# Sensor Transfer Fuction

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)

**sensor**properties.

Logarithmic function:

S = a + b ln s (2)

Exponential function:

S = a e^{ks} (3)

Power function:

S = a_{0} + a_{1}s^{k} (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, s_{0}, 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 ( T_{b}^{4} - T_{s}^{4} ) (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 / ∂T_{b} = 4GT_{b}^{3} (7)