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If [A] Matrix is Incidence matrix then which one of the following is true?
Incidence matrix:
Hence, an Only statement I is correct.
For a network graph having its fundamental loop matrix B_{f} and its submatrices B_{t} and B_{l} corresponding to twigs and links, which of the following statements are correct?
1) B_{l} is always an identity matrix.
2) B_{t} is an identity matrix.
3) B_{f} has rank of b – (n – 1), where b is the number of branches and n is the number of nodes of the graph.
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:
Where, Bt and Bl are the submatrices of tieset matrix (Bf) corresponding to twigs and links of a connected graph, respectively.
The elements of tie set matrix [M] = [aij]
[a_{ij}] = +1, If j^{th} branch current is incident at ith tie set current at oriented in the same direction.
= –1, if j^{th} 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,
Following is not the property of a complete incidence matrix:
Incidence matrix:
The elements of incidence matrix are given by [A] = [a_{ij}]n × b
Where a_{ij} = 1, if ^{jth} branch is incident at ith node and oriented away.
a_{ij} = 1, if j^{th} branch is incident at ith node and oriented towards.
a_{ij} = 0, if j^{th} branch is not incident at i^{th} node
b is total number of branches
n is total number of nodes
Reduced incidence matrix:
Which network topology term got reference directions and marked on the edges of the graph by arrow heads?
Oriented Graph:
If one link fails, the entire network can be disabled in ______.
Concept:
A single cable links all of the included nodes in a bus topology. The primary cable serves as the network's backbone. If the common cable breaks, the entire system will be brought to a halt.
Every device in a ring network has exactly two neighbors for communication purposes. It's called ring topology because it's shaped like a ring. Every computer in this topology is linked to another computer. A single computer failure can disrupt the entire network.
As a result, bus and ring topologies have difficulties, such as the possibility of the entire network being deactivated if one connection fails.
Hence the correct answer is Bus and Ring Topology.
To construct the dual of a fourmesh network how many nodes are required?
Concept:
Duality:
Nodal Analysis:
Nodal analysis is a method of analyzing networks with the help of KCL equations.
For a network of N nodes, the number of simultaneous equations to be solved to get the unknowns
= Number of KCL equations
= N  1
Mesh Analysis:
Mesh analysis is a method of analyzing networks with the help of KVL equations.
For a network having N nodes and B branches, the number of simultaneous equations to be solved to get the unknowns
= Number of KVL equations
= number of independent loop equations
= B  N + 1
Application:
Given: KVL equations = 4,
Therefore, B  N + 1 = 4
B = 3 + N ........(1)
For duality of two networks,
Mesh equations of one network = nodal equations of other
Hence,
B  N + 1 = N  1
N = B + 2 / 2 .....(2)
substitute (1) in (2)
N = N + 5 / 2
N = 5
The number of twigs and links in a connected network graph with 'n' nodes and 'b' branches are, respectively.
Concept:
Tree: A tree is a sub graph of main graph which connects all the nodes without forming a closed loop.
For a graph with ‘n’ nodes, the rank of tree = n – 1
Any tree for a given graph can be constructed with (n–1) branches.
Twig: The branch of a tree is called as twig indicated by thick Line. Any tree with n nodes has (n – 1) twigs.
Cotree: The set of branches in a graph other than tree branches form a co tree.
Link: The branch of a co tree is called link indicated by dotted Line.
For any graph with n nodes and b branches, the numbers of links is given by:
= b – n + 1
Considering the principle of duality, which of the following pair is INVALID dual pair?
Duality:
Some important dual relations are given below.
A loop which does not contain any other loop within it is called _________.
Node:
A point or junction where two or more circuit elements (resistor, capacitor, inductor, etc.) meet is called Node. In other words, a point of connection between two or more branches is known as a Node.
Supernode:
If a voltage source is connected between two nonreference nodes, then we combine the two nodes as to yield super node.
Branch:
That part or section of a circuit locate between two junctions is called the branch. In a branch, one or more elements can be connected, and they have two terminals.
Loop:
A closed path in a circuit where more than two meshes can occur is known as Loop i.e. there may be many meshes in a loop, but a mesh does not contain on one loop.
Mesh:
A closedloop that contains no other loop within it or a path which does not contain other paths is called Mesh.
Super mesh:
If a current source is present at the common boundary of two meshes, then we create a super mesh by avoiding the current source and any element connected to it in series.
