You might wonder why you can't always get the value of resistor you want. 6Ω sounds easy, right? What about capacitors. Why 470uF and not 500uF? Are these chosen at random? Why, why, why do they seem to want to make it hard for me? And why aren't there more resistor values between 1kΩ and 10kΩ than there are between 1Ω and 10Ω. Does that mean the higher value resistors are less accurate or something?
The numbers are not chosen arbitrarily at all. There is some (not a lot) of maths behind it.
The numbers are chosen so that they follow a geometric progression with a certain number of values per decade. A decade is 1 to 10, or 100 to 1000. Having the same values repeated between decades means there is less to remember for you, but more importantly (mathematically) it means that the percentage difference between them remains constant.
For E3 values (the earliest and now unused series, there were 3 values per decade. That is:
A geometric progression is one which is defined by the starting point and a constant. You multiply the previous value with a constant to get the next value. For the E series, the start value can be any decade (1, 10, 100, 1000000, 0.00001, etc). Note that you cant start with 0!
In the case of E3, if the constant is k, then 1 * k * k * k = 10, so k is the cube root of 10. Thus k is 2.15. In the sequence a, b, 10, c, d, 100, e, f, 1000 given above, we can now fill in the missing values:
E3 was used in the era of 50% tolerance. Resistors were so inaccurate that if you tried to make a particular value, it could be anywhere between of that value. The sequence 1, 2.2, 4.7 (or 1, 2, 5) was used to denote nominal values.
Numerically, this series works because
If you're worried by the "almost the same", repeat this with 47% and you'll find the values are much closer (46.78% is even closer)
However, when you're talking about such large tolerances, it doesn't make sense to worry about a few percent either way.
As resistors (and capacitors, etc) began to be able to be manufactured in tighter and tighter tolerances, it made sense to have values between the E3 values.
The next step was E6. These were nominally 20% resistors, and for some time I had some of these which were hand painted with their values! (using the dot rather than stripe scheme).
E6 goes 1, 1.5, 2.2, 3.3, 4.7, 6.8, 10. It is no accident that the values inserted between the E3 values to make E6 as the midpoints seen when calculated above.
E6 is technically a geometric series with the multiplier being the 6th root of 10 -- 10^(1/6) = 1.47, and numerically speaking the tolerance would be the square root of this (21.15%).
Note the relationship between the mathematical tolerance and the multiplier for the next range.
After this E12 was used for 10% resistors, and E24 for 5%, and so on. For a long time after the introduction of these new-fangled 5% resistors, they were commonly only stocked in the E12 series, and even today many designs will use E12 series values exclusively.
For the higher tolerance ranges (E48, E96, and E192) the mathematical fudges which were employed to get values like 4.7 (instead of the closer 4.6 -- remember it was 4.64?) have significant effects. In these series, rather than the intermediate values slipping between the previous values, there are areas where the old values change. If you look at this page, you'll see that E6, E12, and E24 have a 4.7, but it is replaced with (surprise!) 4.64 in the E48, E96, and E192 series. See here.
As a little challenge, can you find someone selling a 470Ω 1% resistor? Why? (hint)
There's lots more information on the web. Here and here would be places to start.
The numbers are not chosen arbitrarily at all. There is some (not a lot) of maths behind it.
The numbers are chosen so that they follow a geometric progression with a certain number of values per decade. A decade is 1 to 10, or 100 to 1000. Having the same values repeated between decades means there is less to remember for you, but more importantly (mathematically) it means that the percentage difference between them remains constant.
For E3 values (the earliest and now unused series, there were 3 values per decade. That is:
1 a, b, 10, c, d, 100, e, f, 1000.
A geometric progression is one which is defined by the starting point and a constant. You multiply the previous value with a constant to get the next value. For the E series, the start value can be any decade (1, 10, 100, 1000000, 0.00001, etc). Note that you cant start with 0!
In the case of E3, if the constant is k, then 1 * k * k * k = 10, so k is the cube root of 10. Thus k is 2.15. In the sequence a, b, 10, c, d, 100, e, f, 1000 given above, we can now fill in the missing values:
1, 2.15, 4.64, 10, 21.5, 46.4, 100, 215, 464, 1000.
E3 was used in the era of 50% tolerance. Resistors were so inaccurate that if you tried to make a particular value, it could be anywhere between of that value. The sequence 1, 2.2, 4.7 (or 1, 2, 5) was used to denote nominal values.
Numerically, this series works because
1 * 150/100 = 1.5 and 2.2 * 100/150 = 1.47 (almost the same)
2.2 * 150/100 = 3.3 and 4.7 * 100/150 = 3.13 (almost the same)
4.7 * 150/100 = 7.05 and 10 * 100/150 = 6.7 (almost the same)
2.2 * 150/100 = 3.3 and 4.7 * 100/150 = 3.13 (almost the same)
4.7 * 150/100 = 7.05 and 10 * 100/150 = 6.7 (almost the same)
If you're worried by the "almost the same", repeat this with 47% and you'll find the values are much closer (46.78% is even closer)
However, when you're talking about such large tolerances, it doesn't make sense to worry about a few percent either way.
The E3 series lives on today in test equipment which has 1, 2, 5 series on the ranges. An oscilloscope might, for example, on the vertical gain settings have 10mV, 20mV, 50mV, 100mV, 200mV, 5000mV, 1V, 2V, 5V, 10V, ... as options for the "volts per division". This series often exists on even the most modern digital oscilloscopes.
As resistors (and capacitors, etc) began to be able to be manufactured in tighter and tighter tolerances, it made sense to have values between the E3 values.
The next step was E6. These were nominally 20% resistors, and for some time I had some of these which were hand painted with their values! (using the dot rather than stripe scheme).
E6 goes 1, 1.5, 2.2, 3.3, 4.7, 6.8, 10. It is no accident that the values inserted between the E3 values to make E6 as the midpoints seen when calculated above.
E6 is technically a geometric series with the multiplier being the 6th root of 10 -- 10^(1/6) = 1.47, and numerically speaking the tolerance would be the square root of this (21.15%).
Note the relationship between the mathematical tolerance and the multiplier for the next range.
After this E12 was used for 10% resistors, and E24 for 5%, and so on. For a long time after the introduction of these new-fangled 5% resistors, they were commonly only stocked in the E12 series, and even today many designs will use E12 series values exclusively.
For the higher tolerance ranges (E48, E96, and E192) the mathematical fudges which were employed to get values like 4.7 (instead of the closer 4.6 -- remember it was 4.64?) have significant effects. In these series, rather than the intermediate values slipping between the previous values, there are areas where the old values change. If you look at this page, you'll see that E6, E12, and E24 have a 4.7, but it is replaced with (surprise!) 4.64 in the E48, E96, and E192 series. See here.
As a little challenge, can you find someone selling a 470Ω 1% resistor? Why? (hint)
There's lots more information on the web. Here and here would be places to start.
