The “coverage” explanation given by the top answer doesn’t make a whole lot of sense. With the manufacturing tolerances involved (especially historically when those values were chosen!) the idea that 22 and 47 gives more useful combinations than 22 and 46 is numerology, not engineering.

Wikipedia only says that the deviation is for “unknown historical reasons”. Maybe a deep explanation doesn’t exist, and it’s a simple historical error that was propagated?

Ultra-precise resistor values have been possible for as long as there have been good meters. It turns out that's a very long time ago.

The numerology makes a lot of sense if you are working with 0.1% tolerance parts. Lower tolerances have actually gotten more popular as electronics have become more cost-sensitive.

> Ultra-precise resistor values have been possible for as long as there have been good meters. It turns out that's a very long time ago.

It's not about metering but stability of the resulting value. 1% or 0.1% doesn't do you much good if few degree temperature change gets it out of spec. Now temperature coefficent is an additional spec on the spec sheet but by definition you kinda need low drift to go in pair with high precision

> The numerology makes a lot of sense if you are working with 0.1% tolerance parts. Lower tolerances have actually gotten more popular as electronics have become more cost-sensitive.

Lower tolerances have just become cheap. Back when I was a kid there was significant difference in price between 1% and 5% resistors. Now they cost basically same (for low power ones at least) so why not ? [1]

You also don't really need that many precision parts in the first palce.

Where before in say a power amplifier you had say an analog preamp driving power IC (or outright discrete power amplifier) you had to have a bunch of precise resistors (or someone tweaking a pot on the production line) to keep the gain same in both tracks. Now you just slap a D-class chip that takes line in and outputs power and you're done, and the few % variance in power supply caps or output filter doesn't matter much.

* [1] https://eu.mouser.com/c/passive-components/resistors/?case%2...

On price - you are looking at low quantity. Buy a few million of them from a manufacturer, and you will find the 5% ones still significantly cheaper (even at standard temp coefficients and power levels). That's why new 5% and 10% resistor products are still sold.

Note that the second part of the argument was that the quantities needed are low. I'd go even further than the GP: even in the "old days" my understanding is that the number of precision parts needed was very low. Most circuitry is of the "pick a component value in this order of magnitude" variety, even on sensitive hardware. Only the handful of parts where it really matters need precision.

But, whereas previously you might spec 5% for a few resistors and 10% for the bulk, now it's no longer worth the added line on your BOM.

> Buy a few million of them from a manufacturer, and you will find the 5% ones still significantly cheaper

That just ain't true. 5% are not really used (at least with consumer electronics). 1% and 5% difference is very tiny, even across few millions.

> That's why new 5% and 10% resistor products are still sold.

If you go on digikey and search for 10% you will not find very many compared to 1%

> you had to have a bunch of precise resistors (or someone tweaking a pot on the production line) to keep the gain same in both tracks

Or match them by hand before they’re placed.

But if you need 0.1% tolerance resistors and the circuit diagram requires a 13.3 (±0.1%) kΩ resistor, you can just order one ...?

I have a hard time imagining situations like "We need a very precise 57kΩ (±0.5kΩ or about 1%) resistance but we can only get three precise resistors: 10kΩ, 22kΩ, and 47kΩ! Oh thank god we can connect 10kΩ and 47kΩ, if the last one was 46kΩ we would have been in trouble."

There are a lot of times when even the 1% and 0.1% ladders don't have the value you need, and you can usually construct a combination of only two of them to get that value, thanks to the way the ladders are set up.

In terms of scale, you cannot simply order a larger factory. It has opportunity cost.

Possible, sure, but isn't the origin of the E12 series, as opposed to using e.g. the Renard R10 series, for specifying ±10% precision parts?

1 ±10% is 0.9 to 1.1

1.2 ±10% is 1.08 to 1.32

1.5 ±10% is 1.35 to 1.65

1.8 ±10% is 1.62 to 1.98

2.2 ±10% is 1.98 to 2.42

And so on.

Note that the linked document https://www.govinfo.gov/content/pkg/GOVPUB-C13-f5fea679df4c3... doesn’t just apply to resistors or electrical components. It is a general framework for deriving “preferred” numbers.

Main reason are costs for mass production, and nuances of technology, that electronic components usually made in very huge batches, then distributed to distribution network and lie on storage up to tens years.

So manufacturers made series of agreements, like "in 2020-2025 make e3; in 2025-2030 make e12".

