A little-known minimum concerning resistors in series and in parallel
- Publisher:
- IOP PUBLISHING LTD
- Publication Type:
- Journal Article
- Citation:
- European Journal of Physics, 2020, 41, (3)
- Issue Date:
- 2020-05-01
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| Filename | Description | Size | |||
|---|---|---|---|---|---|
| Frenkel_2020_Eur._J._Phys._41_035705.pdf | Published version | 1.06 MB |
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© 2020 European Physical Society. 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.
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