Intelligent Transducer for Temperature Measurement with Two-Wire or Three-Wire Platinum RTD

Abstract

The article presents an intelligent temperature transducer (ITT), which can work with a two-wire or a three-wire platinum resistance temperature detector (RTD). The ITT design allowed for compensation of the RTD’s lead wire resistance. The ITT used the author’s auto-calibration procedure, which minimized linearity errors of the ITT and RTD processing characteristics, ITT offset and gain errors, and errors resulting from changes in the ITT operating conditions concerning the nominal conditions. The presented results of a simulation and experimental studies confirmed the high effectiveness of this procedure. The determined uncertainty of temperature measurement using the Monte Carlo method and the obtained experimental results confirmed the possibility of measuring temperatures in the range of 0–200 °C with an expanded uncertainty of 0.02 °C at a 99% confidence level.

1. Introduction

Temperature is one of the most popular non-electrical quantities in terms of the number of measurements performed. It can be measured by non-contact and contact methods. Non-contact measurement can be realized, for example, with infrared thermometers and thermal imaging cameras [1,2,3]. Contact measurement is most often carried out with various types of electrical sensors. Among them, the resistance temperature detector (RTD) made of platinum (Pt) can be distinguished due to its relatively good metrological and operational properties. Its parameters are specified in the standard [4]. RTDs are manufactured, among others, by using microtechnology [5,6,7,8]. They are used primarily in monitoring industrial processes [6,9,10] and the environment [8,11,12]. In particular, the work in [6] concerns the measurement of liquid temperature in a pipeline, while [9] addresses temperature measurement in a vessel. In [10], the authors suggest the use of an RTD for modern industries and process control plants. In turn, in [8], biodegradable detectors are presented, which can be used for environmental measurements. RTDs can also be used in weather stations [11] and smart homes [12].

Temperature measurements using RTDs are carried out using temperature transducers (TTs), which process the sensor resistance into voltage (R_S/V) [13,14,15,16,17,18,19] or into time (R_S/t) [10,20,21]. Next, the voltage (V) or the time (t) are most often converted in TTs into a digital signal related to the temperature measured using an analog-to-digital converter (A/D) or a counter, respectively. TTs with R_S/V circuits are more popular due to the higher accuracy of temperature measurements achieved. A Wheatstone bridge [13,16,17] or a resistive divider [14,15,18,22] is used as an R_S/V circuit. Regardless of the circuit used for signal processing in TTs, the RTD accuracy specified in the IEC-60751 standard [4] must be considered, which depends on the class and measurement range of the given RTD. The TT processing error should be significantly smaller than the RTD error. The accuracy of the TT is influenced by its parameters, such as nonlinearity, gain and zero errors, and accuracy of supply voltage and resistors used in an R_S/V circuit. Additionally, the nonlinearity of the RTD characteristic and the resistance of the RTD wire can have a significant influence on the temperature measurement result.

The RTD linearity error can be minimized by using the calibration methods presented in [23,24,25,26]. To achieve a temperature measurement accuracy of hundredths or even tenths of a degree, it is necessary to minimize the influence of wire resistance. Therefore, circuit solutions with three-wire and four-wire RTDs are often used [12,15,22,27,28,29,30,31,32]. In measurement circuits cooperating with a four-wire detector, the lead wire resistance has a negligible effect on the temperature measurement result. In circuits with a three-wire detector, compensation for the lead wire resistance is used. In the case of a two-wire RTD, measurement circuits are often used that do not minimize the influence of wire resistance, and the temperature measurement accuracy is the lowest. However, it is possible to compensate for the lead wire resistance of a two-wire detector [28,33,34]. One compensation method uses a two-wire RTD with an additional compensation loop (the sensor connection is sometimes referred to in the literature as a four-wire connection) [13,28,35]. It involves appropriately introducing an additional wire (compensating loop) into a measurement circuit with a resistance equal to the resistance of the two lead wires of the RTD.

Ensuring high accuracy of temperature measurement requires the use of appropriate hardware and software solutions that allow for minimizing the remaining processing errors. However, when analyzing the previously mentioned papers on TTs with R_S/V circuits, it should be stated that not all processing errors are minimized in the above-mentioned works.

Solutions are also used to achieve very high accuracy of temperature measurements, assuming that some of the processing errors, such as linearity and gain errors, offset voltages, and errors resulting from the influence of ambient temperature, are very small. This assumption can be met at a constant ambient temperature by the additional use of very precise amplifiers, stable power sources for an R_S/V circuit, and A/D converters with very high resolution in TTs [29,36,37].

