Transfer Function

An ideal or theoretical output–stimulus relationship exists for every sensor. If the sensor is ideally designed and fabricated with ideal materials by ideal workers using ideal tools, the output of such a sensor would always represent the true value of the stimulus. The ideal function may be stated in the form of a table of values, a graph, or a mathematical equation. An ideal (theoretical) output–stimulus relationship is characterized by the so-called transfer function. This function establishes dependence between the electrical signal S produced by the sensor and the stimulus s :

 That function may be a simple linear connection or a nonlinear dependence, (e.g., logarithmic, exponential, or power function). In many cases, the relationship is unidimensional (i.e., the output versus one input stimulus). A unidimensional linear relationship is represented by the equation

  

 where a is the intercept (i.e., the output signal at zero input signal) and b is the slope, which is sometimes called sensitivity. S is one of the characteristics of the output electric signal used by the data acquisition devices as the sensor’s output. It may be amplitude, frequency, or phase, depending on the sensor properties.
Logarithmic function:

 Exponential function:

 Power function:

 where k is a constant number.
A sensor may have such a transfer function that none of the above approximations fits sufficiently well. In that case, a higher-order polynomial approximation is often employed.
For a nonlinear transfer function, the sensitivity b is not a fixed number as for the linear relationship [Eq. (2.1)]. At any particular input value, s0, it can be defined as

 In many cases, a nonlinear sensor may be considered linear over a limited range. Over the extended range, a nonlinear transfer function may be modeled by several straight lines. This is called a piecewise approximation. To determine whether a function can be represented by a linear model, the incremental variables are introduced for the input while observing the output.Adifference between the actual response and a liner model is compared with the specified accuracy limits (see 2.4).
A transfer function may have more than one dimension when the sensor’s output is influenced by more than one input stimuli. An example is the transfer function of a thermal radiation (infrared) sensor. The function connects two temperatures (Tb, the absolute temperature of an object of measurement, and Ts , the absolute temperature of the sensor’s surface) and the output voltage V :

 where G is a constant. Clearly, the relationship between the object’s temperature and the output voltage (transfer function) is not only nonlinear (the fourth-order parabola) but also depends on the sensor’s surface temperature. To determine the sensitivity of the sensor with respect to the object’s temperature, a partial derivative will be calculated as

 The graphical representation of a two-dimensional transfer function of Eq. (2.6) is shown in Fig. 2.1. It can be seen that each value of the output voltage can be uniquely

determined from two input temperatures. It should be noted that a transfer function represents the input-to-output relationship. However, when a sensor is used for measuring or detecting a stimulus, an inversed function (output-to-input) needs to be employed. When a transfer function is linear, the inversed function is very easy to compute. When it is nonlinear the task is more complex, and in many cases, the analytical solution may not lend itself to reasonably simple data processing. In these cases, an approximation technique often is the solution.