Expensive high precision parts, are not in this scheme, they allocated by direct requests, like Toshiba manufacture high-end transistors, and order for them high-end resistors with some exact value.

As I read it, rationalization is one of the core reasons.

Also you can just use one magnitude lower resistor... 10 isn't odd but 1 is.

Honestly the explanation looks like someone tried to reverse-engineer a mistake into logic.

The reason you can not build a 0% tolerance resistor is the laws of physics. A very high precision resistor can certainly be build but it will never be perfect. Increasing precision has a cost to it. For normal resistors the shape and thickness of the film of resisting material is calculated and the tolerance is mainly dictated by the precision of the manufacturing process. Increasing this precision of the process adds cost. High precision resistors can be trimmed to specification. When you manufacture the resistor with a lower resistance you can use a laser to trim some of the resistive material away. This is an extra step and adds extra cost. While this can be very precise you are limited to what you can measure and there is a limit to that. Also precision is limited by environmental factors like heat, humidity and aging.

That's also how many higher end analog chips are made, just blast it with laser till it fits tolerance. Making a bunch of chips in less precise process then trimming them chip by chip ends up cheaper than going to more expensive processes.

We are saying the same thing. For most applications, 5% tolerance is fine if could be cheaper. But for some applications it is worth the extra cost.

It sounded like you were saying that a 0 tolerance resistor would be worth the extra cost, and the reply is pointing out that a 0 tolerance resistor is not actually possible anyway, and as you approach 0, the cost increases beyond whatever your limit is.

It would seem pretty pointless to try to nail the resistor to within 0% tolerance when the solder bridges, wire that connects them to the circuit will have resistance.

You can get really high precision for a resistor at the silicon level but as for a regular resistor, the most you can do is buy the best TCR value you can and laser trim it to the exact spec. You can also buy hermetically sealed that have extremely low TCR and tolerances but they are expensive as hell and are relatively massive. If you can control the temperature of the board really well, you can hit very precise resistance values but again, expensive and not possible for all applications.

Overall, it's much easier to understand where your noise bottlenecks are in your system and cheap out on the parts that won't contribute to any reduction. Like if your ADC has a bit resolution of only 1uV, worrying about noise at the nano volt is just a waste of money.

> But there is no reason why 0 tolerance resistor couldn't exist

I think the laws of physics is a pretty good reason.

If you want 0% tolerance, you can hand-match.

No-one needs 0% tolerance.

Sure, you can handmatch. If the thermal radiation of your hands wouldn't affect the measurement.

This is actually a real problem with semiconductors. I often need carefully matched sets of transistors and diodes, and a quick and easy way to get them "good enough" is just to measure either a diode or the B-E junction of the transistor with a multimeter in "diode check" mode. I pin the paper strip to a piece of wood and let the temperature stabilise for about half an hour before I start, and only touch the component with the meter probes.

You can try this yourself. With any given 1N4148 you'll see the Vf indication on the meter change as the microscopic current warms the junction up a tiny bit. If the room (and thus the diode) is fairly cold, it can detect the heat from your finger at about 1cm away.

The reason is clearly outlined on the wikipedia page: https://en.wikipedia.org/wiki/E_series_of_preferred_numbers

> Since the electronic component industry established component values before standards discussions in the late-1940s, they decided that it wasn't practical to change the former established values. These older values were used to create the E6, E12, E24 series standard that was accepted in Paris in 1950 then published as IEC 63 in 1952.

That's not much of a reason: why were the older values chosen? The article looks back and argues that it was to make more unique values possible.

That seems the same problem of "why train track spacing is like it is" and the answer is "nothing actually technical".

1,2,5 series woud've been much more useful considering how often in electronics you need integer ratio of some 2 values

Well, the 1, 2.2, 4.7 E3 sequence that is embedded in the E12 sequence is plenty close enough for almost all applications. The successive ratios are noticeably more consistent than for the 1,2,5,10 sequence (2.2, 2.14, 2.13 versus 2, 2.5, 2)

They were probably chosen because the decision was made in concert with manufacturers who wanted to be able to continue to buy/build with the values they had before.

I’m surprised so many here are taking issue with the answer on the page. It’s a reasonable answer reflecting a pragmatic process. Just don’t rug-pull our existing components and give us more variation.

And don’t give me “you can get odd numbers with two resistors” — at the time this was done, these things were not cheap, they were not small, and doubling your component count and increasing your product size because of a new government standard would not have gone over well.

You can ask that question, too - but it's irrelevant to the question why the E12 series was changed in this way.

