Preferred values for resistors and capacitors
2017-10-29 composition electronics math
You've probably noticed that although resistors and capacitors come in many different values, some of which seem like they could be randomly chosen, there's nonetheless some sort of logic to it. You'll see power-of-ten sizes like 1kΩ, 10kΩ, 100kΩ and it's understandable that those would be "round" numbers it might be convenient to use and manufacture... but for instance there also seems to be something special about the number 47, so you see many resistor values like 4.7kΩ, 47kΩ, 470kΩ, and capacitors like 0.47µF and 470pF. Why 47? Why not 46 or 48 - especially when 47 is a prime number and 48 has many small divisors, which would seem like it might be convenient? Here are some notes on the commonly-used numbers and where they come from.
First of all, there's the question of having standard numbers of some sort at all. This concept goes back to the Industrial Revolution - and it was one of the important defining factors in what the Industrial Revolution was all about. Moving away from the "craft" model of every manufactured thing being made from start to finish by one skilled individual meant there was a need to measure things and have all the things of a given model be the same as each other. Today we take for granted that idea of "interchangeable parts," where things like screw threads are standardized so you could take the nut off one screw and thread it onto some other screw instead of all nuts and screws being made in pairs that have to be kept together. People like Jean-Baptiste Vaquette de Gribeauval, Eli Whitney, and Marc Isambard Brunel are remembered for contributing to the introduction of interchangeability around the late 18th and early 19th Centuries (mostly for making muskets, cannon, and other military technology); it was revolutionary at the time. Now it makes sense to everyone in the business that it's a good thing for the standard list of resistor and capacitor values available from one manufacturer to be basically the same list of values available from some other manufacturer.
Next there's the idea of having the numbers repeat with each "decade" or factor-of-ten interval. That lines up well with the metric system, which does everything by tens and was standard in electronics for as long as electronics has been around, even in places that still used other measurements for other things. Also relevant is the wide dynamic range of resistors and capacitors - here in my workshop I have stocks of resistors from 1Ω to 10MΩ (a factor of 107) and larger and smaller ones yet exist; capacitors in commercial use range from a fraction of a picofarad to multiple farads, with a 1013 factor between the top and bottom. It's nice to have a system that will provide easily understandable values throughout the range.
Tolerance is the remaining major piece of the puzzle. Today we're accustomed to buying resistors of a given value that really are pretty close to the value they're supposed to be; I routinely specify 1% resistors in North Coast projects even in places where 5% would do, just because the saving in effort and inventory control pays for the slightly higher price of the components themselves. But back in The Day™, even 5% resistors were considered "premium" components (which is why they get the metallic gold tolerance band of luxury) and tighter precision was impractical. It was more common to use 10% or even 20% components. Wide tolerances are still common on some other components, notably electrolytic capacitors (which may have uneven tolerances like "-20%, +80%"). Regardless of whether your tolerances are narrow or wide, there's some benefit to having the tolerance ranges form a nice series. If your components are 20% tolerance, then a resistor labelled 1kΩ will really be somewhere from 0.8kΩ to 1.2kΩ and it's nice for the next larger standard size to have its tolerance range start at about 1.2kΩ. If the ranges overlap, you waste numbers having more standard values (and associated costs) than are needed, and you face confusion if you ever measure a component and can't tell which standard value it is meant to be. If there's a gap, then you might calculate a value you want and not be able to find any standard value close enough.
Putting these pieces together: it makes sense to have some sort of standard values. They should cover the range from 1 to 10 and then repeat scaled by 10 in the next decade. And the numbers should be spaced such that the tolerance ranges of the components will just barely touch each other - maybe with small compromises to avoid needing too many digits of precision. That implies that the numbers will be log-periodic or nearly so: each number is a fixed multiple of the previous one. With 20% tolerances, for instance, we want each number to be 1.5 times the previous one; that would correspond to the tolerance ranges exactly touching.
Note the calculation is not 1.0, add 20% to get 1.2, add another 20% to get 1.44; instead, it is 1.0, add 20% to get 1.2, then 1.5, subtract 20% to get 1.2. The difference comes about because each percentage is measured from the middle of the corresponding tolerance range, not the end. Applying the 1.5 factor with a little bit of fudging to keep it rounded to two digits and distribute the rounding error evenly, and expressing it as integers (because we'll be moving the decimal point to wherever is convenient anyway) gives the sequence 10, 15, 22, 33, 47, 68, before spiralling back to 100 to start the next decade. And that sequence gives you the standard values you can most easily buy for 20% components.
