Standard Resistor Values: Why not a true geometric series?

Also, NO, they do NOT include the values of higher tolerances in the tighter tolerance tables. Only true in the worst, highest numerical tolerance tables of 29% 10% and 5%. after that the table does NOT include the previous table's values.

They are LOGARITHMIC progressions.

I see a lot of things not needing to be miniaturized getting SMD stuff.

As for specifically power supplies - a lot of those have big parts such as made-to-order "magnetic components", high current rectifiers, transistors and/or voltage regulator ICs that need to be bolted to substantial heatsinks, largish capacitors, and low count of little bitty things. I also see persistence of through-hole with audio amplifier ICs that have significant heat dissipation.

- Don Klipstein ( snipped-for-privacy@misty.com)

I get 26-10% = 24.3

It seems to me that one would want the tolerance ranges to overlap. That way, any resistance value you calculate analytically will be within the tolerance band of a commercially available resistor.

The value of resistance which is midway between a pair of resistors on a percent tolerance basis is the harmonic mean of the values; this is 2 times the parallel equivalent of the two values. For example, given a 12 ohm and a 15 ohm resistor, the value 2(12*15)/(12+15) = 13.333333 ohms. This is

11.1111111% less than 15 ohms, and 11.1111111% more than 12 ohms.

The amount (on a percentage basis) by which this middle value differs from each of the bounding values is given by 100*(R2 - R1)/(R1 + R2). In the example just above, this gives 100*(15 - 12)/(15 + 12) = 11.1111111%.

So, to find out if an adjacent pair of resistors have tolerance bands which overlap, just calculate 100*(R2 - R1)/(R1 + R2) and compare it to the tolerance of the E series you're investigating. If it's bigger than the series tolerance, they don't overlap.

If the previous 12 and 15 ohm resistors are taken to be members of the E12 series, then we see that since 11.1111111% > 10%, their tolerance bands don't overlap.

If their tolerance bands don't overlap, this means that there is a zone of values between the two nominal resistance values which are not within rated tolerance for either of the two nominal values of that E series.

I find that for the E12 series, the following pairs have a zone of no-overlap between them:

12 and 15 22 and 27

the tolerance zones for the pairs:

18 and 22 27 and 33

just touch.

For the E24 series, these pairs have a zone of no-overlap:

13 and 15 16 and 18 18 and 20 24 and 27 27 and 30 56 and 62 82 and 91

The E48, E96 and E192 series are rife with no-overlap zones. This is to be expected, since the exact values in the logarithmic sequence are rounded to 3 digits, and sometimes we round up, sometimes down.

I noticed on the Wikipedia page

that on the discussion tab, somebody noticed that in the E192 series, there is one value that doesn't fit the expected logarithmic sequence, namely

920, a fact I first became aware of about 15 years ago. It should be 919; if one uses the formula for the 185th member of the sequence:

Value = 100 * 10^(185/192) one gets 919.478686+ which should be rounded to

919, not 920.

However, I discovered that a very simple change will correctly give all the values in the commercial E192 series, including the oddball 920:

Value = 100 * 10^(185/191.9977), with the 185 replaced with the ordinal member number for other values. This small change allows a simple formula to be used in a program, giving ALL the correct commercially available values.

Why did "they" pick 920? I assume that they made an error. Since the digits after the decimal point are near .5000000, which would be the value above or below which one rounds up or down, maybe their calculation was a little off. Perhaps they used a slide rule!

All this still fails to answer the OP's question, a question I've long wondered about. I'm afraid that the people who made the choices are probably no longer with us.

There was a guest editorial, "Preferred Numbers", published in the Proceedings of the IRE, p. 115 in 1951. There were a couple more items in that year on page 467 and page 1572 in answer. Unfortunately, they don't answer the OP's question.

Could you give an example of what you mean by "chaining errors"?

Two 5% tolerance parts in series can be off of a target chosen design series value by as much as ten percent. Typically, the amount the value is off the mark is always going to be more than 5%.

That is called chaining error.

Umm.. 26 - 2.6 = 23.4

The 920 is wrong, it should have been 919. Here's even more relevant reading:

So it is. I was trying to reply to the one below, and cut and pasted from the wrong place!! I looked away for a moment, and when I looked back, well, you see what happened.

I get 27 - 10% = 24.3

This isn't strictly true. When filling in the values from E6 to E12, the value between 47 and 68 should be 57 if you use the geometric mean method, but the actual value is 56. The E12 to E24 transformation works ok with the geometric mean method.

The calculation of the E96 series from the E48 series using the geometric mean method gives 7 values that do not match the actual E96 series, whereas using the 96th root of 10 method gets them all correctly.

The E96 to E192 transformation by the geometric mean method gets 33 wrong values, whereas the 192nd root of 10 method gets them all right, except for that pesky 920 value.

The series total cannot possibly be off more than 5% if the resistors are within spec. Nor the parallel value, for that matter.

Typically it will be within a percent with modern resistors.

Maybe you're thinking of the ratio, which is limited to 5% error when R1 == R2, but can slightly exceed 10% if the ratio is quite different.

