# Is there a simple proof that Kirchhoff's circuit laws always provide an exactly complete set of equations?

Suppose I have a complicated electric circuit which is composed exclusively of resistors and voltage and current sources, wired up together in a complicated way. The standard way to solve the circuit (by which I mean finding the voltage across, and current through, each circuit element) is to formulate Kirchhoff's laws for both current and voltage, and these will yield linear equations which enable one to solve for all the relevant quantities.

However, there are two problems with these laws:

• There are too many of them. For example, in the simple circuit below, there are three different possible loops one can draw, but only two independent voltages. Similarly,

• The equations are not all independent. In the circuit below, the current conservation equations for the two different nodes turn out to be exactly the same equation. Fortunately, in real life, these problems happen to exactly cancel out, and one gets exactly the correct number of equations to solve the circuit. There are never too many contradicting constraints (the linear system is never overdetermined) and there are always enough equations to pin everything down (the linear system is never underdetermined).

Why is this? Is there a simple proof of this fact? What are the fundamental reasons for it?

• For a detailed discussion and proof, see chapter 12 in volume 2 of a course in mathematics for students of physics. amazon.com/Course-Mathematics-Students-Physics-Bk/dp/0521332451 – Alfred Centauri Feb 18 '15 at 14:10
• Perhaps I am misunderstanding the question, but does not current continuity an energy conservation explain this? Or is it a question of closure? – honeste_vivere Mar 11 '18 at 18:46
• @honeste_vivere Conservation equations are not normally guaranteed to provide a set of equations that fully determines the system's evolution (cf. e.g. an elastic collisions of two particles in 2D where the outgoing angle is unknown). Why would it be obvious that the Kirchhoff equations, coming from conservation laws as they do, are sufficient to fully determine the system, without overdetermining it? – Emilio Pisanty Mar 11 '18 at 18:52
• @EmilioPisanty - Ah okay, I see what you mean now. Hmm... that is odd... I had not thought of it that way. – honeste_vivere Mar 11 '18 at 19:07

The answer is not quite simple, to show this we need some graph theory and matrices. There is a beautiful document explaining this relation in detail http://www2.math.uu.se/~takis/L/Circuits/2000/handouts/graphsandckts/graphsandckts.pdf

I think the "fundamental reason" of this is related with the fact that every loop have different variables, if we can generate a loop using another loop the equations will not be independent, of course this is my opinion, all the math is in the document.

• The math is actually quite accessible. The notion of a spanning tree is fairly intuitive, and from there each edge not in the spanning tree links two nodes from the tree. As those 2 nodes share a single unique ancestor in the tree, this defines a unique cycle. – MSalters Feb 19 '15 at 10:40

Here's a counterexample:

Suppose two identical, ideal batteries (with zero internal resistance) are both connected in parallel across a single resistor; equivalently, replace one of the resistors in your diagram with a second, identical battery. Also assume the conducting wires are ideal (again, no resistance).

Kirchhoff's laws in this case result in an underdetermined system. If the current through the single resistor is I and the voltage across both ideal batteries is V, you cannot find the current through either battery using Kirchoff's laws alone; both loops give the voltage across the resistor as V, and both junctions say the sum of the currents through the batteries must equal I, but do not allow you to calculate either of those currents. For instance, a current of 3 I up through one battery and 2 I down through the other satisfies the system of equations. In this case you have to use a symmetry argument to conclude the current through each battery is I/2.

This is not a problem using real-world equipment, though, as voltage sources always have some amount of associated internal resistance. So if we agree to use non-ideal circuit elements, then I agree with the answer @Hu provided.

This indirectly raises another question; are Kirchoff's laws meaningful in ideal circuits? I'm sure there are many more examples like the one above, where the resulting system of linear equations is underdetermined (though I doubt there are cases which are overdetermined). We use ideal situations to model real systems, but is that a good idea when answers are undetermined in the ideal case?

• And the mirror system (overdetermined) has two current sources in a cycle, but with opposite current directions. – MSalters Feb 19 '15 at 10:36
• In ideal circuit theory, connecting two ideal voltage sources in parallel is, in general, an invalid circuit for the same reason that it is invalid to connect two ideal current sources in series since, in general, a contradiction results, e.g., 1 = 2. There are valid examples of under-determined circuits with dependent (controlled) sources. – Alfred Centauri Feb 19 '15 at 12:11