Re: kirchoff's rules and linear dependence

[James Frysinger <frysingerj@COFC.EDU>]

On Friday 2004 March 05 13:47, Michael Edmiston wrote:
....
> I did think of a situation where the two methods would yield different
> advice, but I have not taken the time to see which is correct.  Draw a
> 3-by-3 array of squares.  Imagine that each line segment contains a
> resistor or some other circuit element.  If you come up with a loop
> equation for each of the small squares you will come up with 9
> equations.  My method would say one of these is not linearly
> independent.  After getting loop equations for the 8 perimeter squares,
> every component in the inner square has been included in a loop.

        That's an interesting model, Michael. Maybe I'll give that a whilrl by the
empirical method of just trying one out for my tutorial. Thanks for the
suggestion! By my method, that inner square would indeed be independent so
I'll probably try this out and then see what happens when I drop that one
equation. If I don't have to provide another "given" value, then your thought
would be correct.

> I am sorry to say that I do not have the time or energy to analyze this
> further in the next day or so.
>
> One thing I don't like about the small loop method is the possibility
> students might think the loop theorem only applies to these small loops.
> Take two squares joined along a side.  Yes, you can use the loop theorem
> on each small loop and then it would be redundant to use it on the big
> perimeter loop.  But you can also use the loop theorem on the big
> perimeter loop then either one (but not both) of the small loops.  The
> "new component" method does not contain a bias against using big loops.

        Good point. My instinct is to teach what "always works" and then extend that
by showing that one can drop a "small loop" equation and replace it with a
"larger loop" equation for a loop that already contains a defined loop. It's
an exercise in degrees of freedom I suppose, but symbolic math logic puzzles
make my brain hurt. I OD'd on five-class-period "necessary and sufficient"
proofs in Calculus back a few decades ago and haven't recovered yet. I'll
leave that to the topologists I think. (If they're the responsible parties,
that is!)

Jim

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James R. Frysinger
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