3.1 Transducer Chapter 3 Transducers and Their Response Instrument Society of America defines a sensor or transducer as a device which provides a usable output in response to a specified measured. Here the measured is a physical quantity and the output may be an electrical quantity, mechanical and- optical. Sensor An element that senses a variation in input energy to produce a variation in another or same form of energy is called a sensor. Transducer Transducer converts a specified measured into usable output using transduction principle. For example, a properly cut piezoelectric crystal can be called a sensor where as it becomes a transducer with appropriate electrodes and input/output mechanisms attached to it. So, the sensor is the primary element of a transducer. Transducers is a devices used to transform one kind of energy to another. When a transducer converts a measurable quantity (temperature, pressure, level, optical intensity, magnetic field, etc) to an electrical voltage or an electrical current we call it a sensor. Energy information conversion is the objective of a sensor. The information available in one energy form must be converted into the same or another energy form, with exactly the same information content as the originating energy form. The sensor or the sensing element is the first element in a measuring system and takes information about the variable being measured and transforms it into a more suitable form to be measured. Sensor is sometimes called a primary measuring element, it can be found simply as a mercury thermometer to measure the temperature. It may be embedded in the transducer to perform its function. That means the transducer consists of a primary element (sensor) plus a secondary Compiled by Yidnekachew M. Page 1 of 28
element (signal conditioning circuit) that transforms the passive change or small voltage signal into active signal range that can be easily used in other chains of the control loop. A transducer can be defined as a device capable of converting energy from one form into another. Transducers can be found both at the input as well as at the output stage of a measuring system. The input transducer is called the sensor, because it senses the desired physical quantity and converts it into another energy form. The output transducer is called the actuator, because it converts the energy into a form to which another independent system can react, whether it is a biological system or a technical system. So, for a biological system the actuator can be a numerical display or a loudspeaker to which the visual or aural senses react respectively. For a technical system the actuator could be a recorder or a laser, producing holes in a ceramic material. The results can be interpreted by humans. Actuators are important in instrumentation. They allow the use of feedback at the source of the measurement. However we will pay little attention to them in this course. The study of using actuators and feedback belongs to a course in Control theory. 3.2 Types of Energy Form We can distinguish six different energy domains: (1) radiant, (2) mechanical, (3) thermal, (4) electrical, (5) magnetic and (6) chemical. If certain information is already available in the electrical domain it can be claimed that it requires no energy conversion, but in general there is 'shape' conversion left and this is just the domain which belongs to the field of electronics and electrical science and engineering. A good example of such a sensor only sensitive to electrical energy is the probe of an oscilloscope, with which a good adaptation to the signal source is realized. In the modifier stage we meet other examples of shape converters, for instance the A/D and D/A converters. Table 1.6 Energy types and corresponding measured Compiled by Yidnekachew M. Page 2 of 28
Energy Mechanical Thermal Electrical Magnetic Radiant Chemical Measurands Length, area, volume, force, pressure, acceleration, torque, mass flow, acoustic intensity and so on. Temperature, heat flow, entropy, state of matter Charge, current, voltage, resistance, inductance, capacitance, dielectric constant, polarization, frequency, electric field, dipole moment, and so on. Field intensity, flux density, permeability, magnetic moment, and so on. Intensity, phase, refractive index, reflectance, transmittance, absorbance, wavelength, polarization, and so on. Concentration, composition, oxidation/reduction potential, Reaction rate, ph and the like. 3.3 Classification of transducers The transducers may be classified based on i. The physical effect employed ii. The physical quantity measured iii. The source of energy i. Classification based on physical effect The physical quantity applied as measurand (quantity to be measured) to the transducer causes some physical changes in its element. By this physical effect the transducer converts the physical quantity in to electrical quantity. For example, a change in temperature to be measured causes variation of resistance (physical change) in a copper wire (element) and this effect could, be used to convert temperature in to an electrical output, The physical effects commonly employed are a. Variation of resistance b. Variation. of inductance c. Variation of capacitance d. Piezo electric effect e. Magnetostrictive effect Compiled by Yidnekachew M. Page 3 of 28
