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Saturday, January 9, 2010

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 = Sm( 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/Sm = 1 - 1 / e = 0.6321        (21)    
S/Sm = 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) / dt + 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)

Reference Books About Sensor



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