# Is this "definition" of "independent loop" even logical / a valid definition?

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?

• Just focus on the independent equation part. If I connect two resistors in parallel to the same supply, each one is in its own independent loop because one resistor cannot affect the other. If the supply now has a series output impedance then the loops are no longer independent. Commented Nov 29, 2021 at 15:02
• The definition makes sense to me. It just has to be really verbose because there are two kinds of loops. Commented Nov 29, 2021 at 15:05
• Wait until you study op amps. We make assumptions about the input voltages based on the expectation that a circuit is linear, and then we use those assumptions to show that the circuit is linear. I think the definition is fine, but you might have to apply it iteratively. Commented Nov 29, 2021 at 15:07
• @DKNguyen "A loop is said to be independent if it contains at least one branch which is not a part of any other independent loop."? But how does this make sense? What you're explaining about connecting two resistors in parallel to the same supply doesn't actually define "independent loop." Commented Nov 29, 2021 at 15:08
• @ElliotAlderson But this definition isn't even logical, which means it isn't even understandable, so how can it at all be acceptable? Commented Nov 29, 2021 at 15:10

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.

• So when it says "in Fig. 2.11, $abca$ with the $2 \Omega$ resistor is independent," what it means is that the loop $abca$ is part of a set of independent loops (which includes the other two loops specified), rather than the loop $abca$ itself being independent? Commented Nov 29, 2021 at 16:11
• Yes............ Commented Nov 29, 2021 at 16:21

There's nothing wrong with the definition. What might be confusing you is that the authors don't provide an algorithm for finding a set of independent loops (which is because doing so is, to say the least, pretty hard for any non-trivial circuit; I'm pretty sure it's an NP-hard problem). But if someone describes a set of loops to you and claims that they're independent, verifying that it is so is fairly easy — you check that:

1. Every loop is, in fact, a cycle (it starts and ends at the same node, and doesn't contain any branch more than once).
2. The set of loops covers the whole graph (every branch is contained in at least one loop).
3. Every loop contains at least one branch that doesn't appear in any of the other loops in the set.

If these tests pass, then you have a set of independent loops, and you can continue with whatever analysis they're talking about. If not, then you don't, and you can try to find one.

• Putting on my computer science hat, I think I'm starting to understand this. But due to the recursive nature of this definition, doesn't this mean that we can't know whether any loop we select is actually a valid independent loop until we iterate all the way down to the last loop? And so, in the worst case, we possibly have to check something like $n!$ possibilities/iterations (where $n$ is the total number of nodes or branches or something like that)? Commented Nov 29, 2021 at 15:38
• @ThePointer to the first bit: yes. As MathKeepsMeBusy says, a loop isn't independent per se, rather a set of loops is independent of one another. As to the second bit: maybe. There might be a more efficient algorithm around, I'm not ruling it out, but it could be close to that bad. Commented Nov 29, 2021 at 15:40
• @ThePointer: As far as I know, the first loop that you draw is a valid independent loop. So that starts you out. In fact, I'm pretty sure that any loop that you draw that goes through a branch that has not yet been addressed is a valid independent loop -- but don't quote me on that. Commented Nov 29, 2021 at 16:32