Clarence-W-de-Silva-Mechatronics-A-Foundation-Course-CRC-Press-2010-(F 2)

7 Actuators Study Objectives • The purpose of actuators in a mechatronic system • Types of actuators • • • • Stepper motors and dc motors (including brushless dc motors) AC motors (induction motors and synchronous motors) Linear actuators Hydraulic and pneumatic actuators • • • • Modeling and analysis of actuators Practical performance and parameters of actuators Sizing and selection of actuators for practical applications Instrumentation, drive hardware, and control of actuators 7.1 Introduction This chapter introduces the subject of actuators, as related to mechatronics The actuator is the device that mechanically drives a mechatronic system Joint motors in a robotic manipulator are good examples of such actuators Actuators may be used as well to operate controller components ( nal control elements), such as servovalves, as well Actuators in this category are termed control actuators Actuators that automatically use response error signals from a process in feedback to correct the operation of the process (i.e., to drive the process to achieve a desired response) are termed servoactuators In particular, the motors that use measurements of position, speed, and perhaps load torque and armature current or eld current in feedback to drive a load to realize a speci ed motion are termed servomotors One broad classi cation separates actuators into two types: incremental-drive actuators and continuous-drive actuators Stepper motors, which are driven in xed angular steps, represent the class of incremental-drive actuators They can be considered to be digital actuators, which are pulse-driven devices Each pulse received at the driver of a digital actuator causes the actuator to move by a predetermined, xed increment of displacement Continuous-drive devices are very popular in mechatronic applications Examples are direct current (dc) torque motors, induction motors, hydraulic and pneumatic motors, and piston-cylinder drives (rams) Microactuators are actuators that are able to generate very small (microscale) actuating forces/torques and motions In general, they can be neither 465 466 Mechatronics: A Foundation Course developed nor analyzed as scaled-down versions of regular actuators Separate and more innovative procedures of design, construction, and analysis are necessary for microactuators Micromachined, millimeter-size micromotors with submicron accuracy are useful in modern information storage systems Distributed or multilayer actuators constructed using piezoelectric, electrostrictive, magnetostrictive, or photostrictive materials are used in advanced and complex applications such as adaptive structures An actuator may be directly connected to the driven load and this is known as the “direct-drive” arrangement More commonly, however, a transmission device may be needed to convert the actuator motion into a desired load motion for the proper matching of the actuator with the driven load The stepper motor, dc motor, alternating current (ac) induction motor, and hydraulic actuator are particularly studied in this chapter The modeling, selection, drive system, and control of various actuators are discussed and the procedures of actuator selection are also addressed 7.2 Stepper Motors Stepper motors are a popular type of actuator They are driven in xed angular steps (increments) Each step of rotation is the response of the motor rotor to an input pulse (or a digital command) In this manner, the stepwise rotation of the rotor can be synchronized with pulses in a command-pulse train, assuming of course that no steps are missed, thereby making the motor respond faithfully to the input signal (pulse sequence) in an open-loop manner Like a conventional continuous-drive motor, a stepper motor is also an electromagnetic actuator in that it converts electromagnetic energy into mechanical energy to perform mechanical work The terms stepper motor, stepping motor, and step motor are synonymous and are often used interchangeably One common feature in any stepper motor is that the stator of the motor contains several pairs of eld windings (or phase windings) that can be switched on to produce electromagnetic pole pairs (N and S) These pole pairs effectively pull the motor rotor in sequence so as to generate the torque for motor rotation By switching the currents in the phases in an appropriate sequence, either a clockwise (CW) rotation or a counterclockwise (CCW) rotation can be produced The polarities of a stator pole may have to be reversed in some types of stepper motors in order to carry out a stepping sequence Although the commands that generate the switching sequence for a phase winding could be supplied by a microprocessor or a personal computer (a software approach), it is customary to generate it through hardware logic in a device called a translator or an indexer This approach is more effective because the switching logic for a stepper motor is xed, as noted in the foregoing discussion Microstepping provides much smaller step angles This is achieved by changing the phase currents by small increments (rather than on, off, and reversal) so that the detent (equilibrium) position of the rotor shifts in correspondingly small angular increments 7.2.1 Stepper Motor Classification Most classi cations of stepper motors are based on the nature of the motor rotor One such classi cation considers the magnetic character of the rotor Speci cally, a variablereluctance (VR) stepper motor has a soft-iron rotor while a permanent-magnet (PM) stepper motor has a magnetized rotor The two types of motors operate in a somewhat similar Actuators 467 manner Speci cally, the stator magnetic eld (polarity) is stepped so as to change the minimum reluctance (or detent) position of the rotor in increments Hence, both types of motors undergo similar changes in reluctance (magnetic resistance) during operation A disadvantage of VR stepper motors is that since the rotor is not magnetized, the holding torque is zero when the stator windings are not energized (power off) Hence, there is no capability to hold the load at a given position under power-off conditions unless mechanical brakes are employed A hybrid stepper motor possesses characteristics of both VR steppers and PM steppers The rotor of a hybrid stepper motor consists of two rotor segments connected by a shaft Each rotor segment is a toothed wheel and is called a stack The two rotor stacks form the two poles of a permanent magnet located along the rotor axis Hence, an entire stack of rotor teeth is magnetized to be a single pole (which is different from the case of a PM stepper where the rotor has multiple poles) The rotor polarity of a hybrid stepper can be provided either by a permanent magnet or by an electromagnet using a coil activated by a unidirectional dc source and placed on the stator to generate a magnetic eld along the rotor axis Another practical classi cation that is used in this book is based on the number of “stacks” of teeth (or rotor segments) present on the rotor shaft In particular, a hybrid stepper motor has two stacks of teeth Further sub-classi cations are possible, depending on the tooth pitch (angle between adjacent teeth) of the stator and tooth pitch of the rotor In a single-stack stepper motor, the rotor tooth pitch and the stator tooth pitch generally have to be unequal so that not all teeth in the stator are ever aligned with the rotor teeth at any instant It is the misaligned teeth that exert the magnetic pull, generating the driving torque In each motion increment, the rotor