f. Elastic effect g. Hall effect a. Variation of resistance The resistance of a length of metallic wire is given by l R a Where, R = Resistance in ohm. =Resistivity (or specific resistance) of the material in ohm-m. l = length of wire in m. a = Area of cross-section in m 2. As resistance is a function of,l, a (i.e) R f(, l, a), with any change in anyone of the physical quantities, a or l due to variation in resistance, a variable resistance transducer can be designed to convert physical quantity. Some of the transducers based on this principle are potentiometer, strain gauge, resistance thermometer, carbon microphone, and photoconductive cell. The resistance thermometer is based upon thermo resistive effect which is the change in electrical resistivity of a metal or semiconductor due to change in temperature co-efficient of resistivity. Carbon microphone works on the principle of change in contact resistance due to applied pressure. Photoconductive cell is based on photoconductive effect which is the change in electrical conductivity due, to incident light. Potentiometer works on the principle of change in resistance due to linear or rotational motion. Strain gauge works on the principle of change in resistance due to applied pressure. b. Variation of inductance Compiled by Yidnekachew M. Page 4 of 28
The inductance of a coil is given by d L N dt L N ora l 2 where, L = inductance in henry N = No., of turns o = absolute permeability r = relative permeability A = area of cross section of the core l = length of magnetic path d = rate of change of magnetic flux. dt As L is a function, of N,, A, l, r (i.e) L = f (N,,A, l ), when anyone of these quantities changes, the inductance changes. r This leads to the design of a variable inductance transducer. Some of the transducers based on variation of inductance are induction potentiometer, linear variable differential transformer (LVDT) and synchros. c. Variation of capacitance Capacitive sensors consist of two parallel metal plates in which the dielectric between the plates is either air or some other medium. The capacitance between two conductor plates is given by o C d r A Compiled by Yidnekachew M. Page 5 of 28
Where C = capacitance in farad o = absolute permittivity r = relative permittivity of the separating medium A = area of cross-section of the plates As C is a function of, Ad, i.e C = (, Ad, ) when anyone of these quantities changes, the r capacitance varies. This leads to the design of a variable capacitance transducer. r Capacitive devices in which the distance between the plates is variable are primarily used as displacement sensors. Motion of the moveable capacitive plate relative to a fixed one changes the capacitance. Such devices can be used directly as a displacement sensor by applying the motion to be measured to the moveable capacitor plate. Capacitive displacement sensors commonly form part of instruments measuring pressure, sound, or acceleration. In the alternative form of capacitor, the distance between the plates is fixed. Variation in capacitance is achieved by changing the dielectric constant of the material between the plates in some way. One application is where the dielectric medium is air and the device is used as a humidity sensor by measuring the moisture content of the air. Another common application is as a liquid level sensor, where the dielectric is part air and part liquid according to the level of the liquid that the device is inserted in. This principle is used in devices to measure moisture content, humidity values, and liquid level, as discussed in later chapters. d. Piezoelectric effect When a piezoelectric crystal like quartz or Rochelle salt is subjected to mechanical stress, an electric charge is generated. This is known as piezoelectric effect. The transducer based on this effect is piezoelectric transducer. They can also operate in the reverse mode where an applied voltage produces an output force. They are frequently used as ultrasonic transmitters and receivers. They are also used as displacement transducers, particularly as part of devices measuring acceleration, force, and pressure. In ultrasonic receivers, the sinusoidal amplitude variations in the ultrasound wave Compiled by Yidnekachew M. Page 6 of 28
received are translated into sinusoidal changes in the amplitude of the force applied to the piezoelectric transducer. In a similar way, the translational movement in a displacement transducer is caused by mechanical means to apply a force to the piezoelectric transducer. Piezoelectric transducers are made from piezoelectric materials. These have an asymmetrical lattice of molecules that distorts when a mechanical force is applied to it. This distortion causes a reorientation of electric charges within the material, resulting in a relative displacement of positive and negative charges. The charge displacement induces surface charges on the material of opposite polarity between the two sides. By implanting electrodes into the surface of the material, these surface charges can be measured as an output voltage. For a rectangular block of material, the induced voltage is given by: kfd V A where F is the applied force in g, A is the area of the material in mm, d is the thickness of the material, and k is the piezoelectric constant. The polarity of the induced voltage depends on whether the material is compressed or stretched. e. Magnetostrictive effect When a magnetic material is subjected to mechanical stress, its permeability changes. This effect is magnetostrictive effect and the transducer based on this effect is magnetostrictive transducer. Magnetic sensors utilize the magnetic phenomena of inductance, reluctance, and eddy currents to indicate the value of the measured quantity, which is usually some form of displacement. f. Elastic effect When an elastic member is subjected to mechanical stress it is deformed. The transducer based on this effect is called elastic transducer. g. Hall effect Basically, a Hall effect sensor is a device that is used to measure the magnitude of a magnetic field. It consists of a conductor carrying a current that is aligned orthogonally with the magnetic field, as shown in Figure 3.1 Compiled by Yidnekachew M. Page 7 of 28