One of the objectives of this paper is to fill the research gap regarding the application of the author’s auto-calibration procedure [38] to other temperature transducer designs. In the work in [13], the authors presented the first version of a temperature transducer based on a Wheatstone bridge circuit and an auto-calibration circuit consisting of five standard resistors. A certain limitation of this solution is the relatively complex structure of this measuring circuit and the difficult software implementation of the auto-calibration procedure. In this paper, a new circuit solution of an intelligent temperature transducer circuit is presented for which the auto-calibration procedure [38] is applied. This new solution was based on a resistive voltage divider circuit [14,22], which consisted of an RDT and a resistor connected in series. In the auto-calibration circuit, it was proposed to use four standard resistors. The use of an auto-calibration procedure in this transducer enabled the automatic minimization of RTD nonlinearity, an R_S/V circuit nonlinearity, offset, gain errors, and errors resulting from changes in the supply voltage and ITT operating conditions concerning the nominal conditions. For this reason, this solution did not require the use of A/D converters with very small gain error and offset, and very accurate power sources, in the measurement circuit. The new ITT was dedicated to cooperation with a two-wire RTD with a compensation loop or a three-wire RTD. The advantages of the new intelligent temperature transducer, compared with the one presented in [13], are a simpler design, easier implementation of the auto-calibration procedure [38], and higher accuracy of the T measurement.

The article consists of four sections and one appendix. Section 2 presents the ITT concept, including its circuit diagram, general operating principle, and metrological properties. Section 3 shows an exemplary ITT implementation, including materials and simulation and experimental results. Section 4 contains the conclusions. Appendix A presents a comparison of the test results of the new ITT with the results obtained for a transducer from [13].

2. The Concept of an Intelligent Temperature Transducer

2.1. Circuit Diagram and General Principle of Operation of the ITT

Figure 1a shows a diagram of the ITT circuit that can work with two- and three-wire RTDs, whose wires have the same resistance. The ITT allows compensation for the detector’s lead wire resistance using a compensation loop. The RTDs are connected to the ITT at points 2, 3, and 4. The two-wire detector is connected to points 2 and 3, and the compensation loop is connected to points 3 and 4. In this case, the compensation loop is made of the same wires of the same length as the detector’s two lead wires. In turn, a three-wire RTD is connected without a separate loop to points 2, 3, and 4. Then the two lead wires of the RTD, connected to points 3 and 4, constitute a compensation loop.

The ITT with an RTD (Figure 1a) consists of four basic circuits. The first two circuits, whose equivalent diagram is shown in Figure 1b, are as follows:

• A circuit for processing the RTD resistance R_S into voltage V_{13_S}, consisting of a P_S switch, which is powered from the voltage source V′ through a resistor R: The value of the V_{13_S} voltage is determined by the formula

$$V_{13\_S}=V\prime {\displaystyle \frac{R_S+R_{l\_S}}{R+R_S+R_{l\_S}}},$$

(1)

where R_{l_S} = R₁₂ + 2R_lw + R_Ps is the resistance of the line connecting the R_S with points 1–3 in the ITT. In this, R₁₂ is the sum of the connection resistance of the P_S switch with points 1 and 2, R_Ps is the resistance of the P_S switch in the on state, and 2R_lw is the resistance of the lead wire between points 2–2′ and 3–3′. In this circuit, the resistance R and the supply voltage V′ should be selected so that for the maximum resistance R_S (corresponding to the maximum measured temperature), the voltage V_{13_S} is matched (close to but not greater) to the range of the A/D converter, which is located in a further part of the ITT. In addition, R and V′ should be selected so that the current flowing through the R_S detector does not cause a significant influence of the detector’s self-heating on the temperature measurement result. It was assumed in this study that this current should not exceed 1 mA. For this current value, the measurement range 0–200 °C, and the mean self-heating coefficient value E_K = 0.15 °C/mW [39], the temperature measurement error caused by self-heating of the Pt1000 detector does not exceed 0.15 m°C.

• An auto-calibration circuit, powered by a voltage source V′ through a resistor R, containing standard resistors (from R_St0 to R_StJ): These resistors are switched on accordingly by switches (P₀…P_J). The number of standard resistors and their resistance values depend on the adopted degree (J) of the polynomial describing the processing characteristics of the entire measurement circuit (Figure 1), as discussed further in this section. A compensation loop is included in series with the standard resistor R_Stj (j = 0…J) between points 3 and 4, whose task is to compensate the sum of the connection line resistance between points 2–2′ and 3–3′ (resistance of the RTD sensor wires) [28]. The V_{13_Stj} voltage value is determined by the equation

$$V_{13\_Stj}=V\prime {\displaystyle \frac{R_{Stj}+R_{l\_Stj}}{R+R_{Stj}+R_{l\_Stj}}},$$

(2)

where R_{l_Stj} = R_(14)j + 2R_lw + R_Pj is the resistance of the line connecting the R_Stj with points 1–3 in the ITT. In this, R_(14)j is the sum of the resistances of the connection of switch P_j and R_Stj with points 1–4, R_Pj is the resistance of the P_j switch in the on state, and 2R_lw is the resistance between points 3 and 4. For the three-wire RTD, as a result of the auto-calibration procedure, the wire resistances (R_lw) between points 3–3′ and 4–3′ (Figure 1) naturally form a compensation loop.

• The other ITT circuits are as follows:

• An analog-to-digital (A/D) converter that converts V₁₃ analog voltages to a digital representation;
• A microprocessor (μP) that properly controls the operation of the R_S/V_{13_S} and auto-calibration circuits and calculates the value of the measured temperature (T′).

