Intro
Two main types of nonidealities can be observed in the output of sense elements: nonlinearity and temperature drift. Temperature drift is the behavior where for the same given physical quantity being measured, the sense element output is different at different temperatures. If this change in output due to temperature is not adequately compensated, it will appear as if the physical quantity being measured is changing.
This article focuses on how to use sensor signal conditioner compensation techniques to address linear temperature drift. Specifically, we show how sensor signal conditioners use polynomials to correct for temperature drift in the output of sense elements.
Sensors
A sensor or transmitter consists of a sense element and a signal conditioner. The sense element is used to convert a specific physical property into electrical signals. The signal conditioner then processes these electrical signals and produces an output signal that is sent to a controller. It is during this processing that the signal conditioner compensates for the sense element output nonidealities, such as temperature drift. Figure 1 shows a block diagram, a sensor, and how implementing a signal conditioner can compensate for temperature drift.
Fig. 1: Block diagram of a sensor showing the physical property, x, sense element output, y, and conditioned sensor output, z. The sense element output varies with temperature while the conditioned sensor output does not.
In Figure 1, one can see how for a given value of x, the sense element output is different at different temperatures. A sensor signal conditioner processes the temperaturevarying sense element output to produce a conditioned output in order to minimize variance in temperature.
Temperature drift
The offset and span of a sense element can change nonlinearly with temperature. However, to effectively compensate for this behavior, linear variation is assumed so that polynomials can be used since they are easier to understand. Linear temperature drift can be modeled using Equation 1:
a_{0} = sense element offset at room temperature, or the sense element output when the physical quantity being measured is at its minimum value and at room temperature.
a_{1} = sense element offset temperature drift coefficient, or the relative change in offset of sense element with a given change in temperature.
b_{0} = sense element span at room temperature, or the difference in the sense element outputs at maximum and minimum physical quantity being measured.
b_{1} = sense element span temperature drift coefficient, or the relative change in span of sense element with a given change in temperature.
T = the difference between the sense element temperature and room temperature.
X = physical quantity being measured.
Y = output of the sense element as a function of x.
Equation 1 has the following characteristics:

The sense element offset has a firstorder temperature coefficient. This implies that the offset changes linearly with temperature.
 The sense element span also has a firstorder temperature coefficient similar to that of offset, so that the span also changes linearly with temperature.
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Output temperature drift
The ideal goal of temperature compensation of the sense element output is to make the curves at different temperatures become the same at all temperatures. In practice, however, it may not be possible to achieve identical curves at all temperatures, resulting in residual inaccuracy in the compensated output.
Sensor signal conditioners, such as the PGA900, are used to correct such temperature drifts. More specifically, signal conditioners, such as the PGA300, implement temperature compensation using polynomials.
Temperature compensation using polynomials
Consider the polynomial equation given by Equation 2.
Where h_{i}, i = 0 to H, and g_{j}, j = 0 to G, are the polynomial coefficients, y is the output of sense element, and z is the output of the sensor signal conditioner. The h_{i} coefficients in Equation 3 are offset drift compensation coefficients and g_{j} are the span drift compensation coefficients, while H and G are the respective orders of the polynomials.
Substituting for y from Equation 1 in Equation 2 yields Equation 3:
The key goal now is to cancel the temperature dependence of z. This goal can be achieved by choosing appropriate values for the polynomial coefficients, h_{i}, i = 0 to H, and g_{j}, j = 0 to G.
Using algebraic manipulations, the coefficients to cancel the temperature dependence of z can be evaluated to the following expressions:
The sensor signal conditioner uses the coefficient values and calculates the temperature compensated output using Equation 2.
Example of compensation
Consider, for example, a sense element that drifts with temperature as described by Equation 1, using these parameters:
 0 ≤ x ≤ 1, that is, the physical quantity has been normalized and is now unitless
 a_{0} = 10 mV
 a_{1} = 0.05 mV/°C
 b_{0} = 50 mV
 b_{1} = –0.1 mV/°C
 –40°C ≤ operating temperature ≤ 150°C
 Room temperature = 25°C
In this case, the sense element output can be described by Equation 4.
