Re: number combinations of components


Abstract Dissonance wrote:
> "Keith A. Lewis" <klewis@xxxxxxxxxxxxxxxx> wrote in message
> news:dq645b$2i3$1@xxxxxxxxxxxxxxxxxxxxxx
> > "Abstract Dissonance" <Abstract.Dissonance.hotmail.com> writes in article
> > <11scmv5sqfp5q1e@xxxxxxxxxxxxxxxxxx> dated Thu, 12 Jan 2006
> > 07:38:46 -0600:
> >>
> >>"Proginoskes" <CCHeckman@xxxxxxxxx> wrote in message
> >>news:1137048997.524227.103850@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> >>>
> >>> Abstract Dissonance wrote:
> >>>> I'm having some serious trouble trying to figure out how to figure out the
> >>>> number of combinations involved when trying to hook up components in serial
> >>>> or parallel combinations with phase considerations.
> >>>
> >>> I'm not sure I understand what "phase" means, even when you give
> >>> results for 1 and 2 components below. My picture is of a diode, where
> >>> the current can flow in one of two directions through the component;
> >>> this matches the 2 ways of putting in 1 component, but "in phase" and
> >>> "out of phase" seem to be confusing me.
> >>>
> >>>> if, say, one has 1 component then there is only 2 ways... "in phrase" or
> >>>> "out of phase"... with 2 components one can have series in phase, series out
> >>>> of phase, parallel in phase, parallel out of phase and the reverse of all
> >>>> those(so 8 total)...
> >>>
> >>> I get 8 (not 4) ways of putting 2 components in series, according to my
> >>> model:
> >>>
> >>> ---C1>>>--- ---C2>>>---
> >>> ---C1>>>--- ---<<<C2---
> >>> ---<<<C1--- ---C2>>>---
> >>> ---<<<C1--- ---<<<C2---
> >>> ---C2>>>--- ---C1>>>---
> >>> ---C2>>>--- ---<<<C1---
> >>> ---<<<C2--- ---C1>>>---
> >>> ---<<<C2--- ---<<<C1---
> >>>
> >>
> >>yeah, if one considers them distinct it would be 8.
> >
> > But from an electrical engineering standpoint, you would consider these two
> > the same:
> >
> > ---C1>>>--- ---C2>>>---
> > ---C2>>>--- ---C1>>>---
> >
> > because you can't tell the difference from outside.
>
> well, not necessarily true. Maybe for simple components but there are
> examples where one could get two different results simply by exchanging
> order(i.e., non commutative "addition")... i.e. A + B != B + A
>
> it would require a nonlinear device though...
>
> >
> > Are you looking for *only* serial and parallel circuits? For example
> >
> > +---C1>--+---C2>---+
> > | | |
> > | C3 |
> > | v |
> > | | |
> > ----+---C4>--+---C5>---+----
> >
> > If you want to include things like this, the mathematical question I think
> > you're asking is "How many connected graphs are there with N directed,
> > distinguishable edges and 2 special nodes called 'input' and 'output'?"
> >
>
> ok, I think that makes sense ;)
>
> > Except that would count permutations of components in series, which you
> > don't want to do. :^(
> >
>
> right. Its just that for my circuits it won't matter if if there is a
> permutation of elements that are in series like you mentioned above.
>
>
> > If you only want to consider parallel and series cases, you might want to
> > think about this notation.
> >
> > series(C1,C2,parallel(C3,C4))
> >
> > Meaning C3 and C4 are in parallel and that subcircuit is in series with C1
> > and C2. You'll never have a series subcircuit in series with something,
> > because that would just simplify to one larger series.
> >
>
> yeah, thats basicaly it... one just have to find all the combinations such
> as
>
> series(C1, C2, parallel(C3,C4))
> series(C1, parallel(C2,C3,C4))
> series(C1, parallel(C2,series(C3,C4)))
> etc...

Okay. So what you're looking for are "series-parallel graphs".

The background for this is graph theory, where a graph G consists of a
set of vertices (dots), and edges (lines)(which connect one dot to
another one). In this case, you're allowing "parallel edges" or
"multiple edges", which means you can have more than one edge from u to
v (where u and v are distinct vertices).

A series-parallel graph (or network) is one that can be built up from a
graph with two vertices and one edge, by doing the following, one step
at a time:

(1) Taking an edge uv, and adding a new edge from u to v. ("Parallel")
(2) Taking an edge uv, deleting it, adding a new vertex w, and adding
edges uw and wv. ("Series")

So if you ignore "phase" for the moment, the problem has been solved,
and a formula is given at:

http://mathworld.wolfram.com/Series-ParallelNetwork.html

Incorporating "phase" into the problem, or assuming that some
components are the same, makes it much harder. The second problem can
probably be tackled by considering "edge-colorings" of series-parallel
graphs, but I'd rather do a literature search first. (I don't have
access from my computer at home.)

--- Christopher Heckman

> > So you decide whether the top level is series or parallel, then partition
> > the set of all components into subsets. The next level below that is
> > parallel, for sets of 2 components there's 1 option, sets of 3 or more
> > components can be sub-partitioned into the next series level, etc.
> >
> > The phase or polarity of the components adds a factor of 2^N.
>
> well, but there are many cases where the phase will produce a circuit that
> is the same as another but out of phase.
>
> the thing is that, say, for some circuit I can make some components out of
> phase and arrive at another circuit that is just out of phase with the
> original. For my application I don't need to consider a circuit that is just
> out of phase with another since it doesn't really add anything..
> i.e.
>
> A-C1--C2--C3-B
>
> where C1-3 are all in phase with each other does not change anything if I
> make them all out of phase with each other... the only important thing is
> the how they are in phase with each other in a relative way. So, I think, at
> most one has 2^(N-1) since one can remove atleast 1/2 the circuits since
> they will be "duplicates"( reverse of some other circuit) but there might be
> more.
>
> >
> > --Keith Lewis klewis {at} mitre.org
> > The above may not (yet) represent the opinions of my employer.
>
> I suppose it is like I have N variables and two functions and I am trying to
> find all the ways I can compose the functions together to get a new
> function..
>
> i.e.
>
> series(X,Y)
> parallel(X,Y)
>
> where they map the "space of components(or even circuits)" into itself. then
> I want all the ways to compose these functions.
>
> Maybe that is a good way to approach it... All have to think about it some
> more later though since I have a headache ;/
>
> Thanks for the help,
> AD

.

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