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Oops! The total error will be, at most, 1%.
Suppose we want a 400 ohm resistance. If we put four (100 ohm, 1%) resistors in series, the maximum possible series resistance is (101+101+101+101) = 404 ohms. The error is ((404 - 400) / 400) = +0.01 = +1%. Likewise the minimum possible series resistance is (99+99+99+99) = 396 ohms, for an error of -1%. I am tempted to write "obviously!" but that would be unkind.
Oops! The total error will be, at most, 1%.
Suppose we want a 400 ohm resistance. If we put four (100 ohm, 1%) resistors in series, the maximum possible series resistance is (101+101+101+101) = 404 ohms. The error is ((404 - 400) / 400) = +0.01 = +1%. Likewise the minimum possible series resistance is (99+99+99+99) = 396 ohms, for an error of -1%. I am tempted to write "obviously!" but that would be unkind.
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But he mentioned that probability of this occurring is very, very small. This means that he himself does not believe that it can actually happen.
Yes, the probability that all parts error in the same direction is very small. In fact you get a smaller error in percent for combined parts in most cases.
That is true if the value errors are truly uncorrelated from part to part, and uniformly distributed across the tolerance window. In some situations that's probably a good assumption.
But if all 4 parts come from the same production batch . . .
- Produced at the same time under the same ambient temperature, humidity, atmospheric pressure, etc
- From the same batch of raw materials, with the same level of impurities
- On the same machine, with the same degree of wear, misalignment, calibration errors, etc
Dale
Self discusses this issue of increasing accuracy with increased numbers of parts.
He also gives some space to selecting the multiple parts to give a guaranteed tolerance that is considerably better than the basic component tolerance.
He also gives some space to selecting the multiple parts to give a guaranteed tolerance that is considerably better than the basic component tolerance.
Actually, it is more likely they are in the same direction if they were made at the same time under the same process. That is the holy grail of manufacturing, consistency. Still, pretty big slip. We do need to always give an author a bit of slack sometimes. These are not reviewed academic papers, but a ton of work.
I do wonder if worrying about tolerance of 1% resistors matters when we are dealing with transistors and caps.
So, everyone gets it. Now, there have been quite a few books written on crossovers. Is there anything here that is advancing what we need to know that say, Jung, Lancaster, and Linkwitz have not covered?
Your post is accurate for the most part for parts that are "as fired" or "as processed". Assuming a gaussian distribution for the end value is a reasonable one.
Of course, the manufacturer may post process the devices. Binning at test is one such thing.
If for example, a manufacturer has .1%, .5%, 1%, 2%, 5%, and 10% product, they would cull the closest to value at test, and of course charge more for them.
Given those tolerance ranges I mentioned, if for example you purchase a 5% resistor, you will never get one which is within 2% of the mark. A 10% will never be better than 5%.. And there will not be a gaussian distribution of values in a batch purchased, but rather, each tolerance bin will have a section of the distribution of the bulk product. A batch of 5% resistors at 100 ohms for example, would have values between 95 ohms to 98 ohms, and 102 ohms to 105 ohms. There would be a +/- 2% hole in the distribution.
If the resistors are laser trimed to value, there are two possibilities..
First, they may actively trim to value with a test/zap/test regimen...in that case, they will stop trimming once they have reached the desired tolerance. If they are making a 100 ohm 1% resistor, they will stop when the value of the resistor has risen to just above 99 ohms.
Second, they may do a measure/calculate/zap to theoretical nominal. If they measure a resistor at 90 ohms, they calculate the amount of trim historically required to get exactly to 100 ohms. In that case, they may get very close to nominal, and the distribution will be gaussian and fairly tight.
cheers, jn
jn,
Not necessarily true that "you purchase a 5% resistor, you will never get one which is within 2% of the mark..."
If there is not much customer demand for tighter-tolerance units, and they are producing more than needed of tight-tolerance parts, extra tight-tolerance parts will be 'downgraded'. So, the distribution of values in loose-tolerance units will depend on how many tight-tolerance parts are needed and the tightness of the production distribution.
You get what you pay for. If you really want tight-tolerance parts and you don't have time, equipment, interest, etc., to buy and sort lots of loose-tolerance parts, then you should spend more money/ea. on tight-tolerance units if you can get them.
Not necessarily true that "you purchase a 5% resistor, you will never get one which is within 2% of the mark..."
If there is not much customer demand for tighter-tolerance units, and they are producing more than needed of tight-tolerance parts, extra tight-tolerance parts will be 'downgraded'. So, the distribution of values in loose-tolerance units will depend on how many tight-tolerance parts are needed and the tightness of the production distribution.
You get what you pay for. If you really want tight-tolerance parts and you don't have time, equipment, interest, etc., to buy and sort lots of loose-tolerance parts, then you should spend more money/ea. on tight-tolerance units if you can get them.
This is not strictly in accordance with my experience, I often found that quite a lot of a batch are within 1% of the marked value or that many of them can be matched at 0.3%. The dispersion may help to get values not existing in the 1% standard range (for example 98.5 Ohm) . This way, I could build a 6 steps - 6 circuits attenuator (40 to 20 dB) with a 0.2 dB difference between circuits.
This broad a range of tolerances in one resistor type seems to me unlikely in practice - because the 10% parts generally have much worse tempcos than the closer tolerance parts. It might well apply to a smaller subsection (say 0.1 - 0.5% - 1%) but generally its too expensive (time consuming) to test every resistor for tempco so that needs to be guaranteed by design. Closer tolerance resistors use lower tempco materials, in general.
@forr - I think you missed that jn's statement is conditional on them selling a range of tolerances in the same type of resistor. Its not that common in practice (in my experience).
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