Control Sensors
`and Actuators
`
`CLARENCE W. DE SILVA
`
`PRENTICE HAll, Englewood Cliffs, New Jersey 07632
`
`1
`
`Micro Motion 1038
`
`

`
`De Silva, Clarence W.
`Control Sensors and actuators.
`Includes bibliographies and index.
`1. Automatic control. 2. Detectors. 3. Actuators.
`I. Title.
`TJ213.D38 1989
`ISBN 0-13-171745-6
`
`88-28998
`
`629.8
`
`To Charmaine, C.J., and Cheryl-as their senses develop
`and as they become increasingly active.
`
`EditorialJproduction supervision and
`interior design: David Ershun
`Cover design: Ben Santora
`Manufacturing buyer: Mary Ann Gloriande
`
`II © 1989 by Prentice-Hall, Inc.
`
`~ A Division of Simon & Schuster
`-
`Englewood Cliffs, New Jersey 07632
`
`All rights reserved. No part of this book may be
`reproduced, in any form or by any means,
`without permission in writing from the publisher.
`
`The publisher offers discounts on this book when ordered
`in bulk quantities. For more information, write:
`
`Special Sales/College Marketing
`Prentice Hall
`College Technical and Reference Division
`Englewood Cliffs, NJ 07632
`
`Printed in the United States of America
`
`10 9 8 7 6 5 4 3 2 1
`
`ISBN 0-13-171745-6
`
`PRENTlCE·HALL INTERNATIONAL (UK) LIMITED, London
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`
`2
`
`

`
`7
`Control,
`Instrumentation, and
`Design
`
`1.1 INTRODUCTION
`
`The demand for servomechanisms in military applications during World War II pro(cid:173)
`vided much incentive and many resources for the growth of control technology.
`Early efforts were devoted to the development of analog controllers, which are elec(cid:173)
`tronic devices or circuits that generate proper drive signals for a plant (process). Par(cid:173)
`allel advances were necessary in actuating devices such as motors, solenoids, and
`valves that drive the plant. For feedback control, further developments in sensors
`and transducers became essential. With added sophistication in control systems, it
`'was soon apparent that analog control techniques had serious limitations. In particu-
`lar, linear assumptions were used to develop controllers even for highly nonlinear
`plants. Furthermore, complex and costly circuitry was often needed to generate even
`simple control signals. Consequently, most analog controllers were limited to on/off
`and proportional-integral-derivative (PID) actions, and lead and lag compensation
`networks were employed to conipenstate for weaknesses in such simple control ac(cid:173)
`tions.
`The digital computer, first developed for large number-crunching jobs, was
`employed as a controller in complex control systems in the 1950s and '1960s. Origi(cid:173)
`nally, cost constraints restricted its use primarily to aerospace applications that re(cid:173)
`quired the manipulation of large amounts of d~ta' (complex models, severa] hundred
`signals, and thousands of system parameters) for control and that did not face serious
`cost restraints. Real-time control requires fast computation, and this speed of com(cid:173)
`putation is determined by the required control bandwidth (or the speed of control)
`and parameters (e.g., time constants, natural frequencies, and damping constants) of
`the process that is being controlled. For instance, prelaunch monitoring and control
`of a space vehicle would require digital data acquisition at very high samplipgrates
`(e.g., 50,000 samples/second). As a result of a favorable decline of computation cost
`(both hardware and software) in subsequent years, widespread application of digital
`computers as control devices (i.e., digital control) has become feasible. Dramatic
`developments in large-scale integration (LSI) technology and microprocessors in the
`
`1
`
`3
`
`

