SENSOR CHARACTERISTICS (6)

19. Application Characteristics

Design, weight, and overall dimensions are geared to specific areas of applications. Price may be a secondary issue when the sensor’s reliability and accuracy are of paramount importance. If a sensor is intended for life-support equipment, weapons or spacecraft, a high price tag may be well justified to assure high accuracy and reliability. On the other hand, for a very broad range of consumer applications, the price of a sensor often becomes a cornerstone of a design.

20. Uncertainty

Nothing is perfect in this world, at least in the sense that we perceive it. All materials are not exactly as we think they are. Our knowledge of even the purest of the materials is always approximate; machines are not perfect and never produce perfectly identical parts according to drawings. All components experience drifts related to the environment and their aging; external interferences may enter the system and alter its performance and modify the output signal. Workers are not consistent and the human factor is nearly always present.

Manufacturers fight an everlasting battle for the uniformity and consistency of the processes, yet the reality is that every part produced is never ideal and carries an uncertainty of its properties. Any measurement system consists of many components, including sensors. Thus, no matter how accurate the measurement is, it is only an approximation or estimate of the true value of the specific quantity subject to measurement, (i.e., the stimulus or measurand). The result of a measurement should be considered complete only when accompanied by a quantitative statement of its uncertainty.We simply never can be 100% sure of the measured value.

When taking individual measurements (samples) under noisy conditions we expect that the stimulus s is represented by the sensor as having a somewhat different value s', so that the error in measurement is expressed as:

     d  = s' - s        (27)

The difference between the error specified by Eq. (27) and uncertainty should always be clearly understood. An error can be compensated to a certain degree by correcting its systematic component. The result of such a correction can unknowably be very close to the unknown true value of the stimulus and, thus, it will have a very small error.Yet, in spite of a small error, the uncertainty of measurement may be very large so we cannot really trust that the error is indeed that small. In other words, an error is what we unknowably get when we measure, whereas uncertainty is what we think how large that error might be.

The International Committee forWeight and Measures (CIPM) considers that uncertainty consists of many factors that can be grouped into two classes or types:

  A: Those evaluated by statistical methods

  B: Those evaluated by other means.

This division is not clear-cut and the borderline between Types A and B is somewhat illusive. Generally, Type A components of uncertainty arise from random effects, whereas the Type B components arise from systematic effects.

Type A uncertainty is generally specified by a standard deviation S_i , equal to the positive square root of the statistically estimated variance S_i² and the associated number of degrees of freedom ν_i  . For such a component, the standard uncertainty is u_i=S_i . Standard uncertainty represents each component of uncertainty that contributes to the uncertainty of the measurement result.

The evaluation of a Type A standard uncertainty may be based on any valid statistical method for treating data. Examples are calculating the standard deviation of the mean of a series of independent observations, using the method of least squares to fit a curve to data in order to estimate the parameters of the curve and their standard deviations. If the measurement situation is especially complicated, one should consider obtaining the guidance of a statistician.

The evaluation of a Type B standard uncertainty is usually based on scientific judgment using all of the relevant information available, which may include the following:

• Previous measurement data

• Experience with or general knowledge of the behavior and property of relevant sensors, materials, and   instruments

• Manufacturer’s specifications

• Data obtained during calibration and other reports

• Uncertainties assigned to reference data taken from handbooks and manuals

For detailed guidance of assessing and specifying standard uncertainties one should consult specialized texts. When both Type A and Type B uncertainties are evaluated, they should be combined to represent the combined standard uncertainty. This can be done by using a conventional method for combining standard deviations. This method is often called the law of propagation of uncertainty and in common parlance is known as “rootsum-of-squares” (square root of the sum-of-the-squares) or RSS method of combining uncertainty components estimated as standard deviations:

u_c = √[u₁² + u₂² + · · · + u_i² + · · · + u_n²]       (28)

where n is the number of standard uncertainties in the uncertainty budget.

Table 1 shows an example of an uncertainty budget for an electronic thermometer with a thermistor sensor which measures the temperature of a water bath. While compiling such a table, one must be very careful not to miss any standard uncertainty, not only in a sensor but also in the interface instrument, experimental setup, and the object of measurement. This must be done for various environmental conditions, which may include temperature, humidity, atmospheric pressure, power supply variations, transmitted noise, aging, and many other factors.