The original question was why E12 10^(n/12) was misrounded. The wikipedia page only explains why the E system was E12 inatead of E10.

Those values make the E24 sequence evenly spaced with adjacent values. Since the other series are more course, any error is less important than manufacturing practicalities.

Pretty much all the numbers can be explained as being the closest number to the geometric mean of the previous and following number. Once you have chosen to round the second number in E3 to 22, 47 is the closest to splitting 22 and 100 evenly. The exception is 33 in the E6 series, that should be 32 when splitting 22 and 47, most of the other "errors" in E12 and E24 are there because 33 pushes the other numbers upwards.

Three of the 33's in series also add up to 100 better than three having a value of 32 each.

I'm electric engineer by first education, and have more than ten years exp in field.

Must say, in many cases, these numbers are not important, in good schemes acceptable +-10%, so 26 could be 23.4-28.6.

Even more, cheap resistors marks have accuracy +-5%.

Exists precision resistors with accuracy +-0.5%, or even 0.1%, but high precision applications, typically used some very different technologies. For example, are high quality multi-turn variable resistors, laser cut resistivity pads, created with high cost materials or even rare-earth materials.

And many current applications use some sort of very high quality reference, in many cases, based on totally different physical principle, and scheme constantly adjusted with closed loop.

> Exists precision resistors with accuracy +-0.5%, or even 0.1%, but high precision applications, typically used some very different technologies. For example, are high quality multi-turn variable resistors, laser cut resistivity pads, created with high cost materials or even rare-earth materials.

Most modern resistors are of either a thin-film type, with sub-micron-thin layers of nickel directly sputtered and onto a ceramic body, or a thick-film type, with a metal oxide/ceramic paste applied and baked onto the ceramic body.

The process window is aimed at a target, but binning operations after production are all that separate precision resistors from cheap resistors.

If you sputter on a little too much nickel, or your paste isn't quite as conductive after baking as you hoped, you just sell that one as a 10% or 5% resistor.

If you get lucky, and produce one that is within 0.01% of an E96 resistor (which might even happen while aiming for an E12 resistance!), you sell it as a precision unit.

Agreed that these aren't that important in most modern designs.

> binning operations after production are all that separate precision resistors from cheap resistors

Absolutely, no.

Cheap resistors not only have low precision of marking. They also suffered from many other issues.

Examples of cheap resistors issues: they have large temperature coefficient; they change resistance permanently on overheat and when under high voltage; they aged fast and this mean, their resistance permanently change; they usually have large parasite inductance and also they usually have coil structure, and under high voltage happen leak between turns.

High voltage, in this case, mean just about 100Volts.

A single measurement does not a 0.01% resistor make. When you want that kind of tolerance, drift and aging properties are hugely important and the resistor is not going to be constructed the same way as your bog standard $5/reel 1% resistors.

> binning operations after production are all that separate precision resistors from cheap resistors.

Really? I would have assumed that resistors are so cheap that testing them is more expensive than designing the process up front around the tolerance.

Yup really. Grab some 5% and 1% resistors and measure them with decent multimeter.

You'll find that their values are almost exactly: stated_value+/-tolerance rather than a range of values.

i.e a 100k 5% resistor will be almost exactly either 105k or 95k not some number in between.

Nope. Not buying it. Just pulled a couple of bog standard thick film 100K 1% 1206s and they are measuring at 99.94, 99.96, 100.02, 99.93 kiloohms.

They're probably batch controlled. There is no way every resistor is being tested for compliance given how cheap SMT resistors are.

Are you miss (one time) programmable resistors? They appear in 2000s and where available to by, but expensive.

I have not tried myself, just don't found case to use them.

And I have few high precision resistors, and even microwave resistors, they very different from cheap film resistors.

Technologies exists different, for different cases, for different pockets.

What really need, is to put analog parts into right mode, and to make it stable, for example vs temperature changes.

As electrical contact points and traces could add up to few Ohms, so resistors need not be exact numbers, but few different from range of planned +-these few Ohms.

When were the resistor series' first invented?

Is it possible that calculating the logarithmic scale numerically was quite a lot of effort, so instead a graphical approach, using a slide rule, was used? If so, small errors of a couple of percent could be expected in the results, especially if the rule wasn't precisely made?