`
`1970s resulted in very significant drops in digital processing costs, which made digi(cid:173)
`tal control a very attractive alternative to analog control. Today, digital control has
`become an integral part of numerous systems and applications, including machine
`tools, robotic manipulators, automobiles, aircraft autopilots, nuclear power plants,
`traffic control systems, and chemical process plants.
`Control engineers should be able to identify or select components for a control
`system, model and analyze individual components or overall systems, and choose
`parameter values so as to perform the intended functions of the particular system in
`accordance with some specifications. Component identification, analysis, selection,
`matching and interfacing, and system tuning (adjusting parameters to obtain the re(cid:173)
`quired response) are essential tasks in the instrumentation and design of a control
`system.
`
`1.2 CONTROL SYSTEM ARCHITECTURE
`
`Let us examine the generalized control system represented by the block diagram in
`figure 1.1. We have identified several discrete blocks, depending on various func(cid:173)
`tions that take place in a typical control system. Before proceeding, we must keep in
`mind that in a practical control system, this type of clear demarcation of components
`might be difficult; one piece of hardware might perform several functions, or more
`than one distinct unit of equipment might be associated with one function. Neverthe(cid:173)
`less, figure 1.1 is useful in understanding the architecture of a general control sys(cid:173)
`tem. This is an analog control system because the associated signals depend on the
`continuous time variable; no signal sampling or data encoding is involved in the sys(cid:173)
`tem.
`
`Plant is the system or "process;' that we are interested in controlling. By con(cid:173)
`trol, we mean making the system respond in a desired manner. To be able to accom(cid:173)
`plish this, we must have access to the drive system or actuator of the plant. We apply
`certain command signals, or input, to the controller and expect the plant to behave
`in a desirable manner. This is the open-loop control situation. In this case, we do not
`use current information on system response to determine the control signals. Infeed(cid:173)
`back control systems, the control loop has to be closed; closed-loop control means
`making measurements of system response and employing that information to gener(cid:173)
`ate control signals so as to correct any output errors. The output measurements are
`made primarily using analog devices, typically consisting of sensor-transducer units.
`
`Reference
`Input Signal
`
`Analog
`Control
`Hardware
`
`I---,.....--Jo-Qutputs
`
`1-*-----; Analog Sensors- i -< f - - - - - - - '
`Transducers
`
`Figure 1.1. Components of a typical analog control system.
`
`2
`
`Control, Instrumentation, and Design
`
`Chap. 1
`
`1.3 01
`
`4
`
`

`
`iigi(cid:173)
`. has
`hine
`mts,
`
`Itrol
`oose
`min
`:ion,
`~ re(cid:173)
`Itrol
`
`min
`unc(cid:173)
`~p in
`lents
`nore
`tbe(cid:173)
`sys(cid:173)
`I the
`sys-
`
`con(cid:173)
`:om(cid:173)
`lpply
`have
`) not
`eed(cid:173)
`.eans
`~ner­
`; are
`nits.
`
`Jtputs
`
`Ip.1
`
`An important factor that we must consider in any practical control system is noise,
`including external disturbances. Noise may represent actual contamination of signals
`or the presence of other unknowns, uncertainties, and errors, such as parameter
`variations and modeling errors. Furthermore, weak signals will have to be am(cid:173)
`plified, and the form of a signal might have to be modified at various points of inter(cid:173)
`action. In these respects, signal-conditioning methods such asjiltering, amplification,
`and modulation become important.
`Identification of the hardware components (perhaps commercially available
`off-the-shelf items) corresponding to each functional block in figure 1.1 is one of the
`first steps of instrumentation. For example, in process control applications off-the(cid:173)
`shelf analog proportional-integral-derivative (PID) controllers may be used. These
`controllers for process control applications have knobs or dials for control parameter
`settings-that is, proportional band or gain, reset rate (in repeats of the proportional
`action per unit time), and rate time constant. The control bandwidth (frequency
`range of operation) of these devices is specified. Various control modes-such as
`on/off, proportional, integral, and derivative, or combinations-are provided by the
`same control box.
`Actuating devices (actuators) include DC motors, AC motors, stepper motors,
`solenoids, valves, and relays, which are also commercially available to various
`specifications. Potentiometers, differential transformers, resolvers, synchros, strain
`gauges, tachometers, piezoelectric devices, thermocouples, thermistors, and resis(cid:173)
`tance temperature detectors (RIDs) are examples of sensors used to measure process
`response for monitoring performance and possible feedback. Charge amplifiers,
`lock-in amplifiers, power amplifiers, switching amplifiers, linear amplifiers, tracking
`filters, low-pass filters, high-pass filters, and notch filters are some of the signal(cid:173)
`conditioning devices used in analog control systems. Additional components, such
`as power supplies and surge-protection units, are often needed in control, but they
`are not indicated in figure 1.1 because they are only indirectly related to control
`functions. Relays and other switching devices and modulators and demodulators may
`also be included.
`
`1.3 DIGITAL CONTROL
`
`Direct digital control (DOC) systems are quite similar to analog control systems.
`The main difference in a DOC system is that a digital computer takes the place of
`the analog controller in figure 1.1. Control computers have to be dedicated machines
`for real-time operation where processing has to be synchronized with plant operation
`and actuation requirements. This also requires a real-time clock. Apart from these
`requirements, control computers are basically no different from general-purpose dig(cid:173)
`ital computers. They consist of a processor to perform computations and to oversee
`data transfer, memory for program and data storage during processing, mass storage
`devices to store information that is not immediately nejXied, and input/output devices
`to read in and send out information. Digital control systems might utilize digital in(cid:173)
`struments and additional processors for actuating, signal-conditioning, or measuring
`functions, as well. For example, a stepper motor that responds with incremental mo-
`
`Sec. 1.3
`
`Digital Control
`
`3
`
`5
`
`

