E1C09 09/14/2010
15:4:54 Page 395
Then,
Mðf ¼ 5Þ ¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À f =f n
ð
Þ
2
h
i 2 þ 2z f =f n
ð
Þ
2
r
¼ 1:03
With a dynamic error of only about 3%, this system should be suitable for the intended
measurements.
Example 9.7
A common method to estimate the data acquisition and reduction errors present in the pressuremeasuring instrumentation of a test rig is to apply a series of replication tests on the M calibrated
pressure transducers used on that rig (when the number of transducers installed is large, a few
transducers can be selected at random and tested). In such a test each transducer is interconnected
through a common manifold to the output of a single deadweight tester (or suitable standard) so that
each transducer is exposed to exactly the same static pressure. Each transducer is operated at its
normal excitation voltage. The test proceeds as follows: record applied test pressure (from the
deadweight tester), record each transducer output N (e.g., N ! 10) times, vent the manifold to local
atmosphere, close the manifold to reapply test pressure, repeat the procedure at least K (e.g., K ! 5)
times. What information is available in such test data?
KNOWN M( j ¼ 1, 2, . . . , M) transducers
N(i ¼ 1, 2, . . . , N) repetitions at a test pressure
K(k ¼ 1, 2, . . . , K) replications of test pressure
SOLUTION The mean value for each transducer is given by the mean of the repetitions for that
transducer,
p jk ¼
1
N
X N
i¼1
p ijk
The mean value for the replications of the jth transducer is given by
hp j i ¼
1
K
X K
k¼1
p jk
The difference between the pooled mean pressure and the known applied pressure would provide an
estimate of the systematic error to be expected from any transducer during data acquisition. On the
other hand, variations between p jk and hp j i are due to random errors in the data acquisition and
reduction procedure. The variation of the test pressure mean, p jk , about the pooled mean, hp j i, for
each transducer is
hs j i ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
P K
k¼1
À p jk À hp j i
Á 2
K À 1
s
The pooled standard deviation of the mean for the M transducers provides the estimate of the random
9.5 Pressure Transducer Calibration 395
15:4:54 Page 395
Then,
Mðf ¼ 5Þ ¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À f =f n
ð
Þ
2
h
i 2 þ 2z f =f n
ð
Þ
2
r
¼ 1:03
With a dynamic error of only about 3%, this system should be suitable for the intended
measurements.
Example 9.7
A common method to estimate the data acquisition and reduction errors present in the pressuremeasuring instrumentation of a test rig is to apply a series of replication tests on the M calibrated
pressure transducers used on that rig (when the number of transducers installed is large, a few
transducers can be selected at random and tested). In such a test each transducer is interconnected
through a common manifold to the output of a single deadweight tester (or suitable standard) so that
each transducer is exposed to exactly the same static pressure. Each transducer is operated at its
normal excitation voltage. The test proceeds as follows: record applied test pressure (from the
deadweight tester), record each transducer output N (e.g., N ! 10) times, vent the manifold to local
atmosphere, close the manifold to reapply test pressure, repeat the procedure at least K (e.g., K ! 5)
times. What information is available in such test data?
KNOWN M( j ¼ 1, 2, . . . , M) transducers
N(i ¼ 1, 2, . . . , N) repetitions at a test pressure
K(k ¼ 1, 2, . . . , K) replications of test pressure
SOLUTION The mean value for each transducer is given by the mean of the repetitions for that
transducer,
p jk ¼
1
N
X N
i¼1
p ijk
The mean value for the replications of the jth transducer is given by
hp j i ¼
1
K
X K
k¼1
p jk
The difference between the pooled mean pressure and the known applied pressure would provide an
estimate of the systematic error to be expected from any transducer during data acquisition. On the
other hand, variations between p jk and hp j i are due to random errors in the data acquisition and
reduction procedure. The variation of the test pressure mean, p jk , about the pooled mean, hp j i, for
each transducer is
hs j i ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
P K
k¼1
À p jk À hp j i
Á 2
K À 1
s
The pooled standard deviation of the mean for the M transducers provides the estimate of the random
9.5 Pressure Transducer Calibration 395