Figure 3.1 Principles of Hall effect sensor This produces a transverse voltage difference across the device that is directly proportional to the magnetic field strength. For an excitation current I and magnetic field strength B, the output voltage is given by V = KIB, where K is known as the Hall constant. The conductor in Hall effect sensors is usually made from a semiconductor material as opposed to a metal, because a larger voltage output is produced for a magnetic field of a given size. In one common use of the device as a proximity sensor, the magnetic field is provided by a permanent magnet that is built into the device. The magnitude of this field changes when the device becomes close to any ferrous metal object or boundary. The Hall effect is also commonly used in computer keyboard push buttons. When a button is depressed, a magnet attached underneath the button moves past a Hall effect sensor. This generates an induced voltage in the sensor which is converted by a trigger circuit into a digital output. Such push-button switches can operate at high frequencies without contact bounce. ii. Classification based on physical quantity measured The transducers may be classified based on the physical quantity they measure as follows: Temperature transducers Transducers used to measure temperature Pressure transducers To measure pressure Flow transducers To measure flow Liquid level transducers To measure liquid level Force/Torque transducers To measure force & Torque Velocity/Speed transducers To measure velocity & speed Compiled by Yidnekachew M. Page 8 of 28
Humidity transducers To measure humidity Acceleration/vibration transducers To measure acceleration & vibration Displacement transducers To measure displacement iii. Classification based on source of energy Transducers may be, classified based on source of energy into two types. Active transducer Passive transducer Passive transducer A component whose output energy is supplied entirely or almost entirely by its input signal is called a passive transducer. A passive transducer is the one which absorbs energy from the input medium and converts it directly into the output signal. Example A Thermocouple extracts heat energy from the input medium and converts it into electrical energy (voltage). Active Transducer An active transducer has an auxiliary source of power which supplies a major part of the output power while the input signal supplies only an insignificant portion (i.e) this transducer uses the energy it absorbs from the input medium as a control signal to transfer energy from the power supply to produce a proportional output. Example Strain gauge The energy extracted from the strained member is very small. The energy for the output signal is supplied by an external power source. Figure 1.1 Active and passive transducers Compiled by Yidnekachew M. Page 9 of 28
Selection of Transducers Transducers are used for the measurement of physical quantities. The selection of transducers for particular measurand is very important. The selection of transducers may be based on the following factors for effective measurement. 1. The physical quantity to be measured (measurand), 2. The range of input quantity, Based on physical quantity to be measured The correct type of transducer should be selected for measuring the physical quantity. The following table shows the physical quantity and the corresponding transducer types available. Table 1.2 classification based on physical quantity No. Physical quantity Transducers available 1 Temperature Bimetallic element Fluid expansion systems i. Liquid-in-steel bulb thermometers ii. Liquid-in-glass thermometers iii. Vapour pre-ssure thermometers Thermoresistive elements i. Resistance Temperature detector (RTD) ii. Thermistor Thermocouple Linear-Quartz thermometer Pyrometry 2 Pressure U-tube and ball type manometers Ring balance manometer Metallic diaphragms Capsules and bellows Bourdon tubes Membranes 3 Force (weight) Spring balance Cantilever Diaphragms Pneumatic and hydraulic load cells Column and proving ring load cells 4 Torque Torsion bar Flat spiral springs Dynamometer Gyroscope Compiled by Yidnekachew M. Page 10 of 28
5 Density of liquids Hydrometer Air bubbler system U-tube weighing system 6 Liquid level Float elements. Manometer system Diaphragms Container weight 7 Viscosity Capillary tube Concentric cylinder system 8 Flow rate of fluids Pitot static tube Flow-obstruction elements Rotating vane system Rotameter float system 9 Displacement Flapper nozzle system 10 Absolute displacement, Velocity and acceleration Vehicle attitude 3.4 Characteristics of Transducers Seismic system Gyroscope The selection of most suitable transducer from commercially available instruments is very important in designing an Instrumentation system. For the proper selection of transducer, knowledge of the performance characteristics of them are essential. The performance characteristics can be classified into two namely i. Static characteristics ii. Dynamic characteristics Static characteristics are a set of performance criteria that give a meaningful description of the quality of measurement without becoming concerned with dynamic descriptions involving differential equations. Dynamic characteristics describe the quality of measurement when the measured quantities vary rapidly with time. Here the dynamic relations between the instrument input and output must be examined, generally by the use of differential equations. For further reading on this subtopic please read the additional material Compiled by Yidnekachew M. Page 11 of 28