turns to the minimum reluctance (stable equilibrium) position corresponding to that particular polarity distribution of the stator In multiple-stack stepper motors, operation is possible even when the rotor tooth pitch is equal to the stator tooth pitch, provided that at least one stack of rotor teeth is rotationally shifted (misaligned) from the other stacks by a fraction of the rotor tooth pitch In this design, it is this inter-stack misalignment that generates the drive torque for each motion step It should be obvious that unequal-pitch multiple stack steppers are also a practical possibility In this design, each rotor stack operates as a separate single-stack stepper motor A photograph of the internal components of a two-stack stepper motor is given in Figure 7.1 7.2.2 Hybrid Stepper Motor Hybrid steppers are arguably the most common variety of stepping motors in engineering applications A hybrid stepper motor has two stacks of rotor teeth on its shaft The two rotor stacks are magnetized to have opposite polarities, as shown in Figure 7.2 There are two stator segments surrounding the two rotor stacks Both rotor and stator have teeth and their pitch angles are equal Each stator segment is wound to a single phase, and accordingly, the number of phases is two It follows that a hybrid stepper is similar in mechanical design and stator winding to a multi-stack, equal-pitch, VR stepper There are some dissimilarities, however First, the rotor stacks are magnetized Second, the inter-stack misalignment is ¼ of a tooth pitch (see Figure 7.3) A full cycle of the switching sequence for the two phases is given by [0, 1], [−1, 0], [0, −1], [1, 0], [0 1] for one direction of rotation In fact, this sequence produces a downward movement (CW rotation, looking from the left end) in the arrangement shown in Figure 7.3, starting from the state of [0, 1] shown in the gure (phase off and phase on with N polarity) For the opposite direction, the sequence is simply reversed; thus, [0, 1], [1, 0], [0, −1], [−1, 0], [0, 1] Clearly, the step angle is given by 468 Mechatronics: A Foundation Course FIGURE 7.1 A commercial two-stack stepper motor (Courtesy of Danaher Motion, Rockford, IL With permission.) Phase Phase N S Stator Rotor Rotor stack stack (magnetized N) (magnetized S) FIGURE 7.2 A hybrid stepper motor ∆θ = θ (7.1) where θ = θr = θs = tooth pitch angle Just like in the case of a PM stepper motor, a hybrid stepper has the advantage providing a holding torque (detent torque) even under power-off conditions Furthermore, a hybrid stepper can provide very small step angles, high stepping rates, and generally good torque–speed characteristics 469 Actuators Rotor teeth N Stator teeth S Pitch Offset Rotor stack Stator Rotor Stator segment stack segment 2 (phase 1) (phase 2) surrounding surrounding stack stack FIGURE 7.3 Rotor stack misalignment (1/4 pitch) in a hybrid stepper motor (schematically shows the state where phase is off and phase is on with N polarity) Example 7.1 The half-stepping sequence for the motor represented in Figures 7.2 and 7.3 may be determined quite conveniently Starting from the state [0, 1], if phase is turned on to state “−1” without turning off phase 2, then phase will oppose the pull of phase 2, resulting in a detent position halfway between the full stepping detent position Next, if phase is turned off while keeping phase in “−1,” the remaining half step of the original full step will be completed In this manner, the halfstepping sequence for CW rotation is obtained as: [0, 1], [−1, 1], [−1, 0], [−1, −1], [0, −1], [1, −1], [1, 0], [1, 1], [0, 1] For CCW rotation, this sequence is simply reversed Note that, as expected, in half-stepping, both phases remain on during every other half step 7.2.3 Microstepping Full-stepping or half-stepping can be achieved simply by using an appropriate switching scheme of the phases (stator poles) of a stepper motor For example, half-stepping occurs when phase switchings alternate between one-phase-on and two-phase-on states Fullstepping occurs when either one-phase-on switching or two-phase-on switching is used exclusively for every step In both these cases, the current level (or state) of a phase is either (off) or (on) Rather than using two current levels (the binary case), it is possible to apply several levels of phase current between these two extremes, thereby achieving much smaller step angles This is the principle behind microstepping Microstepping is achieved by properly changing the phase currents in small steps instead of switching them on and off (as in the case of full-stepping and half-stepping) The principle behind this can be understood by considering two identical stator poles (wound with identical windings), as shown in Figure 7.4 When the currents through the windings are identical (in magnitude and direction), the resultant magnetic eld will lie symmetrically between the two poles If the current in one pole is decreased while the other current is kept unchanged, the resultant magnetic eld will move closer to the pole with the larger 470 Mechatronics: A Foundation Course Equilibrium (detentent) position before the microstep i δθ Equilibrium (dentent) position after the microstep i + δi FIGURE 7.4 The principle of microstepping current Since the detent position (equilibrium position) depends on the position of the resultant magnetic eld, it follows that very small step angles can be achieved simply by controlling (varying the relative magnitudes and directions of) the phase currents Step angles of 1/125 of a full step or smaller could be obtained through microstepping For example, 10,000 steps/revolution may be achieved Note that the step size in a sequence of microsteps is not identical This is because stepping is done through the microsteps of the phase current (and the magnetic eld generated by it), which has a nonlinear relation with the step angle Motor drive units with the microstepping capability are more costly, but microstepping provides the advantages of accurate motion capabilities, including ner resolution, overshoot suppression, and smoother operation (reduced jitter and less noise) even in the neighborhood of a resonance in the motor-load combination A disadvantage is that usually there is a reduction in the motor torque as a result of microstepping 7.2.4 Driver and Controller In principle, the stepper motor is an open-loop actuator In its normal operating mode, the stepwise rotation of the motor is synchronized with the command pulse train Under highly transient conditions near rated torque, “pulse missing” can be a problem A stepper needs a “control computer” or at least a hardware “indexer” to generate the pulse commands and a “driver” to interpret the commands and correspondingly generate the proper currents for the phase windings of the motor This basic arrangement is shown in Figure 7.5a For feedback control, the response of the motor has to be sensed (say, using an optical encoder) and fed back into the controller (see the dotted line in Figure 7.5a) to take the necessary corrective action to the pulse command when an error is present The basic components of the driver for a stepper motor are identi ed in Figure 7.5b It consists of a logic circuit called a “translator” to interpret the command pulses and switch the appropriate analog circuits to generate the phase currents Since suf ciently high current levels are needed for the phase windings, depending on the motor capacity, the drive system includes ampli ers powered by a power supply The command pulses are generated either by a control computer (a desktop computer or a