Uniqueness and existence of simplified equivalent circuits

[Andrew732]

I know this probably sounds weird, but I have a research problem that requires "random" analog circuits. Basically what this means is that I create Spice netlists by randomly adding linear and/or nonlinear components of random types with random node and parameter values. This works fine and I get a variety of interesting, valid circuits with between 5 and 50 components.

This is beyond my electrical engineering knowledge so I'm not quite sure how to ask this, but what I'm wondering is whether there is a way to algorithmically find the guaranteed simplest version of a given random circuit that has equivalent electrical behavior. (A simple example would be applying the rule that replaces two resistors in series with one equivalent resistor.) Under what conditions would such a simplified random circuit be unique i.e., no other circuit of equal or lesser size would have the same electrical behavior? Does the answer depend on whether linear or nonlinear components are used?

Hopefully what I'm asking makes sense. Thanks for any help~

[AlephZero]

You need to define exactly what you mean by "equivalent electrical behaviour".

If you mean equivalent behaviour as a 4-terminal network (2 input and 2 output wires) then for a linear circuit you could find the transfer functions, then synthesize the "simplest" circuit from them. Without doing the math, I would guess the resulting "simplest" circuit would not necessarily be unique, but given that the transfer functions have a finite number of poles and zeros, you could probably define some interpretation of "simplest" that makes sense (e.g. the minimum number of components).

With the same transfer functions, the behaviour may not be "exactly" the same. For example if you have active components, the total power dissipations will probably be different, if only because the original circuit might contain components just comsume power, but don't affect the input and output in any other way.

For nonlinear behaviour, my instincts say all bets are off and anything is possible, if you are creative enough to find right counterexamples.

Your question seems rather similar to the problem in experimental dynamics of measuring the response of a system and constructing a (small) math model that matches the response directly, as compared with constructing a (large) finite element model that represents the actual structure in all its (mostly irrelevant) detail and calculating the response of the large model. Anyway, that analogy is what my "instincts" about your problem are based on.

[Andrew732]

Thanks very much for your reply. I actually am only thinking about 1 input, 1 output circuits. You're absolutely right, this is part of a black box modeling project. Based on your advice, I probably will rethink my entire random circuit idea in terms of random transfer functions instead.

There are a few things that are still unclear to me though. Is there a natural sense in which some linear transfer functions are simpler than others e.g., with fewer poles and zeroes? Could any such linear transfer function in theory be embodied by a linear RLC circuit of sufficient (possibly infinite) size? Do you think are all bets off again if we start talking about nonlinear transfer functions and nonlinear circuits?