RU2507476C1 - Method to tune strain-gauge sensors with bridge measurement circuit by multiplicative temperature error with account of positive non-linearity of temperature characteristic of output signal of sensor - Google Patents

Abstract

FIELD: measurement equipment.

SUBSTANCE: determine a temperature sensitivity coefficient (TSC) of a bridge circuit α⁺ _do and α^- _do at temperature t⁺ and t^-, corresponding to upper and lower limit of the working temperature range, non-linearity of bridge circuit TSC (Δα_do=α⁺ _do-α^- _do). If the produced value Δα_do is positive, they convert the positive non-linearity of bridge circuit TSC into the negative one by inclusion of an heat-independent resistor R_i. For this purpose they determine input resistance and TSC of input resistance, and the input value of TSC value, and the input resistance, TSC of resistance strain gauges α⁺ _d and α^- _d at temperature t⁺ and t^-, calculate non-linearity of bridge circuit TSC (Δα_d=α⁺ _d-α^- _d). If α⁺ _d and Δα^- _d are in the field of conversion of positive non-linearity of bridge circuit TSC into the negative, then they calculate the nominal value of the resistor R_i. They connect the resistor R_i into the diagonal of the bridge circuit power supply. The TCS of the bridge circuit is defined at the temperature t⁺ and t^-, non-linearity of the bridge circuit TCS is calculated as Δα_do. If Δα_do takes the negative value, then compensation of multiplicative temperature error is carried out with account of negative non-linearity of bridge circuit TSC by inclusion of a heat independent resistor R_αoutp, shunted with a heat independent resistor R_doutp, into the output diagonal of the bridge circuit under the resistance of the load R_L≤2 kOm.

EFFECT: higher accuracy of compensation.

3 tbl, 3 dwg

Description

The invention relates to measuring equipment and can be used to configure the strain gauge sensor equipment with a bridge measuring circuit for a multiplicative temperature error.

There is a method of tuning strain gauge sensors with a bridge measuring circuit for a multiplicative temperature error, taking into account the non-linearity of the temperature characteristic of the sensor output signal (see Patent for invention RU 2443973 C1, G01B 7/16 "Method for setting strain gauge sensors with a bridge measuring chain for a multiplicative temperature error, taking into account non-linearity of the temperature characteristics of the sensor output signal ", published on 02.27.2012 in Bull. No. 6), adopted as a prototype, in which to compensate for the mule a multiplicative temperature error with a load resistance of R _n > 500 kOhm is determined by the TCD of the bridge circuit α ⁺ _up and α ^- _up for the temperature range Δt ⁺ = t ⁺ -t ₀ and Δt ^- = t ^- -t ₀ , where t ₀ , t ⁺ , t ^- - normal temperature, upper and lower limits of the operating temperature range, respectively. Calculate nonlinearity TCF bridge circuit (Δα = α ⁺ _{do do} -α ^- _do). If Δα _to takes a negative value, then the sensor is connected to a load R _n <2 kOhm. The output resistance of the bridge circuit and the TCS of the sensor output resistance are determined. Check whether the DC link bridge circuit and its nonlinearity are found in the field of application of the method, if these sensor parameters are in the field of application, the required value of the thermally dependent resistor R _αout and the thermally independent resistor R _{two are calculated} , the resistor R _αout , shunted by the resistor R _two , is _installed in series with the load.

The reasons that impede the achievement of the technical result indicated below when using the known method include the fact that the non-linearity of the DC circuit of the bridge circuit can take both negative and positive values, as shown in the description of the prototype. The prototype allows for full compensation of the multiplicative temperature error, taking into account the negative nonlinearity of the DC circuit of the bridge circuit, satisfying the inequality Δα _to ≤2 · 10 ^-6 1 / ° С.

The description of the prototype shows that the absence of taking into account the nonlinearity of the TCF bridge circuit allows compensation of the multiplicative temperature error at one extreme point of the operating temperature range, for which the values of the compensation resistors R _αout and R _{two were calculated} , which allows to obtain a multiplicative sensitivity of the sensor to a temperature within ± 1 · 10 ^-4 1 / ° С at a given point of the operating temperature range. At the other extreme point of the operating temperature range, the multiplicative sensitivity of the sensor to temperature is of the order of ± 2 · 10 ^-4 1 / ° С and more, which exceeds the permissible value, which is ± 1 · 10 ^-4 1 / ° С.

The invention consists in the following.

The task to which the claimed invention is directed is to develop a method for tuning strain gauge sensors with a bridge measuring circuit by a multiplicative temperature error, which would improve the accuracy of compensating for a multiplicative temperature error in the setup process with a positive non-linearity of the DC-circuit TCD.

The technical result consists in increasing the accuracy in the process of tuning strain gauge sensors with a bridge measuring circuit by a multiplicative temperature error with a positive nonlinearity of the DC circuit of the bridge circuit.

The specified technical result in the implementation of the invention is achieved by preliminarily converting the positive nonlinearity of the DTC of the sensor bridge circuit into negative and subsequent compensation of the multiplicative temperature error in accordance with the prototype.

This is achieved by the fact that a thermally independent resistor R _{i is} included in the diagonal of the bridge circuit power supply, which gives a shift in the nonlinearity of the DC-link circuit of the sensor bridge towards negative values. The value of the resistor R _{i is} selected on the basis of the need to ensure the negative nonlinearity of the TFC of the bridge circuit Δα _to ≤2 · 10 ^-6 1 / ° С, which allows the use of a prototype to compensate for the temperature error. To do this, if the non-linearity of the DC current factor of the bridge circuit is positive, for R _n > 500 kOh, the TCD of the strain gages α ⁺ _d and α ^- _{d are} determined for the temperature range Δt ⁺ and Δt ^- respectively, and the nonlinearity of the TCD strain gauges Δα _d = α ⁺ _d- α ^- _d . Determine the value of the input resistance R _I , TCS input resistance α ⁺ _I , α ^- _I for the temperature range Δt ⁺ and Δt ^- respectively. Check the affiliation of α ⁺ _d and Δα _{d of} the transformation region of the positive nonlinearity of the DC circuit of the bridge circuit to the negative one specified by table 3. If α ⁺ _d and Δα _d satisfy the region specified by table 3, then the value of the thermally independent resistor R _{i is} calculated. Install the resistor R _i in the diagonal of the power supply of the bridge circuit. The DC circuit of the bridge circuit and its nonlinearity are calculated after turning on the thermally dependent resistor Rα _in .

The invention is illustrated by the drawings, in which Fig. 1 shows the influence of a thermally independent resistor R _i on the non-linearity of the DC-link DC circuit, in Fig. 2 - the region of converting the positive non-linearity of the _{DC-link DC} circuit to negative, in Fig. 3 - connection diagram of the resistors R _i , _Rout and R _two .