`
`tion steps when driven by pulse signals can be considered a digital actuator. Further(cid:173)
`more, it usually contains digital logic circuitry in its drive system. Similarly, a two(cid:173)
`position solenoid is a digital (binary) actuator. Digital flow control may be
`accomplished using a digital control valve. A typical digital valve consists of a
`bank of orifices, each sized in proportion to a place value of a binary word
`, i = 0,1,2, ... ,n). Each orifice is actuated by a separate rapid-acting onloff so(cid:173)
`(2i
`lenoid. In this manner, many digital combinations of flow values can be obtained.
`Direct digital measurement of displacements and velocities can be made using shaft
`encoders. These are digital transducers that generate coded outputs (e.g., in binary
`or gray-scale representation) or pulse signals that can be coded using counting cir(cid:173)
`cui try. Such outputs can be read in by the control computer with relative ease. Fre(cid:173)
`quency counters also generate digital signals that can be fed directly into a digital
`controller. When measured signals are in the analog form, an analog front end is
`necessary to interface the transducer and the digital controller. Input/output interface
`boards that can take both analog and digital signals are available with digital con(cid:173)
`trollers.
`A block diagram of a direct digital control system is shown in figure 1.2. Note
`that the functions of this control system are quite similar to those shown in figure
`1.1 for an analog control system. The primary difference is the digital controller
`(processor), which is used to generate the control signals. Therefore, analog mea(cid:173)
`surements and reference signals have to be sampled and encoded prior to digital pro(cid:173)
`cessing within the controller. Digital processing can be conveniently used for signal
`conditioning as well. Alternatively, digital signal processing (DSP) chips can func(cid:173)
`tion as digital controllers. However, analog signals are preconditioned, using analog
`circuitry prior to digitizing in order to eliminate or minimize problems due to alias(cid:173)
`ing distortion (high-frequency components above half the sampling frequency ap(cid:173)
`pearing as low-frequency components) and leakage (error due to signal truncation)
`as well as to improve the signal level and filter out extraneous noise. The drive sys-
`
`Reference input
`
`1.4 !
`
`f---.--__ Outputs
`
`t I JAddress
`II rl---- ....
`I II Analog
`I
`I II multiplexing
`L--------+.L.J
`I L.._
`IL
`: ~ __ i __ ,
`I __
`Digital
`I
`i multiplexer r+
`'-- _____ .J
`
`:
`
`L
`
`__...J
`
`Address
`
`Analog
`sensors/
`transducers
`
`I
`I
`I
`I
`I
`I
`, - - - - -1
`I
`I
`Digital
`I
`- - -i
`........ - - - - - __ ...1
`sensors/
`I transducers I
`L _____ ...1
`
`Figure 1.2. Block diagram of a direct digital control system.
`
`4
`
`Control, Instrumentation, and Design
`
`Chap. 1
`
`6
`
`

