Re: kirchoff's rules and linear dependence

[John Denker <jsd@AV8N.COM>]

Quoting Michael Edmiston <edmiston@BLUFFTON.EDU>:

> (3) The situation described immediately above is what happened to
> Justin's student.  How did this happen?  When writing loop equations, a
> "new" equation will not be independent of equations already obtained if
> the new equation lacks a new circuit element that does not already
                         ^^^^^^^^^^^^^^^^^^^^^
> appear in the existing equations.

That's an ingenious rule ... but I don't think it's 100% reliable.
By way of example, consider the following two-terminal circuit
(don't ask me what it does):

  >-+-----R-----+-----L-----+-----R------->
    |           |           |
    +-----C-----+-----R-----+
    |           |
    +-----R-----+

Now contrast that with the following circuit:

  >-+-----R-----+-----L-----+-----R-----+->
    |           |           |           |
    +-----C-----+-----R-----+           |
    |           |                       |
    +-----R-----+-----------------------+

where I have added a new loop but not added any new *components*.

When analyzing the second circuit, the loop that goes through
the southeast corner is not trivial and is not linearly
dependent on the loops that make up the first circuit.

> Therefore, when looking for new loops, you need to find loops that
> contain at least one resistor or battery or some component that has not
> already appeared in a loop.  A "new" loop for which all components
> already appear in existing loop equations will not contain any new
> information that cannot be algebraically determined from the
> previously-existing loop equations.

Again:  that's ingenious, but I don't think it is a 100% reliable
rule.  Sorry.

So what *should* we tell students?

For circuits that can be laid out in the plane, including the
two examples presented above, I think it suffices to find *small*
loops ... i.e. for each area in the plane, find the smallest
loop that encloses it.  Write down the loop equation.  "Color in"
that area so you know not to visit it again.  Move on to the next
area.  (My "Stokes law" intuition tells me this is a complete
description ... but I haven't proved it.)

For non-planar circuits, I have no comparable advice.  Indeed
I suspect the question of how to factor a network into loops
might be NP-hard in the general case (but I haven't proved
that, either).  For introductory classes, stick to planar
circuits!