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# Uncertainty of measurements and DMM's

Due to some recent questions about test equipment, thought I would do a little post on DMM specifications and how they relate to the real world.
"Accuracy" is really a relative concept. You have to read and understand what the manufacturer of a piece of test equipment is saying when he quotes specifications. The figure that counts is uncertainty, which is a calculated value.
As an example, let's do an uncertainty calculation on a recently recommended piece of equipment, the Protek 608 DMM, measuring 1kohm of resistance. Let's say you wish to match resistors to 0.1%, a not unreasonable figure of 1 ohm. The specification of this machine is 0.2% of SCALE, or .2% of 5k, or 10 ohms, plus 5 digits,or 5 ohms That's going to give us a total spec of +/- 15 ohms. Statistically, we are going to assume a rectangular distribution of this figure, which says that there is 100% probability that it will fall within these bounds. So we are going to divide that number by the square root of 3, or 1.732, or 8.66 ohms. This reduces the spec to 1 sigma, or a 60% probability it lies whithin this figure (+/-).
Now for resolution. The resolution of the 5k scale is 5,000 counts, or 1 ohm. We don't know if the meter is rounding up or down for any measurement, so we are going to assume for calculations that we can read it to half a digit, or .5 ohm. Taking a distribution of this number you get 0.288 ohms.
Now to add all this up, we are going to use a method called root sum of squares, (RSS) or the square root of the sum of the squared value of the uncertainty contributions. The sum of squares is 75.078, the RSS is 8.66 ohms. Now, there is only a 60% probability that ANY ONE MEASUREMENT is going to end up in this 17.3 ohm range (+/- 8.66 ohms). Not good odds. So, we are raise this to a 95% probability by multiplying the number by 2. Thus, we get +/- 17.33 ohms as the uncertainty of any one measurement.
What does all this mean? It means that the protec is NOT going to match resistors to 0.1%. Any measurement can fall within the band of +/- 17.33 ohms AND STILL BE WITHIN THE MANUFACTURER'S SPEC FOR THE METER. And all you can really count on is the spec. Inexpensive meters are spec'ed like this because they really are this bad. Internal components in the meter self heat, giving you different readings each time you make a measurement.
This is why I advocate the purchace of good used benchtop meters from Ebay, or really high quality handhelds from Fluke or HP. Probability theory does not lie, and you absolutly get what you pay for. Do you need my \$9700 Agilent 3458 to make useful measurements? No. But a cheap chinese handheld is not going to perform to the level that we all profess to want to work to.
BTW, I don't work for anyone who manufactures or sells instruments.

Yep - you are definitely a metrologist.

This was a thought-provoking post, and while your arguments are technically sound, I can't help but associate them with this classic teaser question posed by many a statistics prof:

If a coin is flipped 1,000 times and it comes up heads every time, what is the chance it will come up heads on the next flip?

The correct answer, statistically speaking, is 50%, but in the real world only a fool would bet money on the coin coming up tails for that next toss!

My point, in a roundabout way, is that if you measure 10 different 5k resistors over a short period of time and they are all within 5 ohms of each other then you can reliably state that they are matched within 0.1%. Now, if you space the measurements out over several hours, days, weeks, or what have you then, yes, I agree that you will need a much more accurate instrument to do the measuring if you want reliable results!

Also, if the meter uses dual-slope integration - as virtually all seem to, these days - then the biggest determinant of accuracy is the dielectric absorption of the ADC's integration capacitor. That said, teflon sports the lowest dielectric absorption next to air/vacuum.
Quote:
 Originally posted by jeffreyjYep - you are definitely a metrologist.
Gee you say that like it's a BAD thing!
Yes, I agree....for a high quality meter. In my world (and yours, I'm assuming, you sound like an EE too) we don't normally run into things like the Protek. Since our researchers can buy anything they want (and wish them off on my staff for calibration), I have had occasion to deal with these fine pieces of oriental electronic, uh, art. I have seen standard deviations as large as 25% of the spec on closely spaced measurements. This isn't just "I'm right 'cause I'm the metrologist" , This is sad experience of telling people why their \$50,000 experiment won't produce repeatable data.
I'm not sure if this is what Jeffrey meant, but because a meter will tend to be inaccurate the same amount, a relative measurement is easier to make than an absolute measurement. Meaning: one will be less certain that a resistor is actually 1k ohms than one would that one "1k" is close to another "1k". In fact, you wrote: "I have seen standard deviations as large as 25% of the spec on closely spaced measurements." This indicates that a relative measurement is more accurate than the meter specification. Also, if the closely spaced measurements drift, then one can remeasure the resistors X number of times to further improve relative accuracy.

I found your post interesting too. But think it applies more to absolute measurements. Relative measurements are more of a special case.

JF
JF - that's half of what I was saying; the other half was that if the readings are very similar from resistor to resistor then the meter is probably trustworthy for matching resistors. If there is 25% deviation from measurement to measurement then the meter should immediately be tossed into the trash... no, make that a compactor - someone might yank it out of the trash and think they found something great if it isn't promptly crushed into bits

The thing is, one element of good circuit design is to reduce or eliminate the need for components with high initial or absolute accuracy. For example, compare the cost and performance of a Wheatstone bridge to a simple divider in transducer interface circuits... Which is to say, don't confuse accuracy and repeatability; very often, it is the latter quality which is most desirable in electronic components.
Quote:
 Originally posted by jeffreyjWhich is to say, don't confuse accuracy and repeatability; very often, it is the latter quality which is most desirable in electronic components.
Yes, repeatability precedes accuracy.

Along those lines, a spec of 0.2% is over its calibration period (6mos, 1 yr, 2 yr...). If the meter didn’t have better short term accuracy (over 20 minutes), then it's unlikely to achieve its long term accuracy (6 months later).

JF
Quite true, relative measurements are very important when matching components, etc. However, remember that the time will come when you do want to make an absolute measurement ... and a DMM is used for many things in the course of a project.
My point is, you can get a really decent 51/2 or even 61/2 digit HP meter in the range of \$100 - \$200 on ebay, or even from the used equipment houses ... then you pick up the capability of 4 wire resistance measurements. That gets rid of the largest source of error in resistance measurements, lead resitance variability.
Oh, and when I said standard deviations of 25% of spec, that's within the total error band. ...on top of the inaccuracy, or deviation from a cardinal number. If you are wiring a house or working on your car, that's fine ... not when building a piece of very high performance equipment that you have \$300 or so invested in the parts alone.
IMHO, if I had to choose between a brand new Hyundai econobox and a (insert your favorite performance car/SUV/whatever) in gently used condition for the same price......
I agree. If I want a super accurate resistance reading, even up to 10k ohms, I'll use a 4-wire ohm meter.

However, a "two wire" meter measures a resistor in a "four wire" fashion. It's just that the "Kelvin Connections" are inside the meter, rather than out at the end of the test leads.

And most people are familiar with the idea of nulling the lead resistance resistance by noting the reading with the leads connected together. Just something else to improve the accuracy of the measurement with the equipment at hand.