DE102018216131B4 - Method and device for the simultaneous determination of the change in temperature and resistance of sensor resistances of a bridge circuit designed as a quarter or half bridge - Google Patents

Abstract

In a method and a device for the simultaneous determination of the change in temperature and resistance of one or more sensor resistances of a bridge circuit designed as a quarter or half bridge, at least two of the comparison resistors with different temperature coefficients α, γ are selected. Different voltages are measured in the bridge circuit, of which at least one voltage drops across one of the resistors of the bridge circuit and which are selected such that the voltage across the bridge circuit, a voltage across the or one of the sensor resistors, and from one or more of the different voltages a voltage can be determined across a parallel comparison resistor. From the measured voltages, known resistance values of the resistors and known temperature coefficients α, β, γ of the resistors of the bridge circuit, the temperature change ΔT and therefrom the change in resistance dR is calculated or vice versa. The process does not require an additional temperature sensor and enables, among other things. a more precise determination of the measured variable.

Description

Technical application area

The present invention relates to a method and a device for determining the change in temperature and resistance of one or more sensor resistors of a bridge circuit, which is fed by a voltage source with a constant operating voltage or by a constant current source and is designed as a quarter or half bridge with several resistors.

The Wheatstone bridge circuit is used in many applications to determine measured variables such as pressure, force or magnetic fields. Depending on the measured variable, different sensor resistances are used in the bridge circuit, e.g. magnetoresistive sensors, e.g. AMR or GMR sensors, for the determination of magnetic fields. The bridge circuit can be used as a quarter bridge with only one sensor resistor, as a half bridge with two sensor resistors or as a full bridge with four sensor resistors. As a rule, a voltage proportional to the change in the resistance of the transmission elements (change in resistance dR) is output. This also enables zero adjustment (offset adjustment), so that the output voltage of the bridge is 0 for a reference state.

The sensor resistors, also referred to as sensor elements, and the comparison resistors which are also present, depending on the design of the bridge circuit, show a temperature dependency which can be described with a temperature coefficient. This temperature coefficient is often assumed to be constant and thus results in an approximately linear temperature behavior of the resistors. If the temperature behavior of the sensor resistors differs from the temperature behavior of the comparison resistors, this can have a negative effect on the output voltage of the bridge circuit and lead to incorrect measurement signals. Furthermore, in the case of the sensor resistors, the change in the resistance dR is not only dependent on the variable to be measured, but also on the temperature. Here, too, an almost linear behavior with a constant temperature coefficient can be specified in some cases.

Overall, when measuring with bridge circuits, this results in an output voltage that depends not only on the size to be measured, but also on the temperature. Compensation of the temperature influence when measuring with a bridge circuit is therefore essential for accurate measurement results.

State of the art

From the DE 10 2004 026 460 A1 A temperature-compensated measuring circuit arrangement with a magnetoresistive sensor bridge and an adjustment method for adjusting its temperature compensation are known. In this circuit arrangement, an amplifier output is connected via a compensation conductor to a shunt resistor at which a measurement voltage drops. The compensation conductor is arranged on the sensor bridge in such a way that it generates a compensation field opposite to a measuring field. The sensor bridge is powered by a power source.

In the EP 0 565 124 A1 describes a method and a circuit arrangement for the electrical compensation of the temperature influence on the measurement signal from mechanoelectric measurement transducers. Here, the temperature-dependent change in the resistance value of a transducer working with piezoresistive resistors in a bridge circuit is used to obtain a temperature-proportional correction signal, with which temperature-related changes in the transmission factor of the transducer and the offset value of the measurement signal are then compensated for in terms of circuitry. The bridge circuit is connected between a constant current source and a reference resistor.

The DE 88 04 598 U1 describes a circuit arrangement for compensating the temperature influence in a Wheatstone bridge circuit, in which, in addition to the bridge circuit, an essentially temperature-independent compensation resistor is connected, which is intended to compensate for the temperature dependence. When the bridge circuit is operated with a voltage source, this compensation resistor is connected in series with the bridge circuit. In addition, the semiconductor doping of the sensor elements implemented as semiconductors must be selected in a specific form so that the compensation works.

US 2003/0 117 254 A1 describes a sensor arrangement with a Wheatstone bridge circuit, with which a measured value for a temperature can be read out at a first output and a measured value for a magnetic field at a second output. An additional resistor is connected in series with the bridge circuit with magnetoresistive resistors, from which the measured value for the temperature can be tapped.

From the DE 10 2009 028 958 A1 a magnetic field and temperature sensor with a Wheatstone resistance measuring bridge is known, each having two magnetoresistive resistors in at least one half bridge. The sensor has a current mirror circuit with a current-giving branch and a current-reflecting branch, the measuring bridge being arranged in one branch and an ohmic resistor in the other branch.

The object of the present invention is to provide a method and a device for determining the change in temperature and the change in resistance of one or more sensor resistances of a bridge circuit, by means of which a more precise determination of the measurement variable is made possible regardless of the temperature.

Presentation of the invention

The object is achieved with the method and the device according to patent claims 1 and 10. Advantageous embodiments of the method and the device are the subject of the dependent claims or can be found in the following description and the exemplary embodiments.

