The Precise Temperature Measurement System with Compensation of Measuring Cable Inﬂuence

: The article presents an active bridge system that enables the solution of a signiﬁcant problem consisting in ensuring correct indications of temperature values in a wide measuring range for a Pt100 temperature sensor with properties deﬁned by the standard (EN-60751 + A2). The presented active bridge system combines the properties of the measuring ampliﬁer with the stabilization of the current value in the branch in which the Pt100 sensor was placed. The article focuses on the comparison of the temperature measurement in a typical resistance bridge and the measurement made in the developed active bridge, which has also become the subject of a patent. For the performed tests, in which the correctness of the temperature measurement system operation was veriﬁed, and on the basis of the obtained results, the quality of temperature measurements was compared in a wide range of changes.

This article describes an accurate thermometer measurement system that includes resistive temperature detectors RTDs, which are temperature sensors that contain a resistor. The resistor used in an RTD changes its resistance value as the temperature changes. The most common example of an RTD detector are Pt100 series temperature sensors, which have been used for many years to measure temperature in laboratory and industrial processes. The Pt100 is one of the most accurate temperature sensors. It does not only provide good accuracy but also excellent stability and repeatability of the measured values. The Pt100 sensors are also relatively resistant to electrical interference and are therefore well suited for temperature measurements in industrial environments, especially around motors, generators, and other high voltage equipment [14][15][16][17].
The most RTD temperature detectors consist of a thin and coiled wire that is wrapped around a ceramic or glass core. There are also designs of the sensor in the form of platinum sputtered onto a ceramic substrate. The sensor is usually fragile, so it is often placed in a probe sheath. The RTD sensor is made of a specific material, which is the most important element of this sensor in this case.
The performance of the sensor depends on the material it is made of. Most RTD sensors are made of platinum or nickel. The temperature characteristics of these materials precisely documented, which makes it possible to precisely determine the tempera based on them. There are two standards for Pt100 thermoresistors, one of which is European standard, also known as the standard DIN (Deutsches Institut für Norm Berlin, Germany) or the IEC standard (International Electrotechnical Commission, neva, Switzerland) and the American standard ASTM (American Society for Testing Materials, West Conshehocken, PA, USA). The European standard is considered world standard for platinum RTD sensors. DIN/IEC 60,751 (or simply IEC751), requ the RTD to have the electrical resistance of 100.00 Ω at 0 °C and the temperature coeffi of resistance (TCR) of 0.00385 O/O/°C over a temperature range from 0 °C to 100 °C [ The paper presents a method of precise evaluation of a real temperature value large range of temperature TC = 0 ÷ 850 °C using a thermoresistor Pt100. A special a measuring bridge was developed as well as the method of connecting a thermoresi effectively eliminating the influence of the wire's resistance.
An important problem that occurs during temperature measurement using a r tive temperature sensor is the need to limit the influence of the resistance of the wire which such a sensor is connected to the measuring system, and which is the cause o roneous temperature determination by such a sensor. Basically, three approaches are sidered [8], i.e., using two wires [9,10], three wires [11], and the third approach, using wires [13]. It should be mentioned that the described measurement system uses the c pensation of the resistance of the lead wires by the three-wire method [16], which improved and adapted to the system described below in this paper.
The paper presents a method of accurate determination of the actual tempera value of a Pt100 thermoresistor in a wide temperature measurement range TC = 0 ÷ 85 using an active measuring bridge developed for this purpose and taking into accoun compensation of the influence of the wires connecting the sensor. The contents of th ticle are based on the patent descriptions [16,17], of which the author is a co-author.

Active Measuring Bridge with Current Stabilization
In most applications, the Pt100 thermoresistor is used in a passive measuring br system, which is made of matched resistors, as shown in Figure 1 [6,7,14]. The circuit in Figure 1 is a passive bridge for determining the resistance value contains a thermoresistor, represented by resistance Rt, and resistors RB and RA, which placed in the second branch of the bridge. Resistor RX is connected (in the first branc The circuit in Figure 1 is a passive bridge for determining the resistance value R t . It contains a thermoresistor, represented by resistance R t , and resistors R B and R A , which are placed in the second branch of the bridge. Resistor R X is connected (in the first branch of the bridge) in series with the resistor R t . For a specific value of the temperature T C expressed in [ • C], it is possible, by changing the value of the resistance R X , to bring the voltage U A = U B into equilibrium. Then the condition according to Equation (1) A change in temperature causes a change in the resistance value R t , which introduces ( Figure 1) a change in the voltage value U B and it affects the value of the deviation ∆U (A−B) . The change of the U B value will cause the nonzero ∆U (A−B) value. Another adjustment of the R X value will cause the voltage deviation ∆U (A−B) to approach zero.
