Jon Slaughter said:
Jon Slaughter said:
Robert said:
Is there a generalized Thevenin's theorem for arbitrary "black boxes"?
i.e., Suppose I have something like
---> I
V +---[ ]--- 0
where [] is a black box.
I should be able to write something like
V = Z(t, V, I)*I
which sorta resembles ohms law. V and I generally depend on t.
if [ ] is a resistor then Z(t,V,I) = R and in general Z also depends on
a set of parameters.
But what about more complex black boxes?
If its a resistor and a capacitor then what?
V ---||---/\/\/\/\---- 0
Then Z(t,V,I) = ?
For passive components is Z a linear differential equation?
Any other ways to simplify such expressions?
The reason I ask is I have a circuit that has a lot of these "paths"
that are connected in some way but each path is the same configuration
with only the "constants" of the components that are different.
Thanks,
Jon
There's Middlebrook's Theorems but I don't think that's what you're
asking about.
http://en.wikipedia.org/wiki/Extra_element_theorem
Robert
Actually it looks very similar and it seems close to my problem.
Essentially I have a circuit where each "branch" looks identical(Actually
its not but uses identical topology... its almost fractal like) and I am
trying to use that symmetry to make it easier to solve. I'll have to read
up on it to see what exactly it doing though.
hmm... actually it doesn't seem to be what I want. My problem is similar to
his but each branch in the graph has the same "structure".
For example, take any graph and treat an edge as a black box that contains
passive elements. Then each edge is described by a linear differential
equation L_E where E is the edge indicator which really only depends on the
passive elements characteristics. Then is there a way to simplify the
graph/solve the system for the voltages and currents?
In a passive circuit, the complexity of finding the solution grows with the
number of nodes - sorry :-( Think of finding the voltages along a string of
resistors. For two resistors, the voltage at the middle point depends on the
value of two resistors. For three resistors (same structure in each branch...)
both of the two internal voltages depend on the value of three resistors, etc
etc. (However, if you know in advance that all the resistors have the same
value, you can easily calculate the solutions for all the internal nodes.)
Solving the network to find all node voltages / branch currents can certainly be
done the "hard way." This uses admittance matrices, as in i = Y v (with "v" the
vector of internal voltages and "i" the vector of external currents and "Y" the
admittance matrix).
http://users.ecs.soton.ac.uk/mz/CctSim/chap1_2.htm (for instance)
I think there's actually hope for solving your problem if you have a very simple
and regular structure. In that case, you might be able to find the inverse of
the Y matrix (this is what you need in order to solve the circuit equations) in
a way simpler than the general way. An advantage of this approach is that you
don't need to use the actual elements in each branch, you can use a symbol for
each distinct branch admittance and enter the actual branch admittances in the
solution.
Symbolic simplification of matrix inversions is pretty difficult to do, though.
So, the short answer is that going from simple circuits like a resistive divider
to more general circuits is not simple.
[..]
So essentially between any two nodes we have Vb - Va = Z*I which is just
ohms law in some sense but Z is somewhat arbitrary. The problem is that in
general Z depends on the nodes, current, and time along with all the other
components that exist on that branch.
Worse - the voltages at the ends of this branch depend on all other voltages.
You can think of it very similar to solving the graphs of resistors that
most people do in basic electronics or physics courses except I want to
replace them with more general components. In the case of a resistor its
very simple. In the case of a capacitor it is not.
Actually, for a circuit with only capacitors, it's not different from a purely
resistive circuit (just a lot more unknown voltages at DC...).
Since I want to take
into account the transients I get system of integeral equations(because of
the non-constant/non-sinusodial input voltages and unknown node voltages).
Of course these integral equations are equivilent to a system of
differential equations.
Essentially you can think of each branch representing a linear differential
equation. But each linear DE "looks" the same as every other one except for
the constant coefficients and it might depend on different currents and
voltages.
It seems though I should be able to recursively simplify the
circuit until I find all the unknowns. well, this should be obvious but the
issue is, is the size. Just hoping for a way to reduce the complexity
because of the "symmetry" that exists.
For some very special circuits, yes (e.g. resistor strings, delta or star shaped
circuits, etc). In general, I think not.
Anyways, probably no way to do what I want but I can wish...
It's worth thinking about this. I studied in a group where the professor made a
Big Discovery as a young student when hearing about a deceptively simple result
at a conference (for all passive circuits, the sum of voltages in the solution
is always constant - or something[1]) and asking himself if that could really be
true and then wondering what the consequences were. The answer was the companion
circuit and its use for sensitivity analysis
. He thought it was so trivial
that he didn't publish it right away. Tut tut. Two lessons in that anecdote ;-)
Good luck
Best Regards
Jens
[1] You look it up. Annoyingly, I cannot remember right now.