Uncertainty Calculations of Pressure Calibration

Uncertainty means how Uncertain (or certain) we are.

No readings are complete without units and Uncertainty.

Types of Uncertainty

There are two types of Uncertainty

  • Type A uncertainty
  • Type B uncertainty

The Type A

The Type A evaluation of standard uncertainty is the method of evaluating the uncertainty by the statistical analysis of a series of observations.

Type A uncertainty is based on readings taken during calibration.

The Type B

 The Type B evaluation of standard uncertainty is the method of evaluating the uncertainty by means other than the statistical analysis of the repeat measurements.

Type B uncertainty is based on other factors than readings taken during calibration.

Uncertainty can be reported either in % value or absolute value.

% Value can be reported as % of reading (% rdg) or % of Full-Scale deflection (%FSD)

Uncertainty Components of Pressure Calibration to be considered but not limited to:

Type A

  1. Repeatability
  2. Reproducibility

Type B

  1. Uncertainty due to Standard equipment
  2. Accuracy of Standard equipment
  3.  Resolution
  4. Hysteresis
  5. Zero Error

Uncertainty Calculations

For understanding Uncertainty Calculations, we will take an example of Pressure Gauge calibration

Uncertainty Calculations of Pressure Calibration

5 Point Calibration with 1 Up cycle and 1 Down cycle

Consider UUC pressure gauge of range 0 to 10 bar with resolution 0.1 bar calibrated with master pressure Gauge of range 0 to 40 bar with resolution 0.001 bar.

Uncertainty of master pressure Gauge is 0.01 bar with K=2. Accuracy is 0.1% rdg.

Cal PointUUC
Reading
Standard Reading
(UP)
Standard Reading
(DOWN)
Average
BarBarBarBarBar
00.00.0000.0000.000
22.02.0032.0052.004
44.04.0064.0044.005
88.08.0058.0068.006
1010.010.00810.00710.008

Type A Uncertainty

Type A uncertainty is the standard deviation of readings.

Type A Uncertainty Formula

Where:

x:  reading

x̄: Mean (average)

n: no of readings i.e. 2 in the above case

Using excel in can be calculated using formula = STDEV ()

Uncertainty due to Standard Deviation (Ua)

(Ua)= S.D / sqrt (n)

Sqrt: square root

Cal PointType AUa
BarBarBar
000
20.0014140.001
40.0014140.001
80.0007070.0005
100.0007070.0005

Type B Uncertainty

Uncertainty due to Standard equipment

Uncertainty of standard equipment is mention on the calibration certificate. Also, the K value is mention in the calibration certificate.

Uncertainty due to Standard equipment (Ub1) = Uncertainty of standard equipment/ k

Here, in our example Uncertainty of Standard is 0.01 bar and K value is 2.

As this is absolute value directly in Pressure units, it is the same for the whole range.

If Uncertainty is provided in % rdg, it will be different for different calibration points.

Cal PointUncertainty of StandardUb1
BarBarBar
00.010.005
20.010.005
40.010.005
80.010.005
100.010.005

Uncertainty due to Accuracy of Standard Equipment

Accuracy is expressed in % rdg or % full scale or absolute readings direct in units

Uncertainty due to Accuracy of Standard equipment (Ub2)

Ub2 = Accuracy of standard equipment/ Sqrt (3)

Assuming rectangular distribution, the value obtained is divided by a square root of 3

In our example, accuracy is % rdg. Therefore accuracy will be different for every reading.

Cal PointAccuracy of StandardAccuracy of StandardUb2
BarBarBarBar
00.1%*000
20.1%*20.0020.001155
40.1%*40.0040.002309
80.1%*80.0080.004619
100.1%*100.010.005774

Uncertainty due to Resolution

If UUC is set according to calibration point, and Standard reading is changed, resolution of Standard is considered.

If the Standard Pressure gauge is set according to calibration point, and UUC readings are changed, the resolution of UUC is considered.

