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

Designs for angle blocks

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.
design for 5 angle blocks
Block sizes Angle blocks normally come in sets of

1, 3, 5, 20, and 30 seconds

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: 2-3-2-1-2-4-2. Schematically, the calibration design is completed by counter-clockwise rotation of the test blocks about the reference block, one-at-a-time, 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: 2-3-2-1-2-4-2
Series 2: 4-2-4-1-4-3-4
Series 3: 3-4-3-1-3-2-3
Measurements for 5 angle blocks (see diagram)
Series 1: 2-3-2-1-2-4-2
Series 2: 5-2-5-1-5-3-5
Series 3: 4-5-4-1-4-2-4
Series 4: 3-4-3-1-3-5-3
Measurements for 6 angle blocks
Series 1: 2-3-2-1-2-4-2
Series 2: 6-2-6-1-6-3-6
Series 3: 5-6-5-1-5-2-5
Series 4: 4-5-4-1-4-6-4
Series 5: 3-4-3-1-3-5-3
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 least-squares 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 between-series 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: $$ {\large s}_{avg \,\, test} = \sqrt{\frac{1}{4(n-1)}\sum_{i=1}^n \delta_i^2} \,\, .$$
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 between-series 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.
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