The following analysis of the values of voltages U B and U A for the passive bridge shown in Figure 1 is carried out in order to obtain a relationship, from which the temperature of a Pt100 thermoresistor can be easily calculated. The voltage drop U B is expressed by the formula: where U Z -the value of the bridge supply voltage. Whereas the voltage drop U A is given by the relation: If R A = R B , then α in (3) is equal to: α = 1 /2 .
Since the voltage difference ∆U (A−B) is calculated from Equation (1). Then after substituting the relations (2) and (3) into this formula one can obtain: After ordering the right-hand side of Equation (5), the relation for the voltage difference ∆U (A−B) (6) is obtained. This relation is crucial in determining the temperature.
In the circuit of Figure 1, the measurement and calculation of the resistance value R t is performed when the principle of voltages' equality U A = U B is applied. For the case when U A = U B , the resistance value R t corresponds to the resistance value R X . On the basis of the resistance value R t and the knowledge of the temperature characteristics of platinum, the temperature value T C can be calculated based on the formula obtained by solving the quadratic equation included in the European standard EN-60751 + A2 or by using the calculated temperature values, which, in tabulated form, are included in that standard [14][15][16][17].
In the European standard (EN-60751 + A2), the change of a platinum thermoresistor Pt100 resistance value (7) was described analytically as the value dependent on the temperature value in the range from 0 to 850 • C.
Rt-the resistance value of the thermoresistor at temperature T C ( • C), R 0 -the resistance value of the thermoresistor at temperature T C = 0 • C, according to the standard (EN-60751 + A2), R 0 = 100 Ω (or 500 Ω, and also 1000 Ω), A, B-constants connected with the material property of platinum Pt according to the standard (EN-60751 + A2), By solving the quadratic Equation (7) the relation according to Equation (8) is obtained [15]. It determines the temperature value T C of the thermoresistor depending on the resistance value R t ; In the Formula (8), the fragment described by the relation (9) is important; The calculation of the temperature value directly according to Formula (8) introduces a great simplification, whereas if one introduces into Formula (8) the voltage value U P resulting from the measurement of this voltage in the bridge system, a new relation is obtained equal to the product of the proportionality coefficient (S K ) and the measurement voltage (U P ) (10); An important disadvantage of the passive bridge presented in Figure 1 is the fact that the output voltage ∆U (A−B) according to Formula (6) does not represent the relations given in (9) and (10). This passive bridge limitation evoked considerations about an active resistance bridge using an operational amplifier presented in Figure 2 [10,13]. This solution permits stabilizing current I = I const flowing through the thermoresistor R t .
By solving the quadratic Equation (7) the relation according to Equation (8) is obtained [15]. It determines the temperature value TC of the thermoresistor depending on the resistance value Rt; In the Formula (8), the fragment described by the relation (9) is important; The calculation of the temperature value directly according to Formula (8) introduces a great simplification, whereas if one introduces into Formula (8) the voltage value UP resulting from the measurement of this voltage in the bridge system, a new relation is obtained equal to the product of the proportionality coefficient (SK) and the measurement voltage (UP) (10); An important disadvantage of the passive bridge presented in Figure 1 is the fact that the output voltage ΔU(A−B) according to Formula (6) does not represent the relations given in (9) and (10). This passive bridge limitation evoked considerations about an active resistance bridge using an operational amplifier presented in Figure 2 [10,13]. This solution permits stabilizing current I = Iconst flowing through the thermoresistor Rt. Operational amplifiers have been known for many years in electronics, and due to their well-known properties, they are still used in many applications [16][17][18][19][20][21]. The amplifier marked as WOx in Figure 2 acts as a current source, whose task is to stabilize the current in the bridge branch, including the resistor Rt. On the other hand, the instrumental amplifier marked as WI acts as a scaling circuit for the output voltage representing the measured temperature [13,14].
