Resistor Network

Consider a network of n resistors R_i so that R_2 may be connected in series or parallel with R_1, R_3 may be connected in series or parallel with the network consisting of R_1 and R_2, and so on. The resistance of two resistors in series is given by

 R_(net, series)=R_1+R_2,

and of two resistors in parallel by

 R_(net, parallel)=1/(1/(R_1)+1/(R_2)).

The possible values for two resistors with resistances a and b are therefore


for three resistances a, b, and c are


and so on. These are obviously all rational numbers, and the numbers of distinct arrangements for n=1, 2, ..., are 1, 2, 8, 46, 332, 2874, ... (OEIS A005840), which also arises in a completely different context (Stanley 1991).

If the values are restricted to a=b=...=1, then there are 2^(n-1) possible resistances for n 1-Omega resistors, ranging from a minimum of 1/n to a maximum of n. Amazingly, the largest denominators for n=1, 2, ... are 1, 2, 3, 5, 8, 13, 21, ..., which are immediately recognizable as the Fibonacci numbers (OEIS A000045). The following table gives the values possible for small n.

npossible resistances

If the n resistors are given the values 1, 2, ..., n, then the numbers of possible net resistances for 1, 2, ... resistors are 1, 2, 8, 44, 298, 2350, ... (OEIS A051045). The following table gives the values possible for small n.

npossible resistances

See also

Fibonacci Number, Resistance Distance

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Amengual, A. "The Intriguing Properties of the Equivalent Resistances of n Equal Resistors Combined in Series and in Parallel." Amer. J. Phys. 68, 175-179, 2000.Sloane, N. J. A. Sequences A000045/M0692, A005840/M1872, and A051045 in "The On-Line Encyclopedia of Integer Sequences."Stanley, R. P. "A Zonotope Associated with Graphical Degree Sequences." In Applied Geometry and Discrete Mathematics: The Victor Klee Festschrift (Ed. P. Gritzmann and B. Sturmfels). Providence, RI: Amer. Math. Soc., pp. 555-570, 1991.

Referenced on Wolfram|Alpha

Resistor Network

Cite this as:

Weisstein, Eric W. "Resistor Network." From MathWorld--A Wolfram Web Resource.

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