No matter how accurately any individual measurement is made, (i.e., how close the measured temperature is to the true temperature of an object), one never can be sure that it is indeed accurate. The combined standard uncertainty of 0.068°C does not mean that the error of measurement is no greater than 0.068°C. That value is just a standard deviation, and if an observer has enough patience, he may find that individual errors may be much larger. The word “uncertainty” by its very nature implies that the uncertainty of the result of a measurement is an estimate and generally does not have well-defined limits.

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17 Environmental Factors

Storage conditions are nonoperating environmental limits to which a sensor may be subjected during a specified period without permanently altering its performance under normal operating conditions. Usually, storage conditions include the highest and the lowest storage temperatures and maximum relative humidities at these temperatures.

The word “noncondensing” may be added to the relative humidity number.Depending on the sensor’s nature, some specific limitation for the storage may need to be considered (e.g., maximum pressure, presence of some gases or contaminating fumes, etc.).

Short- and long-term stabilities (drift) are parts of the accuracy specification. The short-term stability is manifested as changes in the sensor’s performance within minutes, hours, or even days. The sensor’s output signal may increase or decrease, which, in other terms, may be described as ultralow-frequency noise. The long-term stabilitymaybe related to aging of the sensor materials, which is an irreversible change in the material’s electrical, mechanical, chemical, or thermal properties; that is, the long-term drift is usually unidirectional. It happens over a relatively long time span, such as months and years.

Long-term stability is one of the most important for sensors used for precision measurements. Aging depends heavily on environmental storage and operating conditions, how well the sensor components are isolated from the environment, and what materials are used for their fabrication. The aging phenomenon is typical for sensors having organic components and, in general, is not an issue for a sensor made with only nonorganic materials. For instance, glass-coated metaloxide thermistors exhibit much greater long-term stability compared to poxy-coated thermistors.Apowerful way to improve long-term stability is to preage the component at extreme conditions. The extreme conditions may be cycled from the lowest to the highest. For instance, a sensor may be periodically swung from freezing to hot temperatures.

Such accelerated aging not only enhances the stability of the sensor’s characteristics but also improves the reliability (see Section 18), as the preaging process reveals many hidden defects. For instance, epoxy-coated thermistors may be greatly improved if they are maintained at +150°C for 1 month before they are calibrated and installed in a product.

Environmental conditions to which a sensor is subjected do not include variables which the sensor measures. For instance, an air-pressure sensor usually is subjected not just to air pressure but to other influences as well, such as the temperatures of air and surrounding components, humidity, vibration, ionizing radiation, electromagnetic fields, gravitational forces, and so forth. All of these factors may and usually do affect the sensor’s performance. Both static and dynamic variations in these conditions should be considered. Some environmental conditions are usually of a multiplicative nature; that is, they alter a transfer function of the sensor (e.g., changing its gain). One example is the resistive strain gauge, whose sensitivity increases with temperature.

Environmental stability is quite broad and usually a very important requirement. Both the sensor designer and the application engineer should consider all possible external factors which may affect the sensor’s performance.Apiezoelectric accelerometer may generate spurious signals if affected by a sudden change in ambient temperature, electrostatic discharge, formation of electrical charges (triboelectric effect), vibration of a connecting cable, electromagnetic interference (EMI), and so forth.

Even if a manufacturer does not specify such effects, an application engineer should simulate them during the prototype phase of the design process. If, indeed, the environmental factors degrade the sensor’s performance, additional corrective measures may be required. (e.g., placing the sensor in a protective box, using electrical shielding, using a thermal insulation or a thermostat).

Temperature factors are very important for sensor performance; they must be known and taken into account. The operating temperature range is the span of ambient temperatures given by their upper and lower extremes (e.g., -20°C to +100°C) within which the sensor maintains its specified accuracy. Many sensors change with temperature and their transfer functions may shift significantly.

Special compensating elements are often incorporated either directly into the sensor or into signal conditioning circuits, to compensate for temperature errors. The simplest way of specifying tolerances of thermal effects is provided by the error-band concept, which is simply the error band that is applicable over the operating temperature band.

A temperature band may be divided into sections, whereas the error band is separately specified for each section. For example, a sensor may be specified to have an accuracy of ±1% in the range from 0°C to 50°C,±2% from-20°C to 0°C and from+50°C to 100°C and ±3% beyond these ranges within operating limits specified from -40°C to +150°C.

