In this article, we investigate two methods that are widely used to correct for sense element non-linearity. The two approaches are compared and the trade-offs between them are discussed.

Sense elements are used to convert physical quantities of interest into electrical signals. For example, a Wheatstone bridge can be used to convert pressure into electrical output. Many sense elements are inherently non-linear. In other words, their outputs are not linearly proportional to the physical quantity they are measuring. As the physical quantity of interest changes, the output changes non-linearly.
Sensor signal conditioners are used to correct for non-linearity of sense element outputs. In this article, we investigate two methods that are widely used to correct for sense element non-linearity: 1) look-up table (LUT) or interpolation; and 2) polynomials or curve-fitting. The two approaches are compared and the trade-offs between the two methods are discussed.
**Sense element and sensor signal conditioner**

A sensor or a transmitter consists of a sense element and a signal conditioner. The sense element, such as a Wheatstone bridge or hot-wire anemometer, converts the physical pressure such as pressure or mass air flow, respectively, into electrical signals. The sensor signal conditioner processes the sense element output nonidealities and transmits the signal to a controller. **Figure 1** shows the general block diagram of this ecosystem.

**Figure 1:** Block diagram of a sensor showing the physical quantity of interest, *x, *sense element output,* y, *and conditioned sensor output,* z.*

**Figure 1** shows that the physical quantity being measured is *x*, the output of the sense element is *y*, and the processed output of the sensor signal conditioner is *z*. The sensor signal conditioner processes the sense element output for nonidealities such as non-linearities, temperature variation, and dynamic response. In this article, we discuss the processing of the sense element output for non-linearity.
**Sense element non-linearity**

While sense element non-linearities can have different forms, in this article we discuss a specific form of non-linearity, which is a *second-order non-linearity*. This will allow for an easy understanding of how sensor signal conditioners are used to linearise the output. Note that these concepts can be generalized to arbitrary non-linearities.
A sense element is said to have second order non-linearity, if its input and output have a mathematical relationship as shown in Equation 1:

Where:
*a* = sense element offset, which is defined as the output of sense element when the physical quantity being measured is at its minimum value
*b* = sense element span, which is defined as the difference in the outputs of sense element at maximum and minimum physical quantity being measured
*c* = sense element second order non-linearity
x* = *physical quantity being measured
*y* = output of the sense element as a function of *x*
For example, consider a non-linear sense element with the following example parameters:
• x is between 0 and 1. That is, the physical quantity has been normalized and so has no units
• a = 0
• b = 1
• c = 10 per cent full scale (FS), where FS = b; in other words, the sense element has 10 per cent non-linearity.
In this case, the sense element output as a function of its input is described by Equation 2.

**Figure 2** shows the plot of Equation 2 as the normalized input *x *changes from 0 to 1. From **figure 2**, it can be inferred that the sense element output at x = 0.5 is 0.6. If the sense element output is perfectly linear, then the output would be 0.5. That is, the sense element has a 10 per cent FS deviation at the midpoint. This is the *bump* that is present in the output of sense element with second order non-linearity.

* **Figure 2:** Non-linear sense element output. Notice the bump in the output line.*

**Linearizing the sense element output**

The goal of linearisation of the sense element output is to make the non-linear line linear. In this article, we discuss two common approaches that are used in sensor signal conditioners. Note that this discussion is valid regardless of how the linearisation process is realised. The linearisation process can be realised using analogue circuits as in the PGA309, or using digital logic or firmware as in the PGA900.
*Linearisation using the LUT*

A lookup table (LUT) is a table of measured inputs and outputs. In the context of sensor signal conditioning, the inputs in the table are the sense element outputs at different physical quantities of interest and the outputs are the desired linear outputs. Based on the LUT, for a given input value, the output is *looked up* in the LUT.
Consider the sense element example described in the last section. A three-point LUT can be constructed as shown in **table 1**.