Similarly as in the case of the passive bridge (Figure 1), the analysis of the system in Figure 2 is carried out below. Its aim is to derive the relation, which allows calculating the relation given by the Equation (1).
At the beginning of the analysis, voltage drops UA and UB were determined according to the scheme presented in Figure 2: Operational amplifiers have been known for many years in electronics, and due to their well-known properties, they are still used in many applications [16][17][18][19][20][21]. The amplifier marked as WOx in Figure 2 acts as a current source, whose task is to stabilize the current in the bridge branch, including the resistor Rt. On the other hand, the instrumental amplifier marked as WI acts as a scaling circuit for the output voltage representing the measured temperature [13,14].
Similarly as in the case of the passive bridge (Figure 1), the analysis of the system in Figure 2 is carried out below. Its aim is to derive the relation, which allows calculating the relation given by the Equation (1).
At the beginning of the analysis, voltage drops U A and U B were determined according to the scheme presented in Figure 2: where r p -resistance of the wire used to connect the thermoresistor. If in the Formula (11) R A = R B , then α = 1 /2. The relation ∆U (A−B) for an active measuring bridge is equal to: After reducing it to a common denominator and applying the substitution α = 1 /2, we obtain: For the purpose of stabilizing the current value I in the bridge branch, in which the thermoresistor R t is placed, Formula (15) expresses the dependence of the current value I on the voltage U REF .
Inserting into Equation (14) the expression (17), one can obtain the form: Or where K WI -the gain value of the WI amplifier, U REF -the reference voltage for the adopted measurement range.
In particular, it is important that the relation (9), resulting from the equation solution given in EN-60751 + A2 is equal: On the other hand, if we take the resistor value R X to be the resistor value R 0 from Formula (9), then Formula (20) will take the form (21).
From the analysis presented above, it follows that the structure of the active measuring bridge ( Figure 2) with the stabilization of the current value in the bridge branch, in which the thermoresistor R t is placed, provides the output voltage U out in the form convenient for the substitution into Formula (8).
Comparing Formula (9) with Formula (21), Formula (8) can be written in the form (22) or in the form (23): or where S K -denotes the calibration constant equal to:

Selection of the Partition Coefficient α and the Resistance R 0
In Formulas (3) and (11), the division coefficient α was introduced. Theoretically, in the bridges under consideration, it is possible to assume any resistance value in the range 100 ÷ 1000 Ω, which meets the condition R A = R B = R, while from the point of view of the resistance measurement R t it is more important to be able to tune the values of these resistors, e.g., by laser correction. In practical applications, it is possible to take the resistance value R A and then match the resistance value R B to it very precisely. You can also do the opposite, i.e., take the value of R B resistance and then match the value of the resistance R A to it. The absolute resistance value of R A and R B is not important in this case, but the exact value of the ratio of these resistances is very important and must be equal to α = 1 /2. An accurate value of α contributes to effective influence elimination of the wires resistance r p connecting the Pt100 sensor to the measuring amplifier system ( Figure 2).
Another issue is the selection of the resistor value R 0 (R X ). If the temperature measuring range is to start from T C = 0 • C, the resistor value R 0 must be R 0 = 100.00 Ω. Selecting a different resistor value R 0 allows you to set a different reference point for the initial value of the measured temperature. When selecting the resistor value R 0 , which is not an easy task, one can try to use highly stable MFR resistors with a small coefficient of the resistance change versus temperature. This type of resistors are difficult to obtain and very expensive. Therefore, a different solution was used in the design of the bridge measuring the amplifier. The resistor R 0 is a more complex design. The resistor value R X was initially assumed to be a parallel combination of two MRF 1% (25 ppm) resistors with the respective values of 150 Ω and 330 Ω.
