Concept of Fundamental Loops and Tieset Currents MCQ Quiz - Objective Question with Answer for Concept of Fundamental Loops and Tieset Currents - Download Free PDF

Explanation:

Number of Independent Loops in a Network

Definition: In electrical network analysis, the number of independent loops in a network is a critical concept used to analyze circuit behavior. An independent loop is defined as a closed path in the network that cannot be formed by combining other loops. The number of independent loops is determined by the network's topology, specifically the number of nodes and branches in the circuit.

Formula: The equation for the number of independent loops in a network is:

L = b - n + 1

Where:

• L = Number of independent loops
• b = Number of branches
• n = Number of nodes

Explanation of the Correct Option:

The correct option is:

Option 3: b - n + 1

This formula is derived from the fundamental concepts of graph theory applied to electrical circuits. In a network:

• Nodes represent connection points where two or more branches meet.
• Branches represent the paths connecting the nodes.

The number of independent loops is determined by subtracting the number of nodes from the number of branches and adding 1. This accounts for the inherent redundancy in the network's connectivity, ensuring that the loops identified are truly independent.

Derivation:

To derive the formula, consider the following:

1. A connected network with n nodes and b branches forms a graph.
2. The graph has a certain number of loops, which can be calculated using the concept of a spanning tree. A spanning tree is a subgraph that connects all nodes without forming any loops.
3. The number of branches in a spanning tree is equal to n - 1, where n is the number of nodes.
4. Any additional branches in the network beyond those in the spanning tree contribute to the formation of loops. The number of additional branches is b - (n - 1).
5. Thus, the total number of independent loops is given by:

   L = b - n + 1

Example:

Consider a network with:

• Number of nodes (n) = 5
• Number of branches (b) = 8

Using the formula:

L = b - n + 1 = 8 - 5 + 1 = 4

Therefore, the network has 4 independent loops.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: n - 1

This formula is incorrect as it represents the number of branches in a spanning tree, not the number of independent loops. The number of branches in a spanning tree is equal to the number of nodes minus one, but this does not account for the additional branches that contribute to loop formation.

Option 2: b + n - 1

This formula is incorrect and does not represent the number of independent loops. Adding b and n together and subtracting 1 does not align with the principles of graph theory or electrical network analysis.

Option 4: b - n

This formula is close but incorrect. It fails to account for the additional +1 term that arises from the inclusion of the spanning tree concept.

Option 5: This option is not provided in the question.

Conclusion:

The formula L = b - n + 1 is the correct equation for calculating the number of independent loops in a network. This formula is derived from graph theory and accounts for the network's topology, including nodes and branches. Understanding this formula is crucial for analyzing electrical networks and solving circuit problems efficiently.

Nodal Analysis :

For N nodes number of  KCL equations are required

= N-1

Mesh Analysis :

Number of KVL required 

Number of independent loops= B-N+1

B: Number of branches

N: Number of Nodes

L: Number of independent loops

Given :

B =12 , N = 8

L =  B - N + 1

= 12 - 8 + 1

= 5

Concept:

The number of meshes in a network is given as:

M = B – (N – 1)

B → number of branches

M → number of meshes

N → number of nodes

Analysis:

Given: N = 7, M = 5

The number of branches is calculated as: 

B = M + (N - 1) 

= 5 + 6 = 11

The Number of Node pair voltage = \(\frac{{N\left( {N - 1} \right)}}{2}\)

Concept:

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
• The rank of a tie-set matrix is (B – N + 1)

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)

Calculation:

Given:

B = 5, N = 4

Order of Tie set matrix is = (B – N +1) × B

= (5 – 4 + 1) × 5

= 2 × 5

Order of cut set matrix is = (N - 1) × B

(4 – 1) × 5

= 3 × 5

Concept:

Fundamental loop: A fundamental loop is a closed path of a given graph with only one Link and 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 sub- matrices.
•  

Where, B_t and B_l are the sub-matrices of tie-set matrix (B_f) corresponding to twigs and links of a connected graph, respectively.

The elements of tie set matrix [M] = [a_ij]

[a_ij] = +1, If j^th branch current is incident at i^th tie set current at oriented in same direction.

= –1, if j^th branch current is incident at i^th tie set current at oriented in opposite direction.

= 0, If j^th branch current is not incident with i^th 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 forms an identity matrix as marked in the above Tie set matrix.

For a given network, KVL equations are obtained in terms of tie-set matrix. It is given as

[B][V_b] = 0

Where B is the tie-set matrix and V_b is the branch voltage vector.

