# Why are the “common” resistors and capacitors rated the way they are?

Very newbie question here...one to which I've been unable to find an answer in any of my books or online resources.

Whenever I see a capacitor/resistor assortment that features the most "common" ones (which, I understand can be quite subjective), they tend to come rated in powers of 1, 2.2 and 4.7. What's magical about those numbers?

Is that some mathematical relationship in Ohm's law of which I'm not aware? Is it some logarithmic progression for some OTHER electrical engineering formula that I've not yet encountered? Why is there such a huge gap between 4.7 and 10.0, why isn't there a 7.1 or some such? As someone who a scary uncanny knack for solving "find the pattern in this series of numbers" puzzles, it's killing me that I can't figure this one out. :)

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I don't have time for a proper answer, but take a look at this list of values and at this article. – AndrejaKo Aug 12 '12 at 13:09

## 2 Answers

The values of the resistors are related, they are in a geometric progression. This means each is a fixed multiple of the previous one. For example,

• 1 3.3 10 - multiple is ~3.3
• 1 2.2 4.7 10 - multiple is ~2.2
• 10 12 15 18 22 27 33 39 47 56 68 82 100 - multiple is ~1.2

Yes, there is a large gap between 4.7 and 10. However, when plotted on a logarithmic scale they are evenly spaced.

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A-ha...that's the one thing that I didn't take into account. The fact that results could be `~` rather than `=`. I was looking for 1, 2.2, 4.8, 10.6...not 1, 2.2, 4.7, 11. Of course, that still doesn't answer why there aren't 4.8 and 11 ohm/uf resistors in the standard batch. :) – dwwilson66 Aug 12 '12 at 13:23
Actually after 470 there's 680 which goes before 1000 in E6. – AndrejaKo Aug 12 '12 at 13:27
Because it should include 1, 10, 100, 1K etc. for probably the most basic reason - nice round numbers that are easy to calculate. The reason it is 3.3 and not 3.33 is that more resistor color bands would be required and anyway, that 0.03 extra is not worth worrying about when the tolerance is more than that. :) – geometrikal Aug 12 '12 at 13:29
@geometrikal - I think it's just the latter: no use giving more significant digits than the precision. If you look at what's needed to make a PTH transistor, and then see the price it won't matter much if that painter would need an extra pot of paint :-). – stevenvh Aug 13 '12 at 18:07

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For the E12 series the step size is the 12th root of 10, or about 1.2 larger than the previous one, so 12 steps take you from 10 to 100. That goes with a 10 % tolerance: you can always find an E12 value within 10 % of the desired value. That's because $\sqrt[12]{10} \approx$ 1.2115, and 1.1$^2$ = 1.21.

For example: 18 Ω + 10 % = 19.8 Ω. The next E12 value is 22 Ω. Then 22 Ω - 10 % = 19.8 Ω. (It doesn't always fit that neatly. The blue line shows a small gap between 12 Ω and 15 Ω, but most often there's an overlap.) Nowadays 10 % isn't used much anymore for resistors, 5 % is much more common, and 1 % is not that much more expensive.

That means your desired value won't fall into the 1 % tolerance of the E12 series. For example, if you want a 20 Ω resistor the closest E12 values are 18 Ω and 22 Ω. With a 1 % tolerance they don't come closer than 18.18 Ω and 21.78 Ω, resp. That's why 1 % resistors are offered in a much larger range, typically the E96 range, which includes 20 Ω.

Further reading
EIA resistor values series

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I wonder why the E-series use 33 instead of 32, which would be closer to 31.62, the geometric mean of 10 and 100. – starblue Aug 12 '12 at 16:20
@starblue - the EIA works in mysterious ways :-). I guess it's a cumulation of rounding errors. If you multiply 27 by the 12th root of 10 you get 32.7, which is closer to 33. But I guess you're right, they probably should have calculated each value independently. – stevenvh Aug 12 '12 at 16:30
Yes, it seems to be the result of rounding up several times in a row.  Though it's not done consistently, 33 multiplied by the 12th root of 10 gives almost exactly 40. – starblue Aug 12 '12 at 20:30
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