# Computing effective resistance

I am not so good with electricity and I need this for a computer science project: I want to compute the effective resistance between a set of nodes in a circuit. Now, I know how to compute the effective resistance between 2 nodes: how calculate resistance between two points for arbitrary resistor grid? But is there a way I can modify that circuit so that I can compute more than one resistance at a time? In that example, they compute R_AD. Could I connect C to the ground as well as D, and then get R_AD and R_AC in one go?

Thank you.

No, there is no general way to obtain simultaneously (e.g., by means of simple matrix operations) all possible driving point ("equivalent") resistances of a resistive network.

In general, the driving point resistance of a resistive network is given by the ratio of two determinants. One determinant depends only on the network and not on which resistance you want to determine. The second determinant is a suitable cofactor of the first one, and depends on the nodes across which you want to find the driving point resistances.

This means that if you have a network with $N$ nodes, you have to determine $\binom{N}{2}$ possibly different determinants (some can be equal due to symmetries).

If, instead, you want to consider only the equivalent resistances that can be found by fixing one of the nodes, as in the example you gave, this can be done easily through matrix inversion.

For more on these topics, you can have a look at:

[1] N. Balabanian, T. A. Bickart, Electrical network theory, Chapter 3, John Wiley & Sons, Inc., 1969.

[2] W.-K. Chen, Active network analysis, Chapter 2, World scientific, 1991.

[3] H. W. Bode, Network Analysis and Feedback Amplifier Design, Chapter 1, D. Van Nostrand, 1945.

• Thanks for your reply! I'll try and get those books from the library :) Oct 20 '16 at 3:09
• Hi, I have been looking through [3] and a couple of other books in my uni library. I underdtand now that we can get those equations by nodal analysis or mesh equations, as well as by simply calculating equivalent resistances using the basic formulas (i.e. sum in series, sum of inverse in parallel). Is there any other alternative way? I am trying to justify that the way I am doing it is the only way of doing it that can be expressed as linear equations. Thanks Oct 23 '16 at 22:41
• "as well as by simply calculating equivalent resistances using the basic formulas": this is not a general method: there are resistive networks that cannot be decomposed as series-parallel connections (a basic example is given by the Wheatstone bridge). The only general method is going through nodal/mesh analysis or the indefinite admittance matrix (see [1] and [2]). The good thing is that the writing of nodal equations or of the primitive indefinite admittance matrix can be easily automated. Oct 23 '16 at 22:50
• Thanks very much! (annoyingly I could not find [1] or [2] at the library, so I'll google "indefinite admittance matrix"). Thanks again! --- Oh, I just realised it's the same thing as the Laplacian Matrix. Great :) Oct 23 '16 at 22:54