graph - What algorithm do I use to calculate voltage across a combination circuit? -

i'm trying programmatically calculate voltage changes on large circuit.

*this question may seem geared toward electronics, it's more applying algorithm on set of data.

to keep things simple,
here complete circuit, voltages calculated:

i'm given battery voltage , resistances:

the issue have voltage calculated differently among parallel , series circuits.
a similar question asked on so.

some formulas:

when resistors in parallel:
rtotal = 1/(1/r1 + 1/r2 + 1/r3 ... + 1/rn)

when resistors in series:
rtotal = r1 + r2 + r3 ... + rn

ohm's law:

v = ir
i = v/r
r = v/i

v voltage (volts)
i current (amps)
r resistance(ohms)

every tutorial i've found on internet consists of people conceptually grouping parallel circuits total resistance, , using resistance calculate resistance in series.

this fine small examples, it's difficult derive algorithm out of large scale circuits.

my question:
given matrix of complete paths,
there way me calculate voltage drops?

i have system graph data structure.
of nodes represented(and can looked by) id number.

so example above, if run traversals, i'll list of paths this:

[[0,1,2,4,0] ,[0,1,3,4,0]] 

each number can used derive actual node , it's corresponding data. kind of transformations/algorithms need perform on set of data?

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it's portions of circuit compound, , compound sections may find being in parallel or series other compound sections.

i think problem akin this:
http://en.wikipedia.org/wiki/series-parallel_partial_order

some circuits cannot analyzed in terms of series , parallel, example circuit includes edges of cube (there's code @ bottom of web page might helpful; haven't looked @ it). example can't analyzed series/parallel pentagon/pentagram shape.

a more robust solution thinking series , parallel use kirchhoff's laws.

1. you need make variables currents in each linear section of circuit.
2. apply kirchhoff's current law (kcl) nodes linear sections meet.
3. apply kirchhoff's voltage law (kvl) many cycles can find.
4. use gaussian elimination solve resulting linear system of equations.

the tricky part identifying cycles. in example give, there 3 cycles: through battery , left resistor, battery , right resistor, , through left , right resistors. planar circuits it's not hard find complete set of cycles; 3 dimensional circuits, can hard.

you don't need cycles. in above example, 2 enough (corresponding 2 bounded regions circuit divides plane). have 3 variables (currents in 3 linear parts of circuit) , 3 equations (sum of currents @ top node 3 linear segments meet, , voltage drops around 2 cycles). enough solve system currents gaussian elimination, can calculate voltages currents.

if throw in many equations (e.g., currents @ both nodes in example, , voltages on 3 cycles instead of two), things still work out: gaussian elimination eliminate redundancies , you'll still unique, correct answer. real problem if have few equations. example, if use kcl on 2 nodes in example , kvl around 1 cycle, you'll have 3 equations, 1 redundant, you'll have 2 independent equations, not enough. throw in every equation can find , let gaussian elimination sort out.

and can restrict planar circuits, easy find nice set of cycles. otherwise you'll need graph cycle enumeration algorithm. i'm sure can find 1 if need it.