Mechanical Engineer''''s Reference Book 2011 Part 3 ppsx

Instrumentation 3/25 their paths differ by an odd number of half wavelengths they cancel. Figure 3.43 shows how this can be used for an accurate measurement of movement. As the mirror M moves, the light intensity changes from maximum to minimum and back for successive distances of half a wavelength - a fraction of a micrometre, making the system highly sensitive. Refinements are needed to determine the direction of motion, and to give general stability; a corner cube reflector instead of a simple mirror eliminates the otherwise high sensitivity to the angle of the mirror. A laser is a convenient source of coherent radia- tion. The output signal, going through a succession of peaks, is essentially digital. Moire fringes are sometimes used to measure movement. Figure 3.44 shows two adjacent gratings as seen from above. If they are positioned as in (a) light can pass through, but if one is moved by half a 'wavelength' as in (b) the path is blocked and the combination appears dark. The 'wavelength' or pitch can be very short, as small as a few micrometres (the name Moire comes from the silk weave in which the effect can be observed), giving a high potential accuracy. Again there is a basically digital output and the need to determine the direc- tion of movement. As shown in Figure 3.44, the interrogating light is tran- smitted through the gratings; it is, of course, possible to have a mirror system when the light source and detector are both on the same side of the gratings. The gratings may be at an angle to each other (Figure 3.49, when the alternate bright and dark areas form fringes perpendicular to the gratings; the fringes move bodily with linear displacement of either grating, while the separation between them depends on the angle between the two. Figure 3.41 Super-linear variable capacitor (SLVC) (courtesy ASL) transducers. Audiofrequency (AF) power supplies are used for resistance and inductance measurements, though d.c. is, of course, also effective for resistance. Circuits used for capacitance measurement must take account of the stray capacitance that occurs between nearby conductors unless they are specifically screened from each other. Thus in Figure 3.42 capacitance variations between its lead and earth (either within or outside the screened cable) are indistinguishable (to the measuring circuit) from transducer capacitance changes. Various arrangements can be adopted to overcome this problem. The relatively small capactiance in most transducers corresponds to a very high impedance at lower frequencies and this is an argument for working at higher frequencies, but in fact AF bridge systems with very high sensitivities are available and give good performance when spurious effects are eliminated. Many considerations come into the choice of transducer technique. As a very simple summary, it may be suggested that resistance devices are simple and inexpensive, inductance devices, while tending to be larger and more complicated, have a long history of development and mass production. Capacitance devices, simple and sensitive in principle, need more elaborate circuitry, but may well give the best approach for particualrly onerous requirements. Sometimes the force needed to move a transducer element is important. In general, the force is less for capacitors than for inductors, while with variable resistors it may be less repeatable. 3.5.2.5 Optical methods of position measurement Some classical experiments in physics depend on optical interference. If two coherent light beams are superposed they reinforce or cancel each other, according to whether they are in or out of phase, and this phase difference depends on the different lengths of the paths they have travelled. If they have travelled the same distance or their paths differ in length by an integral number of wavelengths then they reinforce, while if Screened connectinq cable & Figure 3.42 Effects of stray capacitance Reference mirror 1 I/ M Coherent - - light source - / 1 I/ M Coherent - - light source - / Light detector Figure 3.43 Movement measured by optical interference Incident light 1111 0 0'0'0'0 00000 Light detection 'Wave- length' A ooboo 00000 (a) (b) Figure 3.44 Principle of Moir6 fringes 3/26 Microprocessors, instrumentation and control Figure 3.45 Moire gratings at an angle If the gratings do not have quite the same pitch, there are fringes parallel to the grating elements (Figure 3.46). This principle is sometimes used in strain measurement, when the strain to be measured is arranged to alter the pitch. In all these arrangements there is an effective magnification, so that small movements, on the scale of the small pitch of the gratings, give rise to much larger movements of the fringes. Figure 3.46 Unequally spaced Moire gratings 3.5.2.6 Pneumatics Currently, pneumatic instrumentation systems are used less than electronic ones. They have the drawbacks of needing somewhat delicate mechanical devices and of introducing significant delays when signals are transmitted over long distances. However, they are by no means extinct and have the great safety advantage that there