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Consider an electric circuit with dc sources ( voltage and current) and resistors. Write down the equations. In the most general case, the solution of the system is not unique. The set of solutions can be empty or positive dimensional (simple example: 2 points in the graph and two batteries in parallel joining the points).

The dimension of the space of solutions can be computed with two different methods: 1) Mathematically : compute the determinants and the compatibility conditions 2) Physically. Give conditions on the graph to give a meaning for the dimension. For instance a cycle of batteries give an empty set of solutions or make the dimension increase by one.

I am interested in second method and I am looking for references in the litterature.

What I found is the following. In the student books, the unicity is always assumed to be true, eg. in the standard book by nilsson and riedel. In more advanced books, I have found discusions only for particular cases, and with very technical tools. For instance, in Frankel (The geometry of physics), only purely resistive circuits are considered with source of currents in the nodes. And the proof uses ( a simplified version of) Hodge theory.

Now my question: - Is there a book or an article where the very general case ( any graph with dc sources and resistors ) is considered and the dimension of the system described in terms of the graph ? I am interested both in sophisticated answers as above and in answers with basic tools of linear algebra. All references welcome.

I am looking for references, not for the solution. I have already written a solution for my students (an elementary one, with basic linear algebra). I want to compare my solution with the existing litterature. If this is useful and not a waste of time, I will make public my personal notes.

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I remember seeing a rigorous proof that Kirchhoff's laws constitute exactly the number of independent linear constraints on the system as its dimension, based on maximal spanning trees of the graph. If I find it I'll post it here. Good question! – Emilio Pisanty Dec 5 '13 at 16:56
... aaaand I can't find it. I'm pretty sure I know where a hard copy is, but I won't be able to see it for some ten days or so. I'll post it here if this doesn't have a good answer by then. – Emilio Pisanty Dec 5 '13 at 19:22

This may not be a direct answer, but it can be shown by an elementary method that the relation between the number of nodes, branches and loops in a well-posed problem corresponds to Euler's polyhedron formula.

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It must be noted, though that this holds only for non-intersecting branches and thus for planar circuits, which is quite restrictive. – Emilio Pisanty Dec 5 '13 at 18:49
Thak you Doru and Emilio. In the non planar case, one uses the cyclomatic number introduced by Maxwell ( ie in modern terms the first cohomology group of the graph) and the number of equations is correct too. The problem is not to count the number of equations. It is to count the number of INDEPENDENT equations. – user35303 Dec 6 '13 at 14:21

Two rigorous references are:

Leon O. Chua, Charles A. Desore and Ernest S. Kuh: Linear and Nonlinear Circuits.


Martin Hasler and Jacques Neirynck: Nonlinear Circuits.

Both books work with cut- and loop-set spaces together with Tellegen's theorem.

I think the first reference is easier to read while the second one is more comprehensive regarding unique solutions of nonlinear circuits.

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