`
`tem of a plant typically takes in analog signals. Often, the digital output from con(cid:173)
`troller has to be converted into analog form for this reason. Both analog-to-digital
`conversion (ADC) and digital-to-analog conversion (DAC) can be interpreted as
`signal-conditioning (modification) procedures. If more than one output signal is
`measured, each signal will have to be conditioned and processed separately. Ideally,
`this will require separate conditioning and processing hardware for each signal chan(cid:173)
`nel. A less expensive (but slower) alternative would be to time-share this expensive
`equipment by using a multiplexer. This device will pick one channel of data from a
`bank of data channels in a sequential manner and connect it to a common input
`device. Both analog and digital multiplexers are available. In a digital multiplexer,
`the input signals come from a bank of digital sensors, and the output signal itself,
`which would be in digital form, goes directly into the digital controller. High-speed
`multiplexers (e.g., over 50,000 switchings/second) use electronic switching.
`For complex processes with a large number of input/output variables (e.g., a
`nuclear power plant) and with systems that have various operating requirements
`(e.g., the space shuttle), centralized direct digital control is quite difficult to imple(cid:173)
`ment. Some form of distributed control is appropriate in large systems such as man(cid:173)
`ufacturing cells, factories, and multicomponent process plants. A favorite distributed
`control architecture is provided by heirarchical control. Here, distribution of control
`is available both geographically and functionally. An example for a three-level hier(cid:173)
`archy is shown in figure 1.3. Management decisions, supervisory control, and coor(cid:173)
`dination between plants are provided by the management (supervisory) computer,
`which is at the highest level (level 3) of the hierarchy. The next lower level computer
`generates control settings (or reference inputs) for each control region in the corre(cid:173)
`sponding plant. Set points and reference signals are inputs to the direct digital con(cid:173)
`trol (DDC) computers that control each control region. The computers communicate
`using a suitable information network. Information transfer in both directions (up and
`down) should be possible for best performance and flexibility. In master-slave dis(cid:173)
`tributed control, only downloading of information is available.
`
`1.4 SIGNAL CLASSIFICATION IN CONTROL SYSTEMS
`
`A digital control system can be loosely interpreted as one that uses a digital com(cid:173)
`·puter as the controller. It is more appropriate, however, to understand the nature of
`the signals that are present in a control system when identifying it as a digital con(cid:173)
`trol system.
`Analog signals are continuous in time. They are typically generated as outputs
`of a dynamic system. (Note that the dynamic system could be a signal generator or
`any other device, equipment, or physical system.) Analytically, analog signals are
`represented as functions of the continuous time variable t.
`Sampled data are, in fact, pulse amplitude-modulated signals. In this case, in(cid:173)
`formation is carried by the amplitude of each pulse, with the width of the pulses
`kept constant. For constant sampling rate, the distance between adjacent pulses
`is also kept constant. In a physical situation, a pulse amplitude-modulated signal is
`generated through a sample-and-hold operation, in which the signal is sampled at
`
`Sec. 1.4
`
`Signal Classification in Control Systems
`
`5
`
`7
`
`

`
`enter measure-
`
`enter measure(cid:173)
`using equation
`ons, the eccen-
`
`le angle itself,
`jed as
`
`(5.14)
`
`error is given
`
`(5.15)
`
`increases, the
`'0 value for a
`. occurs when
`t is clear that
`
`(5.16)
`
`city error be(cid:173)
`is than r, we
`
`(5.17)
`
`.n.
`
`Example 5.5
`Suppose that in example 5.3, the radius of the code disk is 5 cm. Estimate the maxi(cid:173)
`mum error due to eccentricity. If each track has 1,000 windows, determine whether the
`eccentricity error is significant.
`
`Solution With the given level of confidence, we have calculated the overall eccen(cid:173)
`tricity to be 0.36 mm. Now, from equation 5.16 or 5.17, the maximum angular error is
`
`,Mmax
`
`2 ~~.36 = 0.014 rad
`
`0.83°
`
`Assuming that quadrature signals are used to improve the encoder resolution, we have
`
`3600
`resolution = 4 x 1,000 = 0.090
`
`Note that the maximum error due to eccentricity is more than ten times the encoder
`resolution. Hence, eccentricity will significantly affect the accuracy of the encoder.
`
`Eccentricity also affects the phase angle between the quadrature signals of an
`incremental encoder if a single track and two pick-off sensors (with circumferential
`offset) are used. This error can be reduced using the two-track arrangement, with
`the two sensors positioned along a radial line, so that eccentricity affects the two
`outputs equally.
`
`5.6 DIGITAL RESOLVERS
`
`Digital resolvers, or mutual induction encoders, operate somewhat like analog re(cid:173)
`solvers, using the principle of mutual induction. They are known commercially as
`Inductosyns (Ferrand Controls, Valhalla, N.Y.). A digital resolver has two disks fac(cid:173)
`ing each other (but not touching), one (the stator) stationary and the other (the rotor)
`coupled to the rotating object. The rotor has a fine electric conductor foil imprinted
`on it, as shown schematically in figure 5.14. The printed pattern is a closely spaced
`set of radial pulses, all of which are connected to a high-frequency AC supply of
`voltage Veef. The stator disk has two separate printed patterns that are identical to the
`rotor pattern, but one pattern on the stator is shifted by a quarter-pitch from the
`other pattern. The primary voltage in the rotor circuit induces voltages in the two
`secondary (stator) foils at the same frequency. As the rotor turns, the level of the in(cid:173)
`duced voltage changes, depending on the relative position of the foil patterns on the
`two disks. When the foil pulse patterns coincide, the induced voltage is maximum
`(positive or negative), and when the rotor foil pattern has a half-pitch offset from the
`stator foil pattern, the induced voltage in adjacent parts cancel each other, producing
`a zero output. If the speed of rotation is constant, the output voltages VI and V2 in the
`two foils of the stator become signals that have a carrier frequency (supply fre(cid:173)
`quency) component modulated by periodic and nearly sinusoidal signals with a
`
`Chap. 5
`
`Sec. 5.6
`
`Digital Resolvers
`
`243
`
`8
`
`

