Why are resistors tolerances relative instead of absolute?

Every resistor has a tolerance, this provides the user with an idea of the accuracy of the product. This tolerance is represented by a percentage. This means: a big value resistor will be less accurate than a small resistor with the same tolerance.

$$1\text{ k}Ω 10\% \in [900Ω , 1100Ω] → 100Ω$$ $$100Ω 10\% \in [90Ω , 110Ω] → 10Ω$$

The 100Ω 10% resistor will be closer to 100Ω than a 1kΩ 10% will be close to 1kΩ.

Why is that? Because high value resistors are harder to produce than small ones? If not, why is tolerance a percent and not a fixed amount of Ohms? Why are tolerance relative and not absolute?

These questions are also valid for capacitors, but I'm pretty sure the answer will be the same.

• That's an excellent question as many many things in engineering have absolute tolerances rather than relative... Commented May 21, 2017 at 8:26
• If you have two resistors in a voltage divider then, if percentage tolerances are the same, $\frac{R_1}{R_1+R_2}$ is going to give the same accuracy if the resistor are $\small1 \Omega$ or $\small 1M \Omega$
– Chu
Commented May 21, 2017 at 8:50
• @JImDearden That point of view make sense after all. But, Why is it harder to make a +/- 10 Ohms tolerance on a 1MOhms than on a 200Ohm? Commented May 21, 2017 at 10:19
• Probably because a 1M resistor has a much higher "resistance density" than a 200R resistor? Commented May 21, 2017 at 11:05
• Becasue you do need a 10-ohm resistor that is accurate to 1 ohm but you don't need a 1M-ohm resistor that is accurate to 1 ohm. Commented May 22, 2017 at 6:04

I'll try to simplify this for you... Hopefully successfully.

If you imagine making a resistor just by cutting pieces of a material, let's say a special metallic film;

You want your resistor to fit in a usable box, else it's pointless, so you cannot make super long strips or incredibly short ones. So you use film that's different thickness of the same metal.

Now, say that you have a bunch of thicknesses, each thickness is ten times less resistive than the one that's one step thinner. And they all have to be 10mm long to fit your box, so that you can only cut away from a standard strip width, let's say 5mm.

If you want to make 10 Mohm, you take the thinnest one, and you have to remove half of its width. So you have to remove 2.5mm. If the material works linearly, which we'll assume for the ease of it, that means you "cut away" 10 Mohm in 2.5mm. To remove 10 Ohm more or less, that would mean cutting with an accuracy of (brackets for clarity of order, not because they are needed):

(10 / 10000000) * 2.5mm = 2.5nm.

2.5nm is smaller than what we can do in silicon chip technology. Written in meters that is 0.0000000025m, where for the uninitiated, one meter is close to one yard, or about the size of a long stride of an adult human.

If you wanted to get the same 10 Ohm error on a 100 Ohm resistor, you'd take the foil that's five steps up, which if it's still linear would get you about 50 Ohm (2 bits of 100 Ohm in parallel), so you'd have to cut off 2.5mm again. But this time, you can cut away only accurate to:

(10 / 100) * 2.5mm = 0.25mm.

That's something a practised person could do with a pair of scissors.

See the difference in difficulty there? Scissors versus can't even do it in microchips?

And that's when your resistor's box is allowed to be 10mm x 5mm, which is around 10 times the size of the most commonly used types these days.

Now, obviously resistors aren't made in an elf workshop full of reels of metal film... anymore... We've gotten much better at making more different thicknesses of different materials, so it's gotten better.

But, it does illustrate the point, even if you would use laser-trimming on everything, trimming to one part per million, which is 10 Ohm on 10 Mohm, is going to be a very difficult process to get consistent and it will even then still create a lot of parts that are over or under trimmed.

By accepting that any process in engineering is governed by statistics and percentages, as well as rules of average, we can very easily cope with resistors that are 10%, or 1% or 0.1% accurate, so there is no need to do it better for most cases.

Only when you need a very accurate reference, which is uncommon if your name isn't Fluke, Keysight, Keithley or any of those others, will you want someone to give you a resistor that's better than 0.001% and those are usually large ceramic plates with very accurately applied layers of resistive material, which then get cut to a very accurate recipe and will cost ridiculous amounts of money, even now. Though the 0.01% are finally getting close to affordable.

• You brought me a very nice answer here ! So, if I get it right, the problem come from the manufacturing tools' accuracy. This accuracy decrease as the "metal plate" does thinner Commented May 21, 2017 at 12:22
• @M.Ferru No, the point is that they keep the accuracy the same. They cut both the 100 Ohm and the 10 MOhm in my example with the theoretical scissors, so the 10 MOhm has the same 10% possible error. Commented May 21, 2017 at 12:44
• @M.Ferru, it's just as the UV'd Asmyldof's answer says. The tolerance is about the precision with which they can machine the resistive material. Their machines can get more or less than perfect by a certain percentage of accuracy. They use different resistive materials for different value resistors. Commented May 21, 2017 at 13:07
• Even if all resistors were made using a wrap of wire that had the same nominal resistance per millimeter, any uncertainty in the resistance per millimeter would have 1000 times as much absolute effect in a 1M resistor as in a 1K resistor. Commented May 22, 2017 at 5:19
• "Now, obviously resistors aren't made in an elf workshop full of reels of metal film" Lies I say, lies! Nice answer otherwise. Commented May 22, 2017 at 7:49

This is more of a materials science or manufacturing question. It really depends on the resistor technology also the process used for manufacturing. Source: Chip Resistors Information

The resistance of an element measures its opposition to electric flow, expressed in ohms (Ω). Every material has a specific resistivity, which measures the strength of this opposition. For an even cross section of an element, the resistance ® is proportional to the material's resistivity (ρ) and length (L), and inversely proportional to area (A).

$$R = \rho* (L/A)$$

$\rho$ is fixed, its due to the materials resistivity and probably not a big factor in the tolerance. The area is probably an easily controllable factor, but the amount of material especially the height is not easily controllable. Either way you end up with a 1% or so error, you etch the area with a laser while measuring the resistance to bring the precision in which takes time and the cost of etching. But the process is the same for large and small resistors and you have the same manufacturing error for both large and small.

• I got it now. It wasn't really clear until now. Commented May 22, 2017 at 16:47

The tolerance is fix among resistor regardless of the value because of the manufacturing process. As said in the other answer, this is due to tools or materials used. Theses tools or materials have there own tolerance which echo on the resistor's tolerance.

You can learn a bit more on resitor manufacturing process on this webpage.