need be no question of their introducing electric sparks. The heart of a pneumatic instrument is a flapper adjacent to a nozzle as shown in Figure 3.47. As the separation, d, between these is changed, the air flow through the nozzle changes markedly and hence also the pressure drop across the ‘series’ restrictor. A typical relation between d and the pressu- Pressure gauge Air supply Restrictor Flapper Nozzle Figure 3.47 Pneumatic nozzle and flapper (courtesy Foxboro Company) re P just upstream of the nozzle is shown in Figure 3.48. The effect can be amplified by the introduction of further elements in the shape of valves and a pressure-sensitive diaphragm. The primary behaviour is inherently non-linear, but the use of a pressure-feedback device with levers and a spring-controlled bellows allows a movement of the order of a millimetre to give a proportional pressure change of some tens of kilopascals. 3.5.2.7 Angular displacement The synchro - sometimes called a Magslip or Selsyn - is widely used in the measurement of angles. If ax. is applied to the central element [rotor) of such a device (left-hand side of Figure 3.49) then the voltages induced in the three circumfe- rential windings depend on the angular position of the rotor 120 100 80 I m a YI E E n Q 60 L ._ 40 20 0 Air supply pressure 0 c - I I I 0.2 0.4 0.6 Distanced (mm) Figure 3.48 Relation between pressure and gap for pneumatic device (courtesy of Foxboro Company) Instrumentation 3/27 A. C. power A. C. power II I I’ 1 relative to them. This system has the particular advantage that if a second, identical unit is connected appropriately (right- hand side of Figure 3.49) forces will act within it until the two rotors take up identical angles. This is a robust and widely used technique for telemetering an angular position. Capacitive transducers with variable overlap readily give a measurement of angle. The arrangement is in fact just that of the orthodox variable capacitor. Encoders are used to give a digital signal corresponding to angular position. Moving clockwise round the disc shown in Figure 3.50, it can be seen that successive positions 1,2,3 . . correspond to successive binary numbers if black and white areas give digits 1 and 0, respectively, for powers of 2 starting at the largest radius. Black can be distinguished from white using six optical beams in the example shown, or a single beam can be traversed radially across the encoder. Alternatively, the distinction can be between conducting and insulating material, detected electrically. 31 Figure 3.50 Encoding for angular position A difficulty with this form of coding follows from imperfec- tions of manufacture. Considering, for instance, the move from position 7 to position 8. if the outermost black should turn white slightly before the others, the configuration will momentarily correspond to position 6, while premature changes of the other blacks would indicate 5 or 3, respectively. The problem arises from the need for simultaneous changes at more than one radius. and to overcome this, codes have been devised in which only one change occurs at a time. As indicared previously, small changes of angle can also be detected with Moire fringes. 3.5.2.8 Velocity measurement Angular velocity is commonly measured employing electrical induction. Using the fundamental law that induced voltage is II Figure 3.49 Principle of synchrc proportional to rate of change of flux, generators, either d.c. or ax., can be made for which output voltage gives a direct measure of the speed of rotation. Under a completely diffe- rent principle, a technique is to mount markers on the circumference of a rotor and count the number passing a stationary point in a given time, or alternatively, the time lapse between successive passages, which can be detected optically, magnetically or electrostatically. This system, of course, provides a digital output; it requires a finite time to give an indication. Linear velocity is sometimes deduced from angular velocity as in a car’s speedometer. It can also be calculated as the rate of change of position or as the integral of acceleration, and this is particularly relevant to vibration studies (Section 3.5.4). Measurement of fluid velocity is discussed under Flow in Section 3.5.7. 3.5.3 Volume and level Volume, as such, is a quantity that is seldom measured. Instrumentation for rate of change of volume (or flow) is widely applied and can be integrated to give total volume; this is dealt with in a later section. Volume and mass are simply related through density and mass can be measured as weight. Again, the volume of material in a container can be inferred from the level it reaches, and this is a common measurement. Measuring level, we can distinguish between continuous, normally analogue methods and digital techniques. in which the action is really detection rather than measurement. The presence or absence of the material in question at a particular level is indicated. The second category can be used, as shown in Figure 3.51. to move a ‘follower’ outside the container under study so that it remains opposite the internal interface, allowing the height to be measured in