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Wed Feb 17 18:04:40 EST 2016

## Got lost in thinking about graphs...

Not easy. I'm missing a lot of intuition.
It's not clear how to think of circuits graphs, since there are
terminals and nets, which have more structure than graphs (nodes,
edges).
Circuits are terminals and nets.
It's helpful to draw a fully connected circuit and replacing
electrical symbols with "blank" 2 and 3 terminals.
2-terminals can be represented by edges.
Such a graph is not simple (i.e. parallel connections possible).
However, parallel connections can be eliminated by using composite
objects. The result is then a simple graph.
Using norton/thevenin transformation, even further reduction is
possible (eliminate serial connections) until a single norton/thevenin
circuit remains.
When 3-terminals enter the picture, this is no longer possible.
Modeling 3-terminals as a collection of 2-terminals and some extra
equations (gyrators?) is possible but will "halt" the ability to
construct equivalent circuits, as that is only possible for linears..
Are circuits planar graphs[1]? Not necessarily.
https://en.wikipedia.org/wiki/Planar_graph
For circuit analysis, a problem that arises is which loops to pick to
determine independent KVL equations. Planar graphs seem to make this
easier (as it is easier to construct the dual?)
Circuits (networks) are n-terminals connected to nets. They are not
graphs per se (in the mathematical sense; objects made of nodes and
links).
Circuits of linear 2-terminals can be combined to a single 2-terminal
in the Laplace domain.
Circuits of non-reactive nonlinear 2-terminals can be combined in a
similar way into a single current/voltage relation.
Once nonlinear and active are combined, the behavior becomes more
complex and cannot be condensed into functions: systems of nonlinear
differential equations arise.
N-ports, N>2 complicate matters. However in the application of
interest (analysis of diode + bjt circuits), these can be decomposed
into networks of 2-terminal devices with extra current/voltage
relations (Ebers-Moll, with 2 current relations).
So the objects we're working with are:
- networks of 2-terminals
- KCL, KVL
- passive static (R) and dynamic (C,H) relations
- current/voltage controlled current/voltage sources
Such a circuit is a non-simple, directed graph, where every edge
represents a 2-terminal.

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