The method is as follows.

As shown in the description of the prototype, the nonlinearity of the DC circuit bridge circuit includes two components:

1. non-linearity introduced by strain gauges mounted on an elastic element, which can take both negative and positive values;

2. non-linearity introduced by the measuring circuit, which is always negative when using a bridge circuit.

In accordance with paragraph 2, it is possible to obtain a negative nonlinearity of the DC-link circuit of the bridge circuit by changing the component introduced by the measuring circuit. To do this, a resistor should be included in the bridge circuit, which will increase the negative component of nonlinearity.

When a thermally independent resistor R _i is included in the power supply circuit of the bridge circuit, a decrease in the input resistance when exposed to temperature, due to the negative nonlinearity of the dependence of the input resistance of the bridge circuit on temperature, will lead to a decrease in the supply voltage, introducing negative nonlinearity in the temperature characteristic of the output signal. In accordance with the prototype, the output voltage of the bridge circuit after turning on the resistor R _i when exposed to temperature can be represented as follows:

$$U_{\mathrm{at}sxt}=U_{P\mathrm{and}t}\frac{k}{{\left(k+\mathrm{one}\right)}^2}\frac{R_{\mathrm{at}x}\left(\mathrm{one}+{\alpha }_{\mathrm{at}x}\Delta t\right)\left(\mathrm{one}+{\alpha }_d\Delta t\right)}{R_{\mathrm{at}x}\left(\mathrm{one}+{\alpha }_{\mathrm{at}x}\Delta t\right)+R_i}{\displaystyle \sum _{j=\mathrm{one}}^{\mathrm{four}}{\epsilon }_j,\left(\mathrm{one}\right)}$$

where U _o is the output voltage of the bridge circuit when exposed to temperature;

U _pit is the supply voltage of the bridge circuit;

k = R ₁ / R ₂ = R ₃ / R ₄ is the symmetry coefficient of the bridge circuit;

R _I - input resistance of the bridge circuit of the sensor;

α _I - TCS input resistance;

Δt = tt ₀ - temperature change;

α _d - TKH strain gages;

R _i is the nominal value of a thermally independent resistor included in the power circuit;

t is the acting temperature;

t ₀ - normal temperature;

ε _j is the relative change in shoulder resistance R _{j of the} bridge circuit.

Analysis of the denominator of dependence (1) allows us to conclude that, after turning on the resistor R _i , the dependence of the supply voltage on temperature will have a component inversely proportional to the temperature increase, which will lead to a shift in the dependence of the DC-link circuit of the bridge circuit to negative values.

At normal temperature, the output voltage of the sensor after turning on the resistor R _i taking into account (1) can be represented as follows:

$$U_{\mathrm{at}sx}=U_{P\mathrm{and}t}\frac{k}{{\left(k+\mathrm{one}\right)}^2}\frac{R_{\mathrm{at}x}}{R_{\mathrm{at}x}+R_i}{\displaystyle \sum _{j=\mathrm{one}}^{\mathrm{four}}{\epsilon }_j,\left(2\right)}$$

where U _o is the output voltage of the bridge circuit at normal temperature.

As shown in the prototype, the DC circuit of the bridge circuit can be determined through the output voltage:

$${\alpha }_{d\mathrm{about}}=\frac{U_{\mathrm{at}sxt}-U_{\mathrm{at}sx}}{U_{\mathrm{at}sx}\Delta t}.\left(3\right)$$

Substituting (1) and (2) in (3), we can obtain the dependence of the TFC on the sensor parameters:

$${\alpha }_{d\mathrm{about}}=\frac{R_{\mathrm{at}x}{\alpha }_d\left(\mathrm{one}+{\alpha }_{\mathrm{at}x}\Delta t\right)+R_i\left({\alpha }_{\mathrm{at}x}+{\alpha }_d+{\alpha }_{\mathrm{at}x}{\alpha }_d\Delta t\right)}{R_{\mathrm{at}x}\left(\mathrm{one}+{\alpha }_{\mathrm{at}x}\Delta t\right)+R_i}.\left(\mathrm{four}\right)$$

The nonlinearity of the DC circuit of the bridge circuit can be represented as follows:

$$\begin{array}{c}\Delta {\alpha }_{d\mathrm{about}}={\alpha }_{d\mathrm{about}}^{+}-{\alpha }_{d\mathrm{about}}^{-}=\frac{R_{\mathrm{at}x}{\alpha }_d^{+}\left(\mathrm{one}+{\alpha }_{\mathrm{at}x}^{+}\Delta t^{+}\right)+R_i\left({\alpha }_{\mathrm{at}x}^{+}+{\alpha }_d^{+}+{\alpha }_{\mathrm{at}x}^{+}{\alpha }_d^{+}\Delta t^{+}\right)}{R_{\mathrm{at}x}\left(\mathrm{one}+{\alpha }_{\mathrm{at}x}^{+}\Delta t^{+}\right)+R_i}-\\ -\frac{R_{\mathrm{at}x}{\alpha }_d^{-}\left(\mathrm{one}+{\alpha }_{\mathrm{at}x}^{-}\Delta t^{-}\right)+R_i\left({\alpha }_{\mathrm{at}x}^{-}+{\alpha }_d^{-}+{\alpha }_{\mathrm{at}x}^{-}{\alpha }_d^{-}\Delta t^{-}\right)}{R_{\mathrm{at}x}\left(\mathrm{one}+{\alpha }_{\mathrm{at}x}^{-}\Delta t^{-}\right)+R_i}\end{array},\left(5\right)$$

where Δt ⁺ = t ⁺ -t ₀ , Δt ^- = t ^- -t ₀ - positive and negative temperature range;

t ₀ - normal temperature;

t ⁺ , t ^- - upper and lower limit of the operating temperature range;

α ⁺ _up , α ^- _up - TFC of the sensor bridge circuit at t ⁺ and t ^- respectively;

α ⁺ _d , α ^- _d - TFC of strain gauges at temperature t ⁺ and t ^- respectively;

α ⁺ _in , α ^- _in - TKS input resistance of the sensor bridge circuit at a temperature of t ⁺ and t ^- respectively;

Δα _do - nonlinearity TCF bridge circuit.

To assess the influence of the resistor value R _i on the non-linearity of the DC-link circuit of the bridge circuit, the non-linearity of the DC-link circuit of the bridge circuit was calculated by formula (5) with the following initial data:

1. Input resistance of the bridge circuit: Rin = 1000 Ohm;

2. TKH strain gages takes the following values: α _d = (1, 5, 10) · 10 ^-4 1 / ° C;

3. The non-linearity of the TSC strain gages takes the following values: Δα _d = α ⁺ _d -α ^- _d = (1, 5, 10) · 10 ^-6 1 / ° С;

4. TKS input resistance: α _I = 5 · 10 ^-4 1 / ° C;

5. nonlinearity TCR of the input resistance: Δα = α ⁺ _{Rin Rin} -α ^- _in = -5 · 10 ^-6 1 / ° C;

6. TCS compensation resistor: α _to = 4 · 10 ^-3 1 / ° C;

7. The value of a thermally independent resistor: R _i = (1, 100, 300, 500) Ohms. The collected material is presented in table 1.