We had good logarithm tables in the 1620’s. http://www2.cfcc.edu/faculty/cmoore/LogarithmInfo.htm: “Napier died in 1617. Briggs published a table of logarithms to 14 places of numbers from 1 to 20,000 and from 90,000 to 100,000 in 1624. Adriaan Vlacq published a 10-place table for values from 1 to 100,000 in 1628, adding the 70,000 values“

These will have had errors, but I doubt they had them in the first 4 digits and if they had them, they would be easily spotted. (Edit: https://adsabs.harvard.edu/full/1872MNRAS..32..255G says there were errors)

(https://en.wikipedia.org/wiki/Adriaan_Vlacq)

> One of the modern applications of Egyptian fractions is the request of a specific resistance value needed in the design of an electrical circuit, a problem called in the literature the 2- Ohm problem. College students know well from their physics class, that the equivalent resistance R of two parallel resistances and is given from a law very easy to deduce, based on equating the current passing through the fictitious equivalent resistance R with the two currents passing through both resistances while maintaining same potential. One direct application of this, suppose an engineer wishes to incorporate in one of his designs a resistor of so many ohms which the manufacturer does not produce; for it is impossible that the latter displays in the market all possible ohm-values for his resistors. First, the market cannot possibly sustain it, but more important, one cannot feasibly produce resistors with values as elements of a dense subset of the real line, being, as analysis taught us, an uncountable set. Rather, manufacturers display only in the market what they call an "E12 series", i.e. resistors in sets of 12 different values, namely

> 10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 68, 82 ......

> 100, 120, 150, 180, 220, 270, 330, 390, 470, 560, 680, 820......

> 1000, 1200.......... Ohms, etc.....

> Now suppose an engineer needs in one of his designs a resistor of 7 ohms, then he would resort to a parallel combination from the fraction 1/7=1/10+1/56+1/100+1/120+1/150 in which all the resistors belong to the E12 series, i.e. he will replace his 7 ohms resistor with 5 parallel resistors; this he reaches using a special software (computer programmes exist for such designs, yet the exact solution is by no means trivial). What I did myself instead, is to resort to Ahmes 2/n table and wrote 2/7=1/4+1/28 or that 1/7=1/8+1/56. Decomposing further 1/8 into 1/12+1/24=1/12+1/48+1/48, but then I shall have to use instead of the 48 ohms resistor a resistor of 47 ohms from the E12 table. My final fraction is 1/7=1/12+1/ 47+1/47+1/56, i.e. my resistor of 7 ohms will be simulated by 4 parallel resistors instead of 5 (I am accepting equal fractions). My solution is both minimal and optimal based on Ahmes table. The relative error of my design will not exceed 0.6 per cent, being negligible; especially that any manufactured resistor will itself be subject to some allowed tolerance of the same order.

https://web.archive.org/web/20130625181118/http://weekly.ahr...

So, a thousands of years ago? Ancient Egypt and before? :)

Also:

> The measured values of voltages and currents in the infinite resistor chain circuit (also called the resistor ladder or infinite series-parallel circuit) follow the Fibonacci sequence. The intermediate results of adding the alternating series and parallel resistances yields fractions composed of consecutive Fibonacci numbers. The equivalent resistance of the entire circuit equals the golden ratio.

> then he would resort to a parallel combination from the fraction

Easier to find series combinations and then maybe clean them up with one parallel resistor. e.g. 4.7 + 2.2 ohms = 6.9 ohms; if you want better, 3.9+3.9 =7.8, and a 68 ohm resistor in parallel yields 6.997 ohms, an error of .04%.

Often that parallel resistor will be a trimpot or other adjustable means.

Or the greedy approach: 6.8+.22 in parallel with 2700 gives you 7.0018 with .026% error.

>my resistor of 7 ohms will be simulated by 4 parallel resistors instead of 5 (I am accepting equal fractions).

If we had just 1,2,5 series that would just be 2 resistors tho ? 3 resistors to get 9,8; 2 to get 6,4,3

The whole thing seems to be not that practical for electronics where you don't aim your amplifier to have amplification of "golden ratio" but in most cases some integer like x5 or x100

> one cannot feasibly produce resistors with values as elements of a dense subset of the real line, being, as analysis taught us, an uncountable set

The rational numbers are an example of a countable dense set.

Unfortunately, countably-infinite product catalogs are still a bit unwieldy.

1/7 ≈ 1/8.2 + 1/47 gives about half the error (<.3%) with half the resistor count.

Its funny how the comments section of this question on stackexchange.com are people complaining to @jonk that his answer is irrational when he's just reciting material, yet they double down on their position despite his objection. Doesn't help that @jonk is kind of arrogant, but my point is... people.