microprocessor), the software approach, or by a variable-frequency oscillator (or 471 Actuators Controller/ indexer Driver Motor Response Feedback (a) Controller Position pulse train Computer/ indexer (b) Translator Direction pulse train Drive Power supply Amplifier Stepper motor To load Current to windings FIGURE 7.5 (a) The basic control system of a stepper motor; (b) The basic components of a driver an indexer), the basic hardware approach For bidirectional motion, two pulse trains are necessary: the position-pulse train and the direction-pulse train, which are determined by the required motion trajectory The position pulses identify the exact times at which angular steps should be initiated The direction pulses identify the instants at which the direction of rotation should be reversed Only a position pulse train is needed for unidirectional operation The generation of the position pulse train for steady-state operation at a constant speed is a relatively simple task In this case, a single command identifying the stepping rate (pulse rate), corresponding to the specified speed, would suffice The logic circuitry within the translator will latch onto a constant-frequency oscillator with the frequency determined by the required speed (stepping rate) and continuously cycle the switching sequence at this frequency This is a hardware approach to open-loop control of a stepping motor For steady-state operation, the stepping rate can be set by manually adjusting the knob of a potentiometer connected to the translator For simple motions (e.g., starting from rest and stopping after reaching a certain angular position), the commands that generate the pulse train (commands to the oscillator) can be set manually Under the more complex and transient operating conditions that are present when following intricate motion trajectories, however, a computer-based (or microprocessor-based) generation of the pulse commands, using programmed logic, would be necessary This is a software approach, which is usually slower than the hardware approach Sophisticated feedback control schemes can be implemented as well through such a computer-based controller The translator module has logic circuitry to interpret a pulse train and “translate” it into the corresponding switching sequence for stator eld windings (on/off/reverse state for each phase of the stator) The translator also has solid-state switching circuitry (using gates, latches, triggers, etc.) to direct the eld currents to the appropriate phase windings according to the particular switching state A “packaged” system typically includes both indexer (or controller) functions and driver functions As a minimum, it possesses the capability to generate command pulses at a steady rate, thus assuming the role of the pulse generator (or indexer) as well as the translator and switching ampli er 472 Mechatronics: A Foundation Course functions The stepping rate or direction may be changed manually using knobs or through a user interface The translator may not have the capability to keep track of the number of steps taken by the motor (i.e., a step counter) A packaged device that has all these capabilities, including pulse generation, the standard translator functions, and drive ampli ers, is termed a preset indexer It usually consists of an oscillator, digital microcircuitry (integrated-circuit [IC] chips) for counting and for various control functions, a translator, and drive circuitry in a single package The required angle of rotation, stepping rate, and direction are set manually, by turning the corresponding knobs With a more sophisticated programmable preset indexer, these settings can be programmed through computer commands from a standard interface An external pulse source is not needed in this case A programmable indexer—consisting of a microprocessor and microelectronic circuitry for the control of position and speed and for other programmable functions, memory, a pulse source (an oscillator), a translator, drive ampli ers with switching circuitry, and a power supply— represents a “programmable” controller for a stepping motor A programmable indexer can be programmed using a personal computer or a hand-held programmer (provided with the indexer) through a standard interface (e.g., RS232 serial interface) Control signals within the translator are on the order of 10 mA, whereas the phase windings of a stepper motor require large currents on the order of several amperes Control signals from the translator have to be properly ampli ed and directed to the motor windings by means of “switching ampli ers” for activating the required phase sequence Power to operate the translator (for logic circuitry, switching circuitry, etc.) and to operate phase excitation ampli ers comes from a dc power supply (typically 24 V dc) A regulated (i.e., the voltage is maintained constant irrespective of the load) power supply is preferred A packaged unit that consists of the translator (or preset indexer), the switching ampli ers, and the power supply is what is normally termed a motor-drive system The leads of the output ampli ers of the drive system carry currents to the phase windings on the stator (and to the rotor magnetizing coils located on the stator in the case of an electromagnetic rotor) of the stepping motor The load may be connected to the motor shaft directly or through some form of mechanical coupling device (e.g., harmonic drive, tooth-timing belt drive, hydraulic ampli er, rack, or pinion) 7.2.5 Driver Hardware The driver hardware consists of the following basic components: Digital (logic) hardware to interpret the information carried by the stepping pulse signal and the direction pulse signal (i.e., step instants and the direction of motion) and to provide appropriate signals to the switches (switching transistors) that actuate the phase windings This is the “translator” component of the drive hardware The drive circuit for phase windings with switching transistors to actuate the phases (on, off, and reverse in the uni lar case; on and off in the bi lar case) Power supply to power the phase windings These three components are commercially available as a single package to operate a corresponding class of stepper motors Since there is considerable heat generation in a drive module, an integrated heat sink (or some means of heat removal) is needed as well Consider the 473 Actuators V + Power supply Motor windings STEP (stepping pulses) A HALF/FULL (half stepping) A CW/CCW (direction pulses) B RESET (initialization) B A+ A A– A Translator B+ B B– B G Ground Unipolar drive FIGURE 7.6 Basic drive hardware for a two-phase bi lar-wound stepper motor drive hardware for a two-phase stepper motor The phases are denoted by A and B A schematic representation of the drive system, which is commercially available as a single package, is shown in Figure 7.6 What is indicated is a unipolar drive (no current reversal in a phase winding) As a result, a stepper motor with bi lar windings (two coil segments for each phase) has to be used The motor has ve leads, one of which is the “motor common” or ground (G) and the other four are the terminals of the two bi lar coil segments (A+, A−, B+, B−) There are several pins in the drive module, some of which are connected to the motor controller/computer (driver inputs) and some are connected to the motor leads (driver outputs) There are other pins, which correspond to the dc power supply, common ground, various control signals, etc The pin denoted by STEP (or PULSE) receives the stepping pulse signal (from the motor controller) This corresponds to the