Note: The number of independent loops for a network with n nodes and b branches is = b  n + 1
Tree: A tree is a connected subgraph of a connected graph containing all the nodes of the graph but containing no loops, i.e., there is a unique path between every pair of nodes.
Therefore, the number of closed paths in a tree of the graph is zero.
Twig: The branches of the tree are called twigs.
Link: Those branches of the graph which are not in the tree.
Cotree: All the links of a tree together constitute complement of the tree and is called cotree, in which the number of branches is equal to b  (n  1)
Where b is the number of branches of the graph.
Number of twigs: t = n  1
Number of links: L = b  t = b – n + 1
Consider the following data for twigs and links:
N = Number of nodes
L = Total number of links
B = Total number of branches
The total number of links associated with a tree is
Concept:
Tree: A tree is a sub graph of main graph which connects all the nodes without forming a closed loop.
For a graph with ‘n’ nodes, the rank of tree = N – 1
Any tree for a given graph can be constructed with (N–1) branches.
Twig: The branch of a tree is called as twig indicated by thick Line. Any tree with n nodes has (n–1) twigs.
Cotree: The set of branches in a graph other than tree branches form a co tree.
Link: The branch of a co tree is called link indicated by dotted Line.
For any graph with n nodes and b branches, the numbers of links is given by:
= B – N + 1
The graph of a network has 8 nodes and 5 independent loops. The number of branches of the graph is
Concept:
Nodal Analysis:
Nodal analysis is a method of analyzing networks with the help of KCL equations.
For a network of N nodes, the number of simultaneous equations to be solved to get the unknowns
= Number of KCL equations
= N  1
Mesh Analysis:
Mesh analysis is a method of analyzing networks with the help of KVL equations.
For a network having N nodes and B branches, the number of simultaneous equations to be solved to get the unknowns
= Number of KVL equations
= number of independent loop equations
= B  N + 1
Where, B = no of the branch, N = No of node
Calculation:
Given that,
Number of nodes (N) = 8
Number of independent loops (L) = 5
Let the number of branches are B.
We know that,
L = B – N + 1
⇒ 5 = B – 8 + 1 ⇒ B = 12
Consider the following statements regarding trees:
1. A tree contains all the nodes of the graph.
2. A tree shall contain any one of the loops.
3. Every connected graph has at least one tree.
Which of the above statements are correct?
Let us consider the following electric circuit:
An equivalent graph corresponding to the above electric circuit is shown in the following figure:
Tree:
The tree is a connected subgraph of a given graph, which contains all the nodes of a graph. But, there should not be any loop in that subgraph. The branches of a tree are called as twigs.
This connected subgraph contains all the four nodes of the given graph and there is no loop.
Hence, it is a Tree.
Consider the following with regards to graph as shown in the figure given below:
1. Regular graph
2. Connected graph
3. Complete graph
4. Nonregular graph
Which of the above are correct?
Concept:
Analysis:
A network graph is given as:
From the network graph above, we have:
Total nodes or vertex = 6
= (1, 2, 3, 4, 5, 6)
And Total edges or branches = 9
= (1  2), (2  3), (3  4), (4  5), (5  6), (6  1), (1  4), (2  6), (3  5)
What are the properties of a tree in a network graph?
1. It consists of all the nodes of the graph.
2. If the graph has N number of nodes, the tree will have (N – 1) branches.
3. There will be only one closed path in the tree.
Tree: A tree is a subgraph of the main graph which connects all the nodes without forming a closed loop.
For a graph with ‘n’ nodes, the rank of tree = n – 1
Any tree for a given graph can be constructed with (n–1) branches.
Twig: The branch of a tree is called twig indicated by a thick line. Any tree with n nodes has (n–1) twigs.
Cotree: The set of branches in a graph other than tree branches form a co tree.
Link: The branch of a co tree is called the link indicated by the dotted line. For any graph with n nodes and b branches, numbers of links = b – n + 1
Example:
A network with 4 nodes and corresponding graphical representation is represented below:
Now, if we remove the branches c, a, and f from the circuit, we will get the tree as shown below:
We observe that the tree for the given network contains all the four nodes of the network but does not form any closed path. (Statement 1 is correct, but Statement 3 is incorrect)
We can also observe that there are 3 twigs or branches of a tree in the given electric network. The number of nodes in the network is 4.
∴ The number of branches/twigs = Number of nodes  1. (Statement 2 is correct)
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