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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Thermocouples are the most frequently used sensors for temperature measurement because of their wide applicability, long-term stability and high reliability. However, one of the major utilization problems is the linearization of the transfer relation between temperature and output voltage of thermocouples. The linear calibration equation and its modules could be improved by using regression analysis to help solve this problem. In this study, two types of thermocouple and five temperature ranges were selected to evaluate the fitting agreement of different-order polynomial equations. Two quantitative criteria, the average of the absolute error values |_{ave} and the standard deviation of calibration equation e_{std}, were used to evaluate the accuracy and precision of these calibrations equations. The optimal order of polynomial equations differed with the temperature range. The accuracy and precision of the calibration equation could be improved significantly with an adequate higher degree polynomial equation. The technique could be applied with hardware modules to serve as an intelligent sensor for temperature measurement.

Temperature measurement is basic and important work in a variety of industries. Electrical temperature sensors included resistive temperature detectors, thermistors and thermocouples [

Output voltage tables for various types of thermocouples list the output voltage corresponding to different temperatures [

Because these calibration equations are higher order polynomial equations, Sarma and Boruan [

Hardware modules have been designed to linearize the non-linear signals with hardware linearization [

The theory of the calibration with piecewise linear regression has been discussed [

Some generalized software techniques for linearisation transducers had been used for thermocouples [

Sarma and Boruan [

For determining the optimal order of polynomial equations for temperature measurement, data fitting ability and prediction performance are both important [^{2}). However, the standard values of estimation could be increased with the loss of data freedom. A higher degree polynomial equation may be over-fitted and the predicted ability thus decreased [

In the previous studies, the curves of the relationship of temperature and output voltage were divided into many pieces. Each piece of these curves was assumed as a linear relationship, however, the residual plots of each piece still indicated nonlinear results [

In this study, the data of output voltage for two types of thermocouple were used from the US National Institute of Standards and Technology (NIST) standard. Five temperature ranges were selected to evaluate their calibration polynomial equations, called piecewise polynomial equations. The parameters for these equations were estimated by the least squares technique. The fitting performance of these equations was evaluated by several statistical methods.

The inverse calibration equation was used to describe the relationship between temperature (T) and output voltage of thermocouples (mv). Because the output voltage at 0 °C for thermocouples is zero, the intercept is excluded in a polynomial equation:
_{1}, c_{2} to c_{k} are constants.

Table data for thermocouples [

Two-types of thermocouples were selected in this study for their popularity in industry. The method developed in this study could be used for other thermocouples. The J-type thermocouple is commonly used for higher temperature ranges. In this study, the type of thermocouple was selected to evaluate the improved performance by piecewise polynomial equation.

There were five ranges (a) 0∼100 °C; (b) 0∼200 °C; (c) −50∼50 °C; (d) −100∼0 °C; and (e) −100∼100 °C. They are the ranges for most living systems, included human beings. The distribution of temperature data for temperature

Microsoft Excel 2003 was used to estimate the parameters of the different order polynomial equations. The _{i} is the error of calibration equation, y_{i} is the dependent variable and _{i}

Three statistics, e_{max}, e_{min} and |_{ave} were used as quantitative criteria. The e_{max} is the maximum e_{i} value, e_{min} is the minimum e_{i} value and |_{ave} is the average of the absolute errors:
_{i}_{i} and n is the number of data. The smaller of the |_{ave}, the better the accuracy of the calibration equation.

The other criterion for uncertainty comparing of calibration equations is precision. The precision performance could be calculated from the standard deviation of the calibration equation [

The estimated parameters of calibration equations for five temperature ranges are listed in

The 2nd order polynomial equation produced a clear systematic pattern of residual plots (

The |_{ave} value represents the accuracy of the calibration equation. From _{max}, e_{min} and |_{ave}. The |_{ave} values for the 3rd and 4th order polynomial equations did not differ substantially: 0.00681306 and 0.00676768, respectively.

The e_{std} value represents precision of the calibration equation. The e_{std} values for the 2nd, 3rd and 4th order polynomial equations were 0.05325660, 0.00840050 and 0.00824098, respectively (_{std} values between 2nd and 3rd order polynomial equation was about 1/6.5 but that between 3rd and 4th order polynomial equations was not substantial. The increase in the 4th order (c_{4}x^{4}) of the calibration equation had only a marginal effect on improving performance. The adequate calibration equation for the T-type thermocouple for temperature 0 to 100 °C is as follows:

The 2nd and 3rd order polynomial equations produced a systematic residual pattern and 4th and 5th order polynomial equations revealed a uniform distribution. All residual figures were showed in Supplement Figures A.