`
`Rotor Disk
`
`Stator Disk
`
`Secondary
`Foil 2
`
`Output
`
`Commutator
`with Brush
`
`Supply
`AC
`
`Secondary
`Foil 1
`
`Primary
`Foil
`
`Figure 5.14. Schematic diagram of a digital resolver.
`
`phase shift of 90". The modulation signals can be extracted by demodulation and
`converted into pulse signals. When the speed is not constant, pulse width will vary
`with time. As in the case of an incremental encoder, angular displacement and an(cid:173)
`gular velocity are determined by counting or timing the pulses. The direction of ro(cid:173)
`tation is determined hy the phase difference in the two modulating output signals. (In
`one direction, the phase shift is 90°; in the other direction, it is -90°.) Very fine res(cid:173)
`olutions are obtainable from a digital resolver; it is usually not necessary to use step(cid:173)
`up gear systems or other techniques to improve resolution. Resolutions up to 0.0005°
`can be obtained from these transducers, but they are usually more expensive than
`optical encoders.
`
`5.7 DIGITAL TACHOMETERS
`
`Since shaft encoders are also used for measuring angular velocities, they can be con(cid:173)
`sidered tachometers. In classic terminology, a digital tachometer is a device that em(cid:173)
`ploys a toothed wheel to measure angular velocities. A schematic diagram of one
`such device is shown in figure S.lS. This is a magnetic induction tachometer of the
`variable-reluctance type. The teeth on the wheel are made of ferromagnetic mate(cid:173)
`rial. The two magnetic induction (and variable-reluctance) proximity probes are
`placed facing the teeth radially, a quarter-pitch apart. When the toothed wheel ro(cid:173)
`tates, the two probes generate output signals that are 90° out of phase. One signal
`leads the other in one direction of rotation and lags the other in the opposite direc(cid:173)
`tion of rotation. In this manner, directional readings are obtained. The speed is
`computed either by counting pulses over a sampling period or by timing the pulse
`width, as in the case of an incremental encoder.
`Alternative types of digital tachometers use eddy current proximity probes or
`capacitive proximity probes (see chapter 3). In the case of an eddy current tachome-
`
`244
`
`Digital Transducers
`
`Chap. 5
`
`9
`
`

`
`accurate nonlinear calibration curves are available. Since the proximity sensor is a
`noncontacting device, mechanical loading is negligible. Because a ferromagnetic ob(cid:173)
`ject is used to alter the reluctance of the flux path, the mutual-induction proximity
`sensor is a variable-reluctance device.
`Proximity sensors are used in a wide variety of applications pertaining to non(cid:173)
`contacting displacement sensing and dimensional gaging. Some typical applications
`are
`
`1. Measurement and control of the gap between a robotic welding torch head and
`the work surface
`2. Gaging the thickness of metal plates in manufacturing operations (e.g., rolling
`and forming)
`3. Detecting surface irregularities in machined parts
`4. Angular speed measurement at steady state, by counting the number of rota(cid:173)
`tio~s per unit time
`5. Level detection (e.g., in the filling, bottling, and chemical process industries)
`
`Some mutual-induction displacement transducers depend on relative motion
`between the primary coil and the secondary coil to produce a change in flux linkage.
`Two such devices are the resolver and the synchro-transformer. These are not vari(cid:173)
`able-reluctance transducers because they do not employ a ferromagnetic moving ele(cid:173)
`ment.
`
`The resolver. This mutual-induction transducer is widely used for measur(cid:173)
`ing angular displacements. A simplified schematic diagram of the resolver is shown
`in figure 3.11. The rotor contains the primary coil. It consists of a single two-pole
`winding element energized by the AC supply voltage 'Oref. The rotor is directly at(cid:173)
`tached to the object whose rotation is being measured. The stator consists of two sets
`
`Output
`Vo2 = aVrel sin e
`
`Rottir .
`
`
`AC
`"v Supply
`vref
`
`•
`
`Output
`Vol = aVref cos 9
`
`Figure 3.11. Operating schematic
`diagram of the resolver.
`
`104
`
`Analog Sensors for Motion Measurement
`
`Chap. 3
`
`al
`tc
`
`s
`
`10
`
`

