# Are there gaps in the 2% resistor standard values?

I was reading the Wikipedia entry for E-series resistor values, and I decided to graph the values on a log plot. The plot shows the values nearly evenly spaced, which makes sense because adjacent values are supposed to be related approximately by the factor where N is the resistor series (E6, E12,...,E192).

If I then plot the tolerance range for each value in the E6,E12, E24 ranges, I see overlap for each standard resistor value with the adjacent value – this makes sense because every resistor made can then be used. Here is my plot, which is crude but makes the point – note that I zoomed in a bit, so all values are not shown.

However, if you plot the range of values for the E48, E96, and E192 ranges, I see small gaps. This means that every resistor made may not fall into a standard 2% tolerance value. I show my plot for the E48 resistor series below. I zoomed in quite a bit to make things a bit clearer.

I can also demonstrate that the gaps exist using algebra, but I thought the charts illustrate the issue better.

Are there really gaps in the 2% standard resistor values?

• I do not see problem there. If it does not fit, it will go into 5% tolerance. I think they first choose for most tight tolerance (1%?), then lower... Because of that 2% tolerance usualy mean deviation more than 1% (but less than 2%) from declared value. – Darko Apr 22 '16 at 17:41
• I may be suffering from the belief that there was a resistor value in the E48 series within 2% of any value from 1 to 10 (for example). That appears not to be the case. – ziggle314 Apr 22 '16 at 17:46
• I don't think this is a very good question, when you have proven the answer already. – pipe Apr 22 '16 at 18:19
• There are gaps in the E12 series as well — 12 * 1.1 is less than 15 * 0.9. E24 is gapless because the 24th root of 10 is (very slightly) less than 1.05^2, but E12 and E48 can't cover the whole range without gaps at the conventional tolerances because the 12th root of 10 is more than 1.1^2, and the 48th root of 10 is more than 1.02^2. – hobbs Apr 22 '16 at 18:24
• A dreampt up problem that is not a real problem – Andy aka Apr 22 '16 at 18:24

Resistors are not made to random value and then sorted, they are manufactured to be close to the desired value ( a bit lower) and then trimmed with cutters, abrasive or laser trimming) into the final value (sometimes a bit high, depending on the process), typically well within tolerance (typical error is usually less than 1/3 of the guaranteed tolerance, last I measured a bunch.

So the gaps due to rounding or whatever in the standard series are of no consequence to the manufacturers.

As such, you can get much closer than you might expect, most of the time, to the desired value by paralleling or putting in series standard values to make the nominal value very close to the exact value you want.

• Thanks for response. I had always wondered exactly how the resistance values were set. What motivated this discussion was that I could not find a 2% resistor within 2% of a value that I needed. I now see that I may need to use a combination of resistors. Thanks for the help. – ziggle314 Apr 22 '16 at 19:01

There will quite probably be - in fact you appear to have shown there are.

However you have to ask yourself if it really matters?

• If you need those values exactly you would go with higher tolerance (e.g. 0.5%, 0.1%, etc) resistors.

• If you are relying on tolerance to get exact resistor values, then your design can clearly cope with the resistor not being exact anyway so you pick the one which best suits your design (e.g. if you have a maximum current limit you might go for the closest one above to ensure the limit is met).

• I guess I was just surprised. I have been using these standard resistor values for decades. It is the first time I noticed that I may not be able to get a 2% resistor within 2% of any value that I wish. You learn something new every day. – ziggle314 Apr 22 '16 at 17:54
• As long as what you wish is the color coded value, you can. Even if you get a $93\Omega$ resistor, you put it in the circuit, the temperature changes and the resistor value changes. If you need a more exact value, go 1%. – StainlessSteelRat Apr 22 '16 at 19:07

The gaps are due to roundoff error, e.g. for the least example in your $\rm E\color{#C00}{48}$ graph.

$$8.25 \times 1.\color{#0a0}{02} = 8.415$$

$$8.66 \times 0.98 = 8.487$$

So there is a gap between $8.415$ and $8.487,$ as your graph shows. But more exactly

$$\overbrace{8.2540}^{\Large {10^{\frac{44}{\color{#C00}{48}}}}} \times \overbrace{1.\color{#0a0}{0243}}^{\Large{10^\frac{1}{2\cdot{\color{#C00}{48}}}}} = 8.454$$

$$\underbrace{8.6596}_{\Large 10^{\frac{45}{\color{#c00}{48}}}} \times 0.9757 = 8.449$$

and now there is no gap since $\:\: 8.449 < 8.454$.

Notice that, more exactly, the tolerance is a slightly higher than $2\%,$ i.e. $\color{#0a0}{2.43}\%.$

A little algebra shows that there will never be any gaps when using exact values.

• The problem is that the E48 series of resistor values are rounded. 8.25 and 8.66 are the nominal values, 10^(44/48) and 10^(45/48) are not. – pipe Apr 29 '16 at 20:34
• @pipe Yes, as I said, it is due to roundoff error (both in the resistor values and tolerance values). As I show above, if you use more exact values for both then there is no gap. – Bill Dubuque Apr 29 '16 at 20:39
• But 2% isn't rounded, it's defined. So for the defined tolerance of 2%, there are gaps, even if you use the (arguably) more accurate 10^(44/48). When you write "never be any gaps", isn't that true by definition, in the way you substitute the tolerance as 10^1/96? – pipe Apr 29 '16 at 20:47
• @pipe No, it is not defined. Rather it is derived from the spacing of the 48 intervals that [1,10] is split into by the points 10^(n/48) for n from 0 to 48. Each point is a factor of 1.048 = 10^(1/48) from its neighbors, so no intermediate point will be further away than the square-root of the distance 10^(1/96) = 1.0243. In other words every point is at most 2.43% away from one of the 48 "standard" values. The spec rounds 2.43% to 2% for convenience. – Bill Dubuque Apr 29 '16 at 21:32
• Actually it depends on viewpoint which is defined vs. derived. In any case my point was that you need to use exact (or more precise) values for both the standard points and the scaling factor ("tolerance") to ensure that there are no gaps. – Bill Dubuque Apr 29 '16 at 22:12