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 :
S = f(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:
S = a + bs (1)
S = a + b ln s (2)
S = a eks (3)
S = a0 + a1sk (4)
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. (1)]. At any particular input value, s0, it can be defined as:
b = dS(s0) / dS (5)
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.
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 :
V = G ( Tb4 - Ts4 ) (6)
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:
b = ∂V / ∂Tb = 4GTb3 (7)