# Independent loops confusion

Discussion in 'General Electronics Chat' started by abhiananth, Sep 29, 2014.

1. ### abhiananth Thread Starter New Member

Jul 24, 2014
18
1
Hi,

I am learning circuit theory from “Fundamentals of Electric Circuits 4th ed - C. Alexander” book.

I am confused while learning the independent loops with the example that is provided in the book.

The definition of independent loop given in that book is “A loop is said to be independent if it contains at least one branch which is not a part of any other independent loop”

Example given for that is as follow:

Example circuit

In book, it is mentioned that there are 3 independent loops

With the given definition, I am able to identify 2 independent loops when I take one sort of loops and 3 independent loops when I take other set of loops. (explained below)

2 independent loops:

I took loops as below

2 loops

3 independent loops:

I took loops as below

3 loops

I am confused whether it has, 2 independent loops or 3 independent loops.

In book, one equation is mentioned.

B = L + N – 1

Where B – no. of branches

L – no. of independent loops

N – no. of nodes

There are 5 branches and 3 nodes in this circuit.

Substituting values in that equation,

5 = L + 3 – 1

L = 3

The equation says, there should be 3 independent loops.

My questions are,

1.Whether there are 2 independent loops or 3 independent loops and please explain why?

2.How they formulated this equation?

3.Is there any other rule to identify independent loop?

Thanks in advance for any help.

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2. ### studiot AAC Fanatic!

Nov 9, 2007
5,005
519
An electrical network can be modelled for connectivity purposes as a set of points or nodes connected by branches or links.

The mathematical study of such networks is called graph theory and is part of mathematical topology.

Now any diagram of branches that connects nodes, but does not contain any loops, is called a tree and one which connects all the nodes (including indirectly) is called a spanning tree.

The condition that the tree does not contain any loops means that there is only one path between any two nodes. This is because a loop would introduce a second path between them.

Since electrical circuits are , well circuits, they have loops.

So any electrical circuit will contain a spanning tree. Further this tree will not contain all the branches of the diagram.

The diagram of the ' spare ' branches is called the cotree. These branches need not be connected to each other. They are called links

Each time a branch that is not part of the spanning tree is connected a loop is formed.

A network with N nodes contains trees with ( N-1) branches.

If B is the number of branches in the whole network this is therefore made up of L links plus (N-1) branches in a spanning tree.

B = L + (N-1)

Which is the derivation of your formula.

The trick when doing loop current analysis is to choose a tree containing all the voltage sources.

Last edited: Sep 29, 2014
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3. ### studiot AAC Fanatic!

Nov 9, 2007
5,005
519

A spanning tree has 2 branches say ab and ac as I have sketched.
(note there is an indirect path from c to b through a).

Of the 5 branches in the circuit that leaves 3 so L = 3

So you require 3 equations to solve this circuit.

Where does the (N-1) come from?

You need 1 branch to connect them.
Any more would form a loop.

Thus Tree Branches = 1 = (2-1) = (N-1)

For every node you add, one additional branch will suffice to connect it to the network.

So add the same number of new nodes (n) to both sides of the equation

TB + n = (N-1) + n

and voila TB = (N-1)

Last edited: Sep 29, 2014
Siddharthmanik and abhiananth like this.
4. ### abhiananth Thread Starter New Member

Jul 24, 2014
18
1
Thanks a ton studio