3.5 Calibration Calibration consists of comparing the output of the instrument or sensor under test against the output of an instrument of known accuracy when the same input (the measured quantity) is applied to both instruments. This procedure is carried out for a range of inputs covering the whole measurement range of the instrument or sensor. Calibration ensures that the measuring accuracy of all instruments and sensors used in a measurement system is known over the whole measurement range, provided that the calibrated instruments and sensors are used in environmental conditions that are the same as those under which they were calibrated. For use of instruments and sensors under different environmental conditions, appropriate correction has to be made for the ensuing modifying inputs. Instruments used as a standard in calibration procedures are usually chosen to be of greater inherent accuracy than the process instruments that they are used to calibrate. Because such instruments are only used for calibration purposes, greater accuracy can often be achieved by specifying a type of instrument that would be unsuitable for normal process measurements. For instance, ruggedness is not a requirement, and freedom from this constraint opens up a much wider range of possible instruments. In practice, high accuracy, null-type instruments are very commonly used for calibration duties, because the need for a human operator is not a problem in these circumstances. Instrument calibration has to be repeated at prescribed intervals because the characteristics of any instrument change over a period. Changes in instrument characteristics are brought about by such factors as mechanical wear, and the effects of dirt, dust, fumes, chemicals, and temperature changes in the operating environment. To a great extent, the magnitude of the drift in characteristics depends on the amount of use an instrument receives and hence on the amount of wear and the length of time that it is subjected to the operating environment. However, some drift also occurs even in storage, as a result of aging effects in components within the instrument. A proper course of action must be defined that describes the procedures to be followed when an instrument is found to be out of calibration, that is when its output is different to that of the calibration instrument when the same input is applied. The required action depends very much upon the nature of the discrepancy and the type of instrument involved. In many cases, deviations Compiled by Yidnekachew M. Page 12 of 28
in the form of a simple output bias can be corrected by a small adjustment to the instrument (following which the adjustment screws must be sealed to prevent tampering). In other cases, the output scale of the instrument may have to be redrawn, or scaling factors altered where the instrument output is part of some automatic control or inspection system. In extreme cases, where the calibration procedure shows up signs of instrument damage, it may be necessary to send the instrument for repair of even scrap it. Whatever system and frequency of calibration is established; it is important to review this from time to time to ensure that the system remains effective and efficient. It may happen that a cheaper (but equally effective) method of calibration becomes available with the passage of time, and such an alternative system must clearly be adopted in the interests of cost efficiency. However, the main item under scrutiny in this review is normally whether the calibration interval is still appropriate. Records of the calibration history of the instrument will be the primary basis on which this review is made. It may happen that an instrument starts to go out of calibration more quickly after a period of time, either because of aging factors within the instrument or because of changes in the operating environment. The conditions or mode of usage of the instrument may also be subject to change. As the environmental and usage conditions of an instrument may change beneficially as well as adversely, there is the possibility that the recommended calibration interval may decrease as well as increase. 3.5.1 Control of Calibration Environment Any instrument that is used as a standard in calibration procedures must be kept solely for calibration duties and must never be used for other purposes. Most particularly, it must not be regarded as a spare instrument that can be used for process measurements if the instrument normally used for that purpose breaks down. Proper provision for process instrument failures must be made by keeping a spare set of process instruments. Standard calibration instruments must be totally separate. To ensure that these conditions are met, the calibration function must be managed and executed in a professional manner. This will normally mean setting aside a particular place within the Instrumentation Department of a Company where all calibration operations take place and where all instruments used for calibration are kept. As far as possible this should take the form of a Compiled by Yidnekachew M. Page 13 of 28