**Table 1:** LUT for the sense element example described in the section on sense element non-linearity.

Based on the LUT in Table 1, if the sense element output is 0.6, then the sensor signal conditioner will *lookup* the LUT and the output will be 0.5. What happens if the sense element output is, for example, 0.5 and this value is not in the LUT? In this case, the sensor signal conditioner typically uses linear interpolation to determine the sensor signal conditioner output.
**Figure 3** shows the signal conditioner output for the LUT-based linearisation and **figure 4** shows the percentage of the FS error between the physical quantity of interest, *x*, and the signal conditioner output, *z*. **Figure 4** shows that the sensor signal conditioner has corrected the sense element linearity error from 10 per cent FS to 3 per cent FS.

**Figure 3:** Linearisation using the LUT.

**Figure 4:** A percentage of FS error with linearisation using the LUT.

Note that the non-linearity error can be further reduced by choosing more points in the LUT.**Linearisation using polynomials **

We next consider the linearisation using polynomials. Specifically, we use a three-coefficient polynomial (similar to using a three-point LUT), or a second-order polynomial given by Equation 3:

Where *h*, *g* and *n* are the polynomial coefficients, *y *is the output of the sense element, and *z *is the output of the sensor signal conditioner.
Consider the sense element example we described earlier. Using Equation 2, Equation 3 can be rewritten as Equation 4:

The goal of linearisation is to remove the dependence of the signal conditioner output on *x ^{2}*. This can be achieved by choosing polynomial coefficients

*h*,

*g*and

*n*so that the resultant coefficients of are eliminated or small. Using algebraic manipulations, the three polynomial coefficients to minimise resultant coefficients of

*x*can be evaluated to:

^{2}The sensor signal conditioner uses the above coefficient's values and calculates the linearized output using Equation 3.
**Figure 5** shows the signal conditioner output for polynomial-based linearisation, while **figure 6** shows the percentage of FS error between the physical quantity of interest, *x,* and the signal conditioner output, *z*. **Figure 6** shows that the sensor signal conditioner has corrected the sense element linearity error from 10 per cent of FS to 1.5 per cent of FS.

**Figure 5:** Linearisation using polynomials.

**Figure 6:** A percentage of FS error with linearisation using polynomials.

Note that the non-linearity error can be further reduced by choosing more numbers of coefficients in the polynomial – or by choosing higher order polynomials.
**Practical considerations**

In practice, each sense element has a unique non-linearity characteristic. Because of this, sensor manufacturers calibrate each sensor during manufacturing. The process of calibration involves determining points on the LUT, or polynomial coefficients using actual measurements. Specifically, the sensor is exposed to different physical quantities of interest and its output is measured. Based on this measured data, you can determine the LUT points or polynomial coefficients. The target accuracy determines the number of LUT points or polynomial coefficients which in turn determines the number of measurement points. Note that the higher the number of measurement points, higher is the cost of calibration and, hence, the cost of the sensor is also higher. Keep in mind that a higher number of measurements may be needed to achieve better sensor accuracy.
**Summary**

This article discusses sense element non-linearities and two widely used approaches to linearise the sense element output. A three-coefficient polynomial can help achieve higher accuracy versus a three-point LUT (**figures 4** and **6**). Intuitively, this is likely because the inherent non-linearity of sense elements is a second or higher order polynomial. Another perspective is that when using polynomial-based linearisation, it is possible to achieve similar accuracy with fewer calibration points, which lowers product cost. Alternately, a higher degree of accuracy can be achieved with similar product cost. This assumes that the sensor signal conditioner is capable of performing polynomial computation real-time.
**Reference**

Vemuri, Arun; Valle-Mayorga, Javier. Two-step calibration of sensor signal conditioners, Texas Instruments Analog Applications Journal, Second Quarter 2015, pp. 4-6.
**About the authors**

Arun Vemuri is a systems architect at Texas Instruments where he is responsible for architecting and defining mixed-signal 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.
Chenan "Hawk" Tong is a field applications engineer at Texas Instruments where he supports factory automation customers and applications. Chenan received his bachelor's degree from Guilin University of electronic technology, Guangxi, China, and his master's degree from Xi'an Jiaotong University, Shaanxi, China. Prior to joining TI, Chenan was an R&D engineer for 12 years, which included power supply, data acquisition and sensor applications.