a more accessible place. The level of a liquid conductor can be found from resistance measurement. Figure 3.52 shows two resistive wires that are effectively short circuited where they enter the liquid, so that the resistance seen at their terminals decreases as the level rises. For an insulating liquid, capacitance measurement is appropriate. With the arrangement of Figure 3.53, capaci- tance increases as the level rises and a larger area of the overlapping plates is separated by a dielectric of higher permittivity. A sonar-ranging system can also be used in which the time taken for an echo to return from the surface being studied gives an indication of its position (Figure 3.54). A sophisticated single-point technique involves passing gamma rays through the container. These will be more attenuated if there is a denser material in their path, so the intensity of radiation received at the detector shows whether liquid (or solid) rather than just gas is present. In Figure 3.55 it can be recognized that the detector output will be larger if the level of liquid in the container falls below the line from source to detector. 3/28 Microprocessors, instrumentation and control I I I I I DETECTOR senses FOLL~WER it is above level 0 'vds signal to Controller follower to be lowered Contro I ler Figure 3.51 Level measurement with a follower Measured resistance corresponds to length of wire not immersed Figure 3.52 Level measurement using resistance Figure 3.53 Level measurement using capacitance 0 3 Receiver Duration of flight depends on path length to surface Figure 3.54 Level measurement by sound ranging Detector signal depends on material between it and source collimator Figure 3.55 Level measurement by gamma rays Different types of probe have been devised for detecting the presence of a liquid that depend on refractive index or resistivity or permittivity, all of which may have different values above and below the interface whosc level is to be detected. Yet another technique is to measure level as a differential pressure. If one pressure transducer is mounted internally at the bottom of a vessel and another at the top, then the difference between the pressures they show will depend on how much of the height between them is occupied by liquid and how much by gas (Le. on the level of the former). 3.5.4 Measurement of vibration Wider aspects of vibration. which may be thought of as movement over small distances at comparatively high frequen- cies, are discussed in Chapter 1. Here we touch briefly on techniques of measurement. In sinusoidal motion at frequenty w, linear amplitude s, velocity v and acceleration a are simply related as a = wv = w2s v = 0s Displacement can be measured using techniques described in Section 3.5.2, or velocity with a generator, commonly a coil moving in the field of an electromagnet. If a nearby point is known to be stationary, either measurement can be relative to this. Alternatively, part of the transducer can be an element with sufficient inertia not to move, when the measurement can be made relative to that. The criterion whether the inertia is large enough is that the resonant frequency of the ele- ment - decided by its mass and its flexible mounting - should be much lower than any frequencies in the vibration. Vibrational accelerations are very often measured, using the associated force F, where F = ma, and m is an inertial mass. In this case, the element's resonant frequency must be much greater than the highest vibrational frequency to ensure that a is the same as the acceleration of the part on which the transducer is mounted. A piezoelectric device often forms the link to the intertial mass, the charges excited in it providing the output signal, while its high degree of stiffness gives a high resonant frequency. The wide range of mass and the variety of piezoelectric elements that may be used give scope for a wide range of applicability for vibration pick-ups. We may want to know the frequencies contributing to the vibration studied. For this, some form of spectrum analyser will be desirable. 3.5.5 Force/weight measurement This field of instrumentation gives good examples of some general principles mentioned in Section 3.5.1. Concerning speed of response, in many instances the measurement called for is a static one, but sometimes a quickly varying force is to be studied, and this calls for a different approach. The potential accuracy varies more than a thousandfold - with corresponding price ranges for equipment. Sensitivity to extra- neous influences is also a factor in accurate force measure- ment, where errors from temperature, wrong location of the force and other things must be guarded against. Weight, of course, is a force, and, in general, it is measured in similar ways to other forces, though the measurement is always a static one. Lever-type instruments, such as the classical analytical balance, are basically devices for comparing forces - often the weights of different masses. Unequal lever arms allow widely different forces to be compared; an arm of variable length Instrumentation 3/29 greater sensitivity if mercury is replaced by water. Further movement for