The purpose of the auto-calibration procedure is to minimize errors (linearity of RTD and R_S/V₁₃ circuits, gain error, offset voltage, and supply voltage changes) caused by changes in T measurement conditions from the nominal conditions. The procedure is carried out in two stages. In the first stage of the ITT auto-calibration procedure, the voltages V_{13_S} and V_{13_Stj} (for the successively switched on reference resistors R_Stj and RTD (R_S)) are converted by the A/D converter to obtain the digital values N_S and N_St0, N_St1, N_St2,…, N_StJ. Then these values are stored in the µP memory. In the second stage of the auto-calibration procedure, the saved digital values together with the specified values of standard temperatures T_Stj constitute the basis for calculating the current values of the polynomial coefficients from a₀ to a_J, describing the characteristics of the entire processing path, from the equations [38]

$$\begin{array}{l}{a}_0+a_1{T}_{St0}+a_2{T}_{St0}^2+\dots +a_J{T}_{St0}^J=N_{St0},\\ a_0+a_1{T}_{St1}+a_2{T}_{St1}^2+\dots +a_J{T}_{St1}^J=N_{St1},\\ a_0+a_1{T}_{St2}+a_2{T}_{St2}^2+\dots +a_J{T}_{St2}^J=N_{St2},\\ \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \\ \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \\ a_0+a_1{T}_{StJ}+a_2{T}_{StJ}^2+\dots +a_J{T}_{StJ}^J=N_{StJ}.\end{array}$$

(3)

Standard temperatures T_Stj are values calculated based on the RTD characteristic (R_S = f(T)—detector resistance as a function of the ambient temperature in which it is located) presented in the standard [4] for the resistors R_Stj used. The values of the coefficients a₀, a₁,…, a_J also depend on external factors affecting the parameters of the components used in the ITT and on the failure to meet the condition [13]:

$$R_{l\_Stj}=R_{l\_S}.$$

(4)

Then, based on the calculated values of the coefficients from a₀ to a_J and the measured value of N_S, a non-linear equation [38],

$$a_0+a_{1}T\prime +a_{2}T{\prime }^2+\dots +a_{J}T{\prime }^J=N_S,$$

(5)

is solved in microprocessor (μP), the result of which is the value of the measured temperature T′.

The degree of the polynomial (5), describing the processing characteristics of the transducer’s measurement track, indicates the number of standard resistors needed in the measurement circuit. The number of these resistors (and thus the degree of the polynomial) affects the accuracy of the temperature determination T′, among others, under the nominal conditions of the transducer. Typically, an increase in the number of resistors increases the accuracy of the temperature determination under nominal conditions but leads to a more complex measurement circuit and more difficult mathematical operations to solve Equation (5). In general, the values of the standard resistors are determined based on initially assumed standard temperatures, distributed evenly over the entire measurement range. For these temperatures, the resistances are calculated using the formula R_S = f(T) [4], and then, based on these resistances, resistors are selected from a range of values available in the market.

Generally, the result of the determined temperature can be written

$$T=T\prime \pm {\Delta }_T,$$

(6)

where Δ_T is the error of the measured temperature T′, which depends on the following:

• The adopted number of standard resistors and their values, accuracy (tolerances), and temperature coefficients (TCRs);
• Condition (4) is satisfied;
• The resolution and nonlinearity of the A/D converter and distortion in the form of noise [29] occurring in the ITT.

The tolerance of the resistor R and the zero, gain, and power ITT errors do not significantly influence the temperature measurement results by using of the auto-calibration procedure. The following part of the article discusses temperature measurement errors associated exclusively with the metrological properties of the ITT. The RTD accuracy class is not considered in the temperature measurement accuracy evaluation.

2.2. ITT Properties

2.2.1. Effectiveness of Minimizing the Influence of Linearity, Zero, Gain, and Power Supply Errors on Temperature Measurement Results

In order to show the effectiveness of the auto-calibration procedure and comparing the metrological properties of the ITT with the transducer presented in [13], the following assumptions were made:

• The temperature measurement range was 0–200 °C.
• RTD (Pt1000) resistance was determined by the formula [4]

$$R_S=R_{S0}\left(1+aT+b{T}^2\right),$$

(7)

where R_S0 = 1000 Ω, a = 3.9083·10^–3 °C⁻¹, and b = −5.774·10^–7 °C⁻².

• In the simulation process, the temperature values T were set with an increment of 0.1 °C.
• In the circuits R_S/V_{13_S} and R_S/V_{13_Stj}, a resistor R with a nominal value equal to R = 7060 Ω and TCR_R = 200 ppm/°C was used.
• The nominal supply voltage of the basic part of the measuring circuit was V′ = 5 V.
• The ambient temperature (T_a) of the standard resistors (T_a(RStj)) and the rest of the ITT circuit under nominal conditions was 25 °C.
• Condition (4) was satisfied.

