- #1

swampwiz

- 571

- 83

I was thinking about the famous resistor cube problem, and I've come up with some observations:

- Every branch (I think that's the term - i.e., a branch is any subset of a circuit with the same current) is like an edge in a polyhedron, and thus anything in direct series can be abstracted out into a single branch (i.e., in "series").

- The tightest loop of any set of branches is like a face in a polyhedron.

- A circuit node is like a vertex in a polyhedron

- Any set of branches that go between the same 2 nodes can be abstracted out into a single branch (i.e., in "parallel").

My understanding is that the parallel & series abstraction are done as per Thevenin's Theorem, which is trivial if the circuit elements are of the same type (i.e., R, C, L).

With this model then, every edge has a current (the direction would need to be arbitrarily chosen), and so there is a current continuity EQ for every node/vertex, which results in a simultaneous system of V (# of vertices) equations in E (# of edges) unknowns, and that also has a zero vector for the resultant, which means there must be a nullspace of size 1 between the equations. Similarly, every face is an independent loop, and so that net voltage drop across that loop must be zero, which results in a simultaneous system of F (# of faces) equations in E unknowns - and to give the circuit "life", there must be some EMF element in at least one of the branch/edges, and so via the principle of superposition, each EMF element could be examined one at a time, and such that the EMF element term would wind up as a non-zero term in the resultants, and thus there need not be any nullity between these equations.

Going back to the polyhedron model, topographically any polyhedron can be made into a something similar to a net in which all but one of the faces can be represented in a plane such that the remaining face is on the other side of the plane, and thus there must be a nullspace of size 1 in the simultaneous system of equations for voltage drop. And so taken in total, there are [ ( V - 1 ) + ( F - 1 ) = ( V + F - 2 ) ] equations in total in E unknowns - which wonderfully exactly matches Euler's polyhedron EQ (i.e., without holes), and so if all the non-EMF elements were of the same type, there would be a simple, well-conditioned matrix equation (with real values) to solve for the edge currents. (My hunch is that adding the other type of elements would result in a series of differential equations, but such that if the EMF were a sine function, the resultant equations for the steady-state would be similar well-conditioned matrix equation (but with complex values, as per the impedance phasor).

Does all this sound accurate?

Also, this model only considers a system that is topographically without holes, and I have to wonder that in some contrived circuit system which does have holes, there must be more nullspace in the equations as per Euler's EQ (i.e., [ E = V + F + 2 ( H - 1 ) ], where H is the # of holes). I wonder if any mathematician/EE has done a paper on this; I can't possibly be the originator of this idea.