a given pressure difference can be achieved if such a manometer is at a small angle to the horizontal instead of being held vertical. Care must be taken that the liquid can move freely over the inside surface of the tube (otherwise there will be hysteresis) and that any distortion of the surface from surface tension is the same in both limbs. Note that it is pressure differential, P1-P2, that is of concern. Three situations should be distinguished: 1. Absolute pressure, where P2 is zero, corresponding to a vacuum; 2. Gauge pressure, where P2 is the atmospheric pressure in the neighbourhood of the equipment; 3. Differential measurements where both PI and Pz may vary but it is their difference that is significant. Pressure can be measured in terms of its fundamental definition of force per unit area. In Figure 3.61 if the cross- sectional area of the cylinder with its piston is known, then the pressure, P, is directly given by the force, F(usual1y a weight), needed to balance it. This method can give high accuracy; but there are practical complications, notably to ensure that the piston can move freely without the liquid leaking, so it is used mainly to calibrate other pressure gauges. Neither of these approaches uses compact equipment or leads directly to an output signal, so they are often replaced by the use of transducer elements. Various configurations change shape with pressure and the consequent displacement can be used for measurement. A Bourdon tube has an elliptical (or otherwise unsymme- trical) cross-section and is bent into a circular arc (Figure 3.62). If the pressure inside the tube increases, it tends to make the cross-section more nearly circular, and this in turn straightens the arc. With some further mechanical amplifica- tion, the movement is large enough to be read against a scale. A metal diaphragm distorts according to the difference in pressure of the fluid on either side of it. The sensitivity varies widely with the dimensions. A wide, thin diaphragm moves appreciably under small pressures, and the danger of its rupturing under overload can be greatly reduced by the provision of ‘stops’. The movement can be detected pneuma- tically or by capacitive or inductive devices. An alternative approach is to measure strain in the diaph- ragm. Since different parts are strained in different senses, strain gauges connected in different arms of an electrical bridge can be mounted on a diaphragm so that their pressure- induced outputs sum while the spurious changes from, for example, temperature variation cancel each other out. A development from this is to have the strain gauges integral with the diaphragm. Using appropriate fabrication techniques, a silicon member can serve as diaphragm and can have certain parts modified and electrically insulated so that their strain- sensitive properties can be used to give an electrical output. Stiff diaphragms - with a high natural frequency - allow rapidly changing pressures to be measured. Other devices have much slower responses, but often the measurement required is only of quasi-static pressure. Pressure transducers including piezoelectric force measurement have a quick res- ponse but cannot be used statically. allows a precise ratio to be established without the need for an adjustable force. Tie spring balance is the most familiar member of a large family of instruments in which the force to be measured is balanced by the reaction from an elastically strained member whose distortion can be measured. The proving ring illustrated in Figure 3.56 is a refined form of spring balance. The applied force, which may be compress- ive or tensile. distorts the ring from its intially circular shape. The change in diameter (measured mechanically with a dial gauge or electronically) indicates the force. Much smaller movements are measured than with a coil-spring spring balance: making the total system more compact. In a strain-gauge load cell the process is taken rather further, the elastic strains in a member being directly mea- sured with strain gauges, allowing very compact devices to be constructed. The principle of a simple, columnar load cell is shown in Figure 3.57: the four strain gauges are connected into the four arms of a bridge. Structures in which shear strains are measuresd are also widely used; their readings are less depen- dent on the position where the load is applied. Two instru- ments of this type are shown in Figure 3.58. In hydraulic load cells the unknown force alters the pressure in a liquid system, allowing it to be measured as a pressure. Load cells can be built into the supports of hoppers to weigh their contents, or included in weighbridges. Systems of particular value for dynamic measurement of quickly changing forces include piezoelectric elements, men- tioned in connection with vibration instrumentation. Force balance systems are also used. Figure 3.59 shows how the displacement produced by a force to be measured can control the restoring force in a coil, the current in which gives a direct indication of the first force provided the gain is large and the displacement small - with due attention paid to stability. Table 3.2 summarizes the features of different ways of measuring force. 