application of two comparator in mass calibration using subdivision method
TRANSCRIPT
Unurbileg.D, MASM
APPLICATION OF TWO COMPARATORS for
DISSEMINATION of E2 WEIGHT
BY USING SUBDIVISION METHOD
APMP 2014, TCM meeting,
Daejeon , Korea
22 Sep 2014
For the determination of the conventional mass, in the calibration of
weights of the highest accuracy classes, the subdivision method
and its variants are widely used
The subdivision method has following advantages;
• it minimizes the use or wear of standards;
• it produces a set of data which provides important statistical
information about the measurement and performance of
comparator.
• it offers redundancy of data
However :
For subdivision calibration beside the number of comparison, mass
comparator’s resolution and maximum loading level required by the
design is crucial.
In this respect, due to limited weighing range of mass comparators
available in mass laboratory of MASM two mass comparators with
different readability were used for the second decade of subdivision
comparison (i.e from 100 g to 10 g) of entire 1 kg – 1 mg set
against one reference
INTRODUCTION
INTRODUCTION
E2 (1mg - 50 kg)
E1 (20 kg)
E2 (1 mg -10 kg)
E1 (1mg – 10 kg)
(1,2,2,5) x 10n kg where:
n – is positive negative integer
Design for 1 kg – 100 g decade
Weighing design for the 1 kg – 100 g decade
5 unknown weight, 12 equation, 100* g is being check standard
Applied
comparator
C1000S
d=2 mg
Sp ~ 5 mg
1)( XXV T
12
11
10
9
8
7
6
5
4
3
2
1
*100
100
*200
200
500
1000
110000
110000
111000
111000
110100
110100
001100
001100
101110
011110
101111
011111
y
y
y
y
y
y
y
y
y
y
y
y
g
g
g
g
g
X
MATRIX NOTATION
YMX
variance-covariance matrix of
Sum (diag) =0.26
>>
Sum (off diag) =0.04
Almost orthogonal..
Normal equation
MASS COMPARATOR
Sartorius C1000S
Max=1 kg, Min=100 g, d=2 mg, Sp ~ 5 mg
Used since 2001
Mass comparators in mass lab of MASM
Sartorius CC 50, Max=50 g, d=1 mg,
Sp ~ 2.5 mg
Used since 2001
Design for 100 g – 10 g decade
Weighing design for the 100 g – 10 g decade
7 unknown weights, 10 equations, 10* g is being check standard
CC 50
d=1 mg, Sp ~ 2 mg
Applied
comparators
C1000S
d=2 mg
Sp ~ 5 mg
ggg
gg
ggg
MX
*101020
2050
50100100
110000001110000011010000
00110000011110000111010000001100000011100000110100000011
'
'
YMX
MATRIX NOTATION
is
*2/12/1 RsYWMXW weighing equations
need to be weighed
prior to regression.
mass differencies with
from three STTS cycles
where - W is weight matrix:
mgY
)00086.0(01799.0
)00099.0(030986.0
)00144.0(01266.0
)000764.0(01916.0
)0225.0(034156.0
)00076.0(0588.0
)00099.0(00549.0
)0045.0(00983.0
)00133.0(055.0
)0025.0(0625.0
WEIGHT MATRIX
nis
wi
i .....1,
2
0
W= (wi)
n
i is12
2
01
1 mg0003486.00
*2/1 XXW
*2/1 YYW
*** RsYMX
AFTER WEIGHTING
84417.084417.0000000
31973.031973.031973.000000
31973.031973.0031973.00000
0028146.028146.00000
106315.00106315.0106315.0106315.0000
008561.008561.008561.0008561.000
0000120528.0120528.000
000012051.012051.012051.00
000006489.006489.0006489.0
0000001286.01286.0
*X
11000000
11100000
11010000
00110000
01111000
01110100
00001100
00001110
00001101
00000011
X
mgY
01799.0030986.001266.001916.0034156.0
0588.000549.000983.0055.00625.0
mgY
01168.00009.000773.000563.000154.000191.0
00052.000396.000106.0
00935.0
*
Weighing equation is transformed
-6
-5
-4
-3
-2
-1
0
1
2
3
4
0 50 100
mg
residuals st deviation*** YMXRs
g g
is
gM
9999951.90000132.100000396.200000206.200000204.500000147.500000377.100999979.99
RESULTS
**1** )( YXXXM TT
Solution matrix
variance-covariance matrix of weighted matrix X*
1**2 )( XXSVMT
261076.1 mgVM
mgS 0004039.0DF
Rs
S
n
i
1
2*
2
)(
16.1/ 0 S
mg0003486.00
internal consistency of observed weighting
results or absence of systematic errors.
with degree of freedom 23
Unknown
masses, g
type A, mg ref wt, mg balance, mg
Combined
st. uncertainty,
mg
Expanded
uncertainty, mg,
k=2
100.0000377 0.0026 0.0093 0.0012 0.010 0.019
50.0000147 0.0009 0.00465 0.0012 0.005 0.010
50.0000204 0.0009 0.00465 0.0012 0.005 0.010
20.0000206 0.0007 0.00186 0.0012 0.002 0.005
20.0000396 0.0007 0.00186 0.0012 0.002 0.005
10.0000132 0.00006 0.00093 0.0012 0.002 0.003
9.9999951 0.00006 0.00093 0.0012 0.002 0.003
jM 2/1)( jjjA VMu crjjr uhMu )( )( jba Mu )( jc Mu
UNCERTAINTY BUDGET
kMuU c )(
2/122 )()( ressenjba uuMu
)( jA Mu - is square of diagonal elements of VM matrix
- is comparator’s of smallest accuracy
)( jb Mu - is assumed to be neglected because mass
standards of a set have same density.
Nominal mass
(g)
Subdivision method Direct method
d (mg ) U, (mg) d (mg ) U, (mg)
100 0.0377 0.019 0.0152 0.051
50 0.0147 0.010 0.0146 0.021
20 0.0206 0.005 0.0187 0.010
20* 0.0396 0.005 -0.00052 0.010
10 0.0132 0.003 0.001 0.007
SUMMARY
The comparison results obtained for the E2 weights by
subdivision method for unknown method are better than
those with direct comparison.
It is possible to use two comparators in one decade and its
variance and covariance matrix is dependent from individual
comparison’s standard deviations.
In case of one comparator variance- covariance matrix is
fixed one, it can be orthogonal.
Necessitates of placing group of weights on balance pan
cause poor eccentricity characteristic of mass comparator
THANK YOU FOR YOUR
ATTENTION