To show the effectiveness of the auto-calibration procedure, simulation tests were performed in the Mathcad program, enabling the selection of the degree of the polynomial (J) describing the ITT processing characteristics under nominal conditions with the smallest possible error Δ_T. During the research, the temperature standards values (represented by R_Stj) were also selected appropriately for each degree of the polynomial to achieve the lowest possible error values Δ_T. The test results were consistent for two- and three-wire RTDs. For the nominal conditions, the following results were obtained:

• Δ_T ∈ (−0.11, 0.097) °C for J = 2,
• Δ_T ∈ (−0.0026, 0.0033) °C for J = 3,
• Δ_T ∈ (−0.00013, 0.000093) °C for J = 4.

• A polynomial of degree J = 3 was chosen for further research. For this polynomial, the error values of the presented ITT were smaller than those obtained for the transducer described in [13] (Figure 2) for J = 4 (Δ_T ∈ (−0.011, 0.007) °C). Adopting such a solution (J = 3) allowed simplifying the auto-calibration circuit in the ITT, easier implementation of the auto-calibration procedure in μP, and shortening the running time of the procedure.

The implementation of the adopted degree of the polynomial (J = 3), for the assumed measurement range, was ensured by standard resistors (from R_St0 to R_St3) with the following nominal resistance values and corresponding standard temperatures (from T_St0 to T_St3):

• R_St0 = 1020 Ω→T_St0 = 5.121 °C;
• R_St1 = 1260 Ω→T_St1 = 67.192 °C;
• R_St2 = 1540 Ω→T_St2 = 141.110 °C;
• R_St3 = 1760 Ω→T_St3 = 200.392 °C.

It should be noted that the characteristics of the RTD and the R_s/V_{13_S} circuit were strongly nonlinear. In the temperature range from 0 to 200 °C, such a detector had a linearity error of about 0.5%. The total linearity error of the R_s/V_{13_S} circuit characteristic, taking into account the nonlinear RTD characteristic, was about 2%. The auto-calibration procedure minimized the influence of these nonlinearities on the T measurement result. For the assumed J = 3, the maximum relative error of the T measurement in nominal conditions was about 0.0017% (0.033 °C in the range from 0 to 200 °C). This result confirmed the effectiveness of minimizing the influence of the above-mentioned elements’ linearity errors.

The effectiveness of the auto-calibration procedure for other sources of error or measurement uncertainty was tested in two cases. In the first case, T measurement errors were analyzed when this procedure was not used. In these tests, constant values of the coefficients in Equation (5) were assumed, which were determined for nominal conditions (the characteristic Δ_T = f(T) was identical to that in Figure 2). However, much higher error Δ_T values (Figure 3) were obtained in the following cases.

• For example, see the following cases:

• If ambient temperature (T_a) was changed from the nominal value of 25 °C to 35 °C affecting the resistor R with the value TCR_R = 200 ppm/°C, then the maximum error of Δ_T in the range 0–200 °C was −0.96 °C for T_a = 35 °C and −4.8 °C for T_a = 75 °C.
• If the R_S/V_{13_S} and R_Stj/V_{13_Stj} circuits’ supply voltage (V′) was changed by −0.5% concerning the nominal value, then the maximum error of Δ_T was −3.0 °C.
• If the offset voltage (V₀) was changed from 0 mV to −0.1 mV, then the maximum error of Δ_T was −0.061 °C.
• If the gain error was changed from 0 to –0.2%, then the maximum error of Δ_T was −1.2 °C.

• Zero and gain errors came from the A/D converter.

In the second case, T measurement errors were analyzed using the auto-calibration procedure under the same conditions as in the first case. The proposed procedure effectively reduced the Δ_T error value in the case of the influence of the ambient temperature T_a on the resistor R. For T_a = 75 °C, the Δ_T error in the entire measurement range T, from 0 to 200 °C, varied within the range of −0.0025 °C to 0.0032 °C.

Similarly, after changing the following, the error Δ_T over the entire measurement range for each of these cases varied within the same limits as in nominal conditions (from –0.0026 °C to 0.0033 °C):

• The voltage V′ by –0.5%;
• The A/D converter offset voltage (U₀) from 0 mV to –0.1 mV;
• The A/D converter gain error from 0 to –0.2%.

• The results of the influence of changes in T_a, V′, V₀, and gain on the Δ_T error, presented for all analyzed cases, proved that the auto-calibration procedure worked very well. Obtaining such a high efficiency of the auto-calibration procedure required ensuring that its execution time was short enough so that changes in environmental conditions had a negligible impact on the elements of the measurement track.

2.2.2. The Influence of Measurement Channel Components on the Accuracy of Temperature Measurement

An important issue in the auto-calibration procedure is the influence of T_a on changes in the values of the standard resistors R_Stj, which significantly affects Δ_T (Figure 4).

Similarly to [13], it was proposed to measure the ambient temperature of the standard resistors (T_a) and take it into account in the auto-calibration procedure. The effect of this action was a very significant reduction in the error value Δ_T (Figure 5).

• The choice of lower TCR_RStj values and higher accuracy of measurement of the ambient temperature of the standard resistors (T_a) only slightly changed the accuracy of T measurement. These were changes at the level of ten thousandths of °C.