Temperatures will also affect dynamic characteristics, particularly when they employ viscous damping. A relatively fast temperature change may cause the sensor to generate a spurious output signal. For instance, a dual pyroelectric sensor in a motion detector is insensitive to slowly varying ambient temperature. However, when the temperature changes quickly, the sensor will generate an electric current that may be recognized by a processing circuit as a valid response to a stimulus, thus causing a false-positive detection.

A self-heating error may be specified when an excitation signal is absorbed by a sensor and changes its temperature by such a degree that it may affect its accuracy. For instance, a thermistor temperature sensor requires passage of electric current, causing heat dissipation within the sensor’s body.

Depending on its coupling with the environment, the sensors’ temperature may increase due to a self-heating effect. This will result in errors in temperature measurement because the thermistor now acts as an additional spurious source of thermal energy. The coupling depends on the media in which the sensor operates—a dry contact, liquid, air, and so forth. A worst coupling may be through still air. For thermistors, manufacturers often specify self-heating errors in air, stirred liquid, or other media.

A sensor’s temperature increase above its surroundings may be found from the following formula:

   ΔT° =V² / (ξνc+α)R       (25)

where ξ is the sensor’s mass density, c is specific heat, v is the volume of the sensor, α is the coefficient of thermal coupling between the sensor and the outside (thermal conductivity), R is the electrical resistance, and V is the effective voltage across the resistance. If a self-heating results in an error, Eq. (25) may be used as a design guide.

For instance, to increase a, a thermistor detector should be well coupled to the object by increasing the contact area, applying thermally conductive grease or using thermally conductive adhesives. Also, high-resistance sensors and low measurement voltages are preferable.

18. Reliability

Reliability is the ability of a sensor to perform a required function under stated conditions for a stated period. It is expressed in statistical terms as a probability that the device will function without failure over a specified time or a number of uses. It should be noted that reliability is not a characteristic of drift or noise stability. It specifies a failure, either temporary or permanent, exceeding the limits of a sensor’s performance under normal operating conditions.

Reliability is an important requirement; however, it is rarely specified by the sensor manufacturers. Probably, the reason for that is the absence of a commonly accepted measure for the term. In the United States, for many electronic devices, the procedure for predicting in-service reliability is the MTBF (mean time between failure) calculation described in MIL-HDBK-217 standard. Its basic approach is to arrive at a MTBF rate for a device by calculating the individual failure rates of the individual components used and by factoring in the kind of operation the device will see: its temperature, stress, environment, and screening level (measure of quality).

Unfortunately, the MTBF reflects reliability only indirectly and it is often hardly applicable to everyday use of the device. The qualification tests on sensors are performed on combinations of the worst possible conditions. One approach is 1000 h, loaded at maximum temperature. This test does not qualify for such important impacts as fast temperature changes. The most appropriate method of testing would be accelerated life qualification. It is a procedure that emulates thesensor’s operation, providing real-world stresses, but compressing years into weeks.

Three goals are behind the test: to establish MTBF; to identify first failure points that can then be strengthened by design changes; and to identify the overall system practical lifetime. One possible way to compress time is to use the same profile as the actual operating cycle, including maximum loading and power-on, power-off cycles, but expandedenvironmental highest and lowest ranges (temperature, humidity, and pressure). The highest and lowest limits should be substantially broader than normal operating conditions.

Performance characteristics may be outside specifications, but must return to those when the device is brought back to the specified operating range. For example, if a sensor is specified to operate up to 50°C at the highest relative humidity (RH) of 85% at a maximum supply voltage of +15 V, it may be cycled up to 100°C at 99% RH and at +18 V power supply. To estimate number of test cycles (n), the following

empirical formula may be useful:

n = N [ΔT_max / ΔT_test] ^2.5        (26)

where N is the estimated number of cycles per lifetime, ΔTmax is the maximum specified temperature fluctuation, and ΔTtest maximum cycled temperature fluctuation during the test. For instance, if the normal temperature is 25°C, the maximum specified temperature is 50°C, cycling was up to 100°C, and over the lifetime (say, 10 years), the sensor was estimated to be subjected to 20,000 cycles, then the number of test cycles is calculated as:

n = 20,000 [(50-25) / (100-25)]^2.5 = 1283.