In the proposed method, the change in temperature and the change in resistance of one or more sensor resistances of the bridge circuit are determined by calculation. In a preferred method alternative, the bridge circuit, which can also be a Wheatstone bridge circuit, is fed by a voltage source with a constant operating voltage and can be designed as a quarter or half bridge with several comparison resistors and one or more sensor resistors. In the proposed method, at least two of the comparison resistors with different temperature coefficients α, γ are selected for the construction of the bridge circuit. In the case of a quarter bridge, the at least two comparison resistors with known temperature coefficients that differ from one another are preferably arranged in a branch of the bridge circuit parallel to the sensor resistance. During a measurement with the bridge circuit, different voltages are then measured in the bridge circuit, of which at least one voltage drops across one of the resistors of the bridge circuit and which are selected such that a voltage across the bridge circuit, a voltage across, from one or more of the different voltages the or one of the sensor resistors and a voltage can be determined via a parallel comparison resistor. The voltage across the bridge circuit can thus either be measured directly or determined by suitable measurement of several voltages within the bridge circuit. The resistances of the bridge circuit are to be understood as meaning a sensor resistor and at least three comparison resistors for a quarter bridge and two sensor resistors and at least two comparison resistors for a half bridge. From the measured voltages, possibly a known operating voltage, known resistance _values R _{i0 of} the resistances of the bridge circuit at a reference temperature T ₀ and known temperature coefficients α, β, γ of the resistances of the bridge circuit, the dependencies Rsi = R _Si0 * (1 + β * ΔT + dR / R _Si0 ) for the or the sensor resistances of the bridge circuit and R _Vi = R _Vi0 * (1 + α * ΔT) or R _Vi = R _Vi0 * (1 + γ * ΔT) for the comparison _{resistances of} the bridge circuit with With the help of Kirchhoff's rules, the temperature change ΔT and therefrom the change in resistance dR are calculated and made available for further processing or direct output. The change in resistance dR and then the change in temperature ΔT can also be calculated. From the temperature change ΔT determined in this way, the temperature T = T ₀ + ΔT is then preferably also calculated and also made available for further processing or direct output. The reference temperature T ₀ corresponds to the temperature below which the resistance _values R _i0 would be determined. As a rule, this is a temperature in the range of approximately 20 ° C, to which the resistance values in the data sheets of the bridge circuit or the individual resistors refer. If the resistances of the bridge circuit are not known, they can be measured using known techniques, such as, for example, with the aid of a floating ampere meter in the bridge circuit. The temperature during this measurement must then be measured simultaneously and corresponds to the reference temperature T ₀ .

If the temperature behavior of the resistors used in the bridge circuit is known, the proposed method and the associated device enable the temperature change or temperature and the change in resistance of the sensor resistances caused by the size to be measured to be determined simultaneously solely on the basis of the measurement and evaluation of individual voltages of the bridge circuit. The evaluation itself can be carried out using software on a computer or with the aid of a suitably programmed microcontroller. No temperature sensors are required to determine the temperature. The only prerequisite is knowledge of the individual resistances of the bridge circuit and their preferably constant temperature coefficients. A balancing of the resistances of the bridge is no longer necessary in the proposed method, since offset voltages that may occur are automatically corrected by the simultaneous determination of temperature or temperature change and change in resistance of the sensor resistances. The method and the device thus enable a more precise determination of the measured variable with the possibility of temperature compensation by determining the temperature T. The temperature determined can also be used for other purposes, for example for monitoring systems, and thus possibly replace a previously used temperature sensor.

In an advantageous embodiment of the method, a voltage of one of the resistances of the respective branch is measured as one of the different voltages in each of the two parallel branches of the bridge circuit.

The temperature change ΔT and the resistance change dR are preferably calculated analytically with constant temperature coefficients α, β, γ of the resistances of the bridge circuit. In the case of non-constant temperature coefficients α, β, γ, however, a numerical calculation is required. Basically, the temperature coefficients α and β or γ and β need not necessarily be different in the present invention. They can also be the same (α = β or γ = β). In this way, one of the comparison resistors can also be formed from an "insensitive" or "passive" sensor resistor that does not react to the variable to be measured.

The proposed device for carrying out the method comprises a bridge circuit, a measuring device for measuring different voltages in the bridge circuit and an evaluation device which calculates the temperature change ΔT and the change in resistance dR of one or more sensor resistances of the bridge circuit in accordance with the proposed method. The bridge circuit is designed as a quarter or half bridge with several comparison resistors and one or more sensor resistors, at least two of the comparison resistors having known temperature coefficients α, γ that differ from one another. The measuring device is designed such that it can measure at least one voltage across one of the resistors of the bridge circuit as different voltages. Furthermore, the voltages measured with the measuring device are selected such that, from one or more of the different voltages, the voltage across the bridge circuit, the voltage across the (for the quarter bridge) or one of the sensor resistances (for the half bridge) and the voltage across the parallel one Comparative resistance can be determined. The evaluation device is preferably implemented by a suitably programmed microcontroller which is connected to the measuring device.

The proposed method and the associated device detect the temperature of the one or more sensor resistances of the bridge circuit without an additional temperature sensor and calculate the temperature influence on the measurement results using formulas. The measurement data can then be calculated or corrected on the basis of this data. The method and the device are suitable for all measurement applications with bridge circuits, for example for the precise measurement of force, pressure or magnetic field, the latter in particular with the aid of magnetoresistive sensors. Of course, this is not an exhaustive list.

Figure list

• 1 ein Beispiel für eine Schaltungsanordnung mit einer Viertelbrücke zur Durchführung des vorgeschlagenen Verfahrens;
• 2 ein Beispiel für Spannungsverläufe innerhalb der Schaltungsanordnung der 1 in Abhängigkeit von der Widerstandsänderung dR;
• 3 ein Beispiel für eine Schaltungsanordnung mit einer Halbbrücke zur Durchführung des vorgeschlagenen Verfahrens;
• 4 ein Beispiel für Spannungsverläufe innerhalb der Schaltungsanordnung der 3 in Abhängigkeit der Widerstandsänderung dR; und
• 5 ein Beispiel für die Messung der Widerstände einer Brückenschaltung.