"Your document is at least 40 years younger than E-series components and all the odd/even numerology was not an issue." is a pretty strong criticism! It doesn't matter if jonk is reciting material if he's citing the wrong material.

Neither do I. I'm not sure if that claim is right or wrong but it doesn't seem properly resolved.

Agreed. I don't think a 40 year old document can be trusted. (Side-eye at the US constitution. :)

And why does American coinage and currency not observe the 2-3-5 rule?

We have $1, $2 but it's obsolete, $5, $10, $20, $50, $100 but no $200, $500, $1000 but no $2000 and so on.

Worse yet, coinage! 1 cent, 5, 10, 25(!), 50, 100 and we're done.

Because currency isn't trying to fill out the spectrum. We want to have as few different values as possible, focused on a certain range. And really big bills are rare.

So looking at 1-100, 20 vs. 25 doesn't really matter, people won't bother to carry 50s, and people won't bother to carry 2s. 3s wouldn't help at all.

We barely even need the 10 either. 1,5,20/25,100 works fine.

And above 100 you can use powers of 10 and nobody will really care.

Now that some ATMs let you choose your denominations, I get all my cash in fives. Even $300 (pretty much the outer limit of what I ever want to carry in cash) in fives comfortably fits in my wallet, and you avoid most of the headaches of trying to make change for larger bills.

You can fit a stack of 60 bills in your wallet comfortably!? The internet says a bill is 0.0043 inches thick, so 60 bills would be roughly a quarter inch thick. Folded in half, that's over half an inch just for your cash.

ah, sorry. My "wallet" is one of those folio type things that doubles as a phone case. I don't keep it in my back pocket like a traditional Costanza-style wallet.

Do people keep wallets in their back pockets outside of cartoons and 20th century pickpocket victims in films? I prefer it separate to my phone, but always front pocket, I don't want to sit (lop-sided?!) on it.

That is still the standard practice, as far as I am aware; I've never carried my wallet anywhere else. Not sure what else the back pockets would be used for. Perhaps you'd feel weird about watching other people's asses, as you walk down the street, but if you did, I bet you'd notice a lot of worn-in rectangular patterns - especially on blue jeans - revealing the wearer's wallet location.

I used a regular back-pocket wallet until 2020 when I didn't use it for a few months except to enter my credit card number online. I'm in the Midwest and it seems that this is still the norm. Maybe things are different on the coasts?

Something else to think about is the way different resistors and other components have been graded over the decades.

For instance there were manufacturing approaches where a target value was produced for a large number of components, but the manufacturing tolerance was a very wide +/- 20%.

The parts were then graded individually into the 1%, 5%, 10% and 20% bins, marked and priced accordingly.

If you then specified the lowest-cost 20% parts, none of them were actually any closer than 10% to their nominal value.

I do wish someone would put up the paid for magic documents that allegedly contain the actual reasoning behind the choice of the E-series. I think this SE answer coould be true, but it feels a little contrived- and, at least to me, this is one of electronics history’s greatest mysteries.

The standards docs don't give you the reasoning behind decisions, that's all in the closed door meetings. And most of the time it's "because that's the way it is."

except in this case, given its title, the document almost certainly does exactly that: "ISO 497:1973, Guide to the choice of series of preferred numbers and of series containing more rounded values of preferred numbers" (mentioned in the answer on the post that was used to close the one HN links to).

The fact that ISO documents aren't just free PDF files with all rights past "viewing" locked down, charging businesses money for hard copies, is still one of the most blatant ways in which the ISO has held, and continues to hold back the world.

I'm not saying that standards shouldn't be freely viewable, but this isn't that big of a problem or unique to the ISO. There are hundreds of standards for manufacturing split across dozens of publishers and industry organizations (just for electronics you have JEDEC, IPC, ISO, and then 2-3 more depending on specific application domains).

If you're working in industry your company pays the pittance for membership as an organization then you pay the (relative) pittance for the doc and shove the PDF into your company's network store (unless they're jerks and lock it to a device).

Are you responding to something completely different? I was remarking on the "The standards docs don't give you the reasoning behind decisions" claim, which is almost always true, but in this case seems incorrect, given that there is a standards whose sole purpose is to give the reasoning behind the decisions.

I was commenting on your last sentence

The reason is likely because discrete resistor values are trimmed in circuit by placing a much larger value in parallel to bring the overall value down.