E1C08 09/14/2010
14:53:57 Page 329
Consider first the sensitivity indices u i for each of the variables R T , R 0 , T, and T 0 . These may be
tabulated by computing the appropriate partial derivatives of b, evaluated at each of the three
temperatures, as follows:
T (
C)
u RT K=V
ð
Þ
u R0 K=V
ð
Þ
u T
u T0
100
À0.270
0.0247
À37.77
59.17
125
À0.422
0.0198
À27.18
48.48
150
À0.628
0.0168
À20.57
41.45
The determination of the uncertainty in b; u b , requires the uncertainty in the measured value of
resistance for the thermistor, u R T . But R T is determined from the expression
R T ¼ R 1 ½ðE i =E 1 Þ À 1
and thus requires an analysis of the uncertainty in the resulting value of R T , from measured values of
R 1 , E i , and E 1 . All errors in R T are treated as uncorrelated systematic errors, yielding
b R T ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
qR T
qR 1
b R 1
! 2
þ
qR T
qE i
b E i
! 2
þ
qR T
qE 1
b E 1
! 2
s
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
u R 1 b R 1
½
2 þ u E i b E i
½
2 þ u E 1 b E 1
½
2
q
To arrive at a representative value, we compute b R T at 125
C. The systematic standard uncertainty
in R 1 is 0.75% of 130:5 kV; or 978 V, and in R 0 is 450 V. The systematic standard uncertainties in
E i and E 1 are each 0.001 V, and u R 1 ¼ 0:022, u E i ¼ 85238, and u E 1 ¼ 87076. This gives
b R T ¼ 123:7 V.
An uncertainty for b is determined for each of the measured temperatures. The propagation of
the measurement systematic errors for temperature and resistance is found as
b b ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u T b T
½
2 þ u T 0 b T 0
½
2 þ u R T b R T
½
2 þ u R 0 b R 0
½
2
q
where
b T ¼ 0:18
C b R T ¼ 123:7 V
b T 0 ¼ 0:18
C b R 0 ¼ 450 V
The random standard uncertainty for b contains contributions only from the statistically determined
oven temperature characteristics and is found from
s b ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u T s T
ð
Þ
2 þ u T 0 s T 0
À
Á 2
q
where both s T and s T 0 are 0.19, as determined with n ¼ 19.
The resulting values of uncertainty in b are found from
u b ¼ t n;95
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b
2
b þ s 2
b
q
where n b is sufficiently large (see Eq. 5.39) so that t n;95 % 2. At each temperature the uncertainty in
b is determined as shown in Table 8.3. The effect of increases in the sensitivity indices u i on the total
uncertainty is to cause increased uncertainty in b as the temperature increases.
8.4 Electrical Resistance Thermometry 329
14:53:57 Page 329
Consider first the sensitivity indices u i for each of the variables R T , R 0 , T, and T 0 . These may be
tabulated by computing the appropriate partial derivatives of b, evaluated at each of the three
temperatures, as follows:
T (
C)
u RT K=V
ð
Þ
u R0 K=V
ð
Þ
u T
u T0
100
À0.270
0.0247
À37.77
59.17
125
À0.422
0.0198
À27.18
48.48
150
À0.628
0.0168
À20.57
41.45
The determination of the uncertainty in b; u b , requires the uncertainty in the measured value of
resistance for the thermistor, u R T . But R T is determined from the expression
R T ¼ R 1 ½ðE i =E 1 Þ À 1
and thus requires an analysis of the uncertainty in the resulting value of R T , from measured values of
R 1 , E i , and E 1 . All errors in R T are treated as uncorrelated systematic errors, yielding
b R T ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
qR T
qR 1
b R 1
! 2
þ
qR T
qE i
b E i
! 2
þ
qR T
qE 1
b E 1
! 2
s
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
u R 1 b R 1
½
2 þ u E i b E i
½
2 þ u E 1 b E 1
½
2
q
To arrive at a representative value, we compute b R T at 125
C. The systematic standard uncertainty
in R 1 is 0.75% of 130:5 kV; or 978 V, and in R 0 is 450 V. The systematic standard uncertainties in
E i and E 1 are each 0.001 V, and u R 1 ¼ 0:022, u E i ¼ 85238, and u E 1 ¼ 87076. This gives
b R T ¼ 123:7 V.
An uncertainty for b is determined for each of the measured temperatures. The propagation of
the measurement systematic errors for temperature and resistance is found as
b b ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u T b T
½
2 þ u T 0 b T 0
½
2 þ u R T b R T
½
2 þ u R 0 b R 0
½
2
q
where
b T ¼ 0:18
C b R T ¼ 123:7 V
b T 0 ¼ 0:18
C b R 0 ¼ 450 V
The random standard uncertainty for b contains contributions only from the statistically determined
oven temperature characteristics and is found from
s b ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u T s T
ð
Þ
2 þ u T 0 s T 0
À
Á 2
q
where both s T and s T 0 are 0.19, as determined with n ¼ 19.
The resulting values of uncertainty in b are found from
u b ¼ t n;95
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b
2
b þ s 2
b
q
where n b is sufficiently large (see Eq. 5.39) so that t n;95 % 2. At each temperature the uncertainty in
b is determined as shown in Table 8.3. The effect of increases in the sensitivity indices u i on the total
uncertainty is to cause increased uncertainty in b as the temperature increases.
8.4 Electrical Resistance Thermometry 329