It is assumed that temperature measurements are made under established environmental conditions. If these conditions change, then the measurement procedure should be carried out as briefly as possible so that changes in environmental conditions have a negligible impact on the components of the measurement channel.

The accuracy of the T measurement using the ITT was also determined by the integral non-linearity (INL) of the A/D converter, which operated in the positive voltage range. For precise A/D converters, the INL characteristic in the entire range of positive voltage conversion could be similar to the half-period (e.g., MCP3550/1/3 converter [40]) or a periodic sine function (e.g., AD7766 converter [41]). The analyses of the test results taking into account the linearity error of the A/D converter in the ITT showed that higher error values were obtained for the periodic function. For this case, the values of the V_13′ voltages that were the basis for calculating the measured value T, taking into account the INL of the A/D converter, were determined based on the developed relationship:

$$V_{13}\prime =V_{13}\left(1+{\delta }_{INL}\sin \left(2\pi {\displaystyle \frac{V_{13}}{V_{AD\max }}}\right)\right),$$

(8)

where V₁₃ takes the value of V_{13_S} or V_{13_Stj}, respectively; δ_INL is the limiting linearity error; and V_ADmax is the range of the A/D converter. Figure 6 shows the influence of δ_INL on the temperature T measurement error. The results of this test indicated the possibility of achieving the T measurement error below 0.01 °C with a linearity error δ_INL less than 50 ppm.

Further, ITT research concerned the analysis of the uncertainty of temperature measurement, which was determined using the Monte Carlo (MC) method [42,43]. This method allowed for a more accurate estimate of the inaccuracy of the temperature measurement for the next influential quantities. For this method, quantities related to the scatter of the resistance values of the standard resistors (δ_RStj), the scatter of the resistance values of the lines connecting R_{l_S} and R_{l_Stj} (Δ_{l_Stj,S}), the resolution of the A/D converter (Δ_A/D), and noise (Δ_noise), which may occur in the ITT circuit, were randomly generated M = 10⁶ times. For these data, uncertainties of temperature measurements (U_T) were determined for a confidence level of 99%, based on the equation

$${\Delta }_T\left(i\right)=T_{MC}\left(i\right)-T, i=1,2,\dots ,M,$$

(9)

where T_MC(i) is the temperature determined on the basis of random data, according to the procedure described in Section 2.1.

Figure 7 presents the test results in the form of temperature measurement uncertainty intervals (U_{T_low}, U_{T_high}) as a function of the temperature measured for selected accuracies of the standard resistors. The obtained results indicated that achieving high accuracy in temperature measurement required the use of resistors with values known with high accuracy. For example, achieving T measurement uncertainty of approximately ±0.01 °C required the use of standard resistors with values known with an inaccuracy not greater than ±0.002%.

As mentioned earlier, unfulfilled condition (4) caused inaccuracies in the temperature measurement. Such a situation occurred when the resistance of two wires of the sensor leading resistance R_S to points 2–3 of the R_S/V_{13_S} circuit (2R_{lw1_3}) differed from the resistance of the compensation wire in points 3–4 of the R_S/V_{13_Stj} circuit (2R_{lw3_4}). The influence of the error of both resistances Δ_2Rlw on the result of the temperature measurement uncertainty is shown in Figure 8a. Moreover, the fulfillment of condition (4) is also influenced by the scatter of lines connecting the reference resistors and unequal resistances of the keys in the on state. The cumulative influence of the scatter Δ_{l_Stj,S} of the resistance of the lines connecting the measurement channel components is shown in Figure 8b. The results of these tests indicated the possibility of obtaining T measurement uncertainty below the value of ±0.01 °C if Δ_2Rlw and Δ_{l_Stj,S} was not greater than ±0.01 Ω.

Figure 9 shows the results of simulation tests determining the impact of the A/D converter resolution on the temperature T measurement uncertainty. The results of these tests indicated the possibility of achieving the T measurement expanded uncertainty at the level of ±0.01 °C using an A/D converter with a resolution of 18 bits.

The study of the influence of Gaussian noise with standard deviation σ_noise on the uncertainty of a single measurement T is illustrated in Figure 10. These were the results for the measurement case T = 180 °C, for which the highest uncertainties were obtained. This figure shows that it was possible to achieve T measurement uncertainty below ±0.01% with a standard deviation of noise less than 10 µV. It should be emphasized that for noise, the measurement uncertainty (resulting from the scatter of results) could be reduced by averaging a larger number of T measurements and/or using RC filters before the A/D converter [15].

In Figure 7, Figure 8, Figure 9 and Figure 10 the uncertainty intervals depending on the measured temperature are asymmetric. This is due to the influence of the systematic error for the nominal conditions (Figure 2) on the uncertainty range of the temperature measurement. This asymmetry is more visible with higher accuracy of temperature measurements, which is due to the greater influence of systematic error.