3.5.6 Pressure Pressure is easily measured from the difference in level of the liquid in two arms of a U-tube (Figure 3.60): PI - P2 = hp where pis the density of the liquid. Mercury is commonly used as the working liquid. Its high density means that a large pressure difference can be measured without the equipment becoming too big in order to accommodate a large h. There is Applied i force High grade ring - steel compresoive forces T.) 19 Figure 3.56 Proving ring 3.5.6.1 Vacuum Some widely different methods are available for measuring vacuum (Le. a pressure less than atmospheric). As conditions approach the zero of absolute vacuum, measurements become increasingly difficult. 3/30 Microprocessors, instrumentation and control Table 3.2 Method Type of loading Force range, N Accuracy % Size (appr0x.j (approx.) Lever balance Static 0.001 to 150 k Very high Bulky and heavy Force balance Statiddynamic 0.1 to 1 k Very high Bulky and heavy Hydraulic load cell Statiddynamic 5 k to 5 M 0.25 to 1.0 Compact and stiff Spring balance Static 0.1 to 10 k Low Large and heavy Proving ring Static 2kto2M 0.2 to 0.5 Compact Piezoelectric transducer Dynamic 5ktolM 0.5 to 1.5 Small Strain-gauge load cell Statiddynamic 5 to 40 M 0.01 to 1.0 Compact and stiff Force A B Out-of-balance < yr ) voltage P C on four sides of column Figure 3.57 Principle of load cell Applied force u Figure 3.58 Photograph of load cells (courlesy Defiant Weighing) n Restoring coil / e Output voltage CI e applied force XI I I Pernianent Displacement transducer (capacitive or L V.D T Figure 3.59 Force balance system Instrumentation 3/31 Figure 3.60 U-tube manometer F Figure 21.61 Absolute pressure measurement Badiidon rube adjusting screw connecting link iw8th nut and washer) Figure 3.62 Bourdon tube pressure gauge (courtesy Budenberg Gauge Company) In a low vacuum (i.e. an appreciable fraction of an atmos- phere) instruments described in the previous section can be used. The McLeod gauge is a development from the U-tube manometer, in which a sample of gas is compressed by a known amount before its pressure is measured; this allows much lower initial pressures to be measured. Two other broad techniques are used for high vacua. With thermal conductivity instruments (notably the Pirani) use is made of the fact that the larger number of molecules in a gas at higher pressure increase its heat transfer. so that measurement of the temperature of a heated member can indicate the degree of vacuum surrounding it. In ionization instruments (Buckley, Penning, Bayard-Alpert) the current resulting from ions in the vacuum is measured. This gives the population density of ions and hence the pressure. The ranges over which different techniques can be used are shown in Table 3.3. 3.5.7 Flow An important and widely applied field of instrumentation is the measurement of fluid flow. Sometimes the concern is to measure velocity at a ‘point’. More often the requirement is for a single measurement representing the total volume of fluid passing along a pipe or other container - though this can be achieved by integrating from ‘point’ values. Instantaneous readings for flow are of primary interest: often their time integral (i.e. the total volume that has passed) needs to be known. Gases, as well as liquids, come under study. Occa- sionally, the main interest is with the mass rather than the volume that is passing. Conceptually, the most direct form of instrumentation for flow measurement is the positive displacement meter, in which it is arranged that a known volume is repeatedly filled and emptied and the number of times that this takes place is counted. Of this type is the common, domestic gas meter, whose operation is illustrated in Figure 3.63. The capacities of chambers A and B are altered by a known voiume as the diaphragm between them moves between the limits of its travel. A suitably phased slide valve ensures that the two chambers are connected alternately to the inlet manifold and to the outlet. For smoother operation, A and B are duplicated by C and D. running out of phase with them. In other types, the volumes that are alternately filled and emptied are defined by rotating parts. either on a single axis or by two meshing rotors as shown in Figure 3.64. Care must be taken that sealing - usually by a liquid - is effective while still allowing free movement under the small forces associated with gas at low pressure. 