Table 1 The effect of R _i on the non-linearity of the DC circuit of the bridge circuit α ⁺ _in · 10 ^-4 , 1 / ° С Δα _in · 10 ^-6 1 / ° C α ^- _d · 10 ^-4 , 1 / ° С Δα _d · 10 ^-6 , 1 / ° С R _i Δα _to · 10 ^-6 , 1 / ° С 5 -5 one one one 0.955 one hundred -2,720 300 -6,803 500 -8,537 5 -5 one 5 one 4,955 one hundred 1,298 300 -2.758 500 -4,472 5 -5 one 10 one 9,955 one hundred 6,320 300 2,297 500 0.608 5 -5 5 one one 0,995 one hundred 0.944 300 2,492 500 4,883 5 -5 5 5 one 4,995 one hundred 4,962

Continuation of table 1 α ⁺ _in · 10 ^-4 , 1 / ° С Δα _in · 10 ^-6 , 1 / ° С α ^- _d · 10 ^-4 , 1 / ° С Δα _d · 10 ^-6 , 1 / ° С R _i Δα _to · 10 ^-6 , 1 / ° С 5 -5 5 5 300 6,536 500 8,947 5 -5 5 10 one 9.995 one hundred 9,983 300 11,592 500 14,028 5 -5 10 one one 1,045 one hundred 5,524 300 14,110 500 21,657 5 -5 10 5 one 5,045 one hundred 9,542 300 18,154 500 25,722 5 -5 10 10 one 10,046 one hundred 14,563 300 23,210 500 30,802

Data analysis allows us to draw the following conclusions:

1. With an increase in the value of the resistor R (non-linearity of the DC circuit of the bridge circuit is shifted to the region of negative values (table 1 and figure 1).

2. With an increase in the TFC of the strain gauges and its nonlinearity, the nonlinearity of the TFC of the bridge circuit shifts to the region of positive values (table 1 and figure 1).

3. The inclusion of a thermally independent resistor R _i in the power circuit allows you to get a negative value of the nonlinearity of the DC current factor of the bridge circuit with a positive non-linearity of the DC current strain gauge.

4. For small values of the resistor R _i , of the order of 1 Ohm or less, the nonlinearity of the DC current factor of the bridge circuit is determined by the nonlinearity of the DC current transformer of strain gauges.

5. With an increase in the value of the resistor R (the maximum value of the negative nonlinearity of the TFC of the bridge circuit increases; with R _i = 100 Ohms, the negative nonlinearity can reach Δα _to = -3.7 · 10 ^-6 1 / ° С, with R _i = 500 Ohms nonlinearity will reach Δα _to = -12.8 · 10 ^-6 1 / ° С.

Thus, by selecting the nominal value of a thermally independent resistor R _i, it is possible to convert the positive nonlinearity of the DC circuit of the bridge circuit into negative.

The value of the thermally independent resistor R _i should be determined on the basis of the need to ensure the negative nonlinearity of the DC circuit of the bridge circuit, which satisfies the following system that determines the scope of the prototype:

$$\{\begin{array}{c}{\alpha }_{d\mathrm{about}}^{+}>0,325\cdot {\alpha }_{\mathrm{at}sx}^{+}+0,05\cdot \mathrm{one}{0}^{-\mathrm{four}}\mathrm{one}{/}^{\circ }\mathrm{FROM};\\ \Delta {\alpha }_{d\mathrm{about}}\le -2,0\cdot \mathrm{one}{0}^{-6}\mathrm{one}{/}^{\circ }\mathrm{FROM}.\end{array}.\left(6\right)$$

To determine the value of the resistor R _i in order to obtain the non-linearity of the DC current factor of the bridge circuit Δα _to ≤-2 · 10 ^-6 1 / ° С, it is necessary to solve the following equation taking into account dependence (5):

$$\begin{array}{c}\frac{R_{\mathrm{at}x}{\alpha }_d^{+}\left(\mathrm{one}+{\alpha }_{\mathrm{at}x}^{+}\Delta t^{+}\right)+R_i\left({\alpha }_{\mathrm{at}x}^{+}+{\alpha }_d^{+}+{\alpha }_{\mathrm{at}x}^{+}{\alpha }_d^{+}\Delta t^{+}\right)}{R_{\mathrm{at}x}\left(\mathrm{one}+{\alpha }_{\mathrm{at}x}^{+}\Delta t^{+}\right)+R_i}-\\ -\frac{R_{\mathrm{at}x}{\alpha }_d^{-}\left(\mathrm{one}+{\alpha }_{\mathrm{at}x}^{-}\Delta t^{-}\right)+R_i\left({\alpha }_{\mathrm{at}x}^{-}+{\alpha }_d^{-}+{\alpha }_{\mathrm{at}x}^{-}{\alpha }_d^{-}\Delta t^{-}\right)}{R_{\mathrm{at}x}\left(\mathrm{one}+{\alpha }_{\mathrm{at}x}^{-}\Delta t^{-}\right)+R_i}=-2,0\cdot \mathrm{one}{0}^{-6}\mathrm{one}{/}^{\circ }C\end{array}.\left(7\right)$$

In order to determine the areas for converting the positive non-linearity of the DC current factor of the sensor bridge circuit to negative, the nominal value of the thermally independent resistor R _i was estimated, which is necessary to convert the positive non-linearity of the DC current of the bridge circuit to negative, when possible. For this, equation (7) was solved numerically under the following conditions:

1. The input resistance of the bridge circuit: R _I = 1000 Ohms;

2. TKH strain gauge takes the values: α _d = (0 ... 10) · 10 ^-4 1 / ° C;

3. the nonlinearity of the TSC strain gauge takes the following values: Δα _d = (0 ... 10) · 10 ^-6 1 / ° C;

4. TKS input resistance: α _I = (0, 0.5, 1.3, 10) · 10 ^-4 1 / ° C;

5. nonlinearity TCR of the input resistance: Δα _in = -5 · 10 ^-6 1 / ° C;

6. TCS compensation resistor: α _to = 4 · 10 ^-3 1 / ° C.

When evaluating the transform domain positive nonlinearity TCF bridge circuit in the negative considered one of the limiting values of the nonlinearity of the input resistance TCS (Δα _in = -5 · 10 ^-6 1 / °° C), as a numerical experiment was conducted previously, which revealed that the effect of the nonlinearity of the TCS of the input resistance to the limiting value of the nonlinearity of the TFC of the bridge circuit, at which it is possible to convert the positive nonlinearity of the TFC of the bridge circuit into negative, in the entire range of possible values of the TCS of the input about resistance and its nonlinearity is small (no more than 2%).

