Combining uncertainty for a nominal that is zero. Also greetings and first post.

[ttttrigg3r]

Hello everyone,
This is my first post on this forum. I hope everyone is doing well. I just transferred to the metrology field last year and still learning the basics. Here's one thing that I don't really get and would appreciate some help.
Here's the scenario. I'm taking a current measurement of nominal 0A from a power supply and want to calculate the uncertainty. The shunt standard used is 1Ω and has tolerance of 0.01%. A DMM standard is used to measure the voltage drop across this shunt. When the nominal is not zero, I can do it because I just find the uncertainty of each standards in ppm, then use Root Sum Square to find the combined uncertainty in ppm. Then I can multiply that uncertainty with the nominal to get it in absolute unit.

When the nominal is 0, 0 A in this case, I don't know what to do. The voltage reading across the shunt would be zero. If the accuracy of the DMM is (5ppm of reading +3ppm of 100mV range), that gives me ±0.0003mV. BUT the theoretical voltage reading is 0V, I cannot divide 0.0003mV/0V to get the fractional uncertainty.

Then I run across the same issue when I try to get a combined absolute uncertainty of the current measurement of 0A because it is zero. If I somehow got a combined fractional uncertainty and try to multiply it by the nominal zero, it comes out to zero; doesn't work.

What should be my approach in this case?

[TimFox]

When the expectation value or specification of a random variable is zero, it only makes sense to quote the uncertainty (standard deviation, maximum deviation, etc.) in absolute terms (nA or whatever), not relative terms such as ppm.
0.3 mV across an 1 ohm shunt is an absolute error of 0.3 mA.  Correction for the tolerance of the resistance should be negligible, when added to the uncertainty of that quotient in quadrature (rms).

[ttttrigg3r]

Thanks.
If cannot convert to a relative terms, how would one obtain the combined uncertainty when two or more standards are used? In this example one is in mV and one is in Ohms. I cannot simply combine those that are different in units.

[TimFox]

The tolerance of your shunt's value is a fraction: 0.01%.  Therefore, it contributes (1/10,000) x (0.3 mA) of error to the (0.3 mA) error calculated above.  Added in quadrature, that is very negligible.

[RYcal]

Hello everyone,
This is my first post on this forum. I hope everyone is doing well. I just transferred to the metrology field last year and still learning the basics. Here's one thing that I don't really get and would appreciate some help.
Here's the scenario. I'm taking a current measurement of nominal 0A from a power supply and want to calculate the uncertainty. The shunt standard used is 1Ω and has tolerance of 0.01%. A DMM standard is used to measure the voltage drop across this shunt. When the nominal is not zero, I can do it because I just find the uncertainty of each standards in ppm, then use Root Sum Square to find the combined uncertainty in ppm. Then I can multiply that uncertainty with the nominal to get it in absolute unit.

When the nominal is 0, 0 A in this case, I don't know what to do. The voltage reading across the shunt would be zero. If the accuracy of the DMM is (5ppm of reading +3ppm of 100mV range), that gives me ±0.0003mV. BUT the theoretical voltage reading is 0V, I cannot divide 0.0003mV/0V to get the fractional uncertainty.

Then I run across the same issue when I try to get a combined absolute uncertainty of the current measurement of 0A because it is zero. If I somehow got a combined fractional uncertainty and try to multiply it by the nominal zero, it comes out to zero; doesn't work.

What should be my approach in this case?

You would simply use the 0.01% and the 5pmm of reading +3ppm of range and get them back to 1 sigma then work out your Type A contributor however you like then you could convert that to a ppm of reading value to carry over when you take it from a standard uncertainty after RSS'ing the values to an expanded one.....P.S The 3ppm of range is a floor term so that still applies even if the measurement is zero. and I'm guessing the 0.01% is of the nominal value of the shunt (1 ohm) so that's another floor term regardless these specs still add into your budget even if the measurement is zero. Forget the zero part of the expected measurement is what I'm trying to say here.

I highly doubt you will get consistent measurements of zero anyways due to the DMM specs.

[2N3055]

Thanks.
If cannot convert to a relative terms, how would one obtain the combined uncertainty when two or more standards are used? In this example one is in mV and one is in Ohms. I cannot simply combine those that are different in units.

Well, PPMs are unitless... They are simply ratio, to be applied over units, volts, ohms...

[mendip_discovery]

In very basic terms I see it as you have a 2 part budget,
Using the numbers you have
105ppm + 0.0003mV

So at 0V you should have the combined unc of 0.0003mV because the ppm is the range error, and the mV is you constant error though the range.

Don't forget to add in for
Thermal EMFs - swap leads around if working in DC and repeat the test
Self Heating of the Shunt - I have a note of 20ppm per degC from nominal, I don't know where this has come from, not touching that until I know more.
Repeatability - Standard Deviation
Resolution - 1/2 a digit of meter
Imported Unc - for the shunt
Imported Unc - for the meter

[binary01]

As others have already covered, uncertainty estimate for zero measurements are best done in absolute terms and are as important/valid as uncertainty estimates on non-zero points in my opinion.  Personally I always report my uncertainties in the measurement unit, as I find this easier to assess instrument compliance quickly (i.e. is the error/correction + uncertainty within the expected accuracy), but I know many people like uncertainties in relative terms... no issues there.

Thanks.
If cannot convert to a relative terms, how would one obtain the combined uncertainty when two or more standards are used? In this example one is in mV and one is in Ohms. I cannot simply combine those that are different in units.

For this concept, I would recommend you learn about sensitivity coefficients.  For example: https://www.isobudgets.com/how-to-calculate-sensitivity-coefficients-for-measurement-uncertainty/
Then you can combine uncertainty contributions in any units, determine how much they impact the measurand, and include them in your uncertainty budget.  I work mostly in pressure, and when using Dead Weight Testers (primary reference for pressure), almost none of the uncertainty terms are in the measurand/pressure units (gravity, mass, temperature, area, etc).   Having good maths knowledge is helpful for calculating coefficients (partial derivatives), but there are also some very basic methods that are perfectly adequate so it doesn't need to be intimidating in any way.
Regards, Liam