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I find it hard to simplify complex electrical circuits both 2 and 3 dimensional (cube, prism,etc.). Is there any efficient method to simplify a circuit for the purpose of calculating the equivalent resistance between two points?

I have discovered many techniques by solving variety of questions. They are:

  1. Redraw the circuit by taking the nodes at convenient positions or by morphing the circuit.
  2. Find equi-potent points by symmetry, balanced Wheatstone bridges and blank wires.
  3. $\Delta-Y$ conversion and vice versa.

Are there other efficient techniques like these which help simplify a circuit? Please do not talk of using Kirchhoff's law and solving matrix equation as they lengthy and cumbersome.

MrAP
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1 Answers1

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There are a number of techniques that can be employed to analyse in an efficient way complex passive networks by means of paper-and-pencil calculations. Of course, for a computer, matrix methods are the way to go.

Here are a few:

  1. For certain type of networks (e.g., series-parallel networks, ladder networks), one can apply the regula falsi (false position): you start from a convenient resistor (e.g., the last one in a ladder network) and you assume that the current in that resistor is an arbitrary value, e.g., 1A; from this, you can walk the network up to the input terminals to find a voltage and a current whose ratio yields the equivalent resistance. This method is also useful to calculate the current in the last resistor knowing the voltage or the current at the input: you just have to take a proportion.
  2. Middlebrook's extra element theorem (EET) [1,2] and its extension, the $n$EET [3,4]. These are lesser known theorems of network theory which allow to find solutions in so-called "low-entropy" form, that is, a readable form evidencing the most significant terms. I gave an example of outcome from this theorem in this answer. They are not specific to the calculation of equivalent resistances -- they are much more general, in fact -- but they can be also efficiently applied in this case. In the references given below there are examples of applications.
  3. Bartlett's bisection theorem. This theorem, also, is not specific to the calculation of equivalent resistances, but sometimes it can be applied to this aim when the network has certain symmetries.
  4. Exploit the symmetry of the network, especially when you have networks with equal resistances. This is probably what you mean in your points 1 and 2. There exists also a number of works which give a systematic treatment of symmetrical networks, I'll add the references as soon as I find them in my messy folders.

[1] R. D. Middlebrook, "Null Double Injection and the Extra Element Theorem", IEEE Trans. Edu., 32, 167-180, 1989 online copy

[2] V. Vorperian, Fast Analytical Techniques for Electrical and Electronic Circuits, Cambridge University Press, 2011.

[3] R. D. Middlebrook, V. Vorperian, J. Lindal, "The N Extra Element Theorem", IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 45, 919-935, 1998.

[4] R. W. Erickson, "The $n$ Extra Element Theorem", online.

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Massimo Ortolano
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