required stepping sequence of the motor A transition from a low level to a high level (or rising edge) of a pulse will cause the motor to move by one step The direction in which the motor moves is determined by the state of the pin denoted by CW/ CCW A logical high state at this pin (or open connection) will generate switching logic for the motor to move in the CW direction, and a logical low state (or logic common) will generate switching logic for the motor to move in the CCW direction The pin denoted by HALF/FULL determines whether half stepping or full stepping is carried out Speci cally, a logical low at this pin will generate switching logic for full stepping, and the logical high will generate switching logic for half stepping The pin denoted by RESET receives the signal for initialization of a stepping sequence There are several other pins, which are not necessary for the present discussion The translator interprets the logical states at the STEP, HALF/FULL , and CW/ CCW pins and generates the proper logic to activate the switches in the unipolar drive Speci cally, four active logic signals are generated corresponding to A (Phase A on), _ – A (Phase A reversed), B (Phase B on), and B (Phase B reversed) These logic signals activate the four switches in the bipolar drive, thereby sending current through the corresponding winding segments/leads (A+, A−, B+, B−) of the motor The logic hardware is commonly available as compact chips in the monolithic form If the motor is uni lar-wound (for a two-phase stepper there should be three leads—a ground wire and two power leads for the two phases), a bipolar drive will be necessary 474 Mechatronics: A Foundation Course in order to change the direction of the current in a phase winding A schematic representation of a bipolar drive for a single phase of a stepper is shown in Figure 7.7 Note that when the two transistors marked A are on, the current ows in one direction through the phase winding and when the two _ transistors marked A are on, the current ows in the opposite direction through the same phase winding What is shown is an H-bridge circuit V+ A A Phase A A+ A– 7.2.6 Stepper Motor Selection The selection of a stepper motor for a speci c application cannot be made on the basis of geometric parameters alone Torque and speed considerations are often more crucial in the selection process For example, a faster speed of response is possible if a motor with a larger torque-to-inertia ratio is used 7.2.6.1 Torque Characteristics and Terminology A A R Ground FIGURE 7.7 A bipolar drive for a single phase of a stepper motor (uni lar-wound) The torque that can be provided to a load by a stepper motor depends on several factors For example, the motor torque at a constant speed is not the same as that when the motor “passes through” that speed (i.e., under acceleration, deceleration, or general transient conditions) In particular, at a constant speed, there is no inertia torque Also, the torque losses due to magnetic induction are lower at constant stepping rates in comparison with the variable stepping rates It follows that the available torque is larger under steady (constant-speed) conditions Another factor of in uence is the magnitude of the speed At low speeds (i.e., when the step period is considerably larger than the electrical time constant), the time taken for the phase current to build up or be turned off is insigni cant compared with the step time Then the phase current waveform can be assumed to be rectangular At high stepping rates, the induction effects dominate and as a result a phase may not reach its rated current during the duration of a step As a result, the generated torque will be degraded Furthermore, since the power provided by the power supply is limited, the torque × speed product of the motor is limited as well Consequently, as the motor speed increases, the available torque must decrease in general These two are the primary reasons for the characteristic shape of a speed–torque curve of a stepper motor where the peak torque occurs at a very low (typically zero) speed, and as the speed increases, the available torque decreases Eventually, at a particular limiting speed (known as the no-load speed), the available torque becomes zero The characteristic shape of the speed–torque curve of a stepper motor is shown in Figure 7.8 Some terminology is given as well What is given may be interpreted as experimental data measured under steady operating conditions (and averaged and interpolated) The given torque is called the “pull-out torque” and the corresponding speed is the “pull-out speed.” In industry, this curve is known as the “pull-out curve.” Holding torque is the maximum static torque and is different from the maximum (pullout) torque de ned in Figure 7.8 In particular, the holding torque can be about 40% greater than the maximum pull-out torque, which is typically equal to the starting torque (or stand-still torque) Furthermore, the static torque becomes higher if the motor has more than one stator pole per phase and if all these poles are excited at a time The residual torque 854 Mechanical components dynamic isolation components, 32–33 fasteners, 32 harmonic drives, 38–42 lead screw and nut, 35–38 load bearing/structural components, 32 mechanical actuators and controllers, 33 transmission components, 33–35 Mechanical elements damping (dissipation) element, 21 mass (inertia) element, 18–19 spring (stiffness) element, 20–21 Mechanical systems, impedance matching, 235–23 Mechanical tachometer, 313–314 Mechanical transfer functions de nitions, 154 interconnection laws, 155–156 A-type and T-type transfer functions, 156 Mechanics, 757 Mechatronic design quotient (MDQ) categories, evolutionary design, 8–9 Mechatronic system application areas, 10–11 components, concepts and technologies, 3–4 coupled design, 6–7 design evolution, 8–9 evolution, 9–10 hard-disk drives (HDD) unit, humanoid robot, mechatronic design quotient (MDQ), 7–8 modeling and design, 4–5 (see also Modeling) study, 11–12 Memory access, 615–616 bits, bytes, and words, 614 card design, 617–619 physical form of, 614–615 RAM, ROM, PROM, EPROM, and EEPROM, 613–614 volatile, 614 Memory-mapped I/O, 621–622 Metal-oxide semiconductor FET (MOSFET) CMOS memory, 615 n-channel depletion-type, 70 Method of undetermined coef cients, 174 M- les, 817 Microactuators, 369, 465–466 Microcontrollers architecture, 609–610 Index development/application tool, 627–628 input/output hardware clock, counter, and timer, 623 direct memory access method, 622–623 handshaking operation, 623 interrupt, 622 peripheral device connection, 619–620 pin-out, 620 programmed, 620–622 memory access, 615–616 bits, bytes, and words, 614 card design, 617–619 physical form of, 614–615 RAM, ROM, PROM, EPROM, and EEPROM, 613–614 volatile, 614 microprocessor accumulator and data register, 613 address register, 612 arithmetic logic unit, 611–612 instruction register, 613 operation decoder, 613 program counter, 612 sequencer, 613 minimum hardware requirements, 609 operation, 610–611 programming and execution assembly language, 624–625 branching, 626 high-level languages, 625 instruction set, operation codes, and mnemonics, 624 real-time processing, 626–627 working principle, 625–626 Microprocessor accumulator and data register, 613 address register, 612 arithmetic logic unit, 611–612 instruction register, 613 operation decoder, 613 program counter, 612 sequencer, 613 Microstrain, 381 Microstepping, 469–470 Milling machine, 640 Misalignment, 467 Mnemonics, 624 Mobility function, 158 Modal analysis expressions, 176 Modal control, 