The |_{ave} values for the 2nd, 3rd, 4th and 5th order polynomial equations were 0.36507503, 0.03911083, 0.00718054 and 0.00680437, respectively (

The e_{std} values for the above four equations were 0.41512994, 0.04580101, 0.00940073 and 0.00860020, respectively (_{std} value between 3rd and 4th order equation was approximately 1/5. Comparing with the 4th order equation, the contribution of the 5th order equation was substantial. The reduction in precision was limited. Therefore, the adequate equation for the T-type thermocouple for temperature 0 to 200 °C is as follows:

This temperature range included the activity environment for the most biological system. The residual plots of the 2nd, 3rd and 4th order calibration equations are presented in Supplement Figure B. Only the 4th order equation showed a random distribution in residual plot. The 4th order polynomial had the smallest |_{ave} and e_{std} values (

Only the 4th order polynomial equation had a uniform distribution on residual plots (data not shown) and the smallest value of |_{ave} and e_{std} (

The shape of the data distribution between temperature and thermocouple output voltage is a nonlinear curve. Only a higher order polynomial equation could produce a uniform distribution on residual plot (data not shown). The adequate calibration equation was a 6th order polynomial equation and showed as follows:

The |_{ave} value represents the accuracy and the e_{std} value was used to assess the precision of these equations. By the selection of the adequate polynomial calibration equations, the |_{ave} was < 0.009 °C and the e_{std} value was < 0.012 °C for the T-type thermocouple.

The estimated parameters for calibration equations for five temperature ranges are listed in

All datasets for different temperature ranges were evaluated by regression analysis. The residual plots were used to evaluate the adequateness of models. The |_{ave} and e_{std} values were used to assess accuracy and precision. The adequate equations for different temperature ranges are listed as follows:

1.0∼100 °C

0∼200 °C

−50∼50 °C

−100∼0 °C

−100∼100 °C

With the selection of the adequate polynomial calibration equations, the |_{ave} was <0.005 °C and The e_{std} value was <0.008 °C for the J-type thermocouple. The |_{ave} value presented the accuracy and the e_{std} value showed the precision of these equations. These numeric values indicated the performance improvement for this type thermocouple using in the special temperature range.

Now, the development of microprocessor systems is rapid and the price is dwindling. The nonlinear characteristics of sensing element could be improved by software package techniques. The calculation of the higher order polynomial equation could be treated as rapidly and accurately as linear equations. In this study, the orders of their polynomial equations for adequate calibration equations were lower than that of the NIST Standards. The accuracy and precision of these equations were improved significantly compared to that of a linear equation. They could be adapted to microprocessor systems to enhance the measurement performance of different types of thermocouples. The suggestion of the application of these polynomial calibration equations are as follows:

The analog mv output of the thermocouple is amplified to voltage.

The voltage signal is digitized by A/D converter.

The function of the A/D converter is controlled by a microcomputer.

The software for these calibrations is embedded in the flash ROM of the microcomputer.

The true temperature then is computed by its adequate polynomial calibration equation.

The true temperature could be display in a LCD or send to a PC via RS232 for data display or send to temperature controller.

Thermocouples are the most frequently used sensors for temperature measurement. However, linearizing the transfer relationship between temperature and output voltage is one of their major problems. In this study, two types of thermocouple with five temperature ranges were selected to evaluate the fitting agreement of different order polynomial equations to help solve this problem. The estimated parameters were established by regression analysis techniques. Two quantitative criteria, |_{ave} and e_{std} were used to evaluate the accuracy and precision of these calibrations equations. Residual plots were applied to justify the adequateness of these models.

The adequate order of polynomial calibration equation was affected by the temperature range. The 3rd order polynomial equation was adequate for the 0 to 100 °C temperature range and the higher 6th order polynomial equation was adequate for the −100 °C to 100 °C range.

The |_{ave} value represents the accuracy of these equations. The e_{std} value was used to assess the precision of equations. With the adequate polynomial calibration equation, the |_{ave} was <0.009 °C for the T-type thermocouple and <0.005 °C for the J-type thermocouple. The numeric value of e_{std} was <0.012 °C for the T-type thermocouple and <0.008 °C for the J-type thermocouple.

These polynomial calibration equations are easy to be written as software and be incorporated into an IC circuit as calculated equations. The measured thermocouple output could be transformed into the temperature easily and accurately. The technique could be applied with hard modules to serve as intelligent sensors. The regression analysis technique and criteria for comparison used in this study could be applied to evaluate adequate calibration equations for other thermocouples with different temperature ranges. The piecewise polynomial equation could be established to meet the requirement temperature range for practical applications.