`
`of windings placed 90° apart. If the angular position of the rotor with respect to one
`pair of stator windings is denoted by (), the induced voltage in this pair of windings
`is given by
`
`The induced voltage in the other pair of windings is given by
`
`Vol = aVref cos ()
`
`Co2 = aCrefsin ()
`
`(3.9)
`
`(3.10)
`
`Note that these are amplitude-modulated signals; the carrier signal Cref is'modulated
`by the motion (J. The constant parameter a depends primarily on geometric and ma(cid:173)
`terial characteristics of the device. Either of the two output signals Vol and V o2 may
`be used to determine the angular position in the first quadrant (0 :::;; () :::;; 90°). Both
`signals are needed, however, to determine the displacement (direction as well as
`magnitude) in all four quadrants (0 :::;; () :::;; 360°) without causing any ambiguity.
`For instance, the same sine value is obtained for both 90° + () and 90°
`() (i.e., a
`positive rotation and a negative totation from the 90° position), but the correspond(cid:173)
`ing cosine values have opposite signs, thus providing the proper direction. As for
`differential transformers, transient displacement signals can be extracted by demodu(cid:173)
`lating the modulated outputs. This is accomplished by filtering out the carrier signal,
`thereby extracting the modulating signal.
`
`Example 3.6
`The two output signals Vol and Vo2 of the resolver are termed quadrature signals. Show
`how these quadrature signals could be demodulated to obtain the speed of rotation di(cid:173)
`rectly.
`
`Solntion
`If the angular speed is denoted by w, the quadrature signals, at steady
`speed, may be expressed as
`
`Vol = aVref cos wt
`Co2 = aVref sin wt
`By differentiating these signals, we get
`
`Vol = -aVrefW sin wt
`
`V0 2 = aVrefW cos wt
`
`Hence,
`
`or
`
`Each relation provides direction as well as magnitude of the speed.
`
`An alternative form of resolver uses two AC voltages 90° out of phase, gener(cid:173)
`ated from a digital signal generator board, to power the two stator windings. The ro(cid:173)
`tor is the secondary winding in this case. The phase shift of the induced voltage de-
`
`Sec. 3.4
`
`Variable-Inductance Transducers
`
`105
`
`11
`
`

`
`termines the angular position of the rotor. An advantage of this arrangement is that
`it does not require slip rings and brushes to energize the windings. But it will need
`some mechanism to pick off the output signal from the rotor.
`The output signals of a resolver are nonlinear (trigonometric) functions of the
`angle of rotation. (Historically, resolvers were used to compute trigonometric func(cid:173)
`tions or to resolve a vector into orthogonal components.) In robot control applica(cid:173)
`tions, this is sometimes viewed as a blessing. For computed torque control of robotic
`manipulators, for example, trigonometric functions of the joint angles are needed in
`order to compute the required joint torques. Consequently, when resolvers are used
`to measure joint angles in manipulators, there is an associated reduction in process(cid:173)
`ing time because the trigonometric functions are available as direct measurements.
`The primary advantages of the resolver include
`
`1. Fine resolution and high accuracy
`2. Low output impedance (high signal levels)
`3. Small size (e.g., 10 rum diameter)
`4. Simple and robust construction
`
`Its main limitations are
`
`1. Nonlinear output signals (an advantage in some applications where trigonomet(cid:173)
`ric functions of rotations are needed)
`2. Bandwidth limited by supply frequency
`3. Slip rings and brushes needed (which adds mechanical loading and also creates
`wearout, oxidation, and thermal and noise problems)
`
`The synchro-transformer. The synchro is somewhat similar in operation to
`the resolver. The main differences are that the synchro employs two identical rotor(cid:173)
`stator pairs, and each stator has three sets of windings that are placed 1200 apart
`around the rotor shaft. A schematic diagram for this arrangement is shown in figure
`3.12. Both rotors have single-phase windings, and-contrary to popular belief-the
`synchro is essentially a single-phase device. One of the rotors is energized with an
`
`Output
`Vo = aVref cos (e t -
`
`lir)
`
`Receiver
`(Control Transformer)
`
`Figure 3.12. Schematic diagram of a synchro-transformer.
`
`106
`
`Analog Sensors for Motion Measurement
`
`Chap. 3
`
`,
`
`I
`t
`t
`s
`c
`
`t
`1:
`
`12