In the elementary theory of electrical circuits, the series connection of n identical resistors has an equivalent resistance n 2 greater than the equivalent resistance of their parallel connection. More briefly, the series/parallel ratio is n 2. We show that if the resistors are not all identical, the series/parallel ratio is greater than n 2 and can never be less than n 2. 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 n 2 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.

中文翻译：

---

关于串联和并联电阻的鲜为人知的最小值

在电路的基本理论中，n 个相同电阻的串联连接的等效电阻 n 2 大于它们并联连接的等效电阻。更简单地说，串联/并联比为 n 2。我们表明，如果电阻器不完全相同，则串联/并联比大于 n 2 并且永远不会小于 n 2。这个鲜为人知的最小值已被证明以前，使用非负数的算术平均值总是等于或超过其几何平均值的定理。在这里，我们提出了一个避免使用定理的简单证明。如果 n 个电阻器仅略有不同，则串联/并联比在它们的差异中仍为 n 2 至一阶。由于这种“较弱”形式早已为人所知，因此我们仅简要讨论其对电气计量的重要性。我们展示了两个电阻器的串联/并联比的蒙特卡罗模拟，其中一个电阻器根据由三个公差范围定义的高斯密度分布而变化，并且我们以图形方式将模拟与这些电阻器中的每一个的理论串联/并联比进行比较范围。这个理论比率本质上是高斯变量平方的密度分布图，或者等效于一个自由度上的卡方分布。最后，我们注意到最小值和热力学第二定律之间的有趣联系。我们以图形方式将模拟与这些范围中的每一个的理论级/并行比进行比较。这个理论比率本质上是高斯变量平方的密度分布图，或者等效于一个自由度上的卡方分布。最后，我们注意到最小值和热力学第二定律之间的有趣联系。我们以图形方式将模拟与这些范围中的每一个的理论级/并行比进行比较。这个理论比率本质上是高斯变量平方的密度分布图，或者等效于一个自由度上的卡方分布。最后，我们注意到最小值和热力学第二定律之间的有趣联系。