Since with an increase in the value of the thermally dependent resistor R _i , the sensitivity of the sensor decreases, when calculating the value of the resistor R _i , the smaller of the roots should be chosen.

The calculation results are summarized in table 2, which introduced the smaller of the roots of equation (7), when the conversion of the positive nonlinearity of the DC coupling circuit of the bridge circuit into negative is possible.

table 2 The limits of the region of transformation of the positive nonlinearity of the DC circuit of the bridge circuit into negative α ⁺ _in · 10 ^-4 , 1 / ° С, 1 / ° С Δα _in · 10 ^-6 , 1 / ° С Δα _d · 10 ^-6 , 1 / ° С α ^- _d · 10 ^-4 , 1 / ° С R _i , Ohm 0 -5 0 0,000 666 10,000 800 0 -5 one 0,000 1499 10,000 1999 0 -5 2 0,000 3,998 10,000 7996 0 -5 3 0,000 No roots 0.5 -5 0 0,000 598 1,216 1000 2,857 9660379 2,858 No roots 0.5 -5 one 0,000 1345 1,899 2332103 1,900 No roots 0.5 -5 2 0,000 3603 0.942 3831209 0.943 No roots 0.5 -5 3 0,000 No roots one -5 0 0,000 449

continued table 2 α ⁺ _in · 10 ^-4 , 1 / ° C, 1 / ° C Δα _in · 10 ^-6 , 1 / ° C Δα _d · 10 ^-6 , 1 / ° C α ^- _d · 10 ^-4 , 1 / ° C R _i , Ohm one -5 0 1,000 999 1,464 33614 1,465 No roots one -5 one 0 983 0,020 1000 0.970 601254 0.971 No roots one -5 2 0,000 2542 0.478 18979006 0.479 No roots one -5 3 0,000 No roots 3 -5 0 0,000 102 1,852 488 1,853 No roots 3 -5 one 0,000 170 1,397 653 1,398 No roots 3 -5 5 0,000 996 0.001 999 3 -5 5 0.002 1001 0,085 1534

continuation of table 2 α ⁺ _in · 10 ^-4 , 1 / ° C, 1 / ° C Δα _in · 10 ^-6 , 1 / ° C Δα _d · 10 ^-6 , 1 / ° C α ^- _d · 10 ^-4 , 1 / ° C R _i , Ohm 3 -5 5 0,086 No roots 3 -5 6 0,000 No roots 10 -5 0 0,000 10 8,282 106 8,283 No roots 10 -5 one 0,000 fifteen 7,830 136 7,831 No roots 10 -5 5 0,000 36 6,523 227 6,524 No roots 10 -5 10 0,000 66 5,344 316 5,345 No roots

An increase in the value of the resistor R _i leads to a decrease in the sensitivity of the sensor, compensated by an increase in the supply voltage of the bridge circuit. Due to the complexity of compensating for a decrease in sensitivity by more than a factor of 2 by changing the supply voltage, a thermally independent resistor R _i should be taken with a nominal value not exceeding the input resistance of the sensor bridge circuit (R _i ≤R _in ). Given this limitation, the conversion of the positive nonlinearity of the DC circuit of the bridge circuit into negative is possible in a limited area, presented in figure 2 and table 3.

Analyzing the limitation of the region of transformation of the positive nonlinearity of the DC circuit of the bridge circuit into negative, the following conclusions can be formulated:

1. The scope of the scheme under consideration for converting the positive non-linearity of the DC-type strain gauge strain gauge to negative, taking into account the restrictions R _i ≤R _{I, is} limited from above (see Fig. 3);

2. The scope of the considered circuit is reduced with a decrease in the TCS of the input resistance of the bridge circuit from α _in = 10 ^-3 1 / ° C to α _in = 10 ^-4 1 / ° C (see figure 3);

3. With an increase in the non-linearity of the DTC of strain gauges from Δα _d = 0 1 / ° C to Δα _d = 10-5 1 / ° C, the conversion region of the circuit under consideration is reduced by 35% of the conversion region at Δα _d = 0 1 / ° C (see figure 3);

4. With the growth of the DC coupling factor of the bridge circuit, the region of transformation of the positive nonlinearity of the DC coupling coefficient of the bridge circuit decreases to Δα _d ≥10 ^-4 1 / ° С (see Fig. 3).

Table 3 The area of transformation of the positive nonlinearity of the DC circuit of the bridge circuit into negative The non-linearity of the TSC strain gage Δα _d · 10 ^-6 , 1 / ° C TCS of input resistance, α ⁺ _in · 10 ^-4 , 1 / ° С The maximum value of the TCI of the strain gauge α ⁺ _dmax · 10 ^-4 , 1 / ° C 0,0 0-0.5 -17.5700α _in +10 ^-3 0.5-1.0 -0.4280α ⁺ _in + 1.429 · 10 ^-4 1.0-3.0 0.4255α ⁺ _in + 0.5755 · 10 ^-4 3.0-10.0 0.9186α ⁺ _in -0.903710 ^-4 1,0 1.0-3.0 0.6885α ⁺ _in -0.6685 · 10 ^-4 3.0-10.0 0.9190α ⁺ _in -1.3600 · 10 ^-4

Continuation of table 3 The non-linearity of the TSC strain gage Δα _d · 10 ^-6 , 1 / ° C TCS of input resistance, α ⁺ _in · 10 ^-4 , 1 / ° С The maximum value of the TCI of the strain gauge α ⁺ _dmax · 10 ^-4 , 1 / ° C 2.0 1.8-3.0 0.7767α ⁺ _BX -1.3170 · 10 ^-4 3.0-10.0 0.9191α ⁺ _in -1.7444 · 10 ^-4 3.0 2.2-3.0 0.7488α ⁺ _BX -1.5743 · 10 ^-4 3.0-10.0 0.9197α ⁺ _BX -2.087110 ^-4 4.0 2.7-10.0 0.9239α ⁺ _in -2.4357 · 10 ^-4 5,0 3.0-10.0 0.9317α ⁺ _in -2.794110 ^-4 6.0 3.4-10.0 0.9367α ⁺ _in -3.106710 ^-4 7.0 3.7-10.0 0.9411α ⁺ _in -3.3981.10 ^-4 8.0 4.0-10.0 0.9447α ⁺ _in -3.6677 · 10 ^-4 9.0 4.2-10.0 0.9481α ⁺ _in -3.9240 · 10 ^-4 10.0 4.4-10.0 0.9511α ⁺ _in -4.1667 · 10 ^-4

To verify the correctness of the proposed solution, we will calculate the compensation elements and the multiplicative sensitivity of the sensor after compensation.

