Nonlinearity

Nonlinearity error is specified for sensors whose transfer function may be approximated by a straight line [Eq. (2.1)].Anonlinearity is a maximum deviation (L) of a real transfer function from the approximation straight line. The term “linearity” actually

 Fig. 2.4. Transfer function with hysteresis.

means “nonlinearity.” When more than one calibration run is made, the worst linearity seen during any one calibration cycle should be stated. Usually, it is specified eitherin percent of span or in terms of measured value (e.g, in kPa or degree C). “Linearity,” when not accompanied by a statement explaining what sort of straight line it is referring to, is meaningless. There are several ways to specify a nonlinearity, depending how the line is superimposed on the transfer function. One way is to use terminal points (Fig. 2.5A); that is, to determine output values at the smallest and highest stimulus values and to draw a straight line through these two points (line 1). Here, near the terminal points, the nonlinearity error is the smallest and it is higher somewhere in between.

 Fig. 2.5. Linear approximations of a nonlinear transfer function (A) and independent linearity (B).

Another way to define the approximation line is to use a method of least squares (line 2 in Fig. 2.5A). This can be done in the following manner. Measure several (n) output values S at input values s over a substantially broad range, preferably over an entire full scale. Use the following formulas for linear regression to determine intercept a and slope b of the best-fit straight line: 

In some applications, a higher accuracy may be desirable in a particular narrower section of the input range. For instance, a medical thermometer should have the best accuracy in a fever definition region which is between 37 degree C and 38 degree C. It may have a somewhat lower accuracy beyond these limits. Usually, such a sensor is calibrated in the region where the highest accuracy is desirable. Then, the approximation line may be drawn through the calibration point c (line 3 in Fig. 2.5A). As a result, nonlinearity
has the smallest value near the calibration point and it increases toward the ends of the span. In this method, the line is often determined as tangent to the transfer function in point c. If the actual transfer function is known, the slope of the line can be found from Eq. (2.5).
Independent linearity is referred to as the so-called “best straight line” (Fig. 2.5B), which is a line midway between two parallel straight lines closest together and enveloping all output values on a real transfer function.

Depending on the specification method, approximation lines may have different intercepts and slopes. Therefore, nonlinearity measures may differ quite substantially from one another.Auser should be aware that manufacturers often publish the smallest possible number to specify nonlinearity, without defining what method was used.