separate room, rather than a sectioned-off area in a room used for other purposes as well. This will enable better environmental control to be applied in the calibration area and will also offer better protection against unauthorized handling or use of the calibration instruments. The level of environmental control required during calibration should be considered carefully with due regard to what level of accuracy is required in the calibration procedure, but should not be over specified as this will lead to unnecessary expense. Full air conditioning is not normally required for calibration at this level, as it is very expensive, but sensible precautions should be taken to guard the area from extremes of heat or cold, and also good standards of cleanliness should be maintained. While it is desirable that all calibration functions are performed in this carefully controlled environment, it is not always practical to achieve this. Sometimes, it is not convenient or possible to remove instruments from process plant, and in these cases, it is a standard practice to calibrate them in situ. In these circumstances, appropriate corrections must be made for the deviation in the calibration environmental conditions away from those specified. This practice does not obviate the need to protect calibration instruments and maintain them in constant conditions in a calibration laboratory at all times other than when they are involved in such calibration duties on plant. As far as management of calibration procedures is concerned, it is important that the performance of all calibration operations is assigned as the clear responsibility of just one person. That person should have total control over the calibration function, and be able to limit access to the calibration laboratory to designated, approved personnel only. Only by giving this appointed person total control over the calibration function, can the function be expected to operate efficiently and effectively. Lack of such definite management can only lead to unintentional neglect of the calibration system, resulting in the use of equipment in an out-of-date state of calibration and subsequent loss of traceability to reference standards. Professional management is essential so that the customer can be assured that an efficient calibration system is in operation and that the accuracy of measurements is guaranteed. Compiled by Yidnekachew M. Page 14 of 28
Calibration is an essential process to be undertaken for each instrument and measuring system frequently. A reference standard at least ten times more accurate than the instrument under test is normally used. Calibration is the process where the test instrument (the instrument to be calibrated) is compared with the standard instrument. It consists of reading the standard and test instruments simultaneously when the input quantity is held constant at several values over the range of the test instrument. Generally, certification of an instrument manufactured by an industry is undertaken by the National Physical Laboratory and other authorized laboratories where the secondary standards and the working standards are kept. 3.5.2 Procedure for calibration a) Examine the construction of the instrument, and identify and list all the possible inputs. b) Decide, which of the inputs will be significant in the application for which the instrument is to be calibrated. c) Select the apparatus that will allow you to vary all the significant inputs over the ranges considered necessary. Select standards to measure each input. d) By holding some inputs constant, varying others and recording the outputs develop the desired static input-output relations. 3.6 Mathematical Models of System Response 3.6.1 Basic System Models Modelling is the process of representing the behaviour of a system by a collection of mathematical equations & logics. o Modelling is comprehensively utilized to study the response of any system. o Response of a system is a measure of its fidelity to its purpose. Simulation is the process of solving the model and it performed on a computer. Equations are used to describe the relationship between the input and output of a system. Compiled by Yidnekachew M. Page 15 of 28
Analogy approach is also widely used to study system response 3.6.2 Mechanical System Elements A) Spring F = k x F = Force (tension or compression), x = Displacement (extension or compression), k = Spring constant. The bigger the value of k the greater the forces required to stretch or compress the spring & so the greater the stiffness. B) Dashpot dx F cvc dt F = Force opposing the motion at velocity v, c = Damping coefficient. Larger the value of c the greater the damping force at a particular velocity. Compiled by Yidnekachew M. Page 16 of 28
C) Mass dv F mam m dt dt 2 d x F = Force required to cause acceleration, a, m = Mass of the element that is distributed throughout some volume. However, in many cases, it is assumed to be concentrated at a point. In general, the response of measurement instruments under dynamic conditions can be complex. The fundamental concepts of dynamic response, however, can be understood by studying relatively simple mathematical models. We will consider three mathematical models for dynamic system response: zeroth, first and second order systems. 2 3.6.3 Zeroth Order Systems Imagine a thermometer that measures the temperature in a room of an office building. For practical purposes, the thermometer will indicate the current temperature at the location where it has been installed. The fact that the output of the instrument follows the input exactly is the defining characteristic of zeroth order systems. Mathematically, if we let f(t) be the input to the system as a function of time and y(t) be the output, then the relationship between them is Compiled by Yidnekachew M. Page 17 of 28
