Determination of Optimal Measurement Points for Calibration Equations—Examples by RH Sensors
Abstract
:1. Introduction
- The number and the location of the calibration points.
- The regression equations (linear, poly-nominal, non-linear).
- The regression techniques.
- The standard references and their uncertainties.
2. Materials and Methods
2.1. Relative Humidity (RH) and Temperature Sensors
2.2. Saturated Salt Solutions
2.3. Calibration of Sensors
2.4. Establish and Validate the Calibration Equation
2.5. Different Calibration Points
- Case 1: The data set is for 11 salt solutions and 11 calibration points
- Case 2: The data set is for 9 salt solutions and 9 calibration points
- Case 3: The data set is for 7 salt solutions and 7 calibration points
- Case 4: The data set is for 5 salt solutions and 5 calibration points
2.6. Data Analysis
2.6.1. Tests on a Single Regression Coefficient
2.6.2. The Estimated Standard Error of Regression
2.6.3. Residual Plots
2.7. Measurement Uncertainty for Humidity Sensors
3. Results and Discussion
3.1. The Effect of the Accuracy of Different Calibration Points
3.1.1. THT-B121 Resistive Humidity Sensor
- y = −20.530298 + 2.805196x − 0.049153x2 + 0.000539x3 − 2.07539 × 10−6x4
- (sb = 2.5004 sb = 0.2590 sb = 0.0082 sb = 0.00016 sb = 4.770 × 10−7
- t = −8.2107 t = 11.181 t = −6.005 t = −5.0663 t = −4.3514)
- R2 = 0.992, s = 0.7719
- y = −19.471802 + 2.743833x − 0.047663x2 + 0.0005157x3 − 1.93676 × 10−6x4
- (sb = 2.2789 sb = 0.25086 sb = 0.00869 sb = 0.000117 sb = 5.360 × 10−7
- t = −8.5447 t = 10.9396 t = −5.4849 t = 4.3946 t = −3.6101)
- R2 = 0.991, s = 1.014
3.1.2. HMP 140A Capacitive Humidity Sensor
- y = 3.479518 + 0.833274x + 0.001867x2, R2 = 0.9994, s = 0.6837
- (sb = 0.4805 sb = 0.02028 sb = 0.000187
- t = 7.2408 t = 41.098 t = 10.004)
- y = 2.9113205 + 0.864217x + 0.0015542x2, R2 = 0.9995, s = 0.7890
- (sb = 0.63806 sb = 0.02925 sb = 0.000278
- t = 74.5628 t = 29.543 t = 5.5872)
3.1.3. Evaluation of Accuracy
3.2. The Effect of the Precision of Calibration Points
3.2.1. The Measurement Uncertainty for the Two Humidity Sensors
3.2.2. The Precision of the Two Types of RH Sensors
3.3. Discussion
4. Conclusions
Supplementary Materials
Author Contributions
Acknowledgments
Conflicts of Interest
References
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Salt Solutions | OIMI [19] | Lake [27] | Wadso [28] | Duvernoy [29] | Belhadj [30] | JMS [31] | JISC [32] | CMA [33] | Delta [34] | OMEGA [35] | TA [36] | Vaisala [37] |
---|---|---|---|---|---|---|---|---|---|---|---|---|
LiBr | * | |||||||||||
LiCl | * | * | * | * | * | * | * | * | * | |||
CH3COOK | * | * | * | * | ||||||||
MgCl2·GH2O | * | * | * | * | * | * | * | * | * | * | ||
K2CO3 | * | * | * | * | * | * | * | |||||
Mg(NO3)2 | * | * | * | * | * | * | * | |||||
NaBr | * | * | * | * | ||||||||
KI | * | * | * | |||||||||
SrCl2 | * | |||||||||||
NaCl | * | * | * | * | * | * | * | * | * | * | * | * |
(NH4)2SO4 | * | |||||||||||
KCl | * | * | * | * | * | * | * | * | ||||
KNO3 | * | * | * | * | ||||||||
K2SO4 | * | * | * | * | * | * | * | * |
Resistive Sensor | Capacitive Sensor | |
---|---|---|
Model 1 | THT-B121 | HMP 140A |
Sensing element | Macro-molecule HPR-MQ | HUMICAP |
Operating range | 0–60 °C | 0–50 °C |
Measuring range | 10–99% RH | 0–100% |
Nonlinear and repeatability | ±0.25% RH | ±0.2% RH |
ResolutionTemperature effect | 0.1% RH (relative humidity)none | 0.1% RH0.005%/°C |
Salt Solutions | (n1 = 11) Case 1 | (n2 = 9) Case 2 | (n3 = 7) Case 3 | (n4 = 5) Case 4 | uc |
---|---|---|---|---|---|
LiCl | * | * | * | * | 0.27 |
CH3COOK | * | 0.32 | |||
MgCl2 | * | * | * | * | 0.16 |
K2CO3 | * | * | * | 0.39 | |
Mg(NO3)2 | * | * | 0.22 | ||
NaBr | * | * | * | * | 0.40 |
KI | * | * | 0.24 | ||
NaCl | * | * | * | * | 0.12 |