3. Realization of the Concept of an Intelligent Temperature Transducer

3.1. Materials

A computer system that implements the ITT concept for measuring temperature in the range of 0–200 °C has been developed. The system was built to verify the correct operation of the ITT and to evaluate the uncertainty of temperature measurement for its exemplary implementation, taking into account the properties (test results) presented in Section 2. This measurement system consisted of the following components:

• Four 50 cm long copper wires: Two of them were used to lead R_S to the measurement system (ITT), and the other two connected in series constituted a compensation loop (2R_lw).
• A printed circuit board containing the R_S/V_{13_S} circuit and auto-calibration circuit (R_S/V_{13_Stj}), as follows:

  −

  The R_S/V_{13_S} circuit and auto-calibration circuit (R_S/V_{13_Stj}) were made of a resistor R = 7060 Ω with a tolerance of 1% and standard resistors (from R_St0 do R_St3) type RWE 0207 with a coefficient TCR_RStj = 5 ppm/°C. The accuracy of the resistor R did not influence the measurement results T in the ITT. However, the accuracy of the standard resistors was important. For this reason, the values of the standard resistors were measured in a four-wire configuration with an Agilent 3458A multimeter with a limit error of δ_RStj = ±0.002% (for a two-year calibration cycle). The following results were obtained: 1020.070 Ω, 1259.797 Ω, 1539.392 Ω, and 1759.317 Ω.

  −

  The R_S/V_{13_S} circuit and auto-calibration circuit (R_S/V_{13_Stj}) contained reed switches (DIP05–1A72–12A [44]), which, unlike semiconductor switches, were characterized by low on-state resistance and relatively small scatter of their resistances.

  −

  The resistances R_{l_S} and R_{l_Stj} (including the resistances of the reed switches, sensor wire, compensation loop, and other component connections of the R_S/V_{13_S} circuit and auto-calibration circuit), measured with a Thomson bridge, were each 0.12 Ω, and the scatter between these values (Δ_{l_Stj,S}) did not exceed 0.01 Ω.

• An Inmel SQ10 calibrator powering the R_S/V_{13_S} circuit and the auto-calibration circuit with a voltage of V′ = 5 V with a measured short-time instability of δ_V′ = ±10 ppm.
• An Agilent 3458A multimeter for measuring the V₁₃ voltage with a measurement base error of Δ_V13 = δ₁V₁₃ + Δ₁ = 14∙10⁻⁶ V₁₃ + 0.3 µV on a measuring range of ±1 V. It performed the tasks of an A/D converter (Figure 1). The digital voltmeter was made of many elements. In the measurement path, it contained a filter, an amplifier and an A/D converter. The complex construction of the voltmeter meant that its base error did not only result from the resolution of the A/D converter used. In the T measurement, the influence of the first component of the base error (δ₁), which was of a constant nature, was minimized as a result of the auto-calibration procedure. The second component (Δ₁), which was of a random nature, was more important. If this value were to be compared with the quantization error of the A/D converter used alone, it would correspond to a converter resolution of between 22 and 23 bits for the 0–1 V range. Additional errors of the voltmeter given by the manufacturer were as follows: the RMS noise was 0.02 ppm of range, the gain error was 0.01 ppm of reading, and nonlinearity was 0.3 ppm of reading and 0.1 ppm of range. These errors had a negligible effect on the accuracy of the T measurement at the level of 0.01 °C (see Section 2.2).
• A personal computer (PC) that performed the auto-calibration procedure, including the ability to calculate the temperature based on the results of voltage measurements obtained with the 3458A multimeter.

• The ambient temperature of the standard resistors and the remaining part of the circuit (T_a) during the measurements was 23 °C and did not exceed the error Δ_Ta = ±1 °C.

3.2. Results of Simulation and Experimental Studies

The uncertainty of temperature measurement for the proposed system was estimated using the MC method. Similarly to the previous research (Section 2.2.2), the research was carried out in the Mathcad program for the assumed number of experiments M = 10⁶. Each T_MC(i) temperature was determined using the auto-calibration procedure, based on the input values affecting the measurement results, i.e., δ_RStj(i), TCR_RStj(i), δ_V′(i), δ₁(i), Δ₁(i), Δ_Ta(i), and Δ_{l_Stj,S}(i). The values of these quantities were generated randomly (for i = 1…M) for the adopted rectangular probability distribution and the limit errors defined in the description of the computer circuit.

The voltages V_{13_S}(i) (for the temperature sensor) and V_{13_Stj}(i) (for the standard resistors) at the output of the R_S/V_{13_S} and R_Stj/V_{13_Stj} circuits were determined by the following formulas:

$$V_{13\_S}\left(i\right)=k_S\left(i\right)V{\prime }^{*}\left(i\right)\left(1+{\delta }_1\left(i\right)\right)+{\Delta }_1^{*}\left(i\right),$$

(10)

$$V_{13\_Stj}\left(i\right)=k_{Stj}\left(i\right)V{\prime }^{*}\left(i\right)\left(1+{\delta }_1\left(i\right)\right)+{\Delta }_1^{*}\left(i\right),$$

(11)

where

$$k_S\left(i\right)={\displaystyle \frac{R_S+R_{l\_S}\left(i\right)}{R_S+R_{l\_S}\left(i\right)+R\left(i\right)}},$$

