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If the number of branch in a network is b, thenumber of nodes is n and the number of dependent loop is l, then the number of independent node equations will be
- a)n + l - 1
- b)b - 1
- c)b - n + 1
- d)n - 1
Correct answer is option 'D'. Can you explain this answer?
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If the number of branch in a network is b, thenumber of nodes is n and...
The number of independent node equation are n - 1.
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If the number of branch in a network is b, thenumber of nodes is n and...
The number of independent node equations in a network can be determined using the concept of Kirchhoff's current law (KCL). KCL states that the algebraic sum of currents entering and leaving a node in a network is zero. Each independent node equation represents the application of KCL at a distinct node in the network.
Let's analyze the given options:
a) n - l - 1: This option suggests that the number of independent node equations is equal to the number of nodes (n) minus the number of dependent loops (l) minus one. However, this is not correct.
b) b - 1: This option suggests that the number of independent node equations is equal to the number of branches (b) minus one. This is not correct either.
c) b - n + 1: This option suggests that the number of independent node equations is equal to the number of branches (b) minus the number of nodes (n) plus one. This is also not correct.
d) n - 1: This option suggests that the number of independent node equations is equal to the number of nodes (n) minus one. This is the correct answer.
Explanation:
To understand why option D is correct, let's consider a simple example. Suppose we have a network with 5 nodes (n = 5), 7 branches (b = 7), and 3 dependent loops (l = 3).
- Each branch contributes one equation to the system of node equations.
- Each dependent loop introduces an additional equation that is dependent on the other equations.
- Therefore, the total number of equations will be equal to the number of branches minus the number of dependent loops (b - l), plus the number of additional equations needed to account for the dependent loops (l - 1).
In our example, the number of equations would be (7 - 3) + (3 - 1) = 6.
Now, let's consider the number of nodes. Each node introduces one unknown variable in the system of equations. However, one of the nodes can be chosen as a reference node, and its potential can be assumed to be zero. This means that we only need to solve for the potentials at (n - 1) nodes.
Therefore, the number of independent node equations is equal to the number of nodes (n) minus one, which matches with option D.
In conclusion, the correct answer is option D: n - 1.
Let's analyze the given options:
a) n - l - 1: This option suggests that the number of independent node equations is equal to the number of nodes (n) minus the number of dependent loops (l) minus one. However, this is not correct.
b) b - 1: This option suggests that the number of independent node equations is equal to the number of branches (b) minus one. This is not correct either.
c) b - n + 1: This option suggests that the number of independent node equations is equal to the number of branches (b) minus the number of nodes (n) plus one. This is also not correct.
d) n - 1: This option suggests that the number of independent node equations is equal to the number of nodes (n) minus one. This is the correct answer.
Explanation:
To understand why option D is correct, let's consider a simple example. Suppose we have a network with 5 nodes (n = 5), 7 branches (b = 7), and 3 dependent loops (l = 3).
- Each branch contributes one equation to the system of node equations.
- Each dependent loop introduces an additional equation that is dependent on the other equations.
- Therefore, the total number of equations will be equal to the number of branches minus the number of dependent loops (b - l), plus the number of additional equations needed to account for the dependent loops (l - 1).
In our example, the number of equations would be (7 - 3) + (3 - 1) = 6.
Now, let's consider the number of nodes. Each node introduces one unknown variable in the system of equations. However, one of the nodes can be chosen as a reference node, and its potential can be assumed to be zero. This means that we only need to solve for the potentials at (n - 1) nodes.
Therefore, the number of independent node equations is equal to the number of nodes (n) minus one, which matches with option D.
In conclusion, the correct answer is option D: n - 1.
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If the number of branch in a network is b, thenumber of nodes is n and the number of dependent loop is l, then the number of independent node equations will bea)n + l -1b)b -1c)b -n + 1d)n -1Correct answer is option 'D'. Can you explain this answer?
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