In order to understand uncertainty calculations, it is necessary to know how a calibration result is reached.

By Arnthor Gunnarsson, FORCE Technology

By doing that, we can better identify the parameters which can have a possible influence on a calibration result, and this becomes vital in order to create an uncertainty budget.

Read along, while we explore the importance of specific factors and measurements from an uncertainty perspective.

Read also: Uncertainty explained: Understanding an uncertainty budget.

The technical part: How do calibration laboratories reach a calibration result?

When calculating uncertainty, one starts with a model equation. So, we start by asking the question: How do calibration laboratories reach a calibration result? That is, how do you calculate the error of a meter under test (MUT)? 

To do this, we take a top down approach and start with the basics of how to calculate the calibration result or error of a MUT. A calibration result can be calculated based on flow or volume with the following equation:

Equation 1: Simplified form of model equation.
Equation 1: Simplified form of model equation.
Read also: Undertainty explained How to calculate the laboratory CMC?

In the following calculations, we assume an accurate enough signal collection system, that the signal collection starts and stops at the same time for both the Meter Under Test (MUT) and the working standards (WS) used to calibrate the MUT. Thus, subsequent calculations will all be done based on volume.

Here, laboratories normally use reference meters, also called working standards or (WS). These WS have themselves been calibrated and are set up in series with the MUT. By using these meters, and continuously collecting data from the WS, calibration laboratories know the gas flow through the WS at all times, although of course with an uncertainty, but we will come back to that later.

But to generate flow in a system, a pressure difference is generated and there might also be a temperature difference between the WS and MUT, so knowing the gas flow rate at the WS’s is not enough. This is where the “ideal” gas law comes in. However, as we are not working with ideal gasses, we add the compressibility factor to the ideal gas law equation.

Equation 2: Ideal gas equation with compressibility.
Equation 2: Ideal gas equation with compressibility.

And if there is no pressure leakage in or out, and the amount of gas stays the same between the WS and the MUT, we should get the same mass flow trough the WS and the MUT. Expressed in another way, we should have the same amount of mol (n) through the meter under test and the working standard.

So, for calibration laboratories this means that if there are two positions 1 and 2, where gas flows through both positions and there is no loss or gain of gas between the positions, then the same amount of mol flows through the two positions.

Equation 3: Condition of the gas law if there is constant pressure and temperature between WS and MUT.
Equation 3: Condition of the gas law if there is constant pressure and temperature between WS and MUT.

Using this condition, we can now isolate V1 on one side of the equation and instead of calling it position 1 and 2, we say that these two positions are the positions of the WS and the MUT in the calibration loop. Using this, we can now isolate V1=VMUT on one side of the equation. Since both sides of the equation divide by the constant R, it falls out and we get:

Equation 4: Usage of the condition of the gas law in equation 3.
Equation 4: Usage of the condition of the gas law in equation 3.

Since the WS is itself a calibrated meter, we can obtain a VActual, MUT that is traceable to the WS if we correct for the error of the WS, based on its calibration. So, we get:

Equation 5: More specific for how we can use the condition in equation 3.
Equation 5: More specific for how we can use the condition in equation 3.

We now have all the elements that are used to calculate the error of a MUT in a calibration, so the model equation (equation 1) can now be expanded upon so:

Equation 6: More expanded form of the model equation.
Expanded calibration result equation.

Using this expanded form of the model equation, we have all the factors needed to calculate uncertainty in a calibration.

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Read also: Uncertainty explained - Class Half meters or learn more about Calibration here.

 

Symbols

The equations used in this article use symbols that are based on what is normally used in international standards and documents.

Symbol

Description

MUT

Meter Under Test, also sometimes called DUT (Device Under Test), is the meter/device that is being calibrated.

WS

Working Standards are reference meters, used to calibrate a MUT.

P

Pressure, where PWS would be pressure at the WS and PMUT would be pressure at the MUT. Pressure is always measured in absolute pressure.

T

Temperature, where TWS is temperature at the WS and TMUT is temperature at the MUT. Temperature is always measured in Kelvin.

Z

Compressibility, where ZWS is compressibility at the WS, and ZMUT is the compressibility at the MUT.

V

Volume, where for example VIND, MUT is the indicated reading from the MUT during the calibration, and VActual, MUT is the actual volume at the MUT based on WS readings, pressures, temperatures and compressibility.

Q

Flowrate, that is [Volume/time unit] where for example QIND, MUT is the indicated reading from the MUT over a specific time and QActual, MUT is the actual flowrate at the MUT based on WS readings, pressures, temperatures, compressibility and time.

t

Time in seconds, where during a calibration if a timer starts and stops a calibration, the time is the same for the WS and a MUT.

Table 1: Symbols and their description.