3.5.7.1 Similar devices are available as flow meters for liquids. A rotating piston, mounted eccentrically in a larger cylinder, is a common arrangement. Reciprocating pistons are also used as well as the sort of rotary systems described for gases. A turbine meter (Figure 3.65) may be thought of as a positive displacement instrument, having been designed so that the angle its bladed rotor turns through is proportional to the volume of liquid that has passed (axially) through the meter. With all these meters, the number of rotations or excursions must be counted, the flow rate, of course. being given by the number occurring in a particular time. Information about internal movements must be conveyed through a container wall to the outside, and this is often done by the passage Qf permanent magnetic poles past external pick-ups. Positive displacement for liquid flow 3/32 Microprocessors, instrumentation and control Table 3.3 Comparison of vacuum gauge techniques Technique Pressure Accuracy Costa Advantages Limitations range (i%) (Pa> Bourdon tube Diaphragm Liquid manometer McLeod Thermal conductivity Ionization 105-102 10 A Simple, robust io5-10 5 B Good general-purpose lo5-lo2 5-10 A Simple, direct reading 105-10-3 5-10 C Wide range. Used for 103-10-2 10-20 C Can be robust, with 102-10-8 20 D Sensitive. fast gauge calibration fast response response a Scale of costs: A less than f100. B f10G200, C f20k400, D f40G600 Inlet Outlet Housing -Slide valve Chamber A is el B is filling C is empty D has just filled nptying Chamber A is filling Gas outlet I Gas inlet Figure 3.64 Positive displacement flow meter Poor accuracy below 100 Pa Zero setting varies Vapour may contaminate vacuum Intermittent. Measures gas pressures only Risk of zero variation Care needed in use -=- Chamber A is empty B is full C is filling D is emptying Chamber A is B is empty C is emptying D is filling full Figure 3.63 Principle of gas meter Flow along tube acts on blades to rotate turbine Figure 3.65 Turbine meter Instrumentation 3/33 3.5.7.2 Differential pressure Where the cross-sectional area of a pipe changes, so does the pressure of a liquid flowing in the pipe, and the magnitude of the pressure change depends on the flow rate. This is often used as the principle of a flow meter. Two configurations of changing cross section may be distinguished: the Venturi throat and the orifice plate. In the former, a smooth profile serves to reduce the area (Figure 3.66); in the latter, changes are more abrupt (Figure 3.67). The Venturi has the advantage that less energy is absorbed, but at the cost of greater size and expense. Profiles different from either of these are sometimes used. In the Venturi, the difference in pressure between the throat and a point ulpstream is measured. With an orifice plate, the two relevant pressures are simply those upstream and immediately downstream of the plate, since for some distance downstream the effective area is still that of the orifice. There is a square-law relation between the flow-rate Q (m3/s) and the differential pressure, Ap: e2 a Ap or Q = kw In some circumstances this is an inconvenience. point of restriction: One of the factors in the proportionality is the area A at the Q = cA.m and in one type of flow meter, the Rotameter, Ap is kept constant and A made variable. This is achieved as shown in Figure 3.68 by having the liquid flow up through a tapered tube in which is placed a plummet whose weight causes the differential pressure. Increasing flow carries the plummet to a point where the annular area around it is such as to satisfy the equation. Supel-ficially similar to differential pressure types is the Target Flowmeter, in which the force exerted on a body obstructing the flow is used as an index of that flow. This again follows a square-root law. 3.5.7.3 Open channels The measurement of flow in an open channel (or a closed duct that is riot completely filled with liquid) calls for a different Flow G 1 Pressure- I measurement points Figure 3.66 Venturi throat Flow u n Pressure- measurement points Figure 31.67 Orifice plate Maximum flow rate due to maximum annular area is obtained with float at large end of tube Noting position of edge of float .referred to capacity scale on glass gives flow rate reading Metering float suspended freely ‘~n fluid being metered Tapered transparent metering tube (borosilicate glass) 4 Fluid passes through this annular opening between periphery of float head and I.D. of tapered tube. Of course, flow rate varies directly as area of annular opening varies Figure 3.68 Rotameter (courtesy Fkcher and Porter) approach. Flow over a weir may be measured by noting the level to which the liquid rises in a ‘notch’ in the weir. If there is not an adequate head of liquid to allow construction of a weir, the channel may have a ‘flume’ built into it; this is a construc- tion similar to the throat of a Venturi tube, and again the pressure readings at appropriate points allow the flow rate to be calculated. In either case, the water level corresponding to a particular point in the flow is measured in another chamber - a ‘stilling well’ - connected by a small pipe. 3.5.7.4 Newer flowmeter principles Three techniques may be mentioned that have more recently been developed to measure flow. When an obstruction is mounted in a pipe, the flow can be disturbed so that vortices are shed alternately from its opposite sides, and the frequency of this shedding is accurately proportional to flow rate. Sensitive detectors are needed to detect the vortices: com- monly using their pressure or cooling effects, or their moduia- tion of an ultrasonic beam. The method has the advantage of not being dependent on the exact sensitivity of the detector; in fact it uses a digital signal, namely frequency. Electromagnetic flow meters use the principle of Faraday‘s law of electromagnetic induction. This states that a conductor moving in a magnetic field will give rise to an electromotive force (i.e. potential or voltage). The field, the movement and the potential are all mutually perpendicular. In a conventional electrical generator the conductor is a wire. but it can equally be a conducting liquid such as water. All that is needed is to provide a magnetic field - commonly non-sinusoidal at a low frequency - and suitably insulated electrodes in contact with the liquid. [...]