674–675 Modal damping, 674, 675 Index Modal damping ratio, 177 Modeling analytical model development input–output models, 102 state-space model, 102–111 steps, 101–102 block diagrams causality and physical realizability, 172–173 reduction, 171 simulation block diagrams, 171–172 state-space model, 170 superposition principles, 172 computer simulation, 186–193 dynamic systems and models analytical models, 94 complexity, 93 superposition principles, 95 terminology, 92–93 types, 93–94 equivalent circuits and linear graph reduction mechanical circuit analysis, 162–169 Thevenin’s theorem, 159–162 frequency-domain models Bode diagram and Nyquist diagram, 152–153 electromechanical systems, 153–158 frequency transfer function, 150–152 linear graphs compatibility equations, 131–132 continuity equations, 132 single-port elements, 125–127 state models, 132–145 thermal systems, 146–148 two-port elements, 127–131 variables and sign convention, 123–125 linearization functions, 113–114 operating curves, 120–122 operating point, 111–113 system nonlinearities, 114–120 lumped elements and analogies across variables and through variables, 99–100 natural oscillations, 100 lumped model heavy spring, 96 kinetic energy equivalence, 96–97 natural frequency equivalence, 97–98 response analysis analytical solution, 173–175 855 rst-order systems, 175–176 initial conditions, 182–186 Laplace transform, 180–182 second-order systems, 176–180 transfer functions, 148–150 Model-referenced adaptive control (MRAC), 677–678 Model development, 101–111 Model types, 93–94 Modeling, 4–5, 91–189 Modeling process, 194 Modes, uncoupled, 6–7 Modulating signal, 260 Modulation, 262–264 Modulation theorem, 263 Modulators, 260–267 Modulators and demodulators amplitude modulation (AM), 260, 262 applications, 264–267 fault detection and diagnosis, 266–267 modulation theorem, 263 side frequencies and side bands, 263–264 demodulation, 267 frequency modulation (FM), 260, 261 phase modulation (PM), 262 pulse-code modulation (PCM), 262 pulse-frequency modulation (PFM), 260–262 pulse-width-modulation (PWM), 260, 261 quantization error, 262 Mohr’s circle, 766 Mohr’s circle of plane stress, 766–767 Mohr’s theorems, 772 Moiré fringe displacement sensors formation of, 440 orientation of, 441 patterns of, 439 Monochromatic light, 400 Motion transducers, 350 Motion transmissibility, 221 Motor–compressor unit, 209 Motor control console (MCC), 740–741, 743 Motor torque, 513 Mounting techniques, accelerometer, 373, 381 Moving light beam, 348 Multi-component systems, 303–304, 642 Multi-degree-of-freedom system, 214, 222, 227 Multiplexers (MUX), 285 Multiplier circuit, 355 Multivariable systems, 321 Mutual-induction transducers, 350–351 856 Index N O NAND gates, 589–590 Narrow-band lter, 257 Natural oscillations, 100 Natural response, 178, 657 Negative feedback, 669 Neutral axis, 772 Newton’s second law, 18–19, 26 No load condition, 232 Noise bandwidth, 311 Noise immunity, 415 Noncontacting sensor, 438 Noninverting ampli er, 243 Nonlinear damping matrix, 306 Nonlinear damping torque, 215 Nonlinear model, 94, 117 Nonlinear system, 114–120 Nonlinearity error in differential transformers, 282 Nonlinearity percentage, bridge, 291 Norm, 683 Notch lter, 257–258 Notch frequency, 258 Nozzle, 753–754 N-type strain gages, 384 Nuclear plant equipment, 642 Null balance, strain gage, 379 Null voltage, 353 Numerator polynomial, 172, 173 Numerical model, 92 Nyquist frequency, 315 Nyquist plot, 671–672 Negative numbers one’s complement, 583 signed magnitude representation, 582 two’s complement representation, 582–583 Noninteracting two-tank uid system, 211 Nonlinear feedback control, 675–676 NOR gate, 589–590 Norton equivalent circuit, 162 Number systems and codes ASCII (Askey) code, 584–585 binary coded decimal, 584 binary gray codes, 583 binary multiplication and division, 583 binary representation, 580–581 negative numbers one’s complement, 583 signed magnitude representation, 582 two’s complement representation, 582–583 Nyquist plots, 671–673 One’s complement, 583 Offset, 651 Offset current, op-amp, 241 Offset error, ADC, 282 Offset signals, 249 Op-amp, 245 Op-amp gain, open-loop, 241 Open-coiled helical springs, 773–775 Open-loop gain, op-amp, 246 Operating conditions, 506, 514 Operating environment, 323 Operating frequency range, shaker, 332 Operating point, 111–113 Operation codes, 624 Operation decoder, 613 Operation malfunction, 282 Operational ampli er de nitions, 238–242 Operational ampli ers (op-amps) feedback uses, 241–242 history, 238 output voltages, 240–241 schematic model, 239 Optical coupling, 83 Optical displacement sensors, 326–328 Optical encoders, 416–432 absolute advantages and drawbacks, 432 code conversion logic, 431 code pattern illustration, 429–430 gray coding, 430–431 vs digital (pulse) tachometers, 436 incremental data acquisition hardware, 428–429 digital resolution, 421 direction of rotation, 419 displacement measurement, 421 hardware features, 419–421 incremental encoder disk, 417, 418 interpolation, 423 physical resolution, 422 schematic representation, 416, 417 shaped pulse signals, 417, 418 speed measurement, 425–426 step-up gearing, 422–423, 427–428 velocity measurement, 424 velocity resolution, 424–425 rectilinear, 438–439 Optical potentiometer, 348–350 Optical sensors and lasers ber-optic gyroscope, 403–404 ber-optic position sensor, 401–402 Index laser Doppler interferometer, 404–405 laser interferometer, 402–403 Optical tactile sensors, 396–397 Optimal control, 680 see Linear quadratic Gaussian (LQG) control OR gate, 588–590 Ori ce ow meter, 410, 411 Origin, 580 Oscillators damped, 177 undamped, 176–177 Oscillatory systems, 646 Oscilloscopes, 831 Output, 102, 231 Output impedance, 231 Output port, 231 Output variables, 103 Output vector, 205 Overdamped lter, 178 Owen bridge, 291–292 Oxidation, potentiometer, 68–69 P Paint pumping system, 114–115 Parametric drift, 310 Parametric errors, DAC, 275 Particular solution, 174–175 Passive and active lters advantages and disadvantages, 252–253 number of poles, 253 Passive electrical elements and materials dielectric material and capacitor element capacitor types, 52 color codes, 52 permittivity, 51 piezoelectricity, 52–54 magnetic material and inductor element Hall-effect sensors, 57–58 hysteresis loop, 55 inductance, 59–60 magnetic bubble memory, 58 magnetic materials, 55–57 magnetism and permeability, 55 piezomagnetism, 57 reluctance, 58–59 resistor (dissipation) element color codes, 48–49 conductance and resistance, 44 resistivity, 44–45 strain gages, 46–47 superconductivity, 47–48 temperature effects, 45–46 857 Passive transducers, 343 Path equations, 101 Peak magnitude, 646, 649 Peak time, 646, 649 Peak velocity, 723 Percentage overshoot, 646, 649 Performance characteristics, ADC, 281–284 Performance speci cation, 304, 645–649, 655–656 Permanent-magnet AC tachometers, 364 Permanent-magnet tachometers, 362 Permanent-magnet transducers, 362 Phase angle, 262 Phase lead transfer function, 256 Phase margin, 670–671 Phase modulation (PM), 262 Phase shift and null voltage, 353 Photocells, 82 Photodetector, 80 Photodiodes, 80–81 Photo- eld effect transistor (FET), 81 Photoresistive layer, 326 Photoresistors, 80 Photosensors photocells, 82 photodiodes, 80–81 photo- eld effect transistor (FET), 81 photoresistors, 80 phototransistor, 81 Phototransistor, 81 Physical realizability, 172–173 PID control proportional–integral (PI) control, 821–824 proportional (P) control, 821 Piezoelectric accelerometer, 342, 372–373 Piezoelectric element, 53 Piezoelectric sensors charge ampli er, 374–376 equivalent circuit