Example

Compensate for the multiplicative temperature error and determine the temperature sensitivities of the sensor with an equal arm bridge measuring circuit at temperatures corresponding to the limits of the operating temperature range, taking into account the following initial data:

- resistance of strain gages: R ₁ = R ₂ = R ₃ = R ₄ = 1000 Ohms;

- TCS of a thermally dependent resistor R _αout : α _к = 4 · 10 ^-3 1 / ° С;

- TCS output resistance at temperatures corresponding to the limits of the operating temperature range: α ⁺ _out = 5 · 10 ^-4 1 / ° C, α ^- _out = 5.05 · 10 ^-4 1 / ° C;

- TCS input resistance at temperatures corresponding to the limits of the operating temperature range: α ⁺ _in = 5 · 10 ^-4 1 / ° C, α ^- _in = 5.05 · 10 ^-4 1 / ° C;

- TCC strain gages at temperatures corresponding to the limits of the operating temperature range: α ⁺ _d = 1.75 · 10 ^-4 1 / ° C, α ^- _d = 1.7 · 10 ^-4 1 / ° C;

;- the total relative change in the resistance of the strain gages at the nominal value of the measured parameter: $${\displaystyle \sum _{j=\mathrm{one}}^{\mathrm{four}}{\epsilon }_j=0,0\mathrm{one}}$$

;

- temperature range of operation of the sensor: 20 ± 100 ° C;

- supply voltage Upit = 10V.

Since the non-linearity of the DC current factor of the bridge circuit Δα _d = α ⁺ _d -α ^- _d = 5 · 10 ^-6 1 / ° С and the resistance of the power source is negligible, to ensure the negative non-linearity of the DC current factor of the bridge circuit of the sensor, a thermally independent resistor R _i must be included in the power circuit . To verify the applicability of the proposed circuit method, it is necessary to verify the affiliation of the DC bridge circuit and its non-linearity of the region specified in table 3. In accordance with table 3 and taking into account the fact that α ^- _d = 1.7 · 10 ^-4 1 / ° С, α ⁺ _in = 5 · 10 ^-4 1 / ° С and Δα _d = 5 · 10 ^-6 1 / ° С, the inequality that defines the region of transformation of the positive nonlinearity of the DC-link circuit of the bridge circuit into negative will take the form:

. $$\mathrm{one},7\cdot \mathrm{one}{0}^{-\mathrm{four}}\mathrm{one}{/}^{\circ }\mathrm{FROM}<0,93\mathrm{one}7\cdot {\alpha }_{\mathrm{at}x}^{+}-2,79\mathrm{four}\mathrm{one}\cdot \mathrm{one}{0}^{-\mathrm{four}}=\mathrm{one},86\mathrm{four}\cdot \mathrm{one}{0}^{-\mathrm{four}}\mathrm{one}{/}^{\circ }\mathrm{FROM}$$

.

Fulfillment of the above inequality allows us to conclude that it is possible to obtain a negative nonlinearity of the DC-link TCD of Δα _{up to} ≤2 · 10 ^-6 1 / ° С using the above-described circuit method. The sensor itself must be connected to a load with a rating of R _n > 500 kOhm. To determine the required value of the resistor R _i it is necessary when connecting the sensor to the load with a rating of R _n > 500 kOhm to solve equation (7):

$$\begin{array}{c}\frac{\mathrm{one}{0}^3\cdot \mathrm{one},75\cdot \mathrm{one}{0}^{-\mathrm{four}}\cdot \mathrm{one},05+R_i\left(5\cdot \mathrm{one}{0}^{-\mathrm{four}}+\mathrm{one},75\cdot \mathrm{one}{0}^{-\mathrm{four}}+8,75\cdot \mathrm{one}{0}^{-6}\right)}{\mathrm{one}{0}^3\cdot \mathrm{one},05+R_i}-\\ -\frac{\mathrm{<figure-callout\; id="0"\; label="one"\; filenames="00000028.png"\; state="\{\{state\}\}">one</figure-callout>}{0}^3\cdot \mathrm{one},7\cdot \mathrm{one}{0}^{-\mathrm{four}}\cdot 0,9\mathrm{four}95+R_i\left(5,05\cdot \mathrm{one}{0}^{-\mathrm{four}}+\mathrm{one},7\cdot \mathrm{one}{0}^{-\mathrm{four}}-8,585\cdot \mathrm{one}{0}^{-6}\right)}{\mathrm{one}{0}^3\cdot 0,9\mathrm{four}95+R_i}=-2,0\cdot \mathrm{one}{0}^{-6}\mathrm{one}{/}^{\circ }C.\end{array}$$

The solution to the equation is the nominal R _i = 451.880 Ohms. For the subsequent compensation of the multiplicative error, it is necessary to determine the TCD of the bridge circuit after turning on the resistor R _i , which requires the calculation of the output voltage when the sensor is operating in idle mode. At normal temperature, the output signal of the bridge circuit in accordance with (2) will be:

$$U_{\mathrm{at}sx}=2,5\frac{\mathrm{one}{0}^3}{\mathrm{one}{0}^3+\mathrm{four}5\mathrm{one},880}0,0\mathrm{one}=\mathrm{one}7,2\mathrm{one}905\mathrm{four}m\mathrm{AT}$$

In accordance with (1) at a temperature of t ⁺ = 120 ° C, the output voltage will be:

$$U_{\mathrm{at}sxt}^{+}=2,5\frac{\mathrm{one}{0}^3\cdot \mathrm{one},05\cdot \mathrm{one},0\mathrm{one}75}{\mathrm{one}{0}^3\cdot \mathrm{one},05+\mathrm{four}5\mathrm{one},880}0,0\mathrm{one}=\mathrm{one}7,78396\mathrm{one}m\mathrm{AT}$$

and at t ^- = -80 ° C the output voltage of the sensor will be:

$$U_{\mathrm{at}sxt}^{-}=2,5\frac{\mathrm{one}{0}^3\cdot 0,9\mathrm{four}95\cdot 0,9830}{\mathrm{one}{0}^3\cdot 0,9\mathrm{four}95+\mathrm{four}5\mathrm{one},880}0,0\mathrm{one}=\mathrm{one}6,650703m\mathrm{AT}$$

Given the calculated values of the output voltages of the DC circuit bridge at a temperature t ⁺ = 120 ° C will be:

$${\alpha }_{d\mathrm{about}}^{+}=\frac{U_{\mathrm{at}sxt}^{+}-U_{\mathrm{at}sx}}{U_{\mathrm{at}sx}\Delta t^{+}}=\frac{0,56\mathrm{four}907}{\mathrm{one}7,2\mathrm{one}905\mathrm{four}\cdot \mathrm{one}00}=3,28\mathrm{one}\cdot \mathrm{one}{0}^{-\mathrm{four}}\mathrm{one}{/}^{\circ }\mathrm{FROM}$$