The proposed method and the associated device are explained in more detail below on the basis of exemplary embodiments in conjunction with the drawings. Here show:

• 1 an example of a circuit arrangement with a quarter bridge for performing the proposed method;
• 2nd an example of voltage profiles within the circuit arrangement of the 1 depending on the change in resistance dR;
• 3rd an example of a circuit arrangement with a half bridge for performing the proposed method;
• 4th an example of voltage profiles within the circuit arrangement of the 3rd depending on the change in resistance dR; and
• 5 an example for the measurement of the resistances of a bridge circuit.

Ways of Carrying Out the Invention

The proposed method and the associated device use at least two comparison resistors with different, known temperature coefficients α, γ in a bridge circuit which is designed as a quarter bridge or a half bridge.

In the following, the method is first exemplified using a quarter bridge with only one sensor resistor and at least three comparison resistors. The use of a quarter bridge offers cost advantages, since sensor elements or sensor resistors generally cause higher costs than comparison resistors that can be realized with standard resistors. In the second exemplary embodiment, the bridge circuit is implemented as a half-bridge, in which two sensor resistors and at least two comparison resistors are used. The output voltage of the bridge circuit is usually twice as large compared to the quarter bridge.

In the present patent application, the pure change in resistance of a sensor resistance is expressed by dR and the change in temperature by ΔT, in each case based on a reference value. As a rule, the change in resistance dR also depends on the temperature T = T ₀ + ΔT: dR = dR ₀ * (1 + κ * ΔT), where dR ₀ = dR (T ₀ ) and κ represent a preferably constant temperature coefficient. If the temperature change ΔT is known, the correct value of dR can then be determined or vice versa.

1 shows an example of the proposed method for determining the temperature change and the change in resistance of the sensor resistance of a quarter bridge. For this purpose, the bridge circuit has the sensor resistance R1 and the three comparison resistors R2 , R3 and R4 on, with the two comparison resistors R3 and R4 have different constant temperature coefficients α and γ. The bridge circuit is in this example with a constant voltage source V1 operated like this in the 1 is indicated. The further comparison resistance R2 has the same constant temperature coefficient γ as the comparison resistor R4 on, the sensor resistance R1 the constant temperature coefficient β. The temperature change is represented by dT or ΔT. The three comparison resistors can have different resistance values, as evidenced by the factors k2, k3 and k4 with respect to a reference resistor R0 is expressed. The resistance value of the sensor resistance R1 is expressed in the same way by k1 * R0.

2nd shows two diagrams obtained from simulation calculations, which show the voltage profiles within the 1 Bridge circuit shown depending on the change in resistance dR of the sensor resistance R1 represent. UV represents the voltage of the voltage source V1 and thus also the voltage across the bridge circuit, UD represents the bridge voltage. U1 , U2 , U3 and U4 correspond to the voltages across the resistors R1 , R2 , R3 and R4 . The following applies in this circuit: UD = U1-U3, UV = U2 + UD + U3. The following applies: U2 = UV-U1 = UV-UD-U3.

The simulation was created with the LTSPICE simulation tool for the simulation of electrical circuits. From the top diagram of the 2nd it becomes clear that for U3 only the temperature change dT, but not the change in the sensor resistance dR plays a role. It is therefore possible to determine the temperature using this voltage. On the other hand, with UD the lower diagram becomes the 2nd It is equally clear that the change in temperature dT at this voltage essentially results only from a shift in the characteristic curve.

In the following, the temperature change dT or ΔT and the change in resistance dR are derived purely analytically on the assumption that the sensor resistance and the comparative resistances can show different resistance values under reference conditions (temperature T ₀ ) (factors k1 to k4) and the temperature dependence through a linear behavior can be described with constant temperature coefficients α, β and γ.

The following results for the sensor resistance: $${\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_1={\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot {\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_0\cdot \left(1+\beta \cdot \Delta T\right)+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot dR=={\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot {\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_0\cdot \left(1+\beta \cdot \Delta T+\raisebox{1ex}{$dR$}\!\left/ \!\raisebox{-1ex}{$R_0$}\right.\right)$$

$$R_3=k_3\cdot R_0\cdot \left(1+\alpha \cdot \Delta T\right)$$

$$R_4=k_4\cdot R_0\cdot \left(1+\gamma \cdot \Delta T\right)$$

The following results for the comparison resistors: $$R_{\mathrm{2nd}}=k_{\mathrm{2nd}}\cdot {\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_0\cdot \left(1+\gamma \cdot \Delta T\right)$$

$$R_{\mathrm{3rd}}=k_{\mathrm{3rd}}\cdot {\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_0\cdot \left(1+\alpha \cdot \Delta T\right)$$

$$R_{\mathrm{4th}}=k_{\mathrm{4th}}\cdot {\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_0\cdot \left(1+\gamma \cdot \Delta T\right)$$

$$U_D=\left(\frac{R_1}{R_1+R_2}-\frac{R_3}{R_3+R_4}\right)\cdot U_V=\left(\frac{R_1}{R_1+R_2}-\frac{R_3}{R_3+R_4}\right)\cdot U_V$$

$$U_D=\frac{R_1\cdot \left(R_3+R_4\right)-R_3\cdot \left(R_1+R_2\right)}{\left(R_1+R_2\right)\cdot \left(R_3+R_4\right)}\cdot U_V=\frac{R_1\cdot R_4-R_2\cdot R_3}{\left(R_1+R_2\right)\cdot \left(R_3+R_4\right)}\cdot U_V$$