representation, 369, 370 piezoelectric accelerometer, 372–373 sensitivity, 370–371 types of accelerometers, 371 Piezoelectric stepper motors, 566–567 Piezoresistive tactile sensors, 398 PIN diodes, 65 Piston-cylinder, 541 Planar diffusion technology, 608 Plane strain problem, 759 Plane stress problem constitutive and equilibrium equations, 760 in polar coordinates, 760–761 Plate positioning control system, 316, 317 858 PLC, see Programmable logic controllers P-type strain gages, 383 Pneumatic control systems apper valves multistage servovalve, 555–556 operation of, 555 single-jet, 555 two-stage servovalve with pressure feedback, 555–556 vs hydraulic control systems, 554 Pole, dominant, 661 Polling, 621–622 Poppet valve, 535–536 Position error constant, 654 Position feedback, 499 Position feedback with PID control, 499 Position plus velocity feedback control, 498–499 Position sensor ber-optic, 401–402 laser interferometer, 402 ultrasonic, 406 Pot, 345 Potential energy, 22, 124 elastic, 96 gravitational, 96 Potentiometer, 345–350 Potentiometer circuit, 378 strain gage, 378 Power rating, shaker, 331 Power supply current-regulated, 288 regulated, 288 voltage-regulated, 288 Power, 472 Precision, 322–323 Pressure, 22, 409 Principal stress, 758, 766, 769 Principle of superposition, 95, 172 Probability, 797–811 Probability density function, 797–798 Probability distribution function, 797 Process, 14, 33 Production, 725 Prototype, 727–728 Proximity probe, 435 Proximity sensor mutual induction, 449 self-induction, 359 Pulse-code modulation, 262 Pulse-frequency modulation, 261 Pulse-width modulation, 501 Pump, 202, 533 Index Pressure control valve, single-stage, 547–549 Pressure regulated liquid jet system, 195–196 Pressure sensors, 408–409 Probability and statistics least squares t, 808–811 probability distribution dence intervals, 803–806 cumulative probability distribution function, 797 Gaussian distribution, 802–803 independent random variables, 799–800 mean value, 798 probability density function, 797–798 root-mean-square value, 798 sample mean and sample variance, 800 unbiased estimates, 800–801 variance and standard deviation, 798–799 sign test and binomial distribution, 806–808 Probability density function, 797–798 Probability distribution dence intervals, 803–806 cumulative probability distribution function, 797 Gaussian distribution, 802–803 independent random variables, 799–800 mean value, 798 probability density function, 797–798 root-mean-square value, 798 sample mean and sample variance, 800 unbiased estimates, 800–801 variance and standard deviation, 798–799 Program counter, 612 Programmable logic controllers (PLC) continuous-state control, 640 discrete-state control, 640 hardware, 641–642 operation process, 640–641 Programmable read only memory (PROM), 614 Programmed I/O, 620–622 Programming and execution assembly language, 624–625 branching, 626 high-level languages, 625 instruction set, operation codes, and mnemonics, 624 real-time processing, 626–627 working principle, 625–626 Proportional plus derivative control (PPD/PD control), 499 Pulse-code modulation (PCM), 262 Pulse-frequency modulation (PFM), 260–262 Index Pulse-width modulation (PWM) components, 501–502 duty cycle, 261, 502–503 with IC hardware, 503–504 Pump-controlled hydraulic actuators, 553 Q Q-factor method, 668 Quadratic error function, 678 Quadrature error, 678 Quality factor, 451 Quality function deployment (QFD) method, 718–719 Quality of cut, 738 Quantization error, 262, 281–282 Quartz crystal oscillator, 623 R Radiation, 30 Radius of curvature, 362 Radix-two FFT, 581 Random process, 613 Railway car braking system, 750 Random access memory (RAM), 613 Rapid prototyping, 717 Rate error, 356–357 Rate gyro, 399–400 Rating parameters bandwidth, 310 cross-sensitivity, 308 dynamic range, 308, 309 frequency range, 310 linearity, 309 resolution, 308 sensitivity, 308 zero drift, 309–310 Rational fraction form, 307 Reactive transducer, 365 Read-and-write memory, 219 Read only memory (ROM), 614 Real-time processing, 626–627 Rectilinear motion, 438 Recursive algorithm, 259 Recursive digital lters, 259 Redundancy, component, 726 Reference voltage variation, DAC, 275 Refresh rate, 464 Reliability, 415 Relief valve, 535–536 Reluctance, 58–59 Repeatability, 323, 723 859 Repeatable accuracy, 333 Reset-set (RS) ip- op, 599–600 Resistance bridge, 286 Resistance temperature detector, 413 Resistive feedback, charge ampli er, 374 Resistively coupled transducer, 346 Resistivity, 44–45 Resistor, uid, 22, 27 Resistor element color codes, 48–49 conductance and resistance, 44 resistivity, 44–45 strain effects, 46–47 superconductivity, 47–48 temperature effects, 45–46 Resolver demodulation, 358–359 schematic representation, 357, 358 Resonance-type band-pass lters, 257 Resonant frequency, 310 Response analysis analytical solution convolution integral, 174–175 homogeneous solution, 173–174 particular solution, 174 stability, 175 rst-order systems, 175–176 initial conditions for step response, 182–186 Laplace transform initial conditions, 181–182 step response, 180–182 second-order systems forced response of damped oscillator, 177–179 free response of damped oscillator, 177 free response of undamped oscillator, 176–177 transform techniques, 785–790 Response curve, 647 Response of simple oscillator, 647 Revolute joints, 730–732 Robotics design and development actuator selection/sizing, 729–733 ampli ers and power supplies, 733–734 CAD model of the MDMS prototype, 731–732 control system, 734–736 economic analysis, 736 prototype robot, 727–729 steps, 727 general considerations, 720–722 860 robotic workcells, 725–726 robot selection commercial, 723–725 steps, 722–723 Robotic sewing system linear graphs, 135–136 Simulink model, 190–192 Robotic workcells, 725–726 Root locus design example, 825–827 open-loop poles, 662–663 rules, 664–665 steps of, 665–668 Root-mean-square value, 798 Rotameters, 411, 412 Rotary-motion system, 206 Rotating members cylinders, 764–766 disks, 761–762 thick cylinders equilibrium equations, 763 results, 764 strain equations, 763 stress–strain relations, 763 temperature stresses, 764 Rotatory electromechanical system and armature circuit, 210, 218 Rotatory positioning system, 212 Routh array, 658–659 Routh–Hurwitz stability auxiliary equation, 659–660 relative, 661–662 Routh array, 658–659 zero coef cient problem, 660–661 RS ip- op, see Reset-set ip- op Rubber buf ng machine, 215–216 Riccati equation, 681, 682 Ride quality, 93, 163, 164, 169, 451 Rise time, 68, 246, 646 RMS value, 798 Robustness, 308, 637, 645 Roll-off rate, lter, 254 Roll-up slope, lter, 256 Root-mean-square value, 798 Rotary inertia, see rotatory inertia Rotating machinery, 299, 344 diagnosis, 265, 266 Rotatory-variable differential transformer, 351 Rounding off, ADC, 282 Row vector, 816 Rubber buf ng machine, 216 RVDT, see Rotatory-variable differential transformer Index S Sallen-Key lter, 258 Sample-and-hold (S/H) circuit, 284–285 Sample pulse, 284 Sampled data, 289, 314, 698 Sampling rate, 269, 270, 698 of data, 282 Sampling theorem, 314–315 Satellite tracking system, 329–330 Saturation, 199, 305, 680 of op-amp, 240 Saturation nonlinearity, 305 Scale-factor drift, see Parametric drift Schmitt trigger, 606–607 Schottky barrier diodes, 65 Secondary coil, 129, 351 Secondary windings, see secondary coils Second-order systems forced response of damped oscillator, 177–179 free response damped oscillator, 177 undamped oscillator, 176–177 Seismic mass, 377, 451 Self-induction transducers, 359–360 Semiconductor memory, 615 Semiconductor strain gages components, 382 