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Re: kirchoff's rules and linear dependence



M.E. suggested [off list] restating his loop-finding method
to look for *wires* (in addition to components such as resistors,
capacitors, etc.).

With that emendation, the method has no weaknesses I know of ...
and retains its considerable strengths.

My intuition tells me this method finds complete sets of linearly-
independent loops (but I don't have an ironclad proof).

This method is superior to the color-in-the-areas method I mentioned.

Of course there is considerable non-uniqueness in the answer. Given
one linearly-independent set you can construct lots of other sets.

==========

More generally, Kirchhoff's approach uses the formalism of
_graph theory_ i.e. nodes and arcs (sometimes called nodes
and edges). It's clear that components count as arcs
between nodes. And from the graph-theory point of view it
becomes clear that *wires* also serve as arcs between nodes.
Consider them zero-ohm resistors if you must.

Funny story: I actually saw a whole bunch of zero-ohm
resistors once ... complete with the black-black-black
color code. They were used when somebody had an N-layer
PCB board and didn't want to pay for a N+1-layer board,
yet had some traces that needed to cross other traces.
It turns out it was cheaper to insert a fair number of
zero-ohm resistors than to produce N+1-layer boards.

Quoting Michael Edmiston <edmiston@BLUFFTON.EDU>:

... Once I began
telling students to make sure their "new loops" contain a new component,
and they can quit looking for loops once every circuit component is
included in a loop, the students' circuit analysis made a dramatic
improvement, and their attitude toward these problems also made a
dramatic improvement.

I believe that!