At a temperature of t ^- = -80 ° C, the TFC of the bridge circuit will be:

$${\alpha }_{d\mathrm{about}}^{-}=\frac{U_{\mathrm{at}sx}^{-}-U_{\mathrm{at}sx}}{U_{\mathrm{at}sx}\Delta t^{-}}=\frac{-0,56835\mathrm{one}}{\mathrm{one}7,2\mathrm{one}905\mathrm{four}\cdot \left(-\mathrm{<figure-callout\; id="0"\; label="one"\; filenames="00000028.png"\; state="\{\{state\}\}">one</figure-callout>}00\right)}=3,30\mathrm{one}\cdot \mathrm{one}{0}^{-\mathrm{four}}\mathrm{one}{/}^{\circ }\mathrm{FROM}$$

Thus, the inclusion of a thermally independent resistor R _i in the power circuit made it possible to obtain the required negative nonlinearity of the DC-link TCD of the bridge circuit Δα _to = -2 · 10 ^-6 1 / ° С. Taking into account the TCS of the output resistance (α ⁺ _bx = 5 · 10 ^-4 1 / ° C), the obtained TFC of the bridge circuit (α ⁺ _to = 3.281 · 10 ^-4 1 / ° C) system (6), which determines the scope of the prototype will take the form:

$$\{\begin{array}{c}3,28\mathrm{one}\cdot \mathrm{one}{0}^{-\mathrm{four}}\mathrm{one}{/}^{\circ }\mathrm{FROM}>0,325\cdot {\alpha }_{\mathrm{at}sx}^{+}+0,05\cdot \mathrm{one}{0}^{-\mathrm{four}}=\mathrm{one},675\cdot \mathrm{one}{0}^{-\mathrm{four}}\mathrm{one}{/}^{\circ }\mathrm{FROM};\\ \Delta {\alpha }_{d\mathrm{about}}=-2\cdot \mathrm{one}{0}^{-6}\mathrm{one}{/}^{\circ }\mathrm{FROM}.\end{array}$$

The above system confirms that, in accordance with the prototype, to compensate for the multiplicative temperature error, it is possible to include a thermally dependent resistor R _αout , shunted by a thermally independent resistor R _two , in the output diagonal of the sensor bridge circuit. The load resistance should be Rn≤2kOhm. Suppose that the sensor after switching on the compensation resistors will be connected to the load Rn = 2kOhm. To calculate the values of the compensation resistors, it is necessary to solve the following system of equations in accordance with the prototype:

$$\{\begin{array}{c}\frac{\left(3\cdot \mathrm{one}{0}^3\cdot \left(R_{\alpha \mathrm{at}\text{A.}sx}+R_{d\mathrm{at}sx}\right)+R_{\alpha \mathrm{at}sx}{R}_{d\mathrm{at}sx}\right)\left(R_{\alpha \mathrm{at}\text{A.}sx}\cdot \mathrm{one},\mathrm{four}+R_{d\mathrm{at}sx}\right)\cdot \mathrm{one},0328}{\left(R_{\alpha \mathrm{at}\text{A.}sx}+R_{d\mathrm{at}sx}\right)\left[\left(2\cdot \mathrm{one}{0}^3+\mathrm{one}{0}^3\cdot \mathrm{one},05\right)\left(R_{\alpha \mathrm{at}sx}\cdot \mathrm{one},\mathrm{four}+R_{d\mathrm{at}sx}\right)+R_{\alpha \mathrm{at}\text{A.}sx}{R}_{d\mathrm{at}sx}\cdot \mathrm{one},\mathrm{four}\right]\cdot \mathrm{one}00}-0,0\mathrm{one}=0;\\ \frac{\left(3\cdot \mathrm{one}{0}^3\cdot \left(R_{\alpha \mathrm{at}\text{A.}sx}+R_{d\mathrm{at}sx}\right)+R_{\alpha \mathrm{at}\text{A.}sx}{R}_{d\mathrm{at}sx}\right)\left(R_{\alpha \mathrm{at}\text{A.}sx}\cdot \mathrm{one},\mathrm{four}+R_{d\mathrm{at}sx}\right)\cdot \mathrm{one},0328}{\left(R_{\alpha \mathrm{at}\text{A.}sx}+R_{d\mathrm{at}sx}\right)\left[\left(2\cdot \mathrm{one}{0}^3+\mathrm{one}{0}^3\cdot \mathrm{one},05\right)\left(R_{\alpha \mathrm{at}sx}\cdot \mathrm{one},\mathrm{four}+R_{d\mathrm{at}sx}\right)+R_{\alpha \mathrm{at}\text{A.}sx}{R}_{d\mathrm{at}sx}\cdot \mathrm{one},\mathrm{four}\right]\cdot \mathrm{one}00}-0,0\mathrm{one}-\\ -\frac{\left(3\cdot \mathrm{one}{0}^3\cdot \left(R{}_{\alpha \mathrm{at}\text{A.}sx}+R_{d\mathrm{at}sx}\right)+R_{\alpha \mathrm{at}\text{A.}sx}{R}_{d\mathrm{at}sx}\right)\left(R_{\alpha \mathrm{at}\text{A.}sx}\cdot 0,6+R_{d\mathrm{at}sx}\right)\cdot 0,9670}{\left(R_{\alpha \mathrm{at}\text{A.}sx}+R_{d\mathrm{at}sx}\right)\left[\left(2\cdot \mathrm{one}{0}^3+\mathrm{one}{0}^3\cdot 0,9\mathrm{four}95\right)\left(R_{\alpha \mathrm{at}\text{A.}sx}\cdot 0,6+R_{d\mathrm{at}sx}\right)+R_{\alpha \mathrm{at}\text{A.}sx}{R}_{d\mathrm{at}sx}\cdot 0,6\right]\cdot \left(-\mathrm{one}00\right)}-0,0\mathrm{one}=0.\end{array}$$

The solution to this system of equations is the following values of the compensation resistors: R _αout = 132.735 Ohm, R _double = 51102.770 Ohm. The electrical circuit after turning on the compensation resistors will take the form shown in Fig.3.