The authors would like to thank the National Science Council of the Republic of China for financially supporting this research under Contract No. 98-2313-B-005-032-MY3.

The authors declare no conflict of interest.

Distribution of temperature and output voltage of two types of thermocouples with temperature (0 to 200 °C).

Distribution of temperature and output voltage of two types of thermocouples with temperature (−100 to 100 °C).

Residual plots of polynomial calibration equations for T-type thermocouples with temperature 0 to 100 °C. (

Estimated parameters for several polynomial equations for T-type thermocouples by temperature range.

− |
− |
− | ||||
---|---|---|---|---|---|---|

2nd order | b_{1} |
25.67471979 | 25.07879340 | 26.00810515 | −25.22117110 | 26.44647267 |

b_{2} |
−0.54576494 | −0.39314346 | −0.74516432 | −1.26190490 | −0.78530785 | |

3rd order | b_{1} |
25.86464325 | 25.76378439 | 25.85804386 | −25.90849949 | 25.91435043 |

b_{2} |
−0.69457635 | −0.64264925 | −0.75808252 | −0.59839052 | −0.83435521 | |

b_{3} |
0.026133029 | 0.020317674 | 0.066304477 | −0.14489758 | 0.061098417 | |

4th order | b_{1} |
25.84962602 | 25.09020576 | 25.84551540 | −25.77505075 | 25.81912460 |

b_{2} |
−0.673394463 | −0.73340079 | −0.70994624 | −0.830585167 | −0.74867280 | |

b_{2} |
0.017448349 | 0.037584526 | 0.074689216 | −0.026571395 | 0.077691433 | |

b_{4} |
−1.082962 × 10^{−3} |
−9.9772501 × 10^{−4} |
−0.018167033 | −0.018427604 | −8.8817640 × 10^{−3} | |

5th order | b_{1} |
25.88262726 | 25.86505358 | |||

b_{2} |
−0.71357086 | −0.73577069 | ||||

b_{2} |
0.031114204 | 0.062941133 | ||||

b_{4} |
−1.5600801 × 10^{−4} |
−0.010532441 | ||||

b_{5} |
3.7937780 × 10^{−5} |
9.7149801 × 10^{−4} | ||||

6th order | b_{1} |
25.85453185 | ||||

b_{2} |
−0.72787713 | |||||

b_{2} |
0.067478989 | |||||

b_{4} |
−0.012651926 | |||||

b_{5} |
6.0999501 × 10^{−}^{4} | |||||

b_{6} |
−1.3091201 × 10^{−}^{4} |

Criteria for evaluating of polynomial equations for T-type thermocouples by temperature range. Sacle equation to same font size as table.

− |
− |
− | ||||
---|---|---|---|---|---|---|

2nd order | e_{min} |
−0.07447137 | 0.49823066 | 0.225686384 | −0.36968565 | −1.67100581 |

e_{max} |
0.13074332 | 0.98345135 | 0.13642422 | 0.18487408 | 1.21440524 | |

|_{ave} |
0.04592460 | 0.36507503 | 0.06717221 | 0.12630630 | 0.48771413 | |

3th order | e_{min} |
−0.02072832 | 0.13412573 | 0.06409870 | −0.04361028 | −0.55185176 |

e_{max} |
0.01471193 | 0.07282170 | 0.04472070 | 0.03208275 | 0.285123028 | |

|_{ave} |
0.00681306 | 0.03911083 | 0.02223685 | 0.01384381 | 0.150423885 | |

4th order | e_{min} |
−0.01753270 | 0.03052425 | 0.02023304 | −0.01507971 | −0.07094427 |

e_{max} |
0.01501541 | 0.19169659 | 0.02069277 | 0.01633248 | 0.11870578 | |

|_{ave} |
0.00676768 | 0.00718054 | 0.00763593 | 0.00663725 | 0.027618094 | |

5th order | e_{min} |
0.02534386 | −0.027957131 | |||

e_{max} |
0.01656904 | 0.03741114 | ||||

|_{ave} |
0.00680437 | 0.01217997 | ||||

6th order | e_{min} |
-0.02814230 | ||||

e_{max} |
0.02771649 | |||||

|_{ave} |
0.00986177 |

Measurement precision of polynomial equations for T-type thermocouples by temperature range.

− |
− |
− | |||
---|---|---|---|---|---|

2nd order | 0.05325659 | 0.41512994 | 0.079658494 | 0.14552443 | 0.57348457 |

3rd order | 0.00840050 | 0.04580101 | 0.026619274 | 0.01675441 | 0.18449760 |

4th order | 0.00824098 | 0.00940073 | 0.009181103 | 0.00794493 | 0.03658164 |

5th order | 0.00860020 | 0.01527604 | |||

6th order | 0.01228220 |

Estimated parameters for several polynomial equations for J-type thermocouples by temperature range.

