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When dealing with a circuit diagram of some type, a fairly typical procedure for simplifying the circuit is to redraw the circuit diagram a number of times, each time performing one or more independent atomic simplifications, such as combining series or parallel resistors, doing a source substitution, or other such things.

For instance, one might use this sequence of steps if one was asked to calculate the total current in this circuit:

schematic

simulate this circuit – Schematic created using CircuitLab

(One would then conclude that the total current is 2/3 amps.)

However, these successive diagrams are time- and space-consuming to draw.

Such a circuit could also, however, be represented using some sort of algebraic notation. For instance, something like "(R1 + (R2 | R3))". This could then be used with a set of production rules equivalent to the atomic substitutions used above:

1 Ω + (1 Ω | 1 Ω)
1 Ω + 0.5 Ω
1.5 Ω

While this simple off-the-top-of-my-head notation might not be able to represent every circuit topology, I would imagine that a more developed algebraic structure could be constructed to represent more complicated topologies (like some of the algebraic systems used for some things in graph theory). Even a more limited notation restricted to simple branching and such could be useful, though, as such systems are not uncommon.

I have searched online with Google for any some kind of established algebraic notation, but all the search queries I have attempted have only returned results regarding Boolean algebraic notation.

Is there any sort of existing notation or system like this?

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I often see || for "in parallel with". At the end of the day, though, it can be represented as a mathematical formula, so that is what would normally be used.

For instance, your circuit would be written as: $$ R1 + \frac{1}{\frac{1}{R2}+\frac{1}{R3}} $$

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Is there any sort of existing notation or system like this?

Circuit simulation software often employs netlists.

Here's a list of examples, I'll report one:

enter image description here

v1 1 0 dc 24
v2 3 0 dc 15
r1 1 2 10k
r2 2 3 8.1k
r3 2 0 4.7k
.end

The notation is rather straightforward - v1 1 0 dc 24, for example, means "a voltage source v1 from node 0to node 1, 24VDC".

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