To determine the multiplicative sensitivity of the sensor to temperature, it is necessary to determine the voltage values of the output signal at normal temperature and temperatures corresponding to the limits of the operating temperature range. At normal temperature, the resistance of the resistor K _αout , shunted by the resistor R _two , will be:

$$R_{\mathrm{uh}}=\frac{R_{\alpha \mathrm{at}\text{A.}sx}{R}_{d\mathrm{at}sx}}{R_{\alpha \mathrm{at}\text{A.}sx}+R_{d\mathrm{at}sx}}=\frac{\mathrm{one}32,735\cdot 5\mathrm{one}\mathrm{one}02,770}{\mathrm{one}32,735+5\mathrm{one}\mathrm{one}02,770}=\mathrm{one}32,39\mathrm{one}\mathrm{ABOUT}m$$

Therefore, in accordance with the description of the prototype, the output voltage will be:

$$\begin{array}{c}{U}_{\mathrm{at}sx}=U_{P\mathrm{and}t}\frac{k}{{\left(k+\mathrm{one}\right)}^2}\cdot \frac{R_{\mathrm{at}x}}{R_{\mathrm{at}x}+R_i}\cdot \frac{R_n}{R{}_n+R_{\mathrm{at}sx}+R_{\mathrm{uh}}}{\displaystyle \sum _{i=\mathrm{one}}^{\mathrm{four}}{\epsilon }_i=}\\ =2,5\frac{\mathrm{one}{0}^3}{\mathrm{one}{0}^3+\mathrm{four}5\mathrm{one},880}\cdot \frac{2\cdot \mathrm{one}{0}^3}{2\cdot \mathrm{one}{0}^3+\mathrm{one}{0}^3+\mathrm{one}32,39\mathrm{one}}0,0\mathrm{one}=\mathrm{one}0,99\mathrm{four}\mathrm{one}92m\mathrm{AT}.\end{array}$$

At a temperature of t ⁺ = 120 ° C:

; $$R_{\mathrm{uh}}=\frac{R_{\alpha \mathrm{at}\text{A.}sx}\left(\mathrm{one}+{\alpha }_{\mathrm{to}}\Delta t^{+}\right)R_{d\mathrm{at}sx}}{R_{\alpha \mathrm{at}\text{A.}sx}\left(\mathrm{one}+{\alpha }_{\mathrm{to}}\Delta t^{+}\right)+R_{d\mathrm{at}sx}}=\frac{\mathrm{one}32,735\cdot \mathrm{one},\mathrm{four}\cdot 5\mathrm{one}\mathrm{one}02,770}{\mathrm{one}32,735\cdot \mathrm{one},\mathrm{four}\cdot 5\mathrm{one}\mathrm{one}02,770}=\mathrm{one}85,\mathrm{one}56\mathrm{ABOUT}m$$

;

$$\begin{array}{c}{U}_{\mathrm{at}sxt}^{+}=U_{P\mathrm{and}t}\frac{k}{{\left(k+\mathrm{one}\right)}^2}\cdot \frac{R_{\mathrm{at}x}\left(\mathrm{one}+{\alpha }_{\mathrm{at}x}^{+}\Delta t^{+}\right)\left(\mathrm{one}+{\alpha }_d^{+}\Delta t^{+}\right)}{R_{\mathrm{at}x}\left(\mathrm{one}+{\alpha }_{\mathrm{at}x}^{+}\Delta t^{+}\right)+R_i}\cdot \frac{R_n}{R_n+R_{\mathrm{at}sx}\left(\mathrm{one}+{\alpha }_{\mathrm{at}sx}^{+}\Delta t^{+}\right)+R_{\mathrm{uh}}}{\displaystyle \sum _{i=\mathrm{one}}^{\mathrm{four}}{\epsilon }_i=}\\ =2,5\frac{\mathrm{one}{0}^3\cdot \mathrm{one},05\cdot \mathrm{one},0\mathrm{one}75}{\mathrm{one}{0}^3\cdot \mathrm{one},05+\mathrm{four}5\mathrm{one},880}\cdot \frac{2\cdot \mathrm{one}{0}^3}{2\cdot \mathrm{one}{0}^3+\mathrm{one}{0}^3\cdot \mathrm{one},05+\mathrm{one}85,\mathrm{one}56}0,0\mathrm{one}=\mathrm{one}0,99\mathrm{four}\mathrm{one}9\mathrm{one}m\mathrm{AT}.\end{array}$$

At a temperature of t ^- = -80 ° C:

; $$R_{\mathrm{uh}}=\frac{R_{\alpha \mathrm{at}\text{A.}sx}\left(\mathrm{one}+{\alpha }_{\mathrm{to}}\Delta t^{-}\right)R_{d\mathrm{at}sx}}{R_{\alpha \mathrm{at}\text{A.}sx}\left(\mathrm{one}+{\alpha }_{\mathrm{to}}\Delta t^{-}\right)+R_{d\mathrm{at}sx}}=\frac{\mathrm{one}32,735\cdot 0,6\cdot 5\mathrm{one}\mathrm{one}03}{\mathrm{one}32,735\cdot 0,6+5\mathrm{one}\mathrm{one}03}=79,5\mathrm{one}7\mathrm{ABOUT}m$$

;

$$\begin{array}{c}{U}_{\mathrm{at}sxt}^{-}=U_{P\mathrm{and}t}\frac{k}{{\left(k+\mathrm{one}\right)}^2}\cdot \frac{R_{\mathrm{at}x}\left(\mathrm{one}+{\alpha }_{\mathrm{at}x}^{-}\Delta t^{-}\right)\left(\mathrm{one}+{\alpha }_d^{-}\Delta t^{-}\right)}{R_{\mathrm{at}x}\left(\mathrm{one}+{\alpha }_{\mathrm{at}x}^{-}\Delta t^{-}\right)+R_i}\cdot \frac{R_n}{R_n+R_{\mathrm{at}sx}\left(\mathrm{one}+{\alpha }_{\mathrm{at}sx}^{-}\Delta t^{-}\right)+R_{\mathrm{uh}}}{\displaystyle \sum _{i=\mathrm{one}}^{\mathrm{four}}{\epsilon }_i=}\\ =2,5\frac{\mathrm{one}{0}^3\cdot 0,9\mathrm{four}95\cdot 0,983}{\mathrm{one}{0}^3\cdot 0,9\mathrm{four}95+\mathrm{four}5\mathrm{one},880}\cdot \frac{2\cdot \mathrm{one}{0}^3}{2\cdot \mathrm{one}{0}^3+\mathrm{one}{0}^3\cdot 0,9\mathrm{four}95+79,5\mathrm{one}7}0,0\mathrm{one}=\mathrm{one}0,99\mathrm{four}\mathrm{one}30m\mathrm{AT}.\end{array}$$

Multiplicative temperature sensitivity of the sensor at t ⁺ = 120 ° С:

. $$S{}_{kt}{}^{+}=\frac{U_{\mathrm{at}sxt}^{+}-U_{\mathrm{at}sx}}{U_{\mathrm{at}sx}\Delta t^{+}}=\frac{-\mathrm{one}{0}^{-6}}{\mathrm{one}0,99\mathrm{four}\mathrm{one}92\cdot \mathrm{one}00}=-9,096\cdot \mathrm{one}{0}^{-\mathrm{<figure-callout\; id="0"\; label="one"\; filenames="00000028.png"\; state="\{\{state\}\}">one</figure-callout>}0}\mathrm{one}{/}^{\circ }\mathrm{FROM}$$

.

