SENSOR TRANSDUCER DETECTOR: SENSOR CHARACTERISTICS (1)

From the input to the output, a sensor may have several conversion steps before it produces an electrical signal. For instance, pressure inflicted on the fiber-optic sensor first results in strain in the fiber, which, in turn, causes deflection in its refractive index,which, in turn, results in an overall change in optical transmission and modulation of photon density. Finally, photon flux is detected and converted into electric current.

In this article, we discuss the overall sensor characteristics, regardless of its physical nature or steps required to make a conversion. We regard a sensor as a “black box” where we are concerned only with relationships between its output signal and input stimulus.

The characteristics we discuss are:

Transfer Function
Span (Full-Scale Input)
Full-Scale Output
Accuracy
Calibration
Calibration Error
Hysteresis
Nonlinearity
Saturation
Repeatability
Dead Band
Resolution

1. 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 :

      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)

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:

       S = a + b ln s    (2)

Exponential function:

       S = a e^ks        (3)

Power function:

   S = a₀ + a₁s^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, 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 ( T_b⁴ - T_s⁴ ) (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_b³ (7)

2. Span (Full-Scale Input)

A dynamic range of stimuli which may be converted by a sensor is called a span or an input full scale (FS). It represents the highest possible input value that can be applied to the sensor without causing an unacceptably large inaccuracy. For the sensors with a very broad and nonlinear response characteristic, a dynamic range of the input stimuli is often expressed in decibels, which is a logarithmic measure of ratios of either power or force (voltage). It should be emphasized that decibels do not measure absolute values, but a ratio of values only. A decibel scale represents signal magnitudes by much smaller numbers, which, in many cases, is far more convenient.

Being a nonlinear scale, it may represent low-level signals with high resolution while compressing the high-level numbers. In other words, the logarithmic scale for small objects works as a microscope, and for the large objects, it works as a telescope. By definition, decibels are equal to 10 times the log of the ratio of powers:

1 dB = 10 log ( P₂ / P₁) (8)

In a similar manner, decibels are equal to 20 times the log of the force, current, or voltage:

1 dB = 20 log ( S₂ / S₁) (9)

3. Full-Scale Output

Full-scale output (FSO) is the algebraic difference between the electrical output signals measured with maximum input stimulus and the lowest input stimulus applied. This must include all deviations from the ideal transfer function. For instance, the FSO output in Fig. 1 is represented by SFS.