As a result, the accelerated life test requires about 1300 cycles instead of 20,000. It should be noted, however, that the 2.5 factor was derived from a solder fatigue multiple, because that element is heavily influenced by cycling. Some sensors have no solder connections at all, and some might have even more sensitivity to cycling substances other than solder, (e.g, electrically conductive epoxy). Then, the factor

should be selected to be somewhat smaller. As a result of the accelerated life test, the reliability may be expressed as a probability of failure. For instance, if 2 out of 100 sensors (with an estimated lifetime of 10 years) failed the accelerated life test, the

reliability is specified as 98% over 10 years.

A sensor, depending on its application, may be subjected to some other environmental effects which potentially can alter its performance or uncover hidden defects. Among such additional tests are:

• High temperature/high humidity while being fully electrically powered. For instance, a sensor may be subjected to its maximum allowable temperature at 85–90% RH and kept under these conditions for 500 h. This test is very useful for detecting contaminations and evaluating packaging integrity. The life of sensors, operating at normal room temperatures, is often accelerated at 85°C and 85% RH,

which is sometimes called an “85–85 test.”

• Mechanical shocks and vibrations may be used to simulate adverse environmental conditions, especially in the evaluation wire bonds, adhesion of epoxy, and so forth.Asensor may be dropped to generate high-level accelerations (up to 3000g of force). The drops should be made on different axes. Harmonic vibrations should be applied to the sensor over the range which includes its natural frequency. In the United States military standard 750, methods 2016 and 2056 are often used for mechanical tests.

• Extreme storage conditions may be simulated, for instance at +100 and -40°C while maintaining a sensor for at least 1000 h under these conditions. This test simulates storage and shipping conditions and usually is performed on nonoperating devices. The upper and lower temperature limits must be consistent with the sensor’s physical nature. For example, TGS pyroelectric sensors manufactured in the past by Philips are characterized by a Curie temperature of +60°C. Approaching and surpassing this temperature results in a permanent destruction of sensitivity. Hence, the temperature of such sensors should never exceed +50°C, which must be clearly specified and marked on its packaging material.

• Thermal shock or temperature cycling (TC) is subjecting a sensor to alternate extreme conditions. For example, it may be dwelled for 30 min at -40°C, then quickly moved to +100°C for 30 min, and then back to cold. The method must specify the total number of cycling, like 100 or 1000. This test helps to uncover die bond, wire bond, epoxy connections, and packaging integrity.

• To simulate sea conditions, sensors may be subjected to a salt spray atmosphere for a specified time, (e.g., 24 h). This helps to uncover its resistance to corrosion and structural defects.

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16. Dynamic Characteristics

Under static conditions, a sensor is fully described by its transfer function, span, calibration, and so forth. However, when an input stimulus varies, a sensor response generally does not follow with perfect fidelity. The reason is that both the sensor and its coupling with the source of stimulus cannot always respond instantly. In other words, a sensor may be characterized with a time-dependent characteristic, which is called a dynamic characteristic. If a sensor does not respond instantly, it may indicate values of stimuli which are somewhat different from the real; that is, the sensor responds with a dynamic error.Adifference between static and dynamic errors is that the latter is always time dependent. If a sensor is a part of a control system which has its own dynamic characteristics, the combination may cause, at best, a delay in representing a true value of a stimulus or, at worst, cause oscillations.

The warm-up time is the time between applying electric power to the sensor or excitation signal and the moment when the sensor can operate within its specified accuracy. Many sensors have a negligibly short warm-up time. However, some detectors, especially those that operate in a thermally controlled environment (a thermostat) may require seconds and minutes of warm-up time before they are fully operational within the specified accuracy limits.

In a control system theory, it is common to describe the input–output relationship through a constant-coefficient linear differential equation. Then, the sensor’s dynamic (time-dependent) characteristics can be studied by evaluating such an equation. Depending on the sensor design, the differential equation can be of several orders. Azero-order sensor is characterized by the relationship which, for a linear transfer function, is a modified Eq. (1) where the input and output are functions of time t :

       S(t) = a + bs(t)        (18)

The value a is called an offset and b is called static sensitivity. Equation (18) requires that the sensor does not incorporate any energy storage device, like a capacitor or mass. Azero-order sensor responds instantaneously. In other words, such a sensor does not need any dynamic characteristics.

A first-order differential equation describes a sensor that incorporates one energy storage component. The relationship between the input s(t) and output S(t) is the differential equation:

       b₁[ dS(t) / dt ] + b₀S(t) = s(t)       (19)

Atypical example of a first-order sensor is a temperature sensor for which the energy storage is thermal capacity. The first-order sensors may be specified by a manufacturer in various ways. Typical is a frequency response, which specifies how fast a first-order sensor can react to a change in the input stimulus.