In the example above f(t) would be the actual temperature of the room and y(t) would be the indicated temperature. K is a constant that multiplies the input to generate the output. If y(t) is the temperature as displayed in a readout device and the thermometer is calibrated correctly, then K would ideally be equal to one. On the other hand, if the output of the thermometer is, for example, an electrical signal, then K would be a constant with units of Volts per degree Fahrenheit. The electrical signal could be used to actuate a valve that directs either cold or hot air from the airconditioning system to the room. K is usually called the static sensitivity. Note that the output of zeroth order systems is not affected by the speed at which f(t) changes. The above Equation is always valid, so the results of static calibration are sufficient to characterize the response of the system. 3.6.4 First Order Systems Let s consider the example of an oral thermometer used at a clinic to measure body temperature in patients. Prior to use, the thermometer is at room temperature. When the thermometer is put in the patient s mouth it experiences a sudden increase in temperature. Generally, we have to wait for a while before reading the temperature. Unlike the case of a thermometer monitoring the examination room s temperature, the situation at hand cannot be represented by a zeroth order model. Why? Let s consider a common glass bulb thermometer to explain. The thermometer in originally at room temperature, which will be denoted by To. Compiled by Yidnekachew M. Page 18 of 28
Figure 1.2 Thermometer measuring body temperatures It is then put in the mouth of a patient represented by the area inside the dashed line which is at temperature T1. In order for the thermometer to work, the mercury in the bulb must be heated to T1. The thermal expansion of the mercury will cause the column of mercury in the stem of the thermometer to increase in length. Measuring this length with the scale marked in the glass gives a temperature reading T(t). It takes a while, however, for the temperature of the mercury to reach the value T1, so we must wait that long before the thermometer indicates the correct temperature. In order to write an equation to model the response of the thermometer we need a little background in thermodynamics and heat transfer. Heat is a form of energy. We will represent it by Q. It flows from a hot place to a cold one. The energy in the mercury in the bulb of the thermometer, which we will call E, increases as heat travels from the mouth of the patient to the bulb. Due to conservation of energy, the rate of change of energy in the bulb with respect to time is equal to how fast heat is flowing in. In mathematical terms: As E increases, the temperature of the mercury, T, rises in proportion. How fast the temperature increases depends on how much mercury the bulb holds (the mass, m, of the mercury) and a property of mercury called the specific heat, cv. Compiled by Yidnekachew M. Page 19 of 28
The increase in energy is related to the increase in temperature by Heat must travel through the glass walls of the bulb on its way from the patient s mouth to the mercury. How fast heat can flow through the walls of the bulb depends on a property of glass called the convection heat transfer coefficient, h, the surface area of the bulb, A, and the current temperature difference between the mercury and the mouth of the patient. In equation form, Rearranging the above two equation This is a differential equation that governs what the temperature of the mercury is at any time. Since the length of the column of mercury is proportional the temperature, the equation also governs what the indicated temperature is. In general, the equation of a first order system is given For the example above, we have Compiled by Yidnekachew M. Page 20 of 28
As you can see, the equation governing the behavior of a first order system is a first order differential equation, so called because the highest derivative of the output variable in the equation is the first with respect to time. 3.6.5 Second Order Systems When you go to the grocery store, note that weight scales are available at the produce department so you can get an idea how many pounds of potatoes you will have to pay for. The scales generally have a pointer, which indicates weight on a big dial. Figure 1.3 Spring Balance If you drop a bag with potatoes on the scale, chances are that the pointer will oscillate a bit before settling and indicating the correct weight. The reason for the oscillation of the pointer is that the scale has mass, hence inertia, and its behavior is dictated by Newton s laws of motion. You may now see where the second order label comes from if you recall that acceleration is the second derivative of displacement... Let s analyze the balance and see what we get. The balance can be represented by the mass-spring-dashpot system shown in Figure Compiled by Yidnekachew M. Page 21 of 28