The resultant resistance value R X is R X ≈ 150 Ω//330 Ω ≈ 103 Ω. Then an additional resistor R 1 with a potentiometer P 1 connected in series was connected in parallel to R X . By changing the value of the potentiometer P 1 one can vary the resultant value of R 0 around the value R 0 = 100 Ω ±1 Ω. By setting the adjustable resistor decade to R t = 100.00 Ω it is possible to obtain a voltage value of ∆U (A−B) = 0.00 V (measured with the use of a 6 1 /2-digit voltmeter). Moreover, the potentiometer P 1 makes it possible to change the sign of the voltage value ∆U (A−B) . In this way, the resistance value R 0 can be determined precisely.
The value of the current I 0 flowing through the thermoresistor Pt100 depends on the type of the environment, in which the measuring sensor is located. It is connected with the phenomenon of self-heating of the thermoresistor R t , because the electrical power P[W] is emitted on the thermoresistor and converted into heat: When the temperature value t [ • C] is measured in an air (gas) environment, the heat transfer between the environment and the thermoresistor R t (heated due to I 0 current flow) is weak. In an aqueous environment (liquids), the heat transfer from the heated sensor is more intensive. Therefore, lower values of I 0 current are assumed for the measurements in a gaseous environment. In the liquid environment, the value of I 0 current may be higher. In the project, it was initially assumed (for the analysis) that the measurement of temperature t ( • C) takes place in the water environment. (e.g., in district heating systems). For this case, the safe current value I 0 is I 0 = 7 mA. Figure 3 presents the sensor Pt100 and two wires connecting its terminals (x1 i x2) to wire terminals denoted as z1 i z2, which were connected to the operational amplifier input. The measurement system using the thermoresistor Pt100 generates two following problems:

•
The resistance r p of connecting wires is never precisely known; • The environment temperature influences a change of the resistance r p value. input. The measurement system using the thermoresistor Pt100 gene problems: • The resistance rp of connecting wires is never precisely known; • The environment temperature influences a change of the resista  In the bridge circuit of the amplifier (Figure 4) between terminals of the measuring resistance defined by Formula (26) must also be tak With the long connection wires between the sensor location bridge, the resistance value rp starts to be comparable to the resistanc   Figure 3 presents the sensor Pt100 and two wires connecting its terminals (x1 i x2) to wire terminals denoted as z1 i z2, which were connected to the operational amplifier input. The measurement system using the thermoresistor Pt100 generates two following problems:

Problem of Connecting a Pt100 Sensor to a Measuring Amplifier
• The resistance rp of connecting wires is never precisely known; • The environment temperature influences a change of the resistance rp value.   In the bridge circuit of the amplifier (Figure 4) between terminals z1 and z2, the value of the measuring resistance defined by Formula (26) must also be taken into account [22].
With the long connection wires between the sensor location and the measuring bridge, the resistance value rp starts to be comparable to the resistance value of the Pt100 In the bridge circuit of the amplifier (Figure 4) between terminals z1 and z2, the value of the measuring resistance defined by Formula (26) must also be taken into account [22].
With the long connection wires between the sensor location and the measuring bridge, the resistance value r p starts to be comparable to the resistance value of the Pt100 sensor. As a reminder, the resistance value of the Pt100 sensor is R Pt100 ≈ 100.00 Ω. Figures 5 and 6 show the 3-wire method of connecting the Pt100 sensor to the measuring amplifier. The applied 3-wire connection system eliminates the influence of changes in the resistance value r p on the measurement result of the resistance value of the Pt100 sensor R t . Figure 7 shows the method of connecting the Pt100 sensor in a measuring amplifier circuit with the structure of a resistive measuring bridge. The resistance r p of the connection wire between the z2 and x2 terminals was connected to the bridge branch, in which the Pt100 measuring thermoresistor was located. The resistance r p of the connection wire between the z1 and x1 terminals was connected to the bridge branch, in which the reference resistor R X = R 0 was located. This method of connection effectively eliminates the influence of the wires resistance r p . This property is due to the fact that the resistance of the connection wires r p in this connection adds simultaneously to the resistance of the thermoresistor (R t + r p ) and the reference resistance (R 0 + r p ) in the corresponding branches of the measuring bridge. On the other hand, such a connection does not disturb the balance of the measuring bridge because the changes in the resistance r p value (also due to temperature changes) take place simultaneously in both branches of the measuring bridge included in the measuring amplifier circuit (Figure 2). sensor. As a reminder, the resistance value of the Pt100 sensor is RPt100 ≈ 100.00 Ω. Figures  5 and 6 show the 3-wire method of connecting the Pt100 sensor to the measuring amplifier. The applied 3-wire connection system eliminates the influence of changes in the resistance value rp on the measurement result of the resistance value of the Pt100 sensor Rt. Figure 5. The 3-wire circuit to connect the Pt100 sensor takes into account the wire's resistance rp between the x1, x2 and z1 ÷ z3 terminals incorrect connection of the Pt100 sensor to the measuring bridge system.  