A little-known minimum concerning resistors in series and in parallel

Abstract

In the elementary theory of electrical circuits, the series connection of n identical resistors has an equivalent resistance n2 greater than the equivalent resistance of their parallel connection. More briefly, the series/parallel ratio is n2. We show that if the resistors are not all identical, the series/parallel ratio is greater than n2 and can never be less than n2. This little-known minimum has been demonstrated previously, using the theorem that the arithmetic mean of non-negative numbers always equals or exceeds their geometric mean. Here we present a simple proof that avoids using the theorem. If the n resistors differ only slightly, the series/parallel ratio is still n2 to first order in their differences. Because this ‘weaker’ form has long been known, we discuss only briefly its significance for electrical metrology. We present a Monte Carlo simulation of the series/parallel ratio for two resistors, one of which varies in accordance with a Gaussian density distribution defined by three tolerance ranges, and we compare the simulation graphically with the theoretical series/parallel ratio for each of these ranges. This theoretical ratio is essentially a plot of the density distribution of the square of a Gaussian variable, or equivalently of a chi-squared distribution on one degree of freedom. Finally, we note an interesting connection between the minimum and the second law of thermodynamics.