... expressed as Time delay - k 1 -t k + (3. 29) TS Dividing top and bottom by (1 + k ) results in = e-sT = e-jwT (3. 34) Consider the open-loop response of a first-order system incorporating a time delay as illustrated in Figure 3. 106 The open loop transfer function becomes (3. 30) k, k =- l t k (3. 31) and T$ = l+k k Amplitude ratio = We define the following terms as (3. 32) where k, is the closed-loop gain... transfer function is given by equation (3. 18) Substituting equations (3. 18) and (3. 48) into equation (3. 39) results after some manipulatior., in ~- 4 PV( (kK/7T1)(1 ST,) + + [(UT) (kK/.r)]s+ (~KITT,)} + (3. 49) SP(s) Comparing the denominator with that for the generalized second order system (i.e equation (3. 25) it can be shown that {s’ (3. 50) and w: = (~K/T,T) (3. 51) For the system being controlled,... shown in Figure 3. 1 13 Any system which has a response similar to that given in the figure has a transfer function which approximates to a first-order system with a time delay, i.e I Process response t I Time / Time Figure 3. 1 13 Open-loop system response to a step input Classical control theory and practice 31 53 Table 3 5 Control action K T, Three-ierm controller with a first-order system 3. 6.5.4 Optimum... discrete system (Figure 3. 1 23) The transfer function for a zero-order hold is 1 - e-sT (3. 53) S Figure 3. 1 19 ModeVactual error generation The zero-order hold is simply included in the evaluation of the closed-loop transfer function for the discrete time system The next step is to replace the Laplace transforms with their Classical control theory and practice 3/ 57 Figure 3. 120 Model reference adaptive control... necessary to reduce the closedloop gain in order to obtain a stable response The penalty to be paid for increasing the stability in this manner is an increase in the steady-state error (3. 33) Equations (3. 31) and (3. 32) show, respectively, that both the closed-loop system gain and the time constant are less than those associated with the open-loop system This means that the closed-loop response is faster... burnt out 3/ 38 Microprocessors, instrumentation and control Ceramic tube P i co Platinum wire, Ckrarnic rod (C) Figure 3. 77 Construction of resistance thermometers (courtesy Rosemount Engineering) A g 105 Conventional (NTC) thermistor - Switching (PTC) thermistor E io4 m i - - IT io3 - P t resist thermometer 102 - 0 100 200 - * 30 0 Temperature ("C) Figure 3 7 8 Thermistor characteristics 3. 5.8.5 Radiation... process Equation (3. 20), in fact, describes the steady-state output from a first-order, open-loop control system The transient part of the output, which the time-domain solution illustrated in Figure 3. 91, is not apparent in the frequency-domain solution The Laplace operator, s, may be replaced withjw, wherej is the complex operator g-1 Equation (3. 18) then becomes Figure 3. 95 (3. 21) Equation (3. 21) shows... replacement after a reading has been taken 3. 5.8 .3 Figure 3. 73 Thermoelectric potential Resistance thermometers One of the important properties that varies with temperature and so is used in thermometers is electrical resistance This Temperature-sensitive resistor in thermal contact with cold junction Cold 5- 0 Figure 3. 74 Cold-junction compensation Instrumentation 3/ 37 Insulated hot junction Metal sheath... Laplace variable, s Thus dY/dt becomes sY(s) and dZY/d? is replaced with s2Y(s) 3. 6 .3. 1 First-order systems The governing equation is rewritten with the appropriate Laplace transforms replacing the differential operators Thus equation (3. 5) becomes T s Y(s)+ Y(s) = kX(s) (3. 16) Hence Y(s)[l+ Ts] = kX(s) (3. 18) Equation (3. 18) enables the convenient facility of incorporating the transfer function within... ( S= C(S) G(s) E ( s ) ) = (3. 38) C(S) G(s) [SP(s) - PV(s)] Hence PV(s) ~- C ( S ) G(s) - + C ( S ) G(s) SP(S) 1 3. 6.5.1 (3. 39) ONIOFF control In many applications a simple ONiOFF strategy is perfectly adequate to control the output variable within preset limits The ON/OFF control action results in either full or zero power being applied to the process under control A mechanical type of thermostat . is filling full Figure 3. 63 Principle of gas meter Flow along tube acts on blades to rotate turbine Figure 3. 65 Turbine meter Instrumentation 3/ 33 3. 5.7.2 Differential pressure. acceleration, and this is particularly relevant to vibration studies (Section 3. 5.4). Measurement of fluid velocity is discussed under Flow in Section 3. 5.7. 3. 5 .3 Volume and level Volume,. Time Figure 3. 87 Ramp input A I w Time Figure 3. 88 Sinusoidal input Classical control theory and practice 3/ 43 c 3 Q 3 U c rn 4- 2 Q 4- - expressed as a differential