RBR stellt hierbei den Widerstand der Brücke dar. $$I_{34}=\frac{U_3}{R_3}=\frac{U_V}{R_3+R_4}\Rightarrow U_3\cdot \left(R_3+R_4\right)=U_V\cdot R_3\Rightarrow R_3\cdot \left(U_V-U_3\right)=U_3\cdot R_4$$

$$\left[k_3\cdot R_0\cdot \left(1+\alpha \cdot \Delta T\right)\right]\cdot \left(U_V-U_3\right)=U_3\cdot \left[k_4\cdot R_0\cdot \left(1+\gamma \cdot \Delta T\right)\right]$$

$$\Delta T=\frac{k_3\cdot U_V-U_3\cdot \left(k_3+k_4\right)}{U_3\cdot \left(\alpha \cdot k_3+\gamma \cdot k_4\right)-\alpha \cdot k_3\cdot U_V}$$

Außerdem: $$I_{12}=\frac{U_2}{R_2}=\frac{U_V}{R_1+R_2}\Rightarrow U_2\cdot \left(R_1+R_2\right)=U_V\cdot R_2\Rightarrow R_2\cdot \left(U_V-U_2\right)=U_2\cdot R_1$$

$$\left[k_2\cdot R_0\cdot \left(1+\gamma \cdot \Delta T\right)\right]\cdot \left(U_V-U_2\right)=U_2\cdot \left(k_1\cdot R_0\cdot \left(1+\beta \cdot \Delta T\right)+k_1\cdot dR\right)$$

$$\Delta T=\frac{k_1\cdot U_2+k_1\cdot U_2\cdot \frac{dR}{R_0}+k_2\cdot \left(U_2-U_V\right)}{k2\cdot \gamma \cdot \left(U_V-U_2\right)-k_1\cdot \beta \cdot U_2}$$

und somit: $$\Delta T=\frac{k_3\cdot U_2-U_3\cdot \left(k_3+k_4\right)}{U_3\cdot \left(\alpha \cdot k_3+\gamma \cdot k_4\right)-\alpha \cdot k_3\cdot U_V}=\frac{k_1\cdot U_2+k_1\cdot U_2\cdot \frac{dR}{R_0}+k_2\cdot \left(U_2-U_V\right)}{k2\cdot \gamma \cdot \left(U_V-U_2\right)-k_1\cdot \beta \cdot U_2}$$

$$\begin{array}{l}\frac{dR}{R_0}=\frac{\left[k_3\cdot U_V-U_3\cdot \left(k_3+k_4\right)\right]\cdot \left[k_2\cdot \gamma \cdot U_1-k_1\cdot \beta \cdot \left(U_V-U_1\right)\right]}{\left(U_V-U_1\right)\cdot k_1\cdot \left[U_3\cdot \left(\alpha \cdot k_3+\gamma \cdot k_4\right)-\alpha \cdot k_3\cdot U_V\right]}-\\ -\frac{\left(k_1\cdot \left(U_V-U_1\right)-k_2\cdot U_1\right)\cdot \left[U_3\cdot \left(\alpha \cdot k_3+\gamma \cdot k_4\right)-\alpha \cdot k_3\cdot U_V\right]}{\left(U_V-U_1\right)\cdot k_1\cdot \left[U_3\cdot \left(\alpha \cdot k_3+\gamma \cdot k_4\right)-\alpha \cdot k_3\cdot U_V\right]}\end{array}$$

The following applies: $$R_{BR}=\left(R_1+R_{\mathrm{2nd}}\right)\Vert \left(R_{\mathrm{3rd}}+R_{\mathrm{4th}}\right)=\frac{\left(R_1+R_{\mathrm{2nd}}\right)\cdot \left(R_{\mathrm{3rd}}+R_{\mathrm{4th}}\right)}{R_1+R_{\mathrm{2nd}}+R_{\mathrm{3rd}}+R_{\mathrm{4th}}};$$

$$U_D=\left(\frac{R_1}{{\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_1+R_{\mathrm{2nd}}}-\frac{R_{\mathrm{3rd}}}{R_{\mathrm{3rd}}+R_{\mathrm{4th}}}\right)\cdot U_V=\left(\frac{R_1}{{\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_1+R_{\mathrm{2nd}}}-\frac{R_{\mathrm{3rd}}}{R_{\mathrm{3rd}}+R_{\mathrm{4th}}}\right)\cdot U_V$$

$$U_D=\frac{{\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_1\cdot \left(R_{\mathrm{3rd}}+R_{\mathrm{4th}}\right)-R_{\mathrm{3rd}}\cdot \left({\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_1+R_{\mathrm{2nd}}\right)}{\left({\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_1+R_{\mathrm{2nd}}\right)\cdot \left(R_{\mathrm{3rd}}+R_{\mathrm{4th}}\right)}\cdot U_V=\frac{{\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_1\cdot R_{\mathrm{4th}}-R_{\mathrm{2nd}}\cdot R_{\mathrm{3rd}}}{\left({\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_1+R_{\mathrm{2nd}}\right)\cdot \left(R_{\mathrm{3rd}}+R_{\mathrm{4th}}\right)}\cdot U_V$$