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Figure 2 shows the sense element output at two different temperatures. From this figure one can infer that the sense element output transfer function is dramatically different at the two different temperatures.
Fig 2: Sense element output changes with temperature.
Figure 3 shows the desired output of the sensor. Ideally, the sensor output should be independent of temperature.
Fig. 3: Ideal output of the sensor signal conditioner.
Temperature compensation using G = 2
Equation 5 describes the polynomial with G = 2.
Figure 4 shows the signal conditioner output for the polynomial modeled by Equation 5, which shows the percentage of fullscale (%FS) error of conditioned output, z, with respect to the physical quantity of interest, x. Note that the %FS error is a measurement of compensation accuracy.
Fig. 4: Temperature compensated output using polynomial with G = 2.
Figure 5 shows that the sensor signal conditioner has corrected the temperature drift to less than 1.6%FS.
Fig. 5: %FS error of temperature compensated output using polynomial with G = 2.
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Temperature compensation using G = 3
Equation 6 describes the polynomial with G = 3:
By comparing Equation 5 and Equation 6, one can infer that Equation 6 is a higher order polynomial. Figure 6 shows the signal conditioner output for a polynomial modeled by Equation 3 with G = 3.
Fig. 6: Temperature compensated output using polynomial with G = 3.
Figure 7 shows the %FS error of conditioned output, z, with respect to the physical quantity of interest, x. It also shows that the sensor signal conditioner has corrected the temperature drift to less than 0.4%FS.
Fig. 7: %FS error of temperature compensated output using polynomial with G = 3.
By comparing Figure 5 and Figure 7, one can infer that by increasing the order of the polynomial by one results in a four times improvement in accuracy. Note that the temperature drift error can be further reduced by either choosing additional numbers of coefficients in the polynomial, or by choosing higher order polynomials.
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Practical considerations
In practice, each sense element has a unique temperature drift characteristic. Furthermore, the drift can be nonlinear in nature. Because of this, sensor manufacturers "calibrate" each sensor during the manufacturing process. As discussed in reference [1], the process of calibration involves determining a unique set of polynomial coefficients using actual measurements for each sense element. Specifically, the sensor is exposed to different physical quantities of interest and its output is measured. Based on this measured data, the polynomial coefficients are determined. The number of polynomial coefficients determines the number of measurements points. Note that the higher number of measurement points increases the cost of calibration which, in turn, increases the sensor's overall cost. Of course, a higher number of measurements may be needed to achieve better temperature drift compensation.
Summary
Understanding temperature drift behavior of sense elements and how to correct for this behavior using polynomials can be of high value to sensor manufacturers. Referring to Figure 5 and Figure 7, one can see how using higher order polynomials can improve temperature drift compensation and accuracy. Note that compensation for temperature drift using polynomials is possible, but only if the sensor signal conditioner is capable of performing polynomial computation in realtime.
References
Vemuri, Arun; ValleMayorga, Javier; Twostep calibration of sensor signal conditioners, Texas Instruments Analog Applications Journal, Second Quarter 2015.
For questions about this article, you can submit your questions to the TI E2E™ Pressure Sensing forum.
About the Authors
Arun Vemuri is a systems architect at Texas Instruments where he is responsible for architecting and defining mixedsignal signal conditioner ICs for automotive and industrial sensors. Arun has been involved with the development of signals conditioners for pressure, ultrasonic, temperature and linear variable differential transformer (LVDT) position sensors. Arun received his Ph.D. in electrical engineering from the University of Cincinnati, Ohio; his MS in systems science from IISc Bangalore, India; and his BSEE in electrical engineering from IIT Roorkee, India.
Javier ValleMayorga is an applications engineer at Texas Instruments where he provides application support related to the Enhanced Industrial's line of signal conditioners. Javier received his Ph.D. in electrical engineering from the University of Arkansas in Fayetteville, Arkansas; his MSEE from the Aichi Institute of Technology in Toyota, Japan; and his BE in electrical and mechanical engineering from John Brown University in Siloam Springs, Arkansas.
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