Application:

For the given tree,

Twigs = 4, 5, 6

Links = 1, 2, 3

Fundamental loops = 1, 5, 6; 2, 4, 5; 3, 4

Now tie-set matrix is,

[B] [V_b] = 0

\(\Rightarrow \left[ {\begin{array}{*{20}{c}} 1&0&0&0&1&{ - 1}\\ 0&1&0&1&{ - 1}&0\\ 0&0&1&{ - 1}&0&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{V_1}}\\ {{V_2}}\\ {{V_3}}\\ {{V_4}}\\ {{V_5}}\\ {{V_6}} \end{array}} \right] = 0\)

Nodal Analysis :

For N nodes number of  KCL equations are required

= N-1

Mesh Analysis :

Number of KVL required 

Number of independent loops= B-N+1

B: Number of branches

N: Number of Nodes

L: Number of independent loops

Given :

B =12 , N = 8

L =  B - N + 1

= 12 - 8 + 1

= 5

Concept:

The number of meshes in a network is given as:

M = B – (N – 1)

B → number of branches

M → number of meshes

N → number of nodes

Analysis:

Given: N = 7, M = 5

The number of branches is calculated as: 

B = M + (N - 1) 

= 5 + 6 = 11

The Number of Node pair voltage = \(\frac{{N\left( {N - 1} \right)}}{2}\)

Explanation:

Number of Independent Loops in a Network

Definition: In electrical network analysis, the number of independent loops in a network is a critical concept used to analyze circuit behavior. An independent loop is defined as a closed path in the network that cannot be formed by combining other loops. The number of independent loops is determined by the network's topology, specifically the number of nodes and branches in the circuit.

Formula: The equation for the number of independent loops in a network is:

L = b - n + 1

Where:

• L = Number of independent loops
• b = Number of branches
• n = Number of nodes

Explanation of the Correct Option:

The correct option is:

Option 3: b - n + 1

This formula is derived from the fundamental concepts of graph theory applied to electrical circuits. In a network:

• Nodes represent connection points where two or more branches meet.
• Branches represent the paths connecting the nodes.

The number of independent loops is determined by subtracting the number of nodes from the number of branches and adding 1. This accounts for the inherent redundancy in the network's connectivity, ensuring that the loops identified are truly independent.

Derivation:

To derive the formula, consider the following:

1. A connected network with n nodes and b branches forms a graph.
2. The graph has a certain number of loops, which can be calculated using the concept of a spanning tree. A spanning tree is a subgraph that connects all nodes without forming any loops.
3. The number of branches in a spanning tree is equal to n - 1, where n is the number of nodes.
4. Any additional branches in the network beyond those in the spanning tree contribute to the formation of loops. The number of additional branches is b - (n - 1).
5. Thus, the total number of independent loops is given by:

   L = b - n + 1

Example:

Consider a network with:

• Number of nodes (n) = 5
• Number of branches (b) = 8

Using the formula:

L = b - n + 1 = 8 - 5 + 1 = 4

Therefore, the network has 4 independent loops.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: n - 1

This formula is incorrect as it represents the number of branches in a spanning tree, not the number of independent loops. The number of branches in a spanning tree is equal to the number of nodes minus one, but this does not account for the additional branches that contribute to loop formation.

Option 2: b + n - 1

This formula is incorrect and does not represent the number of independent loops. Adding b and n together and subtracting 1 does not align with the principles of graph theory or electrical network analysis.

Option 4: b - n

This formula is close but incorrect. It fails to account for the additional +1 term that arises from the inclusion of the spanning tree concept.

Option 5: This option is not provided in the question.

Conclusion:

The formula L = b - n + 1 is the correct equation for calculating the number of independent loops in a network. This formula is derived from graph theory and accounts for the network's topology, including nodes and branches. Understanding this formula is crucial for analyzing electrical networks and solving circuit problems efficiently.

Concept:

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
• The rank of a tie-set matrix is (B – N + 1)

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)

Calculation:

Given:

B = 5, N = 4

Order of Tie set matrix is = (B – N +1) × B

= (5 – 4 + 1) × 5

= 2 × 5

Order of cut set matrix is = (N - 1) × B

(4 – 1) × 5

= 3 × 5

Nodal Analysis :

For N nodes number of  KCL equations are required

= N-1

Mesh Analysis :

Number of KVL required 

Number of independent loops= B-N+1

B: Number of branches

N: Number of Nodes

L: Number of independent loops

Given :

B =12 , N = 8

L =  B - N + 1

= 12 - 8 + 1

= 5

Concept:

Fundamental loop: A fundamental loop is a closed path of a given graph with only one Link and 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 sub- matrices.
•  

Where, B_t and B_l are the sub-matrices of tie-set matrix (B_f) corresponding to twigs and links of a connected graph, respectively.

