2.
Measurement Process Characterization
2.3. Calibration 2.3.4. Catalog of calibration designs


Purpose  The purpose of this section is to explain why calibration of angle blocks of the same size in groups is more efficient than calibration of angle blocks individually.  
Calibration schematic for five angle blocks showing the reference as block 1 in the center of the diagram, the check standard as block 2 at the top; and the test blocks as blocks 3, 4, and 5. 
A schematic of a calibration scheme for one reference block, one
check standard, and three test blocks is shown below.
The reference block, R, is shown in the
center of the diagram and the check standard, C,
is shown at the top of the diagram.


Block sizes 
Angle blocks normally come in sets of
1, 3, 5, 20, 30 minutes 1, 3, 5, 15, 30, 45 degrees and blocks of the same nominal size from 4, 5 or 6 different sets can be calibrated simultaneously using one of the designs shown in this catalog. 

Restraint  The solution to the calibration design depends on the known value of a reference block, which is compared with the test blocks. The reference block is designated as block 1 for the purpose of this discussion.  
Check standard  It is suggested that block 2 be reserved for a check standard that is maintained in the laboratory for quality control purposes.  
Calibration scheme  A calibration scheme developed by Charles Reeve (Reeve) at the National Institute of Standards and Technology for calibrating customer angle blocks is explained on this page. The reader is encouraged to obtain a copy of the publication for details on the calibration setup and quality control checks for angle block calibrations.  
Series of measurements for calibrating 4, 5, and 6 angle blocks simultaneously  For all of the designs, the measurements are made in groups of seven starting with the measurements of blocks in the following order: 2321242. Schematically, the calibration design is completed by counterclockwise rotation of the test blocks about the reference block, oneatatime, with 7 readings for each series reduced to 3 difference measurements. For n angle blocks (including the reference block), this amounts to n  1 series of 7 readings. The series for 4, 5, and 6 angle blocks are shown below.  
Measurements for 4 angle blocks 
Series 1: 2321242 Series 2: 4241434 Series 3: 3431323 

Measurements for 5 angle blocks (see diagram) 


Measurements for 6 angle blocks 
Series 1: 2321242 Series 2: 6261636 Series 3: 5651525 Series 4: 4541464 Series 5: 3431353 

Equations for the measurements in the first series showing error sources 
The equations explaining the seven measurements for the first series
in terms of the errors in the measurement system are:
\begin{array} \(
Z_{11} = B + X_1 + \,\,\,\,\,\,\,\,\,\,\,\,\,\, error_{11} \\
Z_{12} = B + X_2 + \,\,\, d + error_{12} \\
Z_{13} = B + X_3 + 2d + error_{13} \\
Z_{14} = B + X_4 + 3d + error_{14} \\
Z_{15} = B + X_5 + 4d + error_{15} \\
Z_{16} = B + X_6 + 5d + error_{16} \\
Z_{17} = B + X_7 + 6d + error_{17}
\end{array}
with B a bias associated with the instrument, d is a linear drift factor, X is the value of the angle block to be determined; and the error terms relate to random errors of measurement. 

Calibration procedure depends on difference measurements  The check block, C, is measured before and after each test block, and the difference measurements (which are not the same as the difference measurements for calibrations of mass weights, gage blocks, etc.) are constructed to take advantage of this situation. Thus, the 7 readings are reduced to 3 difference measurements for the first series as follows: \begin{array} \( Y_{11} = (Z_{11}  2Z_{12} + Z_{13}) / 2 \\ Y_{12} = (Z_{13}  2Z_{14} + Z_{15}) / 2 \\ Y_{13} = (Z_{15}  2Z_{16} + Z_{17}) / 2 \\ \end{array} For all series, there are 3(n  1) difference measurements, with the first subscript in the equations above referring to the series number. The difference measurements are free of drift and instrument bias.  
Design matrix 
As an example, the design matrix for n = 4 angle blocks is
shown below.
1 1 1 1 0 1 1 0 1 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 0 The design matrix is shown with the solution matrix for identification purposes only because the leastsquares solution is weighted (Reeve) to account for the fact that test blocks are measured twice as many times as the reference block. The weight matrix is not shown. 

Solutions to the calibration designs measurements 
Solutions to the angle block designs are shown on the following pages.
The solution matrix and factors for the repeatability standard deviation
are to be interpreted as explained in solutions to
calibration designs . As an example, the solution for the
design for n=4 angle blocks
is as follows:
The solution for the reference standard is shown under the first column of the solution matrix; for the check standard under the second column; for the first test block under the third column; and for the second test block under the fourth column. Notice that the estimate for the reference block is guaranteed to be R*, regardless of the measurement results, because of the restraint that is imposed on the design. Specifically, $$ \widehat{Reference} = 0 + R^* $$ $$ \widehat{Check} = \frac{1}{24} \left( \begin{array}{l} \,\,\,\,\, 2.272 Y_{11} + 9.352 Y_{12} + 2.272 Y_{13} \\ 5.052 Y_{21} + 7.324 Y_{22}  1.221 Y_{23}\\ 1.221 Y_{31} + 7.324 Y_{32}  5.052 Y_{33} \end{array} \right) + R^* $$ $$ \widehat{Test}_1 = \frac{1}{24} \left( \begin{array}{l} 5.052 Y_{11} + 7.324 Y_{12}  1.221 Y_{13} \\ 1.221 Y_{21} + 7.324 Y_{22}  5.052 Y_{23}\\ \,\,\,\,\, 2.272 Y_{31} + 9.352 Y_{32} + 2.272 Y_{33} \end{array} \right) + R^* $$ $$ \widehat{Test}_2 = \frac{1}{24} \left( \begin{array}{l} 1.221 Y_{11} + 7.324 Y_{12}  5.052 Y_{13} \\ \,\,\,\,\, 2.272 Y_{21} + 9.352 Y_{22} + 2.272 Y_{23}\\ 5.052 Y_{31} + 7.324 Y_{32}  1.221 Y_{33} \end{array} \right) + R^* $$ Solutions are correct only for the restraint as shown. 

Calibrations can be run for top and bottom faces of blocks  The calibration series is run with the blocks all face "up" and is then repeated with the blocks all face "down", and the results averaged. The difference between the two series can be large compared to the repeatability standard deviation, in which case a betweenseries component of variability must be included in the calculation of the standard deviation of the reported average.  
Calculation of standard deviations when the blocks are measured in two orientations  For n blocks, the differences between the values for the blocks measured in the top ( denoted by "t") and bottom (denoted by "b") positions are denoted by: $$ \delta_i = \widehat{X}_i^{\, t}  \widehat{X}_i^{\, b} \hspace{0.25in} \mbox{for} \,\,\, i=I, \, \ldots , \, n\,.$$ The standard deviation of the average (for each block) is calculated from these differences to be: $$ {\large s}_{avg \,\, test} = \sqrt{\frac{1}{4(n1)}\sum_{i=1}^n \delta_i^2} \,\, .$$  
Standard deviations when the blocks are measured in only one orientation  If the blocks are measured in only one orientation, there is no way to estimate the betweenseries component of variability and the standard deviation for the value of each block is computed as $$ {\large s}_{test} = K_1 {\large s}_1 $$ where \( K_1 \) is shown under "Factors for computing repeatability standard deviations" for each design and \( {\large s}_1 \) is the repeatability standard deviation as estimated from the design. Because this standard deviation may seriously underestimate the uncertainty, a better approach is to estimate the standard deviation from the data on the check standard over time. An expanded uncertainty is computed according to the ISO guidelines. 