(12)

$$k_{Stj}\left(i\right)={\displaystyle \frac{R_{Stj}+R_{l\_Stj}\left(i\right)}{R_{Stj}+R_{l\_Stj}\left(i\right)+R\left(i\right)}},$$

(13)

$$R_{Stj}\left(i\right)=R_{Stj}\left(1+{\delta }_{R\hspace{0.17em}Stj}\left(i\right)\right)\left(1+TC{R}_{R\hspace{0.17em}Stj}\left(i\right){\Delta }_{Ta}\left(i\right)\right),$$

(14)

$$R_{l\_Stj}\left(i\right)=R_{l\_Stj}+{\Delta }_{l\_Stj}\left(i\right),$$

(15)

$$R_{l\_S}\left(i\right)=R_{l\_S}+{\Delta }_{l\_S}\left(i\right),$$

(16)

$$V{\prime }^{*}\left(i\right)=V\prime \left(1+{\delta }_{V\prime }^{*}\left(i\right)\right).$$

(17)

In addition, R_S is the resistance of the temperature sensor calculated based on Equation (7). Moreover, * means that the values of a given input quantity were generated independently for each connected standard resistor and Pt1000 sensor in the auto-calibration and R_S/V_{13_S} circuits.

Based on the T_MC(i) value and Formula (9), the T measurement uncertainty interval (U_{T_low_MC}, U_{T_high_MC}) for a confidence level of 99% was determined following the recommendations resulting from [42]. The obtained uncertainty results as a function of the measured temperature T are presented in Figure 11 with experimental results to compare. The limit values (endpoints) determined based on these tests were U_{T_low_MC_min} = −0.019 °C and U_{T_high_MC_max} = 0.02 °C. In addition, the temperature differences Δ_T obtained in the simulation are presented in

Table 1. Temperature differences Δt and Δ_{T_meas} obtained in the simulation and experimental tests for the ITT with the voltage divider.

Tset (°C)	ΔT (°C) (Simulation)	ΔT_meas (°C) (Measurement)
0	0.0028	−0.00066
25	0.0032	0.0056
75	−0.00087	0.0035
130	−0.0012	0.0014
180	0.0032	0.0048
200	0.000057	0.0042

. The obtained results of the absolute values of Δ_T were less than 0.004 °C. More detailed analyses of simulation test results showed that the influence of the voltmeter on the obtained uncertainty result U_{T_high_MC_max} = 0.02 °C was negligibly small. The component of the base error of the voltmeter Δ₁ = 0.3 µV, which was not minimalized as a result of the auto-calibration procedure, had a negligibly small influence in relation to the influences δ_RStj = ±0.002% and Δ_{l_Stj,S} = 0.01 Ω. Only an increase in the value of Δ₁ to 7 µV (7 ppm in relation to the 1 V range) caused an increase in the uncertainty of the T measurement to U_{T_high_MC_max} = 0.021 °C. It can therefore be assumed that for the ITT presented in Figure 1, the total influence of linearity, quantization, and noise errors of the A/D converter should not exceed 6 ppm if we want to keep the expanded measurement uncertainty at the level of 0.02 °C. These errors varied over the entire A/D converter range, so they were additive in nature (they added to the voltage V₁₃), similarly as in the case of Δ₁ of the voltmeter.

Experimental studies were conducted to verify the results of the simulation studies of the measurement system presented in Section 3.1. In the experimental tests, 10 temperature measurements were performed for each set standard temperature T_set using the developed measurement system. Based on these measurements, the average T_meas value and the standard uncertainty u_{T_meas} were determined using the formulas

$$T_{meas}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{T^{\prime }}_i},$$

(18)

$$u_{T\_meas}=\sqrt{{\displaystyle \frac{1}{N\left(N-1\right)}}{\displaystyle \sum _{i=1}^N{\left({T^{\prime }}_i-T_{meas}\right)}^2}},$$

(19)

where N = 10 is the number of measurements and T_i’ is the ith temperature measurement result.

Then the error of temperature measurement was determined,

$${\Delta }_{T\_meas}=T_{meas}-T_{set}$$

(20)

and the expanded uncertainty interval at a 99% confidence level in which the true value of the measured temperature was located. The limits of this interval (endpoints) were determined based on the Guide [45], from the formulas

$$U_{T\_high\_meas}={\Delta }_{T\_meas}+\sqrt{3}\cdot 0.99\sqrt{u_{T\_meas}^2+{\left({\displaystyle \frac{{\Delta }_{T\_set}}{\sqrt{3}}}\right)}^2},$$

(21)

$$U_{T\_low\_meas}={\Delta }_{T\_meas}-\sqrt{3}\cdot 0.99\sqrt{u_{T\_meas}^2+{\left({\displaystyle \frac{{\Delta }_{T\_set}}{\sqrt{3}}}\right)}^2},$$

(22)

where Δ_{T_set} is the error of the standard temperature setting, and $\sqrt{3}$·0.99 is the coverage factor for the uncertainty Δ_{T_set}/$\sqrt{3}$ with a rectangular distribution, which dominates in the measurement.