RBR represents the resistance of the bridge. $${\mathrm{I.}}_{34}=\frac{U_{\mathrm{3rd}}}{R_{\mathrm{3rd}}}=\frac{U_V}{R_{\mathrm{3rd}}+R_{\mathrm{4th}}}\Rightarrow U_{\mathrm{3rd}}\cdot \left(R_{\mathrm{3rd}}+R_{\mathrm{4th}}\right)=U_V\cdot R_{\mathrm{3rd}}\Rightarrow R_{\mathrm{3rd}}\cdot \left(U_V-U_{\mathrm{3rd}}\right)=U_{\mathrm{3rd}}\cdot R_{\mathrm{4th}}$$

$$\left[k_{\mathrm{3rd}}\cdot {\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_0\cdot \left(1+\alpha \cdot \Delta T\right)\right]\cdot \left(U_V-U_{\mathrm{3rd}}\right)=U_{\mathrm{3rd}}\cdot \left[k_{\mathrm{4th}}\cdot {\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_0\cdot \left(1+\gamma \cdot \Delta T\right)\right]$$

$$\Delta T=\frac{k_{\mathrm{3rd}}\cdot U_V-U_{\mathrm{3rd}}\cdot \left(k_{\mathrm{3rd}}+k_{\mathrm{4th}}\right)}{U_{\mathrm{3rd}}\cdot \left(\alpha \cdot k_{\mathrm{3rd}}+\gamma \cdot k_{\mathrm{4th}}\right)-\alpha \cdot k_{\mathrm{3rd}}\cdot U_V}$$

Furthermore: $${\mathrm{I.}}_{12}=\frac{U_{\mathrm{2nd}}}{R_{\mathrm{2nd}}}=\frac{{\mathrm{<figure-callout\; id="1"\; label="U\; V\; R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">U\; </figure-callout>}}_V}{R_{}}\Rightarrow U_{\mathrm{2nd}}\cdot \left({\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_1+R_{\mathrm{2nd}}\right)=U_V\cdot R_{\mathrm{2nd}}\Rightarrow R_{\mathrm{2nd}}\cdot \left(U_V-U_{\mathrm{2nd}}\right)=U_{\mathrm{2nd}}\cdot {\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_1$$

$$\left[k_{\mathrm{2nd}}\cdot {\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_0\cdot \left(1+\gamma \cdot \Delta T\right)\right]\cdot \left(U_V-U_{\mathrm{2nd}}\right)=U_{\mathrm{2nd}}\cdot \left({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot {\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_0\cdot \left(1+\beta \cdot \Delta T\right)+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot dR\right)$$

$$\Delta T=\frac{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot U_{\mathrm{2nd}}+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot U_{\mathrm{2nd}}\cdot \frac{dR}{{\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_0}+k_{\mathrm{2nd}}\cdot \left(U_{\mathrm{2nd}}-U_V\right)}{k\mathrm{2nd}\cdot \gamma \cdot \left(U_V-U_{\mathrm{2nd}}\right)-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot \beta \cdot U_{\mathrm{2nd}}}$$

and thus: $$\Delta T=\frac{k_{\mathrm{3rd}}\cdot U_{\mathrm{2nd}}-U_{\mathrm{3rd}}\cdot \left(k_{\mathrm{3rd}}+k_{\mathrm{4th}}\right)}{U_{\mathrm{3rd}}\cdot \left(\alpha \cdot k_{\mathrm{3rd}}+\gamma \cdot k_{\mathrm{4th}}\right)-\alpha \cdot k_{\mathrm{3rd}}\cdot U_V}=\frac{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot U_{\mathrm{2nd}}+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot U_{\mathrm{2nd}}\cdot \frac{dR}{{\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_0}+k_{\mathrm{2nd}}\cdot \left(U_{\mathrm{2nd}}-U_V\right)}{k\mathrm{2nd}\cdot \gamma \cdot \left(U_V-U_{\mathrm{2nd}}\right)-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot \beta \cdot U_{\mathrm{2nd}}}$$

$$\begin{array}{l}\frac{dR}{{\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_0}=\frac{\left[k_{\mathrm{3rd}}\cdot U_V-U_{\mathrm{3rd}}\cdot \left(k_{\mathrm{3rd}}+k_{\mathrm{4th}}\right)\right]\cdot \left[k_{\mathrm{2nd}}\cdot \gamma \cdot U_1-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot \beta \cdot \left(U_V-U_1\right)\right]}{\left(U_V-U_1\right)\cdot {\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot \left[U_{\mathrm{3rd}}\cdot \left(\alpha \cdot k_{\mathrm{3rd}}+\gamma \cdot k_{\mathrm{4th}}\right)-\alpha \cdot k_{\mathrm{3rd}}\cdot U_V\right]}-\\ -\frac{\left({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot \left(U_V-U_1\right)-k_{\mathrm{2nd}}\cdot U_1\right)\cdot \left[U_{\mathrm{3rd}}\cdot \left(\alpha \cdot k_{\mathrm{3rd}}+\gamma \cdot k_{\mathrm{4th}}\right)-\alpha \cdot k_{\mathrm{3rd}}\cdot U_V\right]}{\left(U_V-U_1\right)\cdot {\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot \left[U_{\mathrm{3rd}}\cdot \left(\alpha \cdot k_{\mathrm{3rd}}+\gamma \cdot k_{\mathrm{4th}}\right)-\alpha \cdot k_{\mathrm{3rd}}\cdot U_V\right]}\end{array}$$

The temperature T or the temperature change ΔT can thus be determined, for example, from equation 1-1. The pure change in resistance dR or the relative change in resistance dR / R0 for determining the measured variable can be calculated from equation 2-1. To do this, the temperature T or temperature change ΔT must be calculated beforehand. To determine the exact measured variable, the temperature influence on the sensor element can then be compensated or calculated out with knowledge of the temperature. This is possible even if the individual resistors have different values (k1 * R0 to k4 * R0), so that no adjustment of the bridge circuit is necessary.

