About Calculation the Resistance of Two-dimensional Infinite Grid Systems

Abstract

The paper considers the problem of calculating the resistance between nodes of infinite grid resistor systems with square and triangular cells. There has long been a question about the resistance between the nearest nodes of an infinite grid of resistances with square cells with the same resistance <i>r</i>. Here, earlier, by the method of symmetry and superposition, a result was obtained <i>r</i>/2 that is striking in its simplicity. However, this result is only approximate, although many physicists consider this result to be accurate. New examples are presented proving what the results obtained earlier by the superposition and symmetry method is only approximate. The result <i>r</i>/2 gives only the lower limit of the correct resistance value. In our work, the correctness of using the equivalent resistance method to calculate the resistance between nearest nodes of infinite grid systems is proved. Using this method, for the resistance between the nearest nodes of an infinite grid of resistances with square cells, a result is obtained about 0.5216 <i>r</i> that only slightly differs from <i>r</i>/2. The results differ from the previously obtained values by about 10%. The resistance between the diagonal points of an infinite grid of identical resistors <i>r</i> with square cells is calculated. For the value of this resistance, a value founded about 0.7071 <i>r</i> that differs from the value 2<i>r</i>/<i>π</i> obtained previously by the superposition and symmetry method.