Supposed we want to establish a set of standard resistor values between 10 ohms and 100 ohms. Our first thought might be to space them out equally: 10 ohms, 20, 30, 40, 50, 60, 70, 80, 90, and finally 100 ohms. This way we cover all the values between 10 and 100 with equal tolerance: 10 ohms either way.
But is it really equal spacing? The difference between 10 ohms and 20 ohms is 100 percent! 20 ohms is twice as big as 10 ohms. Don't we need another resistor in between those? Like maybe 15 ohms?
At the other end of the scale, the difference between 90 and 100 ohms is only about 10 percent. For lower-precision applications, maybe 10 percent is not enough to warrant the extra resistor. Maybe we should leave out the 90 ohm resistor.
Many values in electronics follow a geometric progression instead of a linear progression. Think about extending our progression to the resistors between 100 and 1000 ohms. Should we continue with 10 ohms per gradation? 100, 110, 120, 130, 140, and so on? Or should we instead use 100, 200, 300, 400, and so on up to 1000 ohms? The fact that we instinctively think that progressing in 100 ohm increments at this larger scale, but at 10 ohm increments at the previous lower scale, indicates that we are already thinking geometrically.
What exactly is a geometric progression? We have already seen one. We first wanted to figure out resistance values between 10 and 100 ohms. Then we thought about values between 100 and 1000 ohms. This is a geometric progression:
| 1 | 10 | 100 | 1000 | 10,000 | 100,000 |
It is a geometric progression because each value is 10 times the previous value. If we want to create a geometric progression, we need to choose a number that indicates the number we will multiply the previous number by in all cases. Here is another example of a geometric progression used often in computers:
| 1 | 2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 |
This is a geometric progression based on the number 2. Each number in the series is 2 times the previous number.
Suppose we want our resistors have 20% tolerance. If we start with a 10 ohm resistor, it could be used anywhere a resistor from 8 ohms to 12 ohms is needed, because 8 ohms is 20% less, and 12 ohms is 20% more. After 12 ohms, we need another resistor. What resistor do we need?
Do we need a 12 ohm resistor? No, we do not. If we had a 12 ohm resistor with 20% tolerance, its actual measured resistance could be anywhere between 9.6 ohms and 14.4 ohms. So it actually overlaps the area covered by the 10 ohm resistor, which is 8 to 12 ohms. So we need to choose a higher value than 12 ohms for its nominal value. The nominal value is the marked value, which might be off by up to 20%.
12 ohms is the top end of the tolerance range for the 10 ohm resistor. But 12 ohms is also the bottom of the tolerance range for some other resistor. You might be tempted to think we should add 20 percent to 12 ohms to come up with a 14.4 ohm nominal value. But that is not quite correct. Instead we want to solve this equation for x:
x * 0.8 = 12
... because x will give us a value, that when we check the bottom end of its tolerance range (-20% or 80% of nominal value) the result will be 12 ohms. To solve this equation, we multiply both sides by 1.25, which gives us
x = 12 * 1.25 = 15
So, if we choose a 15 ohm resistor, the bottom end of its 20% tolerance range will e 12 ohms, which is what we were looking for, and its top end will be 18 ohms. From 18 ohms, we move on to find the next resistance value.
Since this is a geometric progression, and the multiplier remains the same throughout the series, we have already calculated the value we will need to use in all cases. That value is 1.5, because a 15 ohm resistor is 1.5 times the value of a 10 ohm resistor. Continuing from there, we find this geometric series:
| 10 | 15 | 22.5 | 33.75 | 50.625 | 75.9375 | 113.90625 |
So we see that by multiplying by 1.5 each time, we get from 10 to 100 ohms (and a little over) in 6 steps. To find the multiplier to get from 10 to exactly 100 ohms in 6 steps, we use this calculation:
multiplier = 10 (1/6)
or 10 to the 1/6 power. This number works out to be approximately 1.468, which is more than close enough for our purposes. If we use these values, we come up with the following alternative series:
| 10 | 14.68 | 21.55 | 31.64 | 46.44 | 68.18 | 100.08 |
In practice, because we don't need all these significant figures for 20% precision, we can hold down the number of digits to two. So this is the series that the engineers finally decided on:
| 10 | 15 | 22 | 33 | 47 | 68 | 100 |
So we get from 10 to 100 ohms in 6 steps, and with 20% tolerances, we cover all the possible values between 10 and 100 ohms. We multiply all these values by 10 to get the values in the next set, from 100 to 1000 ohms:
| 100 | 150 | 220 | 330 | 470 | 680 | 1000 |
Here are the standard resistor values with their 20% tolerance ranges:
| Low end | Nominal value | High end |
|---|---|---|
| 8 | 10 | 12 |
| 12 | 15 | 18 |
| 17.6 | 22 | 26.4 |
| 26.4 | 33 | 39.6 |
| 37.6 | 47 | 56.4 |
| 54.4 | 68 | 81.6 |
In practice, 20% tolerance resistors are not used much any more; they used to be used more. Currently, 10% and 5% are common, and 1% is used in some applications that must be very precise.
10% and 5% tolerance resistors have their own standard values that are extensions of the series above, and they are calculated in a similar fashion. The 10% standard values are these:
| 10 | 12 | 15 | 18 | 22 | 27 | 33 | 39 | 47 | 56 | 68 | 81 |
So now you know why I occasionally refer to certain resistor values or color code combinations as "illegal."
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