In or above example, UUC is set and Standard readings are changing,

Therefore, the Resolution of a standard pressure gauge is considered.

The resolution of the Standard pressure gauge is 0.001 bar

Uncertainty due to Resolution (Ub3) = ((Resolution of Changing pressure gauge/2)) / SQRT (3))

Assuming rectangular distribution, the value obtained is divided by a square root of 3

Cal PointResolutionUb3
BarBarBar
00.0010.000289
20.0010.000289
40.0010.000289
80.0010.000289
100.0010.000289

Uncertainty due to Hysteresis

Hysteresis is the difference between Down Cycle reading and UP cycle reading

For 2 up and 2 down cycles. Hysteresis is the maximum difference between Down Cycle reading and UP cycle reading. 

Uncertainty due to Hysteresis (Ub4) = (Down reading- up reading)/ Sqrt (3)

Assuming rectangular distribution, the value obtained is divided by a square root of 3

Cal PointHysteresisUb4
BarBarBar
000
20.0020.001155
4-0.002-0.00115
80.0010.000577
10-0.001-0.00058

Uncertainty due to Zero Error

The Reading is noted when the pressure of the instrument is completely released.

The Zero Error is calculated as follows:

Zero Error= l x2,0 – x1,0 l

I.e. Zero readings of down cycle – Zero readings of up cycle

For Multiple cycles. The maximum value is Zero error.

Uncertainty due to Zero Error (Ub5) = Zero error/ Sqrt (3)

Assuming rectangular distribution, the value obtained is divided by a square root of 3

Cal PointZero ErrorUb5
BarBarBar
00.0000.000
20.0000.000
40.0000.000
80.0000.000
100.0000.000

Combined Uncertainty

Combined uncertainty is not simple addition.

Step 1: All individual uncertainty Ua, Ub1, Ub2, Ub3, Ub4, Ub5  are squared.

Step 2: All individual uncertainty is added after squaring

Step 3:  Square root of the value obtained in Step 2

Cal PointUaUb1Ub2Ub3Ub4Ub5Combine Uncertainty
BarBarBarBarBarBarBarBar
000.00500.00028900.0000.005
20.0010.0050.0011550.0002890.0011550.0000.005
40.0010.0050.0023090.000289-0.001150.0000.006
80.00050.0050.0046190.0002890.0005770.0000.007
100.00050.0050.0057740.000289-0.000580.0000.008

Combined Uncertainty is also called Standard Uncertainty

Cal PointCombine UncertaintyCombine Uncertainty
BarBarBar
0Sqrt (02+0.0052+02+0.002892+02+02)0.005
2Sqrt (0.0012 +0.0052  + 0.0011552  +0.0002892 +   0.0011552 +02)0.005
4Sqrt (0.0012 +0.0052  + 0.0023092  +0.0002892 +   0.0011552 +02 )0.006
8Sqrt (0.00052 +0.0052  + 0.0046192  +0.0002892 +   0.0005772 +02)0.007
10Sqrt (0.00052 +0.0052  + 0.0057742  +0.0002892 +   0.000582 +02)0.008

Expanded Uncertainty

Standard Uncertainty is multiplied by the k factor to obtained Expanded Uncertainty.

Where k is the coverage factor corresponding to the effective degree of freedom.

k factor is generally 2.

Cal PointExpanded Uncertainty
BarBar
00.010
20.011
40.011
80.014
100.015

Expanded uncertainty is reported in the calibration certificate.

A similar type of calculation is done for 9 Point calibration with 2 up and 2 down cycles.

Author: Saurabh Vaidya

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1 thought on “Uncertainty Calculations of Pressure Calibration”

  1. For pressure transmitter according to DKD, 9 points is standard procedure. Up down up.
    5 point calibration is for low level pressure gauges. Above CL 1

    Reply

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Uncertainty Calculations of Pressure Calibration

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