Multiplicative temperature sensitivity of the sensor at t ^- = -80 ° С:

$$S_{kt}^{-}=\frac{U_{\mathrm{at}sxt}^{-}-U_{\mathrm{at}sx}}{U_{\mathrm{at}sx}\Delta t^{-}}=\frac{-6,2\cdot \mathrm{one}{0}^{-5}}{\mathrm{one}0,99\mathrm{four}\mathrm{one}92\cdot \left(-\mathrm{<figure-callout\; id="0"\; label="one"\; filenames="00000028.png"\; state="\{\{state\}\}">one</figure-callout>}00\right)}=5,639\cdot \mathrm{one}{0}^{-8}\mathrm{one}{/}^{\circ }\mathrm{FROM}$$

Thus, the sensitivity obtained after compensation is much less than the maximum permissible temperature sensitivity (S _ktdop = 10 ^-4 1 / ° С).

The proposed method for the full compensation of the multiplicative temperature error showed a high accuracy of compensation with positive nonlinearity of the DC circuit of the bridge circuit, which depends only on the accuracy of manufacturing compensation resistors and the accuracy of determining the physical characteristics of strain gauges.

Claims (1)

A method for tuning strain gauge sensors with a bridge measuring circuit by a multiplicative temperature error, taking into account the positive non-linearity of the temperature characteristic of the sensor output signal, namely, when the load resistance R _n > 500 kOhm, the temperature sensitivity coefficient (TCR) of the bridge circuit is determined α ⁺ _to and α ^- _up to a temperature range Δt ⁺ = t ⁺ -t ₀ and Δt ^- = tt _0, where t ₀ ⁺ t, t ^- - normal temperature, the upper and lower limit of the operating temperature range, respectively, calculated nelineynos s TCF bridge circuit _to Δα = α ⁺ _to -α ^- _before, if the nonlinearity TCF bridge circuit takes a negative value, the load resistance R k _n ≤2 determine the output resistance bridge circuit, the output TCR resistance bridge circuit for temperature range ⁺ Δt and Δt ^- , check the presence of the _{DC-link TCD} and the non-linearity of the DC link of the bridge in the field of application and, if the sensor parameters are in the field of application, calculate the values of the resistors R _{α out} and R _two , set the _temperature- dependent resistor R _{α out} , shunted by thermo by an independent resistor R _two , into the output diagonal of the sensor bridge circuit, characterized in that if the non-linearity of the TFC of the bridge circuit takes a positive value, then after determining the non-linearity of the TFC of the bridge circuit and before determining the output resistance of the bridge circuit, as well as the TCS of the output resistance of the bridge circuit, the positive nonlinearity of the DC circuit of the bridge circuit into negative by including a thermally independent resistor R _i in the diagonal of the power supply of the bridge circuit, for which it is determined when R _n > 500 kOhm TCR of the strain gages α ⁺ _d and α ^- _d for the temperature range Δt ⁺ and Δt ^- respectively, calculate the non-linearity of the TFC of the strain gages Δα _d = α ⁺ _d -α ^- _d , determine the value of the input resistance R _in , TKC of the input resistance α ⁺ _in , α ^- _in for the temperature range Δt ⁺ and Δt ^- respectively, identify the presence of α ⁺ _d and Δα _d in the area specified by table 3 and, if α ⁺ _d and Δα _d satisfy the conditions given in the table
The non-linearity of the TSC strain gage Δα _d · 10 ^-6 , 1 / ° C TKC of input resistance, α ⁺ _in · 10 ^-4 , 1 / ° С The maximum value of the TCI of the strain gauge α ⁺ _dmax · 10 ^-4 , 1 / ° С 0,0 0-0.5 -17.5700α _in +10 ^-3 0.5-1.0 -0.4280α ⁺ _in + 1.429 · 10 ^-4 1.0-3.0 0.4255α ⁺ _in + 0.5755 · 10 ^-4 3.0-10.0 0.9186α ⁺ _in -0.903710 ^-4 1,0 1.0-3.0 0.6885α ⁺ _in -0.6685 · 10 ^-4 3.0-10.0 0.9190α ⁺ _in -1.3600 · 10 ^-4 2.0 1.8-3.0 0.7767α ⁺ _BX -1.3170 · 10 ^-4 3.0-10.0 0.9191α ⁺ _in -1.7444 · 10 ^-4 3.0 2.2-3.0 0.7488α ⁺ _BX -1.5743 · 10 ^-4 3.0-10.0 0.9197α ⁺ _BX -2.087110 ^-4 4.0 2.7-10.0 0.9239α ⁺ _in -2.4357 · 10 ^-4 5,0 3.0-10.0 0.9317α ⁺ _in -2.794110 ^-4 6.0 3.4-10.0 0.9367α ⁺ _in -3.106710 ^-4 7.0 3.7-10.0 0.9411α ⁺ _in -3.3981.10 ^-4 8.0 4.0-10.0 0.9447α ⁺ _in -3.6677 · 10 ^-4 9.0 4.2-10.0 0.9481α ⁺ _in -3.9240 · 10 ^-4 10.0 4.4-10.0 0.9511α ⁺ _in -4.1667 · 10 ^-4
determine the value of the thermally independent resistor R _i by solving the equation:
$$\begin{array}{c}\frac{R_{\mathrm{at}x}{\alpha }_d^{+}\left(\mathrm{one}+{\alpha }_{\mathrm{at}x}^{+}\Delta t^{+}\right)+R_i\left({\alpha }_{\mathrm{at}x}^{+}+{\alpha }_d^{+}+{\alpha }_{\mathrm{at}x}^{+}{\alpha }_d^{+}\Delta t^{+}\right)}{R_{\mathrm{at}x}\left(\mathrm{one}+{\alpha }_{\mathrm{at}x}^{+}\Delta t^{+}\right)+R_i}-\\ -\frac{R_{\mathrm{at}x}{\alpha }_d^{-}\left(\mathrm{one}+{\alpha }_{\mathrm{at}x}^{-}\Delta t^{-}\right)+R_i\left({\alpha }_{\mathrm{at}x}^{-}+{\alpha }_d^{-}+{\alpha }_{\mathrm{at}x}^{-}{\alpha }_d^{-}\Delta t^{-}\right)}{R_{\mathrm{at}x}\left(\mathrm{one}+{\alpha }_{\mathrm{at}x}^{-}\Delta t^{-}\right)+R_i}=-2\cdot \mathrm{one}{0}^{-6}\mathrm{one}{/}^{\circ }C,\end{array}$$

include the thermally independent resistor R _i in the diagonal of the power supply of the sensor bridge circuit, determine the DC current factor of the sensor bridge circuit and its nonlinearity after turning on the thermally independent resistor R _i .

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