3rd shows an example in which the bridge circuit as a half bridge with two sensor resistors R1 , R4 and two comparison resistors R2 , R3 is trained. The sensor resistances R1 and R4 can be different (different coefficients k1, k4). The relative change in resistance dR / R0 is assumed to be the same, as can also be expected if the variable to be measured acts in the same way on both sensor resistances. Alignment of the resistors is not necessary here either, as in the previous exemplary embodiment.

$$R_4=k_4\cdot R_0\cdot \left(1+\beta \cdot \Delta T\right)+k_4\cdot dR==k_4\cdot R_0\cdot \left(1+\beta \cdot \Delta T+\raisebox{1ex}{$dR$}\!\left/ \!\raisebox{-1ex}{$R_0$}\right.\right)$$

The 4th shows just like 2nd the voltage curves resulting from a simulation calculation within the bridge circuit as a function of the change in resistance dR. The diagram above again shows that the expression U2-U3 is only dependent on the temperature change dT, regardless of dR. The following also applies to the sensor resistances: $${\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_1={\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot {\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_0\cdot \left(1+\beta \cdot \Delta T\right)+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot dR=={\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot {\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_0\cdot \left(1+\beta \cdot \Delta T+\raisebox{1ex}{$dR$}\!\left/ \!\raisebox{-1ex}{$R_0$}\right.\right)$$

$$R_{\mathrm{4th}}=k_{\mathrm{4th}}\cdot {\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_0\cdot \left(1+\beta \cdot \Delta T\right)+k_{\mathrm{4th}}\cdot dR==k_{\mathrm{4th}}\cdot {\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_0\cdot \left(1+\beta \cdot \Delta T+\raisebox{1ex}{$dR$}\!\left/ \!\raisebox{-1ex}{$R_0$}\right.\right)$$

und $$R_3=k_3\cdot R_0\cdot \left(1+\alpha \cdot \Delta T\right)$$

The following results for the comparison resistors: $$R_{\mathrm{2nd}}=k_{\mathrm{2nd}}\cdot {\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_0\cdot \left(1+\gamma \cdot \Delta T\right)$$

and $$R_{\mathrm{3rd}}=k_{\mathrm{3rd}}\cdot {\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_0\cdot \left(1+\alpha \cdot \Delta T\right)$$

$$\left[k_2\cdot R_0\cdot \left(1+\gamma \cdot \Delta T\right)\right]\cdot \left(U_V-U_2\right)=U_2\cdot \left(k_1\cdot R_0\cdot \left(1+\beta \cdot \Delta T\right)+k_1\cdot dR\right)$$

$$\frac{dR}{R_0}=\frac{k_2\cdot U_1\cdot \left(1+\gamma \cdot \Delta T\right)-k_1\cdot U_2\cdot \left(1+\beta \cdot \Delta T\right)}{k_1\cdot \left(U_V-U_2\right)}$$

Außerdem: $$I_{34}=\frac{U_3}{R_3}=\frac{U_V}{R_3+R_4}\Rightarrow U_3\cdot \left(R_3+R_4\right)=U_V\cdot R_3\Rightarrow R_3\cdot \left(U_V-U_3\right)=U_3\cdot R_4$$

$$\left[k_3\cdot R_0\cdot \left(1+\alpha \cdot \Delta T\right)\right]\cdot \left(U_V-U_3\right)=U_3\cdot \left[k_4\cdot R_0\cdot \left(1+\beta \cdot \Delta T\right)+k_4\cdot dR\right]$$

$$\frac{dR}{R_0}=\frac{k_3\cdot \left(U_V-U_3\right)\cdot \left(1+\alpha \cdot \Delta T\right)-k_4\cdot U_3\cdot \left(1+\beta \cdot \Delta T\right)}{k_4\cdot U_3}$$

und somit: $$\frac{dR}{R_0}=\frac{k_2\cdot U_1\cdot \left(1+\gamma \cdot \Delta T\right)-k_1\cdot U_2\cdot \left(1+\beta \cdot \Delta T\right)}{k_1\cdot \left(U_V-U_2\right)}=\frac{k_3\cdot \left(U_V-U_3\right)\cdot \left(1+\alpha \cdot \Delta T\right)-k_4\cdot U_3\cdot \left(1+\beta \cdot \Delta T\right)}{k_4\cdot U_3}$$

$$\Delta T=\frac{k_4\cdot U_3\cdot \left[k_2\cdot U_1-k_1\cdot \left(U_V-U_1\right)\right]-k_1\cdot \left(U_V-U_1\right)\cdot \left[k_3\cdot \left(U_V-U_3\right)-k_4\cdot U_3\right]}{k_1\cdot \left(U_V-U_1\right)\cdot \left[k_3\cdot \alpha \cdot \left(U_V-U_3\right)-k_4\cdot \beta \cdot U_3\right]+k_4\cdot U_3\cdot \left[k_1\cdot \beta \cdot U_2-k_2\cdot \gamma \cdot U_1\right]}$$

$$\Delta T=\frac{k_2\cdot k_4\cdot U_3\cdot U_1-k_1\cdot \left(U_V-U_1\right)\cdot \left[k_4\cdot U_3+k_3\cdot \left(U_V-U_3\right)-k_4\cdot U_3\right]}{k_1\cdot \left(U_V-U_1\right)\cdot \left[k_3\cdot \alpha \cdot \left(U_V-U_3\right)-k_4\cdot \beta \cdot U_3\right]+k_4\cdot U_3\cdot \left[k_1\cdot \beta \cdot U_2-k_2\cdot \gamma \cdot U_1\right]}$$