nonlinear behavior, 383, 384 properties of, 383 quadratic strain–resistance relationship, 384–385 Sensitivity accelerometer, 373 bridge, 379 capacitive, 368 charge, 370, 371 circuit, 370 eddy-current probe, 361, 362 strain gage, 376–385 tachometer, 364 voltage, 370, 371 Sensitivity drift, see Parametric drift Sensors and transducers digital transducers, 414 absolute optical encoders, 429–432 advantages of, 415 binary transducers, 441–443 encoder error, 432–433 Hall-effect sensors, 436–438 incremental optical encoder, 416–429 linear encoders, 438–439 Index Moiré fringe displacement sensors, 439–441 resolvers, 434–435 shaft encoders, 415–416 tachometers, 435–436 types of sensors, 444 gyroscopic sensors, 398 Coriolis force devices, 400 rate gyro, 399–400 image sensor applications, 447 image-based sensory system, 445–446 image processing, 446–447 image processing and computer vision, 444–445 motion sensors and transducers, 344–345 optical sensors and lasers, 400 ber-optic gyroscope, 403–404 ber-optic position sensor, 401–402 laser Doppler interferometer, 404–405 laser interferometer, 402–403 piezoelectric sensors charge ampli er, 374–376 equivalent circuit representation, 369, 370 piezoelectric accelerometer, 372–373 sensitivity, 370–371 types of accelerometers, 371 potentiometer optical potentiometer, 348–350 performance considerations, 346–347 schematic representation, 345, 346 strain gages measurement equations, 376–382 semiconductor, 382–385 tactile sensing conductive-elastomer, 395–396 construction and operation of, 394–395 dexterity, 398 optical, 396–397 piezoresistive, 398 sensor requirements, 391–394 terminology, 342–344 thermo- uid sensors, 407 ow, 409–412 pressure, 408–409 temperature, 412–413 torque sensors applications of, 385–386 force sensors, 390–392 strain gages, 386–390 ultrasonic sensors, 405 magnetostrictive displacement, 407 ultrasonic position, 406 861 uses of, 341 variable-capacitance transducers capacitance bridge circuit, 368–369 capacitive displacement sensor, 366–368 capacitive rotation sensor, 366–367 schematic representation, 365, 366 variable-inductance transducers AC induction tachometer, 364–365 DC tachometer, 362–363 eddy current transducers, 360–362 linear-variable differential transformer (LVDT), 351–357 mutual-induction transducers, 350–351 permanent-magnet AC tachometer, 364 permanent-magnet tachometers, 362 resolver, 357–359 self-induction transducers, 359–360 Sequential logic devices D ip- op excitation table, 602–603 gray-code counter, 604–606 shift register, 602–603 symbol, 602 JK ip- op excitation table, 602 symbol, 601 truth table, 601–602 latch, 600 reset-set (RS) ip- op, 599–600 Schmitt trigger, 606–607 synchronous and asynchronous operations, 599 T ip- op and counters, 603–604 Series opposition connection, 351, 352 Servoactuators, 465 Servomotors, 465 Servo-valve, 532, 742–745 Sets, 333, 512, 685, 686 Settling time DAC, 275 S/H circuit, see Sample-and-hold circuit Shaft encoders, 415–416 Shaker, 207, 337, 338 Shannon’s sampling theorem, 626 Shear strain, 373, 757 Shear stress, 768, 769 Shift register, 602–603 Side frequencies and side bands, 263–264 Signal acquisition, 395 Signal conditioning, 353–357, see also Component interconnection and signal conditioning Signal conversion, 230 862 Signal modi cation, 230, 720 Signal modulation, 260 Signal sampling, see Aliasing distortion Signed magnitude representation, 582 Sign test and binomial distribution, 806–808 Signum function, 680 Simple oscillator, 153, 226, 227, 647–649 damped, 153, 177, 647 harmonic response, 179, 181 undamped, 178 Simulation block diagram, 171–173, 191, 217 Sine, 152, 423, 781–782 Sine dwell, 152 Sine sweep, 152 Slew rate ampli er, 246, 247, 295, 296 op-amp, 246 Slip ring and brush, 365 Simulink application building, 189 elements, 188–189 robotic sewing machine, 190–192 running a simulation, 189–193 starting Simulink, 188 time-domain model, 190 Single-degree-of-freedom robot, 209 Single-degree-of-freedom systems, 213–215 Single-phase AC motors, 525–526 Single-phase inverter circuit, 518–519 Single-pole high-pass lters, 255–256 Single-pole low-pass lters, 253, 254 Single-port elements electrical system elements, 126 mechanical system elements, 125 T-type source, 126–127 A-type source, 126–127 SISO design tool, 820 Sliding mode control, 678–680 Software control, 551 Software lters see digital lters Software tools control systems toolbox compensator design example, 817–821 PID control, 821–824 root locus design example, 825–827 fuzzy logic toolbox, 825 command line–driven FIS design, 828–829 graphical editors, 828 practical stand-alone implementation, 829 LabVIEW, 829 block diagrams, 832–834 controls palette, 834 Index front panel, 832 functions palette, 834–836 key concepts, 830–831 tools palette, 834 MATLAB® computations, 813–815 linear algebra, 816–817 M- les, 817 relational and logical operations, 815–816 Simulink®, 813 Solenoids duty cycle, 528 ow control valve, 577 hold-in circuit, 528–529 relay, 528 rotary, 529 Solid mechanics beams in bending and shear Castigliano’s rst theorem, 773 elastic energy of bending, 773 Maxwell’s theorem of reciprocity, 772–773 Mohr’s theorems, 772 circular plates with axisymmetric loading boundary conditions, 778 equilibrium equations, 777 moments, 776 strains and stresses, 776 elasticity problem compatibility equations, 758 constitutive equations, 757–758 equilibrium equations, 758 strain components, 757 Mohr’s circle of plane stress, 766–767 open-coiled helical springs, 773–775 plane strain problem constitutive equations, 759 equilibrium equations, 759 plane stress problem constitutive equations, 760 equilibrium equations, 760 in polar coordinates, 760–761 rotating members particular cases of cylinders, 764–766 rotating disks, 761–762 rotating thick cylinders, 762–764 torsion circular members, 767–769 torque sensor, 769–771 Sound waves, 405 Space robotics, 727 Speci c heat, 31 Speed of response, 246 ampli er, 246 Index s-plane application in circuit analysis, 794–795 interpretation of Laplace and Fourier transforms, 794 performance speci cation, 655–656 Splines, 38–40 Spool valves, four-way actuation methods, 538 schematic diagram, 536–537 steady-state characteristics, 538–540 Wheatstone bridge circuit, 552 Spray painting robot, 114–115 Spring element, 20, 21 Spring, heavy, 96–98 Square-root of sum of squares (SRSS) error, 325 Stability in frequency domain Bode and Nyquist plots, 671–673 1, condition, 668–670 gain margin, 670 marginal stability, 668 phase margin, 670–671 root locus method open-loop poles, 662–663 rules, 664–665 steps of, 665–668 Routh–Hurwitz auxiliary equation, 659–660 relative, 661–662 Routh array, 658–659 zero coef cient problem, 660–661 State model, 95 State of inertia element, 19 State of spring element, 20 State-space models ampli ers, 139–140 analytical model development, 102–111 block diagrams, 170 development steps, 104 diagramatic representation, 170 linear graph representation dc motors, 140–142 thermal systems, 146–148 linear state equations, 102–104 observations, 134–135 robotic sewing system, 135–139 sign convention, 133 system order, 133 State variable, 93 State vector, 104 Static calibration curve, 309 Static gain, 313 Static memory, 615 863 Statistical process control (SPC) control limits/action lines, 329 de nition, 328 satellite tracking system, 329–330 steps of, 329 Steady operating conditions, 120 Steady-state error, 654, 655 Steady-state response, 152 Steady-state value, 175, 654 Steel rolling mill automatic gage control (AGC) system, 546 hydraulic control system diagram, 547 mechanical-dynamic equations, 546 Stefan-Boltzmann