Figure 1.4 Free body diagram of spring balance The balance can be represented by the mass-spring-dashpot system shown in Figure. To simplify things a bit, the dial has been replaced by the straight scale at the left. We can write the equation of motion for the balance using basic dynamics as follows: The mass m represents the object weighted so that the weight W is equal to mg where g is the acceleration of gravity. The spring, attached near the left edge of the mass, is a mechanical element which develops force in proportion to how much it has been stretched. Since the top of the spring is fixed to the ceiling, the force is proportional to the displacement y of the mass. If the zero value of y corresponds to the position of the spring when it is unloaded, then the force Fs required to stretch the spring a distance y is given by where k is called the spring constant. The balance needs a way to dampen the oscillations of the pointer after a weight is dropped. The damping is provided by the dashpot, which is attached to the mass near the right. Dashpots are like shock absorbers in cars. Compiled by Yidnekachew M. Page 22 of 28
You can imagine one as a piston inside a closed cylinder. The cylinder is filled with a viscous fluid. As the piston moves, a small gap between the piston and the cylinder allows the fluid to flow from one side of the piston to the other. A force is needed to move the piston due to the viscosity of the fluid. The net result is that dashpots produce a force, Fd which is proportional to the speed of the piston relative to the cylinder. This force-speed relation can be written as where c is called the damping coefficient. The free body diagram of the mass is shown in Figure. Applying Newton s second law we obtain the equation which can be re-written as This is the differential equation that governs the motion of the scale. Since the weight is indicated by the displacement of the scale, the equation also governs the indicated weight. In general, the equation for a second order system is given by Compiled by Yidnekachew M. Page 23 of 28
Where 3.6.6 Solution to the Differential Equations In order to see how these systems behave for a given input we need to solve them. The solutions to the differential equations depend, of course, on how the input varies with time. Does it increase, decrease or cycle? Smoothly or abruptly? If smoothly, are the changes slow or fast? Although an infinite number of possible inputs exist, we will consider only two, which are most telling about the characteristics of the system response. These two inputs are: the step input and the harmonic input. We will not go over the method of solution of the differential equations, instead we will just quote the results. If you know how to solve differential equations, it is reasonably straight forward to obtain the solutions. If not, there is no need to learn now, just make sure you can use the solutions presented below. 3.6.7 First Order Systems 3.6.7.1 Step input response A step input is used to represent situations when the input changes from one, constant, value to another constant value in a very short period of time. Using Laplace transform for first order equation Compiled by Yidnekachew M. Page 24 of 28
Note that at t = 0, y(t) = yo, and that as The transition between the initial and the final values of y(t) is exponential as shown in Figure. Figure 1.5 Instrument responses for first order step input The parameter is called the time constant of the system. It indicates how fast the system responds to a change in the input. Note that after a time of one time constant, y(t) is 63.2% of the way to its final value. Generally, y(t) is considered to have achieved its final value after five time constants. As an example, suppose that an oral thermometer takes at least 3 minutes to indicate a patient s temperature. Then we can guess that the time constant of the thermometer is in the order of 0.6 minutes. 3.6.7.2 Harmonic input response Compiled by Yidnekachew M. Page 25 of 28
It means that the input is a sine wave with constant frequency and amplitude. In other words the input is of the form(s) All three definitions are equivalent. In the first, T is the period in seconds. In the second, f is the frequency in cycles per second or Hertz. In the third, is the circular frequency in radians per second. Why should we be concerned with the response of measurement systems to harmonic inputs? The answer is that many natural and man-made processes are periodic in nature. For example, think about day and night, the seasons repeating every year, the waves at the beach, etc. In the engineering world think about the vibration of a motor, the movement of pistons in an internal combustion engine, the turning of wheels, the air conditioning system at your house turning on and off, and so on. The steady state solution to this equation with the input Where is called the magnitude ratio and is called the phase angle. Compiled by Yidnekachew M. Page 26 of 28
The magnitude ratio tells us how the amplitude of the output of the system depends on the frequency of the input. The phase angle tells us how far the system output is falling behind the input. Figure 1.6 Instrument responses for first order harmonic input 3.6.7.3 Second order systems Now, let us consider the response of a second order system to a step input. In other words we need the solution to In fact, we need to consider three different cases depending on the value of the parameter called the damping ratio. Case 1. Underdamped system 1 Case 2. Critically damped case 1 Compiled by Yidnekachew M. Page 27 of 28
Case 3. Over damped case 1 Figure 1.7 Instrument responses for second order Compiled by Yidnekachew M. Page 28 of 28