KCl | * | * | * | 0.26 | |
KNO3 | * | 0.55 | |||
K2SO4 | * | * | * | * | 0.45 |
Linear | 2nd Order | 3nd Order | 4th Order | |
---|---|---|---|---|
b0 | 0.028672 | −2.74999 | −11.0702 | −20.5303 |
b1 | 1.008985 | 1.13766 | 1.780025 | 2.805196 |
b2 | −0.0011437 | −0.01432 | −0.0491534 | |
b3 | 7.81681 × 10−5 | 5.39281 × 10−4 | ||
b4 | −2.07539 × 10−6 | |||
R2 | 0.9967 | 0.9974 | 0.9987 | 0.9993 |
s | 1.6098 | 1.4612 | 0.982 | 0.7719 |
Residual plots | clear pattern | clear pattern | clear pattern | uniform distribution |
Linear | 2nd Order | 3nd Order | 4th Order | |
---|---|---|---|---|
b0 | −0.970118 | −3.1191770 | −12.201481 | −19.471802 |
b1 | 1.0155235 | 1.12632754 | 1.8869907 | 2.743833 |
b2 | −0.001007316 | −0.01685101 | −0.04766345 | |
b3 | 9.34623 × 10−5 | 5.15689 × 10−4 | ||
b4 | −1.93676 × 10−6 | |||
R2 | 0.9969 | 0.9974 | 0.9994 | 0.9991 |
s | 1.8109 | 1.7146 | 0.7984 | 1.084 |
Residual plots | clear pattern | clear pattern | clear pattern | uniform distribution |
Case 1 (n1 = 11) | Case 2 (n2 = 9) | Case 3 (n3 = 7) | Case 4 (n4 = 5) | |
---|---|---|---|---|
b0 | −20.530297 | −23.41845561 | −23.904948 | −19.4718019 |
b1 | 2.8051965 | 3.5861653 | 3.243023015 | 2.743832845 |
b2 | −0.04915334 | −0.06230766 | −0.06426625 | −0.047663446 |
b3 | 5.39281 × 10−4 | 7.0951 × 10−4 | 7.34202 × 10−4 | 5.15689 × 10−4 |
b4 | −2.07539 × 10−6 | −2.81734 × 10−6 | −2.92042 × 10−6 | −1.93676 × 10−6 |
R2 | 0.9993 | 0.9994 | 0.9994 | 0.9991 |
s | 0.7719 | 0.6951 | 0.8039 | 1.084 |
Linear | 2nd Order | |
---|---|---|
b0 | −0.414520 | 3.479518 |
b1 | 1.031003 | 0.833274 |
b2 | 0.00186718 | |
R2 | 0.9975 | 0.9994 |
s | 1.4002 | 0.6837 |
Residual plots | clear pattern | Uniform distribution |
Linear | 2nd Order | |
b0 | 0.226512 | 2.911321 |
b1 | 1.023088 | 0.814217 |
b2 | 0.00155423 | |
R2 | 0.9981 | 0.9995 |
s | 1.4386 | 0.7890 |
Residual plots | clear pattern | Uniform distribution |
Case 1 (n1 = 11) | Case 2 (n2 = 9) | Case 3 (n3 = 7) | Case 4 (n4 = 5) | |
---|---|---|---|---|
b0 | 3.479580 | 3.156891 | 2.871078 | 2.9113205 |
b1 | 0.833274 | 0.844157 | 0.862302 | 0.8142171 |
b2 | 0.00186718 | 0.00176878 | 0.00161775 | 0.00155423 |
R2 | 0.9975 | 0.9992 | 0.9994 | 0.9995 |
s | 0.6837 | 0.7127 | 0.7490 | 0.7890 |
Description | Estimate Value (%) | Standard Uncertainty u(x), (%) |
---|---|---|
Reference standard, Uref | N1 = 11, uref = 0.3311 N1 = 9, uref = 0.2983 N1 = 7, uref = 0.3151 N1 = 5, uref = 0.3084 | |
Non-linear and repeatability, Unon | ±0.3 | 0.00866 |
Resolution, Ures | 0.1 | 0.00290 |
The combined standard uncertainty of Type B = 0.1926 |
Description | Estimate Value (%) | Standard Uncertainty u(x), (%) |
---|---|---|
Reference standard, Uref | N1 = 11, uref = 0.3311 N1 = 9, uref = 0.2983 N1 = 7, uref = 0.3151 N1 = 5, uref = 0.3084 | |
Nonlinear and repeatability, Unon | ±0.1 | 0.0058 |
Resolution, Ures | ±0.1 | 0.0029 |
Temperature effect, Utemp | ±0.005 | 0.0043 |
The combined standard uncertainty of Type B = 0.1924 |
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Chen, H.-Y.; Chen, C. Determination of Optimal Measurement Points for Calibration Equations—Examples by RH Sensors. Sensors 2019, 19, 1213. https://doi.org/10.3390/s19051213
Chen H-Y, Chen C. Determination of Optimal Measurement Points for Calibration Equations—Examples by RH Sensors. Sensors. 2019; 19(5):1213. https://doi.org/10.3390/s19051213
Chicago/Turabian StyleChen, Hsuan-Yu, and Chiachung Chen. 2019. "Determination of Optimal Measurement Points for Calibration Equations—Examples by RH Sensors" Sensors 19, no. 5: 1213. https://doi.org/10.3390/s19051213
APA StyleChen, H.-Y., & Chen, C. (2019). Determination of Optimal Measurement Points for Calibration Equations—Examples by RH Sensors. Sensors, 19(5), 1213. https://doi.org/10.3390/s19051213