constant, 30 Stepper motors advantages and disadvantages, 485–486 applications, 484–486 classi cation two-stack stepper motor, 467, 468 variable-reluctance (VR) stepper motor, 466–467 driver and controller basic control system, 470–471 components, 470–471 programmable indexer, 472 translator module, 471–472 driver hardware basic components, 472 bipolar drive, single phase stepper motor, 474 two-phase bi lar-wound stepper motor, 473 features, 466 hybrid, 561–562 inter-stack misalignment, 467–469 rotor stacks, 467–468 microstepping, 469–470 piezoelectric, 566–567 selection process positioning (x–y) tables, 478–479 steps for, 476–477 torque characteristics, 474–476 variable-reluctance (VR), 466–467 Stiffness element, 20 Strain gages measurement equations, 376 accelerometer, 377 accuracy considerations, 381–382 bridge constant, 379–380 bridge sensitivity, 379 calibration constant, 380–381 data acquisition, 381 nomenclature, 378 864 semiconductor components, 382 nonlinear behavior, 383, 384 properties of, 383 quadratic strain–resistance relationship, 384–385 torque sensors bending elements, 390 circular shaft, 387 direct-drive robotic arm, 388–389 torque-sensing element, design criteria, 388 Stroke, 407 Subsystems, design, 6, Successive-approximation ADC, 283 Synchronous motors control of, 527 drawback, 526–527 rotor, 526 System matrix, 104 System order, 133 T Tachometers AC induction, 364–365 DC, 362–363 de nition, 703–704 digital, 435–436 mechanical, 313–314 permanent-magnet, 362 permanent-magnet AC, 364 Tactile sensing conductive-elastomer, 395–396 construction and operation of, 394–395 dexterity, 398 optical, 396–397 piezoresistive, 398 sensor requirements, 391, 393–394 Taylor series expansion, 324 Temperature drift, ampli er, 246 Test input characteristics, 654 T ip- op and counters, 603–604 Thermal capacitor, 28 Thermal elements Biot number, 32 capacitor, 28 resistor conduction, 29 convection, 30 radiation, 30–31 three-dimensional conduction, 31–32 Thermal resistance, 28–31 Index Thermal systems, 27–32 Thermistors, 45, 46, 377, 413 Thermocouples, 412 Thermo- uid sensors ow, 409 uid ow measurement, 410 pitot tube, 410, 411 rotameter, 411, 412 pressure, 408–409 temperature, 412–413 Thevenin’s theorem, 159–162 Through variable, 99–100 Thyristors, 65–67, 518–519 Time constant ampli er, 246 charge ampli er, 374–376 DAC, 275 lter, 254, 256, 257 piezoelectric sensor, 375 Time domain, 654–649 Time-domain models, 94, 95 Timer, 623 Torque-sensing element design criteria, 388 Torque sensors applications of, 385–386 force sensors, 390–392 strain gages, 386–390 bending elements, 390 circular shaft torque sensor, 387 direct-drive robotic arm, 388–389 torque-sensing element, design criteria, 388 torsion, 769–771 Torsion circular members, 767–769 torque sensor, 769–771 Tracking lters, 251–252 Transducers variable-capacitance capacitance bridge circuit, 368–369 capacitive displacement sensor, 366–368 capacitive rotation sensor, 366–367 schematic representation, 365, 366 variable-inductance AC induction tachometers, 364–365 DC tachometers, 362–363 Eddy current transducers, 360–362 linear-variable differential transformer (LVDT), 351–357 mutual-induction transducers, 350–351 permanent-magnet AC tachometers, 364 permanent-magnet tachometers, 362 resolver, 357–359 self-induction transducers, 359–360 865 Index Transfer function models Bode diagram, 152–153 electrical and mechanical elements, 156–157 frequency transfer function, 150–152 induction motors, 523–525 mechanical transfer functions de nitions, 154 interconnection laws, 155–156 A-type and T-type transfer functions, 156 Nyquist diagram, 152–153 oscillator, 153–154 Transfer matrix, 682 Transform, Fourier, 149, 792–794 frequency-response function, 792–793 s-plane interpretation, 794 transform techniques, 792 Transform, Laplace constant, 780 convolution integral equation, 790 derivative, 782–783 exponential, 780–781 functions, 783–785 impulse response function, 790–791 initial conditions, 181–182 inverse relation, 779 properties and results, 785 response analysis, 785–786 sine and cosine, 781–782 step response, 180–182 time derivatives and integrator, 149 transfer-function models, 148–158 Transform techniques Fourier transform, 792–793 Laplace transform constant, 780 derivative, 782–783 exponential, 780–781 functions, 783–785 inverse relation, 779 properties and results, 785 sine and cosine, 781–782 transfer function, 790–791 response analysis, 785–786 s-plane application in circuit analysis, 794–795 interpretation of Laplace and Fourier transforms, 794 Transistors bipolar junction transistors (BJT), 67–69 eld-effect transistors (FET), 69–70 junction eld effect transistor (JFET), 71–74 metal-oxide semiconductor FET (MOSFET), 70 Transistor-transistor logic (TTL) NAND gate, 591–592 Trapezoidal rule, 187 T-type elements, 22 Tunnel diodes, 64 Two-channel tracking lters, 252 Two-port elements electrical transformer, 129 gyrator, 129–131 ideal transformer, 127–129 Two’s complement, 582–583 U Ultrasonic sensors magnetostrictive displacement, 407 ultrasonic position, 406 Undamped oscillator, 176–177, 181 V Valve-controlled hydraulic actuators, 552–553 Variable-capacitance transducers capacitance bridge circuit, 368–369 capacitive displacement sensor, 366–368 capacitive rotation sensor, 366–367 schematic representation, 365, 366 Variable-inductance transducers AC induction tachometers, 364–365 DC tachometers, 362–363 eddy current, 360–362 linear-variable differential transformer (LVDT) operating curve, 351, 352 phase shift and null voltage, 353 signal conditioning, 353–357 mutual-induction, 350–351 permanent-magnet AC tachometers, 364 permanent-magnet tachometers, 362 resolver demodulation, 358–359 schematic representation, 357, 358 self-induction transducers, 359–360 Variable-reluctance (VR) stepper motor, 466–467 Variable-reluctance transducers, 350 Vehicle suspension system, 213 Velocity error constant, 654 Velocity feedback control, 498 Volatile memory, 614 Voltage and current ampli ers, 242–243 Voltage control, 520 Voltage sensitivity, 370–371 Voltage variable capacitor (VVC) diodes, 64 866 Index W X Wave interference, 404–405 Weighted resistor DAC, 271–272 Welding robot, 751 White noise, 311 Wheatstone bridge, 286–288 Wien-bridge oscillator, 292–293 Wood cutting machine, 203–204 Wood strander, 751–752 XOR gates, 589–590 Z Zener diodes, 63–64 Zero drift, 309–310 Zero-order hold, 285 Units and Conversions (Approximate) cm = 1/2.54 in = 0.39 in rad = 57.3° rpm = 0.105 rad/s g = 9.8 m/s2 = 32.2 ft/s2 = 386 in./s2 kg = 2.205 lb kg · m2 (kilogram-meter-square) = 5.467 oz · in.2 (ounce-inch-square) = 8.85 lb · in · s2 N/m = 5.71 × 10−3 lbf/in N/m/s = 5.71 × 10 −3 lbf/in./s N · m (Newton-meter) = 141.6 oz · in (ounce-in.) J = N · m = 0.948 × 10 −3 Btu = 0.278 kWh hp (horse power) = 746 W (watt) = 550 ft · lbf kPa = × 103 Pa = × 103 N/m = 0.154 psi = × 10−2 bar gal/min = 3.8 L/min Metric Prefixes Giga Mega Kilo Milli Micro Nano Pico G M k m μ n p 109 106 103 10−3 10−6 10−9 10−12 ... Stator teeth S Pitch Offset Rotor stack Stator Rotor Stator segment stack segment 2 (phase 1) (phase 2) surrounding surrounding stack stack FIGURE 7.3 Rotor stack misalignment (1/4 pitch) in a hybrid... employed in this step Speci cally, the required torque rating is given by T = TR + J eq ω max ∆t (7 .2) where TR is the net resistance torque Jeq is the equivalent moment of inertia (including rotor,... Motor Coupler Bearings Bearing preload nut and lockwasher (a) Anti-backlash adjustment screws (2) from Lead screw opposite side Lead screw Stepper motor Lead screw nut TR (b) FIGURE 7.10 (a)