Figure 7 shows the method of connecting the Pt100 sensor in a measuring amplifier circuit with the structure of a resistive measuring bridge. The resistance rp of the connection wire between the z2 and x2 terminals was connected to the bridge branch, in which the Pt100 measuring thermoresistor was located. The resistance rp of the connection wire between the z1 and x1 terminals was connected to the bridge branch, in which the reference resistor RX = R0 was located. This method of connection effectively eliminates the influence of the wires resistance rp. This property is due to the fact that the resistance of the connection wires rp in this connection adds simultaneously to the resistance of the thermoresistor (Rt + rp) and the reference resistance (R0 + rp) in the corresponding branches of the measuring bridge. On the other hand, such a connection does not disturb the balance of the measuring bridge because the changes in the resistance rp value (also due to temperature changes) take place simultaneously in both branches of the measuring bridge included in the measuring amplifier circuit (Figure 2). Figure 5. The 3-wire circuit to connect the Pt100 sensor takes into account the wire's resistance r p between the x1, x2 and z1 ÷ z3 terminals incorrect connection of the Pt100 sensor to the measuring bridge system.
The applied 3-wire connection system eliminates the influence of changes in the resis value rp on the measurement result of the resistance value of the Pt100 sensor Rt. Figure 5. The 3-wire circuit to connect the Pt100 sensor takes into account the wire's resist between the x1, x2 and z1 ÷ z3 terminals incorrect connection of the Pt100 sensor to the mea bridge system.  Figure 7 shows the method of connecting the Pt100 sensor in a measuring am circuit with the structure of a resistive measuring bridge. The resistance rp o connection wire between the z2 and x2 terminals was connected to the bridge bran which the Pt100 measuring thermoresistor was located. The resistance rp of the conn wire between the z1 and x1 terminals was connected to the bridge branch, in whic reference resistor RX = R0 was located. This method of connection effectively elim the influence of the wires resistance rp. This property is due to the fact that the resis of the connection wires rp in this connection adds simultaneously to the resistance thermoresistor (Rt + rp) and the reference resistance (R0 + rp) in the corresponding bra of the measuring bridge. On the other hand, such a connection does not distur balance of the measuring bridge because the changes in the resistance rp value (als to temperature changes) take place simultaneously in both branches of the meas bridge included in the measuring amplifier circuit (Figure 2).   The resistance rp of the connection cable between the terminals z3 and x1 serves to transfer the potential of the x1 point to the z3 terminal of the measuring bridge. The use of the terminal z3 requires the use of a bridge measuring amplifier WI with a high value of the input resistance Rin. The value of the input resistance Rin of the instrumental amplifier WI (Figure 8) cannot cause a significant voltage value drop on the resistance rp between the terminal x2 and z3. In the actual circuit of the instrumental amplifier WI, the resistance value Rin is very high. Therefore, the problem raised here does not exist in it. The resistance r p of the connection cable between the terminals z3 and x1 serves to transfer the potential of the x1 point to the z3 terminal of the measuring bridge. The use of the terminal z3 requires the use of a bridge measuring amplifier WI with a high value of the input resistance R in . The value of the input resistance R in of the instrumental amplifier WI (Figure 8) cannot cause a significant voltage value drop on the resistance r p between Energies 2021, 14, 8214 9 of 17 the terminal x2 and z3. In the actual circuit of the instrumental amplifier WI, the resistance value R in is very high. Therefore, the problem raised here does not exist in it. The resistance rp of the connection cable between the terminals z3 and x1 serves to transfer the potential of the x1 point to the z3 terminal of the measuring bridge. The use of the terminal z3 requires the use of a bridge measuring amplifier WI with a high value of the input resistance Rin. The value of the input resistance Rin of the instrumental amplifier WI (Figure 8) cannot cause a significant voltage value drop on the resistance rp between the terminal x2 and z3. In the actual circuit of the instrumental amplifier WI, the resistance value Rin is very high. Therefore, the problem raised here does not exist in it.

