Circuit analysis via polyhedron

[swampwiz]

(NOTE: I should state that my circuit analysis knowledge is up to the level of a single comprehensive EE course, as per ME students.)

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.

 

[DaveE]

(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).

Yes. Once you are familiar with the Laplace transform, you can treat all impedances the same: R for resistors, sL for inductors, etc. The DEs are then just algebraic equations in the s domain. It's an essential tool for analog EEs, including network analysis.

I wonder if any mathematician/EE has done a paper on this; I can't possibly be the originator of this idea.

Books, in fact. I HAD to take a semester long course in network theory in grad school because the dean wrote the text. It was awful. It all seemed quite pointless to me. A rather mechanical algorithm to solve any network with matrix equations. It's great if you're the first guy to write a simulator program, but dull as heck if you care about electronics.

BTW, I don't recall much discussion of geometry. Everything can be developed from a systematic compilation of the DEs with expression in matrix form.

Anyway, I'm still not interested enough to check your work. It all sounded good at first reading. Sure... OK... Euler's equation... whatever.

 

[DaveE]

But.. don't let my grad school PTSD discourage you. Your approach is good, I like your thinking here. If you'd done it 75 years ago you'd get a PhD, write a textbook and maybe be dean of an EE department. Now days, it just isn't taught anymore. Network analysis means something very different than it did in 1950. You'll do better searching for "graph theory", like this book:

As you said, any loop with only two nodes can be reduced to a single branch. Any node with only two branches can be reduced to a single branch. Most EEs do this immediately by inspection. But, you can also leave them in the analysis and get higher dimensional matrix equations. Then they can be eliminated in the solution a la Gaussian elimination.

 

[jasonRF]

As DaveE mentioned, there certainly are treatments of circuit analysis that are built on graph theory and they may be the place to look. I took an undergrad-level course that used this approach 30ish years ago in my junior year as an EE major. A very good and accessible text that includes this material is
Basic circuit theory by Desoer and Kuh
which was written for 3rd year EE students at Berkely in the 1960s. It has been out of print for years but used copies can be found for reasonable prices.

edit: note that it does not talk about 3-D geometry - but the 3D graph can be collapsed into a 2D representation of the graph. jason

 

[Baluncore]

Basic circuit theory by Desoer and Kuh

876 pages. Published by McGraw-Hill. ©1969.
"Basic circuit theory" is available new. ISBN: 9780070165755 / 0070165750
https://www.bookfinder.com/isbn/9780070165755/

Chapter 10. "Node and Mesh Analyses", rationalised the approach.
Following which, the first "Berkeley SPICE" was released in April 1973.
That was 50 years ago, next month.

 

[swampwiz]

I've pondered my analysis here, and while the general idea is the same, there are some tweaks to the linear dependencies. The general idea is that due to the span restrictions in the EQs, there is a single linear dependence with the set of node/vertex EQs, no dependence of the loop/face EQs with either node/vertex EQs or with other loop/face EQs, and that the EQ for the remaining face would be the sum of all the other loop/faces, and hence a linear dependence. So there would be ( V + F ) EQs with 2 linear dependencies, or ( V + F - 2 ) independent EQs in E unknowns.

@@@@

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) EQs in E (# of edges) unknowns. And because every branch/edge is between 2 node/vertices in which it will be considered positive in one and negative in the other, the net sum of all the EQs must be zero, and hence there is a linear dependence between the EQs, and so any one of these must be removed to get an independent set of EQ.

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) EQ 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. However, unlike the case for the EQs for the node/vertices, there must topographically be branch/edges on the outside of the resulting graph, and since these branch/edges will only be used one time, the EQs for the outer loop/faces must be linearly independent. If there are any loop/faces that are not outer, they will have branch/edges that are only used once aside from the outer loop/faces (i.e., they would be the next level of inner branch/edges) - and because these branch/edges had only been used once in the outer branches, these loop/faces EQs cannot be linearly dependent on the outer loop/face EQs, the net result is that they must linearly independent just as for the outer loop/faces. This can be done for any level of inner looping, and so the overall net result is that all these loop/face EQs must be linearly independent of each other - and since a branch/edge is only shared by a pair of loop/faces, and any branch/edge only has a span in a pair of node/vertex EQs that are linearly independent themselves, and such that EQs for node/vertices at most share a single branch/edge (i.e. the one between any pair of node/vertices), the rest of the span for the node/vertex EQs in terms of branch/vertices must be zero - and thus it must be that there is no way for there to be linear dependence of a loop/face EQ on any node/vertex EQ. Hence, the final net result must be that the # of EQs is ( V + F - 1 ) in E # of unknown edge/branch current values.

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 loop/face is on the other side of the plane, and thus a loop/face EQ for this remaining face would be a loop that contains all the other loops - and since the loop/faces have only 2 opposite direction conventions (i.e., CW or CCW), if all loop/faces use the same convention, each branch/edge that is not along the edge of the entire figure will be used in a pair of equal & opposite terms, and so the net result is that the sum of the EQs for the original loop/faces must be the EQ for the reamaining loop/face, and thus including this remaining loop/face would introduce another linear dependence.

And so taken in total (i.e., including this final loop/face), there are ( V + F ) EQs with 2 linear dependencies, or a set of ( V + F - 2 ) linearly independent EQs 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.