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06-1 Error *
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Error is the difference between an ideal (or correct) value and an actual value.
- Several different types of error can be measured.
- An error type can be expressed in several ways.
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06-2 Expression of Error *
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Notation
I denotes an ideal value.
A denotes an actual value.
Absolute error defined jI Aj.
Percent error defined 100 jI__Aj__Ifor I 6= 0.
Consider a transducer designed to measure process variables in the range I 2 [xmin ; xmax ].
Percent-full-scale error defined 100 jI__Aj__xfor xmax 6= 0.
max
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06-3 Types of Error *
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- Model Error.
Error in transducer model, Ht.
- Repeatability Error.
Transducer change from occasion to occasion.
- Stability Error.
Transducer change during use.
- Calibration Error.
Difference between two transducers of same kind.
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06-4 Model Error *
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Let y = Ht(x) denote a transducer output, response, and process variable.
The accuracy of Ht(x) depends upon how well the transducer is understood and how
complex a transfer function can be tolerated.
For example, the following are all for the same transducer:
Okay: Ht1(x) = Ro(1 + ax).
Good: Ht2(x) = Ro(1 + ax + bx2).
Better: Ht3(x) = Ro(1 + ax + bx2 + cx3).
Best: Ht4(0 ffiC) = 100 , Ht4(0:01 ffiC) = 100:15 , : :.:(Called a lookup table.)
Model error quantifies the accuracy of the transfer function.
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_ Definition of Model Error Quantities *
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_ Test conditions: a single measurement. Let Ht(x) denote the transducer response, x denote *
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_ the process-variable value, and y the quantity measured at at the transducer outputs. *
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_ Then: Ideal: I = x, Actual: A = H1 (y). *
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06-5 Model Error Example *
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What is the absolute model error of a transducer having response Ht(x) = (10x25) V
under test conditions, with process variable x = 2:130 and measured transducer output
y = 34:90 V.
The ideal quantity is I = 2:130.
r ______________ij
H1t (y) = 1__10y_V+ 5 .
Based on the transducer A = H1t (34:9 V) = 1:998.
The absolute error is then, 0:1325.
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06-6 Repeatability *
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Measures how well a transducer performs over time.
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_ Definition of Repeatability Error Quantities *
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_ Test conditions: *
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_ Let Ht(x) denote the transducer response. *
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_ Let x(t) denote the value of the process variable at time t. *
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_ Two measurements are made, at times t1 and t2, t1 < t2. *
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_ The test is set up so that x(t1) = x(t2) = x and x(t1:5) 6= x for some t1 < t1:5 < t2. *
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_ Let y1 and y2 denote the quantities read at the transducer outputs at times t1 and t2. *
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_ Then: Ideal: I = H1 (y ). Actual: A = H1 (y ). *
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_ t 1 t 2 *
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06-7 Stability *
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Measures how well the a transducer measures a steady quantity.
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_ Definition of Stability Error Quantities *
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_ Test conditions: *
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_ Let Ht(x) denote the transducer response. *
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_ Let x(t) denote the value of the process variable at time t. *
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_ Two measurements are made, at times t1 and t2, t1 < t2. *
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_ The test is set up so that x(t1) = x(t2) = x(t1:5) = x for all t1 < t1:5 < t2. *
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_ Let y1 and y2 denote the quantities read at the transducer outputs at t1 and t2. *
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_ Then: Ideal: I = H1 (y ). Actual: A = H1 (y ). *
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_ t 1 t 2 *
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06-8 Calibration *
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Measures how well two transducers of the same type compare.
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_ Definition of Calibration Error Quantities *
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_ Test conditions: *
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_ Let H (x) denote the transducer response and x denote the value of the process variable. *
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_ A measurement is made with each transducer. *
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_ Let y1 and y2 be the quantities read at the transducers' outputs. *
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_ Then: Ideal: I = H1 (y ). Actual: A = H1 (y ). *
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_ t 1 t 2 *
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06-9 Example *
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A
A type of integrated temperature sensor has a response of Ht(x) = 7x _____. Tests
K
were performed on two such sensors by exposing the sensors to a known temperature,
x, and measuring their response, y, as follows:
At time t1 sensor A exposed to x = 295 K; output y = 2050 A.
At time t2 sensor A exposed to x = 300 K; output y = 2085 A.
At time t3 sensor A exposed to x = 295 K; output y = 2052 A.
At time t4 sensor A exposed to x = 295 K; output y = 2053 A.
At time t5 sensor B exposed to x = 295 K; output y = 2040 A.
Temperature is held constant from t3 to t5. Find model error, repeatability error, sta-
bility error, and calibration error.
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06-10 *
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Inverted Model Function
x = H1t (y) = y _K___7.A
Model Error
Use measurement at t1.
I = 295:0 K and A = H1t (2050 A) = 292:9 K.
Percent model error: j295:0_K__292:9_Kj__295:0=K0:71%.
Could have used any time to compute model error.
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06-11 *
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Repeatability Error
Use measurements at t1 and t3 (since temperature different at t2).
I = H1t (y(t1)) = H1t (2050 A) = 292:9 K:
A = H1t (y(t3)) = H1t (2052 A) = 293:1 K:
Percent repeatability error: j292:9_K__293:1_Kj__292:9=K0:06828%.
Note, actual and ideal quantities could be reversed in this example.
Also possible to use t1 and t4.
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06-12 *
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Stability Error
Use measurements at t3 and t4 (since temperature held constant in this time range).
I = H1t (y(t3)) = H1t (2052 A) = 293:1 K:
A = H1t (y(t4)) = H1t (2053 A) = 293:3 K:
Percent stability error: j293:1_K__293:3_Kj__293:1=K0:06824%.
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06-13 *
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Calibration Error
Use measurements at t4 and t5.
I = H1t (y(t4)) = H1t (2053 A) = 293:3 K:
A = H1t (y(t5)) = H1t (2040 A) = 291:4 K:
Percent calibration error: j293:3_K__291:4_Kj__293:3=K0:6478%:
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06-14 Miscellany *
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Typically, error specially defined for each type of transducer.
The definition includes the exact test circuit and test conditions.
Error measures can be applied to conditioning circuits and anything else that trans-
forms a process variable value.
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| David M. Koppelman - koppel@ee.lsu.edu | Modified 25 Jan 1999 12:40 (18:40 UTC) |