Implementation of the Active Measurement Bridge on the LM723 IC
The LM723 integrated circuit and the WI instrumental amplifier were used to build the measurement amplifier. The LM723 integrated circuit is a voltage stabilizer containing in its structure (Figure 9) all components appearing in Figure 2, i.e.,:

Implementation of the Active Measurement Bridge on the LM723 IC
The LM723 integrated circuit and the WI instrumental amplifier were used to build the measurement amplifier. The LM723 integrated circuit is a voltage stabilizer containing in its structure (Figure 9) all components appearing in Figure 2 The output amplifier WOx has short-circuit protection which can occur between the wires connecting the Pt100 sensor and the measuring amplifier. The value of limiting current I ZW is set by the resistor value R ZW ; • The WOx amplifier is equipped with compensation (capacitor C8), which effectively eliminates the excitation possibility of the WOx amplifier. Figure 9 shows a schematic diagram of an active measuring bridge with I value stabilization in the branch, where a Pt100 thermoresistor is placed. An important limitation arising from the use of the LM723 integrated circuit is that it is adapted to be supplied only with a single positive voltage. Therefore, the voltage value on the pins number 4 and 5 of this I C must be higher than 2 V.
The design of the LM723 chip allows the use of short-circuit protection in the form of a current limiter. A short circuit occurs when the failure of the insulation occurs. It happens when the wires connecting the Pt100 sensor to the measuring bridge fell in damage. Such a case is not so rare when there is a large distance between the place of installation of the Pt100 sensor, e.g., on the pipeline, and the place of installation of the measuring amplifier.    As soon as the electronic modules have been completed, the entire measurement path should be calibrated. The calibration procedure should be as follows:  As soon as the electronic modules have been completed, the entire measurement path should be calibrated. The calibration procedure should be as follows: potentiometer ( Figure 4) should be set in the position K WI ensuring the gain required for a given measurement range; (e) Set the resistor value R t (calibration resistor decade) R t = 313.71 Ω, which corresponds to the temperature value T C_MAX = 850 • C; (f) On the output of the measuring bridge amplifier set the value of the output voltage U out = 5.00 V; the fine adjustment is made with the potentiometer P 3 ; (g) Repeat the adjustment process from pt. 3 ÷ 5; (h) Enter into the calculation program the value of the calibration constant S K given by the Formula (24).
It is obvious that the exact value of the calibration constant S K is never known. This is so because the actual platinum used to build the Pt100 thermoresistor may have a different value of the calibration constants A and B (1). Therefore, the tuning of the calibration constant S K should be performed as follows: • Using the calculation program based on Formula (2) it is necessary to achieve the indication value of the temperature T C = 850 • C. This value can be reached by entering the pre-calculated value of the calibration constant S K . For example, for the temperature value T C = 850 • C and the voltage U out = 5 V the value of the calibration constant S K will be S K = −1,005,991.341991(341991); • Then, by correcting the value of the calibration constant S K , the value of T C = 850.0 • C on the digital indicator should be obtained; • Decreasing (e.g., using the calibration decade) the values of the resistor settings R t according to the values in Table 1; check the correctness of the temperature indication T C ( • C) calculated by the program. The temperature indication value must correspond to the values given in Table 1.  For other ranges of the maximum measured temperature values T C_MAX ( • C), different values of the calibration constant S K must be adopted. The calibration constant always refers to the maximum voltage value U out = 5 V, which is present at the output of the measuring amplifier. The analog-to-digital converter (A/D converter) used in the project can process the analog input voltage of the maximum value U IN−A/D = 5 V. Therefore, the gain value K WI of the measurement amplifier (representing the resistance/voltage converter-R/U) is regulated in such a way that at its output (for the R t value resulting from the maximum temperature value) the voltage value U out = 5 V is obtained.