When performing measurements, the standard temperatures T_set were set by connecting standard resistors instead of the temperature sensor. The standard resistors represented the R_S resistances, thereby eliminating the influence of RTD inaccuracies on ITT studies. Before being connected to the measurement circuit, the resistance of each standard resistor was measured with four wires using an Agilent 3458A multimeter to obtain the highest possible accuracy of the T_set setting calculated from Formula (7). Based on the measurement error of the resistance of the resistors representing the resistance of the R_S sensor, which was ±0.002%, the temperature error Δ_{T_set} was estimated at ±0.01 °C.

The results of the calculated measurement uncertainties are presented in Figure 11. The experimentally obtained limit values in the entire range of measured temperatures did not exceed the values U_{T_low_meas_min} = −0.012 °C and U_{T_high_meas_max} = 0.016 °C. These were the values that did not exceed the limit values obtained by simulation. This indicated a high agreement between the simulation and the experimental results. It is worth noting that the error Δ_{T_set} = 0.01 °C had a large impact on the measurement uncertainty results (U_{T_low_meas}, U_{T_high_meas}). If this component was not considered in the measurement uncertainty calculations ((21), (22)), then the obtained uncertainty values (U_{T_low_meas}^*, U_{T_high_meas}^*) were much smaller than the uncertainties determined by simulation (Figure 11) in the entire range of the measured temperature.

This may mean that the real measurement inaccuracy is smaller than that resulting from the limit errors of the measurement uncertainty sources. This is also an indication to the authors that in further research aimed at increasing the accuracy of temperature measurement, it will be necessary to use temperature standards with even greater accuracy than 0.01 °C. The effectiveness of the applied auto-calibration procedure was confirmed by the summary of the results of temperature differences Δ_T and Δ_{T_meas} obtained as a result of simulation and experimental tests in Table 1. The differences in the results were at the level of thousandths of °C.

4. Conclusions

The presented results of the simulation and experimental studies showed that the auto-calibration procedure presented in [38] could also be used in a temperature transducer in which the R_S/V circuit was in the form of a resistive divider. The advantage of the new solution was primarily the higher accuracy of temperature measurement compared with a transducer with a Wheatstone bridge [13] (Appendix A). Other advantages of the new transducer were the simpler structure of the R_S/V circuit and the possibility of using a lower degree polynomial J = 3 in the auto-calibration procedure compared with the transducer with a bridge in which J = 4 was adopted. Consequently, this translated into the possibility of reducing the number of standard resistors in the ITT and easier software implementation of the procedure due to the smaller number of required coefficients in Equation (3).

The obtained simulation test results showed that the new ITT could achieve a T measurement accuracy of 0.0033 °C under nominal conditions. It was resistant to the influence of RTD and R_S/V₁₃ circuit nonlinearity, zero and gain errors, and supply voltage changes. The T measurement accuracy was influenced by the parameters of standard resistors, resistance scatter of connecting lines (R_{l_S}, R_{l_Stj}), resolution and nonlinearity of the A/D converter, and noise in the ITT.

The obtained results of the simulation and experimental tests of the developed measurement system, performing ITT tasks, showed that the new ITT enabled the processing of the Pt1000 sensor resistance with an accuracy of 0.02 °C. This could be achieved by using the following in the ITT:

• Four standard resistors (R_Stj) with an accuracy of ±0.002% and TCR_RStj = 5 ppm/°C along with a sensor measuring the ambient temperature with an error of no more than ±1 °C;
• A connection circuit in which the line resistances fulfill the condition R_{l_S} = R_{l_Stj} with an error of no more than ±0.01 Ω;
• A circuit for converting analog voltage V₁₃ to a digital form with a total error (resulting from noise, quantization, and nonlinearity) no greater than 6 ppm.

• It should be mentioned that the use of standard resistors with an accuracy of ±0.002% in the ITT does not require the purchase of expensive resistors with a tolerance of ±0.002%. As presented in the work, these may be resistors with a much smaller tolerance, the resistances of which should be measured with an accuracy of ±0.002%. The 3458A multimeter used for this purpose makes it possible to obtain this accuracy during its two-year calibration cycle. More frequent calibration allows for even higher accuracies, e.g., ±0.00125% for 1 year.

The developed ITT makes it possible to perform accurate T measurements using two- and three-wire platinum RTDs. Achieving high temperature measurement accuracy is possible by minimizing the influence of lead wire resistance and nonlinearity of the RTD and also nonlinearity, offset, gain error, supply voltage changes, and ambient temperature of the ITT on the temperature measurement result. The obtained test results indicate that the ITT can cooperate with the RTD about accuracy corresponding to class AA [4].

The research results presented in the paper indicate that it is possible to achieve ITT accuracy below 0.02 °C, which is required, for example, in calibration laboratories. This means, above all, solving in the future the problem of being able to reproduce standard temperatures with higher accuracy. Moreover, further research is required to develop an R_S/V circuit in which the spread of the resistance values of the R_{l_S} and R_{l_Stj} connection lines would be smaller than ±0.01 Ω.