The frequency response is expressed in hertz or rads per second to specify the relative reduction in the output signal at a certain frequency (Fig. 9A).Acommonly used reduction number (frequency limit) is -3 dB. It shows at what frequency the output voltage (or current) drops by about 30%. The frequency response limit fu is often called the upper cutoff frequency, as it is considered the highest frequency a sensor can process.

The frequency response directly relates to a speed response, which is defined in units of input stimulus per unit of time. Which response, frequency or speed, to specify in any particular case depends on the sensor type, its application, and the preference of a designer.

Another way to specify speed response is by time, which is required by the sensor to reach 90% of a steady-state or maximum level upon exposure to a step stimulus. For the first-order response, it is very convenient to use a so-called time constant. The time constant, τ , is a measure of the sensor’s inertia.

In electrical terms, it is equal to the product of electrical capacitance and resistance: τ = CR. In thermal terms, thermal capacity and thermal resistances should be used instead. Practically, the time constant can be easily measured. A first-order system response is:

       S = S_m( 1 - e^-t/τ )       (20)

where Sm is steady-state output, t is time, and e is the base of natural logarithm.

Substituting t = τ, we get:

S/S_m = 1 - 1 / e = 0.6321        (21)    

S/S_m = 1 - 1 / e = 0.6321        (21)

In other words, after an elapse of time equal to one time constant, the response reaches about 63% of its steady-state level. Similarly, it can be shown that after two time constants, the height will be 86.5% and after three time constants it will be 95%.

The cutoff frequency indicates the lowest or highest frequency of stimulus that the sensor can process. The upper cutoff frequency shows how fast the sensor reacts; the lower cutoff frequency shows how slow the sensor can process changing stimuli.

Figure 9B depicts the sensor’s response when both the upper and lower cutoff frequencies are limited. As a rule of thumb, a simple formula can be used to establish a connection between the cutoff frequency, fc (either upper and lower), and time constant in a first-order sensor:

fc ≈ 0.159 / τ       (22)

The phase shift at a specific frequency defines how the output signal lags behind in representing the stimulus change (Fig. 9A). The shift is measured in angular degrees or rads and is usually specified for a sensor that processes periodic signals. If a sensor is a part of a feedback control system, it is very important to know its phase characteristic. Phase lag reduces the phase margin of the system and may result in overall instability.

A second-order differential equation describes a sensor that incorporates two energy storage components. The relationship between the input s(t) and output S(t) is the differential equation:

b₂(d²S_(t)) / dt² + b₁dS(t) / d_t + b₀S(t) = s(t)       (23)

An example of a second-order sensor is an accelerometer that incorporates a mass and a spring. A second-order response is specific for a sensor that responds with a periodic signal. Such a periodic response may be very brief and we say that the sensor is damped, or it may be of a prolonged time and even may oscillate continuously.

Naturally, for a sensor, such a continuous oscillation is a malfunction and must be avoided. Any second-order sensor may be characterized by a resonant (natural) frequency, which is a number expressed in hertz or rads per second. The natural frequency shows where the sensor’s output signal increases considerably.

Many sensors behave as if a dynamic sensor’s output conforms to the standard curve of a second-order response; the manufacturer will state the natural frequency and the damping ratio of the sensor. The resonant frequency may be related to mechanical, thermal, or electrical properties of the detector.

Generally, the operating frequency range for the sensor should be selected well below (at least 60%) or above the resonant frequency. However, in some sensors, the resonant frequency is the operating point. For instance, in glass-breakage detectors (used in security systems), the resonant makes the sensor selectively sensitive to a narrow bandwidth, which is specific for the acoustic spectrum produced by shattered glass.

Damping is the progressive reduction or suppression of the oscillation in the sensor having higher than a first-order response. When the sensor’s response is as fast as possible without overshoot, the response is said to be critically damped (Fig. 10). An underdamped response is when the overshoot occurs and the overdamped response is slower than the critical response. The damping ratio is a number expressing the quotient of the actual damping of a second-order linear transducer by its critical damping.

For an oscillating response, as shown in Fig. 10, a damping factor is a measure of damping, expressed (without sign) as the quotient of the greater by the lesser of a pair of consecutive swings in opposite directions of the output signal, about an ultimately steady-state value. Hence, the damping factor can be measured as:

     Damping factor = F / A = A / B = B / C = etc.      (24)

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