The standard numbers are written up in a document from the International Electrotechnical Commission, IEC 60063, which you can buy from their Web storefront for 40 Swiss francs. This standard is pretty much universally followed because what the IEC did when they made the standard was write up what manufacturers had already been doing for many years anyway, instead of starting with a good idea and trying to lead the horses to water. The numbers in this standard are called "E-series" numbers or "preferred numbers." They're used pretty much universally for the values of resistors and capacitors; often also for other things that have electrical values, such as inductors, Zener diodes (breakdown voltage), and so on. A different system called Renard series is used for fuses and some non-electronic engineering applications. There are different E-series for different tolerances, but there's also a bigger division into two-digit and three-digit sequences. Here is the two-digit sequence:
10 11 12 13
15 16 18 20
22 24 27 30
33 36 39 43
47 51 56 62
68 75 82 91
The entire table is called the E24 sequence, 24 numbers between the top and bottom of the decade, appropriate for 5% components. If you take just the first and third columns (alternate entries, shown in bold) you get the E12 sequence, which is appropriate for 10% components. The first column alone, alternate entries from E12, is the E6 sequence - the same one I mentioned before, for 20% components. There is also defined an E3 sequence (10, 22, and 47, which are alternate entries from E6) for very wide-tolerance components like large electrolytic capacitors; and an E1 sequence which just consists of the number "1."
The E24 sequence includes some adjacent two-digit integer values - there's no integer between 10 and 11 - so to go any further we need another digit of precision. And the rounding errors introduced by trying to keep even E24 down to two digits, mean that it doesn't work well to just add a zero to all the E24 values and continue subdividing in between them. So there's an E192 series for 0.5% components (which is the furthest you can extend the scheme without needing yet a fourth digit) and alternate subsets of it form the E96 (1%) and E48 (2%) series; but some of the E24 values, including that of the 39.0kΩ 1% resistor shown above, are not included in the E192 series. Here is the E96 series; I'll omit E192 as not sufficiently popular to be worth boring you. The E48 series is just alternate entries from this one.
100 102 105 107 110 113 115 118
121 124 127 130 133 137 140 143
147 150 154 158 162 165 169 174
178 182 187 191 196 200 205 210
215 221 226 232 237 243 249 255
261 267 274 280 287 294 301 309
316 324 332 340 348 357 365 374
383 392 402 412 422 432 442 453
464 475 487 499 511 523 536 549
562 576 590 604 619 634 649 665
681 698 715 732 750 768 787 806
825 845 866 887 909 931 953 976
Do we really need this many standard resistor values? Well, if we want to have full coverage - every possible number within 1% of some 1% resistor nominal value - then yes, we do. But especially for old timers like myself who cut our teeth on the 5% and worse components, it's often only necessary that the value be precisely known and repeatable, not that it be very close to a specific calculated value. Rather than using three-digit numbers off the E96 list, I usually design with E24 components but specify that they should be 1% precision; and I think that's a very common practice. For that reason, manufacturers still make 1% components in E24 values, with the third stripe in the colour code always black, even though they don't mesh well with the standard of the E96 list. Not many of us really want to invest in stocking a full set of 96 values per decade for development or resale purposes.
Still, it's nice that the longer list does exist and bigger distributors do stock them; on those occasions when one needs a really specific fixed-resistor value, okay, it can be done. Some places I've used non-E24 resistors include the 140kΩ voltage divider resistors in the Leapfrog VCF, and the 499kΩ 0.1% resistors that set the quantizer steps in the Octave Switch.
I posted this entry partly to have the standard values in a convenient place for my own reference; there are many posted sets of E-series numbers on the Web already, of which my favourite is the one at Logwell, but the original maintainer of that page is deceased, and we can't count on it remaining available forever. As a contribution of reference material that I haven't seen anywhere else, and also because it was fun, I did some ray-tracing to generate portraits of what pretty much all the colour codes look like on real resistors. You can find those linked from the new Resistor Gallery.
The way that the standard resistor values attempt to divide up the decade (factor of 10) in equal intervals is a lot like the way that tuning systems such as 12-EDO divide up the octave (factor of 2) into equal intervals. That suggests a question: could we make music using standard resistor values as a microtonal scale? Back in 2015 I explored that, using simulated annealing to automatically generate some music that would be reasonably consonant while having all the frequencies chosen from the E24 series, and here is the result. It's performed not on the hardware modular, but with a piano-ish Csound patch.
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