The following applies: $${\mathrm{I.}}_{12}=\frac{U_{\mathrm{2nd}}}{R_{\mathrm{2nd}}}=\frac{{\mathrm{<figure-callout\; id="1"\; label="U\; V\; R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">U\; </figure-callout>}}_V}{R_{}}\Rightarrow U_{\mathrm{2nd}}\cdot \left({\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_1+R_{\mathrm{2nd}}\right)=U_V\cdot R_{\mathrm{2nd}}\Rightarrow R_{\mathrm{2nd}}\cdot \left(U_V-U_{\mathrm{2nd}}\right)=U_{\mathrm{2nd}}\cdot {\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_1$$

$$\left[k_{\mathrm{2nd}}\cdot {\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_0\cdot \left(1+\gamma \cdot \Delta T\right)\right]\cdot \left(U_V-U_{\mathrm{2nd}}\right)=U_{\mathrm{2nd}}\cdot \left({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot {\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_0\cdot \left(1+\beta \cdot \Delta T\right)+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot dR\right)$$

$$\frac{dR}{{\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_0}=\frac{k_{\mathrm{2nd}}\cdot {\mathrm{<figure-callout\; id="1"\; label="U"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">U</figure-callout>}}_1\cdot \left(1+\gamma \cdot \Delta T\right)-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot U_{\mathrm{2nd}}\cdot \left(1+\beta \cdot \Delta T\right)}{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot \left(U_V-U_{\mathrm{2nd}}\right)}$$

Furthermore: $${\mathrm{I.}}_{34}=\frac{U_{\mathrm{3rd}}}{R_{\mathrm{3rd}}}=\frac{U_V}{R_{\mathrm{3rd}}+R_{\mathrm{4th}}}\Rightarrow U_{\mathrm{3rd}}\cdot \left(R_{\mathrm{3rd}}+R_{\mathrm{4th}}\right)=U_V\cdot R_{\mathrm{3rd}}\Rightarrow R_{\mathrm{3rd}}\cdot \left(U_V-U_{\mathrm{3rd}}\right)=U_{\mathrm{3rd}}\cdot R_{\mathrm{4th}}$$

$$\left[k_{\mathrm{3rd}}\cdot {\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_0\cdot \left(1+\alpha \cdot \Delta T\right)\right]\cdot \left(U_V-U_{\mathrm{3rd}}\right)=U_{\mathrm{3rd}}\cdot \left[k_{\mathrm{4th}}\cdot {\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_0\cdot \left(1+\beta \cdot \Delta T\right)+k_{\mathrm{4th}}\cdot dR\right]$$

$$\frac{dR}{{\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_0}=\frac{k_{\mathrm{3rd}}\cdot \left(U_V-U_{\mathrm{3rd}}\right)\cdot \left(1+\alpha \cdot \Delta T\right)-k_{\mathrm{4th}}\cdot U_{\mathrm{3rd}}\cdot \left(1+\beta \cdot \Delta T\right)}{k_{\mathrm{4th}}\cdot U_{\mathrm{3rd}}}$$

and thus: $$\frac{dR}{{\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">R</figure-callout>}}_0}=\frac{k_{\mathrm{2nd}}\cdot {\mathrm{<figure-callout\; id="1"\; label="U"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">U</figure-callout>}}_1\cdot \left(1+\gamma \cdot \Delta T\right)-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot U_{\mathrm{2nd}}\cdot \left(1+\beta \cdot \Delta T\right)}{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot \left(U_V-U_{\mathrm{2nd}}\right)}=\frac{k_{\mathrm{3rd}}\cdot \left(U_V-U_{\mathrm{3rd}}\right)\cdot \left(1+\alpha \cdot \Delta T\right)-k_{\mathrm{4th}}\cdot U_{\mathrm{3rd}}\cdot \left(1+\beta \cdot \Delta T\right)}{k_{\mathrm{4th}}\cdot U_{\mathrm{3rd}}}$$

$$\Delta T=\frac{k_{\mathrm{4th}}\cdot U_{\mathrm{3rd}}\cdot \left[k_{\mathrm{2nd}}\cdot U_1-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot \left(U_V-U_1\right)\right]-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot \left(U_V-U_1\right)\cdot \left[k_{\mathrm{3rd}}\cdot \left(U_V-U_{\mathrm{3rd}}\right)-k_{\mathrm{4th}}\cdot U_{\mathrm{3rd}}\right]}{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot \left(U_V-U_1\right)\cdot \left[k_{\mathrm{3rd}}\cdot \alpha \cdot \left(U_V-U_{\mathrm{3rd}}\right)-k_{\mathrm{4th}}\cdot \beta \cdot U_{\mathrm{3rd}}\right]+k_{\mathrm{4th}}\cdot U_{\mathrm{3rd}}\cdot \left[{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot \beta \cdot U_{\mathrm{2nd}}-k_{\mathrm{2nd}}\cdot \gamma \cdot U_1\right]}$$

$$\Delta T=\frac{k_{\mathrm{2nd}}\cdot k_{\mathrm{4th}}\cdot U_{\mathrm{3rd}}\cdot U_1-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot \left(U_V-U_1\right)\cdot \left[k_{\mathrm{4th}}\cdot U_{\mathrm{3rd}}+k_{\mathrm{3rd}}\cdot \left(U_V-U_{\mathrm{3rd}}\right)-k_{\mathrm{4th}}\cdot U_{\mathrm{3rd}}\right]}{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot \left(U_V-U_1\right)\cdot \left[k_{\mathrm{3rd}}\cdot \alpha \cdot \left(U_V-U_{\mathrm{3rd}}\right)-k_{\mathrm{4th}}\cdot \beta \cdot U_{\mathrm{3rd}}\right]+k_{\mathrm{4th}}\cdot U_{\mathrm{3rd}}\cdot \left[{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102018216131B4\_0029.png,DE102018216131B4\_0030.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\cdot \beta \cdot U_{\mathrm{2nd}}-k_{\mathrm{2nd}}\cdot \gamma \cdot U_1\right]}$$