− |
− |
− | ||||
---|---|---|---|---|---|---|

2nd order | b_{1} |
19.71440273 | 19.45794480 | 19.92321865 | 19.53896507 | 20.15400867 |

b_{2} |
−0.14280031 | −0.08790950 | −0.24205613 | −0.42806222 | −0.25366061 | |

3rd order | b_{1} |
19.82859586 | 19.76305984 | 19.84718826 | 19.91333650 | 19.85023043 |

b_{2} |
−0.21497883 | −0.18247776 | −0.24479520 | −0.16293062 | −0.26518721 | |

b_{3} |
0.01024941 | 0.006582481 | 0.019753541 | 0.042391433 | 0.02046607 | |

4th order | b_{1} |
19.84344081 | 19.82989561 | 19.84610586 | 19.83020340 | 19.82940836 |

b_{2} |
−0.23187152 | −0.21979759 | −0.23898495 | −0.26902673 | −0.23785577 | |

b_{3} |
0.01584881 | 0.012647965 | 0.020179476 | 2.9937101 × 10^{−3} |
0.02258792 | |

b_{4} |
−5.6514610 × 10^{−4} |
−3.0010410 × 10^{−3} |
−1.2941520 × 10^{−3} |
−4.5105150 × 10^{−3} |
−1.5837460 × 10^{−3} | |

5th order | b_{1} |
19.85185466 | 19.84739765 | |||

b_{2} |
−0.22599582 | −0.23586796 | ||||

b_{3} |
0.030341877 | 0.019194036 | ||||

b_{4} |
−2.5509630 × 10^{−3} |
−1.7327180 × 10^{−3} | ||||

b_{5} |
6.2928705 × 10^{−4} |
1.2591410 × 10^{−4} | ||||

6th order | b_{1} |
19.84959392 | ||||

b_{2} |
−0.23844914 | |||||

b_{3} |
0.018639399 | |||||

b_{4} |
−1.4776299 × 10^{−3} | |||||

b_{5} |
1.5145010 × 10^{−4} | |||||

b_{6} |
−1.2754301 × 10^{−5} |

Criteria for evaluating of polynomial equations for J-type thermocouples by temperature range.

− |
− |
− | ||||
---|---|---|---|---|---|---|

2nd order | e_{min} |
0.05557809 | 0.24933012 | 0.13616157 | −0.28775249 | −1.18173165 |

e_{max} |
0.08928586 | 0.47674018 | 0.11595334 | 0.14466809 | 0.85074564 | |

|_{ave} |
0.03400235 | 0.18601820 | 0.04196053 | 0.09499065 | 0.33772693 | |

3rd order | e_{min} |
0.01317127 | 0.068261782 | 0.02101240 | −0.04169692 | −0.30645416 |

e_{max} |
0.01187325 | 0.046436731 | 0.01547939 | 0.02779399 | 0.14204194 | |

|_{ave} |
0.00481871 | 0.021161537 | 0.00574589 | 0.01190433 | 0.07435890 | |

4th order | e_{min} |
0.01009384 | 0.013460643 | 0.01074795 | −0.01449671 | -0.06051327 |

e_{max} |
0.00925146 | 0.013254812 | 0.00786120 | 0.01273097 | 0.03879727 | |

|_{ave} |
0.00429203 | 0.004711247 | 0.00438609 | 0.00532295 | 0.01359488 | |

5th order | e_{min} |
−0.01239944 | −0.02524355 | |||

e_{max} |
0.01075262 | 0.01869201 | ||||

|_{ave} |
0.00507465 | 0.00580843 | ||||

6th order | e_{min} |
−0.01393513 | ||||

e_{max} |
0.01228582 | |||||

|_{ave} |
0.00482716 |

Measurement precision of polynomial equations for J-type thermocouples by temperature range.

− |
− |
− | |||
---|---|---|---|---|---|

2^{nd} order |
0.03903280 | 0.21139799 | 0.049680 | 0.10986868 | 0.39874169 |

3^{rd} order |
0.00585086 | 0.02519590 | 0.007350308 | 0.01429016 | 0.09409667 |

4^{th} order |
0.00522916 | 0.00582213 | 0.005281434 | 0.00641780 | 0.016754189 |

5^{th} order |
0.00537354 | 0.00612658 | 0.00723238 | ||

6^{th} order |
0.00581152 |