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Equivalent resistance for infinite grids of resistors

One of the classic ``hard'' questions for early Electrical Engineering students is finding the equivalent resistance for an arrangement of resistors connected, in a regular way, without bound. The one-dimensional case, a 2-sided infinite ``ladder'' of resistors, can be solved with some insight and the usual rules for working with series and parallel resistances. For infinite grids (2-dimensions), cubes (3-dimensions), and higher dimensional arrangements, a different approach must be taken. This alternate approach can also be used for the 1-dimensional case, which is a useful example to compare this new technique with the well-known resistor reduction technique.

The concept of superposition, which holds for linear systems (of which resistor arrangements is but one example), states that the system response for 2 (or more) stimuli is simply the sum of the responses for each individual stimulus. In this case, the stimuli are two current sources - 1A at one node, and -1A at another node. The resistance between two nodes is simply the voltage which results from running a 1A current source between the two nodes (with no other sources anywhere in the network). Superposition comes into play by connecting a 1A current source between one node and ground, and a -1A current source between the other node and ground, then measuring the resulting voltage between them. The resistor network is considered to be connected to a ground which is infinitely far away from the nodes. This enables the use of symmetry to describe the resulting voltages that arise from the sources.

However, for dimensions of 2 and higher, a simple use of superposition is not enough most cases. It turns out that it is also important to transform the problem into a new space which can utilize the periodic nature of the resistor network. The transform used in this exposition is a discrete-space Fourier transform, which has use in image processing and solid-state physics (referred to as the reciprocal lattice). By using the Fourier transform, the space consisting of infinite (but discrete) nodes gets mapped into a finite continuous space which is easier to handle. This technique results in an integral which needs to be evaluated to get the resistance. An analytical evaluation of this integral can be quite onerous, though one example of this with the 2-D grid is shown in detail.

This approach is based on [1], which used a variant of the Fourier transform. This variant allowed one of the dimensions to be avoided in the integration, at the expense of the elegance provided by the Fourier transform itself. I suspect that the references in [1] are likely go with the Fourier transform approach.




Next: Infinite ladder of resistors
frooha@yahoo.com 2004-04-30
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