Formula (28) is the universal calculation formula of the calibration constant value S K . Formula (28) assumes the value of B, R 0 , and U out = 5 V. For the maximum temperature value t MAX ( • C), the corresponding resistance value R t is assumed according to the calculation results. Table 2 shows the calculated values of the calibration constant S K , referring to the maximum value of the measuring temperature range t MAX ( • C), at which at the measuring amplifier output there will be the voltage of the value U out = 5 V.  Figure 11 shows a diagram illustrating the operation of the calculation program that was used in the study. In order to check the correctness of the calculations carried out by the program, an analysis of the algorithm enabling the temperature defining was carried out. The Formulas (44 ÷ 51) present the analysis carried out.

Calculation Software
For the output voltage value U out = 5 V, the temperature value t MAX ( • C) will be: For the assumed temperature range T C = 0 ÷ 850 • C the value of the calibration constant S K can be calculated according to Formula (28). Then, based on Formula (37), the intermediate values of the output voltages U out should be estimated.
By applying the voltage within the range U out = 0 ÷ 5 V to the input of the A/D converter, it is possible to control the correct intermediate indications of the T C temperature values. For example, for calibration of the calculating system indications, the following relation was assumed:   Figure 12 shows the change in the resistance value R t (Ω) = f(T C ) for a temperature value change using the linear and quadratic approximation. As the T C temperature value increases, the linear approximation will cause an overestimation of the resistance R t value. The EN-60751 + A2 standard introduces the possibility of presenting the characteristics of the Pt100 thermoresistor using the linear approximation [14][15][16][17]. Formula (38) defines the coefficient α. where: • R 100 -the resistance value of the thermoresistor Pt100 at temperature T C = 100 • C, R 100 = 138.51 Ω, • R 0 -the resistance value of the thermoresistor Pt100 at temperature T C = 0 • C, R 0 = 100.00 Ω.
The value of α according to the definition given by Formula (38): After the transformation of Formula (39), the resistance value R t is determined for the range of the temperature value changes T C = 0 ÷ 100 • C. Formula (39) should be applied for the temperature measurement T C_MAX = 100 • C. This is the so-called linear approximation. For the temperature measurements T C_MAX ≥ 100 • C, even up to 850 • C, Formula (7) should be used-the so-called quadratic approximation [14][15][16][17]. Figure 13 shows the course of the temperature indication difference ∆T C for the linear and quadratic approximation.

TEMPERATURE INDICATION ERROR ΔT C = f ( T C [ 0 C ] )
Δt = linear apporoximtion -quadratic apporoximtion Figure 13. Distribution of the temperature indication error ∆t for the linear and quadratic approximation.

Conclusions
The material offered in the paper presents a ready to use, measurement circuit of the temperature meter based on Pt100 transducer. The novelty of the solution is proved by the patent PL226444B1 [16,17]. Really, passive and active bridges are well known but are still useful and are not out of the question. Authors suppose that using such a circuit, in order to design a precise and relatively cheap measuring device is probably an optimal decision. They are right that stabilizing the bias current in the device is also known, but it was mandatory to use such a solution in order to obtain the highest accuracy.
The article presents an active bridge system, which makes it possible to solve an important problem consisting in ensuring the correct indications of temperature values in a wide measuring range T C = 0 ÷ 850 • C. The temperature sensor (thermoresistor) Pt100 used in the system has the properties defined by the standard (EN-60751 + A2). The presented active bridge system integrates the properties of a bridge measuring amplifier with stabilization of the current value in the branch, in which the Pt100 sensor is placed.
According to the standard EN-60751 + A2 the Pt100 resistance temperature dependence has a form of a quadratic function. Solving this function, one can receive the relation between temperature value T C and the resistance value R t . The root of the equation has a form (8). In this Formula, one can find a very characteristic term (9). Since the output voltage value U P is proportional to this term (10), then the temperature Tc is directly related to the measurement voltage U P . A real advantage of the proposed solution is no additional components to correct the temperature value, and it concerns the total measurement range (T C = 0 ÷ 850 • C) of Pt100.
The paper focuses on the comparison of the temperature measurement in a typical resistive bridge and in the developed active bridge. For the performed tests the correctness of the temperature measuring system using the Pt100 thermoresistor was verified. The highest quality of the measurements in a wide temperature range was analyzed and confirmed the gathered results.