The elements of tie set matrix [M] = [a_ij]

[a_ij] = +1, If j^th branch current is incident at i^th tie set current at oriented in same direction.

= –1, if j^th branch current is incident at i^th tie set current at oriented in opposite direction.

= 0, If j^th branch current is not incident with i^th 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 forms an identity matrix as marked in the above Tie set matrix.

For a given network, KVL equations are obtained in terms of tie-set matrix. It is given as

[B][V_b] = 0

Where B is the tie-set matrix and V_b is the branch voltage vector.

Application:

For the given tree,

Twigs = 4, 5, 6

Links = 1, 2, 3

Fundamental loops = 1, 5, 6; 2, 4, 5; 3, 4

Now tie-set matrix is,

[B] [V_b] = 0

\(\Rightarrow \left[ {\begin{array}{*{20}{c}} 1&0&0&0&1&{ - 1}\\ 0&1&0&1&{ - 1}&0\\ 0&0&1&{ - 1}&0&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{V_1}}\\ {{V_2}}\\ {{V_3}}\\ {{V_4}}\\ {{V_5}}\\ {{V_6}} \end{array}} \right] = 0\)

The number of twigs on a tree is always one less than the number of nodes.

Number of twigs = n – 1

Number of links, L = b – n + 1

No of tie set currents = number of links = b – n + 1

No of cut set = number of twigs = n – 1

Concept:

The number of meshes in a network is given as:

M = B – (N – 1)

B → number of branches

M → number of meshes

N → number of nodes

Analysis:

Given: N = 7, M = 5

The number of branches is calculated as: 

B = M + (N - 1) 

= 5 + 6 = 11

The Number of Node pair voltage = \(\frac{{N\left( {N - 1} \right)}}{2}\)

Concept:

Fundamental loop: A fundamental loop is a closed path of a given graph with only one Link and 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

Application:

The fundamental tie-set matrix of a graph is

\(\left[ B \right] = \left[ {\begin{array}{*{20}{c}} 1&0&0&0&{ - 1}&{ - 1}&0&0\\ 0&1&0&0&0&{ - 1}&1&0\\ 0&0&1&0&0&0&1&1\\ 0&0&0&1&{ - 1}&0&0&1 \end{array}} \right]\)

For the given matrix, number of rows = number of links, l = 4

Number of loops = 4

Number of branches = number of columns, b = 8

Number of nodes, n = 5

• In the above matrix, branches 1, 2, 3, 4 forms an identity matrix. So, the branches 1, 2, 3, 4 are links, and the branches 5, 6, 7, 8 are twigs.
• From row 1 of the given matrix, the branches 1, 5, 6 form a loop. The direction of the currents flows through the branches 5, 6 are opposite to the direction of the current flows through the branch 1.
• From row 2 of the given matrix, the branches 2, 6, 7 form a loop. The direction of the current flows through the branch 6 is opposite to the direction of the current flows through the branch 2 and the direction of current flows through the branch 7 is same as the direction of current flows through the branch 2.
• From row 3 of the given matrix, the branches 3, 7, 8 form a loop. The direction of the currents flows through branches 7, 8 are the same as the direction of the current flows through branch 3.
• From row 4 of the given matrix, the branches 4, 5, 8 forms a loop. The direction of the current flows through the branch 5 is opposite to the current flows through the branch 4 and the direction of current flows through the branch 8 is same as the direction of current flows through the branch 4.

From the above analysis, the oriented graph of the network can be drawn as

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, B_t and B_l are the sub-matrices of tie-set matrix (B_f) corresponding to twigs and links of a connected graph, respectively.

The elements of tie set matrix [M] = [a_ij]

[a_ij] = +1, If j^th branch current is incident at i^th tie set current at oriented in the same direction.

= –1, if j^th branch current is incident at i^th tie set current at oriented in the opposite direction.

= 0, If j^th branch current is not incident with i^th 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 (B_t)_­ is not an identity matrix
• The submatrix corresponding to links (B_l)_­ is an identity matrix
• The rank of tie set matrix is b – (n – 1) i.e. 3 for the above matrix