With knowledge of the individual resistances R1 to R4 , the temperature coefficients α, β and γ and the measurement of UV, U1 and U3 the temperature change ΔT can first be exactly determined using equation 1-2, whereby the temperature influence on dR can be compensated for later. The change in resistance dR can then be determined using equation 2-2a or 2-2b with knowledge of the voltages at the bridge ( U1 , U3 , UV) can be determined.

5 shows another example of the measurement of the resistances of a bridge circuit if these are not known when the proposed method or the proposed device is implemented. For this purpose, an ungrounded ampere meter is used, as is known, for example, from Tietze, U., Schenk, Ch., Semiconductor circuit technology, 9th edition, Springer-Verlag, 1990, pages 864 ff. The 5 shows the bridge circuit with the corresponding potential points in the left partial illustration E1 and E2 such as U1 and B. Between E1 and E2 is used to measure resistance by negative feedback R7 and R8 (crosswise) creates a virtual short circuit, as can be seen in the middle partial illustration. The flow through R3 respectively. R4 then corresponds to the current through R7 or R8 (since R5 = R6). R3 thus corresponds to U (E2) / I (R8). The same applies to the nodes U1 and B to determine R2 , as can be seen from the right part of the illustration. The following applies: R2 = (U (E1) -U1) / I (R12) = (UV-U1) / I (R12). The sensor resistance can also be adjusted accordingly R1 can be determined in an already completed bridge circuit. Then the offset for UD can be measured under reference conditions. The offset can also be determined using the determined resistances R1 to R4 be determined.

Claims (11)

Method for the simultaneous determination of the temperature and resistance change of one or more sensor resistors of a bridge circuit, which is designed as a quarter or half bridge with several comparison resistors and one or more sensor resistors, in which - when the bridge circuit is constructed, at least two of the comparison resistors with different temperature coefficients α, γ can be selected, during a measurement with the bridge circuit, several different voltages are measured in the bridge circuit, of which at least one voltage drops across one of the resistors of the bridge circuit and which are selected such that a voltage across the bridge circuit, a voltage across, from one or more of the different voltages the or one of the sensor resistances and a voltage can be determined via a parallel comparison resistor, and - from the measured voltages, known resistance _values R _{i0 of} the resistances of the bridge circuit at a reference temperature T ₀ and known temperature coefficients α, β, γ of the resistances of the bridge circuit via the Dependencies R _Si = R _Si0 * (1 + β * ΔT + dR / R _Si0 ) for the sensor resistance or resistances of the bridge circuit and R _Vi = R _Vi0 * (1 + α * ΔT) or R _Vi = R _Vi0 * (1 + γ * ΔT) for the comparison resistances of the bridge circuit using the Kirchhoff's rules the temperature change ΔT and from this the W change in resistance dR is calculated or vice versa. Procedure according to Claim 1 , characterized in that in a quarter bridge the at least two comparison resistors with different known temperature coefficients are arranged in a branch of the bridge circuit parallel to the sensor resistance. Procedure according to Claim 1 or 2nd , characterized in that the bridge circuit is fed by a voltage source with a constant known operating voltage. Procedure according to Claim 1 or 2nd , characterized in that the bridge circuit is fed by a constant current source and the voltage across the bridge circuit is measured as one of the different voltages. Procedure according to one of the Claims 1 to 4th , characterized in that in each of two parallel branches of the bridge circuit, a voltage of one of the resistances of the respective branch is measured as one of the different voltages. Procedure according to one of the Claims 1 to 5 , characterized in that the calculation of the change in temperature ΔT and the change in resistance dR at constant temperature coefficients α, β, γ of the resistors of the bridge circuit is carried out analytically. Procedure according to Claim 6 , characterized in that when the bridge circuit is designed as a quarter bridge, the voltage across the bridge circuit and the voltage across a comparison resistor parallel to the sensor resistance in the bridge circuit are used to calculate the temperature change ΔT. Procedure according to Claim 6 , characterized in that when the bridge circuit is designed as a half bridge for calculating the temperature change ΔT, the voltage across the bridge circuit, the voltage across one of the sensor resistors and the voltage across a comparison resistor corresponding to this sensor resistor in the bridge circuit are used. Procedure according to one of the Claims 1 to 8th , characterized in that the temperature T = T ₀ + ΔT is calculated from the temperature change ΔT. Device with a bridge circuit which is designed as a quarter or half bridge with a plurality of comparative resistors and one or more sensor resistors, at least two of the comparative resistors having different known temperature coefficients α, γ, - A measuring device which is designed to measure a plurality of different voltages in the bridge circuit, of which at least one voltage drops across one of the resistors of the bridge circuit and which are selected such that a voltage across the bridge circuit, a voltage, from one or more of the different voltages can be determined across the or one of the sensor resistances and a voltage across a parallel comparison resistor, and - An evaluation device which is designed such that it calculates a temperature change ΔT and therefrom a change in resistance dR of the one or more sensor resistances of the bridge circuit or vice versa according to the method of one or more of the preceding claims. Device after Claim 10 , characterized in that the evaluation device is designed as a microcontroller.

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