcorresponding to twigs and links, which of the following statements are correct?1) Blis always an identity matrix.2) Btis an identity matrix.3) Bfhas rank of b – (n – 1), where b is the number of branches and n is the number of nodes of the graph.a)1 and 2 onlyb)2 and 3 onlyc)1 and 3 onlyd)1, 2 and 3Correct answer is option 'C'. Can you explain this answer?.
For a network graph having its fundamental loop matrix Bfand its sub-matrices Btand Blcorresponding to twigs and links, which of the following statements are correct?1) Blis always an identity matrix.2) Btis an identity matrix.3) Bfhas rank of b – (n – 1), where b is the number of branches and n is the number of nodes of the graph.a)1 and 2 onlyb)2 and 3 onlyc)1 and 3 onlyd)1, 2 and 3Correct answer is option 'C'. Can you explain this answer? for Electrical Engineering (EE) 2025 is part of Electrical Engineering (EE) preparation. The Question and answers have been prepared according to the Electrical Engineering (EE) exam syllabus. Information about For a network graph having its fundamental loop matrix Bfand its sub-matrices Btand Blcorresponding to twigs and links, which of the following statements are correct?1) Blis always an identity matrix.2) Btis an identity matrix.3) Bfhas rank of b – (n – 1), where b is the number of branches and n is the number of nodes of the graph.a)1 and 2 onlyb)2 and 3 onlyc)1 and 3 onlyd)1, 2 and 3Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Electrical Engineering (EE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For a network graph having its fundamental loop matrix Bfand its sub-matrices Btand Blcorresponding to twigs and links, which of the following statements are correct?1) Blis always an identity matrix.2) Btis an identity matrix.3) Bfhas rank of b – (n – 1), where b is the number of branches and n is the number of nodes of the graph.a)1 and 2 onlyb)2 and 3 onlyc)1 and 3 onlyd)1, 2 and 3Correct answer is option 'C'. Can you explain this answer?.
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