Eg. two values a, b nominal,

Worst case ratio is a * 0.95/(a * 0.95 + b * 1.05)

= 1/(1+ b*1.05/a*0.95)) = 1/(1+ (a/b)* 1.105)

The nominal ratio is a/(a+b), of course, so the error could approach

10.5% for very large ratios.

Best regards, Spehro Pefhany

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^^^^^^^^^^^

^^^^^^

WrongAgain, twice this time!

The total error can't be more than 5%.

And most "5%" resistors these days are much closer than 5%. In fact, the total will NEVER be off more than 5%, and will usually be within

1%. That's why 1% and 5% resistors cost about the same.

Long ago, moulded carbon resistors had a huge production spread, and were culled into 20%, 10% and 5% categories. Nowadays, nearly all resistors are production trimmed to value, by spiral cutting or laser trimming, so they tend to be very close. 5% resistors are usually within a few tenths of a per cent these days. Measure some and see.

John

All interesting comments, and I thank you for them, but I do want to say I don't immediately agree with the premise that I want the 10% bands to overlap. Well, OK, I don't use 10% resistors, or even 5% resistors, but same applies to the 1% parts. I assume that the 1% parts I use will very seldom be close to the edge of their tolerance band. Measurement of lots of parts tells me that _usually_ they are within half a percent, and if my design doesn't stress them they'll stay put pretty darned well for a long time. I'm pretty careful to design things so that parts that are anywhere within their 1% band will work correctly, but if I really want a value that's different from the nominal value, I'll either pick a different part that guarantees to be close enough to the value I really want, or I trim using series or parallel combinations. My first preference, though, is to design the circuit so it works properly with E12 values, and trims are unnecessary. We do a lot with calibration of one sort or another, almost all run under processor control these days.

In general, though, I don't see that overlap of the n% bands makes a whole lot of difference; if I need to have a component be within some specific tolerance of a particular value, that defines what I need to do. I either use a part (or combination of parts) that gives the desired result, or I re-design for a different value and/or tolerance. Especially given that we don't find it economical to design using the full E96 range, but use only the "E24" set out of the E96 values (except in very rare instances), we are quite used to having desired values fall outside the tolerance range of the available parts.

In any event, this is a pretty minor point. We have what we have to work with, and it's pretty unlikely that our discussion here will alter the accepted set of E6, E12, ... E192 values. If you're willing to pay enough for a custom value, you can get it.

Cheers, Tom

As I told you before, you don't take the geometric mean of the actual E series values, you take it from the exact values they are rounded from. My statements are exactly right.

Actually. SMT resistors are trimmed at production run time.

Axial TH std resistors are still mass produced, and culled by automated measuring machinery, and seldom get sent through trimming processes.

Leaded *film* resistors are helically cut to value using a simple mechanical (diamond tipped cutter) or laser process. Often they just set the value on the controller to the nominal so the resistors tend to end up a bit higher than the nominal value.

Film SMT resistors are laser trimmed with much better equipment.

Best regards, Spehro Pefhany

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"it\'s the network..."                          "The Journey is the reward"
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Do they punch out the CU traces as well?

Robert

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When did you tell me this before?

That's because if "you don't take the geometric mean of the actual E series values, you take it from the exact values they are rounded from.", it is a mathematical tautology. The results from taking the geometric mean of the exact values of two adjacent elements of an E series and making that the exact value of the intermediate value of the next series (and then rounding), is mathematically identical to just using the root of 10 method to compute the exact values (before rounding) of intermediate values in the next series.

For example, in the E48 series, the exact values of the 2nd and 3rd members would be:

104.913972914 and 110.069417125

the geometric mean of these two is:

107.460782832

Using the root of 10 method, we would get as an E96 value intermediate between the same two E48 members:

100 * 10^(3/96) = 107.460782832, exactly the same.

There's no need to even consider the geometric mean method if you use exact values, because it gives no different results than the root of 10 method.

If you apply the geometric mean method to the rounded values, it might explain some things in the E6, E12 and E24 series, but not in the higher series.

Unlike the values in the lower series, the values in the higher (E48, E96, E192) are the properly rounded values from the root of 10 method (except for

920), so there's no question where they came from.

Yes, I understand this philosophy. Some years ago when the good quality 5% film resistors started showing up from Japan, I noticed that, without fail, they were within 1% of the nominal E24 value. This meant that if I designed some filters, for example, that needed just those values, I could count on the performance 1% resistors would give, for less money.

But, when a committee selects the values for an E series, I think one would like to get "full coverage". Say you need a single resistor for a series dropping resistor, perhaps. Now, you can't play with ratios, you just have to have that value, within 5%, say. One would like almost every possible resistance value to be within 5% (or 10%, or whatever) of a commercially available value in an E series .But this just isn't going to happen in a series like the E96 because there are so many values to round that sometimes the rounding goes up, sometimes down, giving either a little overlap or a llitle underlap (is that a word?). But the coverage is good enough.

As you say, it's a small point.

I still wonder what the answer is to the OP's question. There's a lot of speculation and reverse engineering, but I've never found an authoritative answer.