I'm currently studying the textbook Fundamentals of Electric Circuits, 7th edition, by Charles Alexander and Matthew Sadiku. Chapter 2.3 Nodes, Branches, and Loops says the following:
A branch represents a single element such as a voltage source or a resistor.
In other words, a branch represents any two-terminal element. The circuit in Fig. 2.10 has five branches, namely, the 10-V voltage source, the 2-A current source, and the three resistors.
A node is the point of connection between two or more branches.
A node is usually indicated by a dot in a circuit. If a short circuit (a connecting wire) connects two nodes, the two nodes constitute a single node. The circuit in Fig. 2.10 has three nodes \$a\$, \$b\$, and \$c\$. Notice that the three points that form node \$b\$ are connected by perfectly conducting wires and therefore constitute a single point. The same is true of the four points forming node \$c\$. We demonstrate that the circuit in Fig. 2.10 has only three nodes by redrawing the circuit in Fig. 2.11. The two circuits in Figs. 2.10 and 2.11 are identical. However, for the sake of clarity, nodes \$b\$ and \$c\$ are spread out with perfect conductors as in Fig. 2.10.
A loop is any closed path in a circuit.
A loop is a closed path formed by starting at a node, passing through a set of nodes, and returning to the starting node without passing through any node more than once. A loop is said to be independent if it contains at least one branch which is not a part of any other independent loop. Independent loops or paths result in independent sets of equations.
It is possible to form an independent set of loops where one of the loops does not contain such a branch. In Fig. 2.11, \$abca\$ with the \$2 \Omega\$ resistor is independent. A second loop with the \$3 \Omega\$ resistor and the current source is independent. The third loop could be the one with the \$2 \Omega\$ resistor in parallel with the \$3 \Omega\$ resistor. This does form an independent set of loops.
I don't understand the authors' definition of "independent loop":
A loop is said to be independent if it contains at least one branch which is not a part of any other independent loop.
This doesn't even seem to be logical / a valid definition. The authors attempt to define an independent loop by describing it as a loop containing "at least one branch which is not a part of any other independent loop." But this reasoning is circular, since, in order to understand the definition of an "independent loop," one must use/understand the definition of … an independent loop. And so, since this explanation of "independent loop" appeals to independent loops, it isn't even a valid definition.
Am I misunderstanding something here?
Best Answer
A set of independent loops contains loops such that each loop in the set contains a branch which is not part of any other loop in the set.
Some circuits may be divided up into a sets of independent loops in multiple ways. Therefore, an independent loop is not independent in itself, but only in relationship to a set of other loops.