Showing posts with label Electric Machinery. Show all posts
Showing posts with label Electric Machinery. Show all posts
Tuesday, December 21, 2010
Table of Constants and Conversion Factors for SI Units
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical Energy Conversion Principles
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
Engineering Aspects of Practical Electric Machine Performance and Operation
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical Energy Conversion Principles
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
The dq0 Transformation
In this appendix, the direct- and quadrature-axis (dq0) theory introduced in Section 5.6 is formalized. The formal mathematical transformation from three-phase stator quantities to their direct- and quadrature-axis components is presented. These transformations are then used to express the governing equations for a synchronous machine in terms of the dq0 quantities.
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical Energy Conversion Principles
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical Energy Conversion Principles
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Both amplitude and waveform of the generated voltage and armature mmf's in machines are determined by the winding arrangements and general machine geometry. These configurations in turn are dictated by economic use of space and materials in the machine and by suitability for the intended service. In this appendix we supplement the introductory discussion of these considerations in Chapter 4 by analytical treatment of ac voltages and mmf's in the balanced steady state. Attention is confined to the time-fundamental component of voltages and the space-fundamental component of mmf's.
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical Energy Conversion Principles
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical Energy Conversion Principles
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
Three Phase Circuits
Generation, transmission, and heavy-power utilization of ac electric energy almost invariably involve a type of system or circuit called apolyphase system or polyphase circuit. In such a system, each voltage source consists of a group of voltages having related magnitudes and phase angles. Thus, an n-phase system employs voltage sources which typically consist of n voltages substantially equal in magnitude and successively displaced by a phase angle of 360°/n. A three-phase system employs voltage sources which typically consist of three voltages substantially equal in magnitude and displaced by phase angles of 120 ° . Because it possesses definite economic and operating advantages, the three-phase system is by far the most common, and consequently emphasis is placed on three-phase circuits in this appendix.
The three individual voltages of a three-phase source may each be connected to its own independent circuit. We would then have three separate single-phase systems. Alternatively, as will be shown in Section A. 1, symmetrical electric connections can be made between the three voltages and the associated circuitry to form a three-phase system. It is the latter alternative that we are concerned with in this appendix. Note that the word phase now has two distinct meanings. It may refer to a portion of a polyphase system or circuit, or, as in the familiar steady-state circuit theory, it may be used in reference to the angular displacement between voltage or current phasors. There is very little possibility of confusing the two.
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical Energy Conversion Principles
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
The three individual voltages of a three-phase source may each be connected to its own independent circuit. We would then have three separate single-phase systems. Alternatively, as will be shown in Section A. 1, symmetrical electric connections can be made between the three voltages and the associated circuitry to form a three-phase system. It is the latter alternative that we are concerned with in this appendix. Note that the word phase now has two distinct meanings. It may refer to a portion of a polyphase system or circuit, or, as in the familiar steady-state circuit theory, it may be used in reference to the angular displacement between voltage or current phasors. There is very little possibility of confusing the two.
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical Energy Conversion Principles
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
Speed and Torque Control
This chapter introduces various techniques for the control of electric machines. The broad topic of electric machine control requires a much more extensive discussion than is possible here so our objectives have been somewhat limited. Most noticeably, the discussion of this chapter focuses almost exclusively on steady-state behavior, and the issues of transient and dynamic behavior are not considered.
Much of the control flexibility that is now commonly associated with electric machinery comes from the capability of the power electronics that is used to drive these machines. This chapter builds therefore on the discussion of power electronics in Chapter 10.
The starting point is a discussion of dc motors for which it is convenient to subdivide the control techniques into two categories: speed and torque control. The algorithm for speed control in a dc motor is relatively straight forward. With the exception of a correction for voltage drop across the armature resistance, the steadystate speed is determined by the condition that the generated voltage must be equal to the applied armature voltage. Since the generated voltage is proportional to the field flux and motor speed, we see that the steady-state motor speed is proportional to the armature voltage and inversely proportional to the field flux.
An alternative viewpoint is that of torque control. Because the commutator/brush system maintains a constant angular relationship between the field and armature flux, the torque in a dc motor is simply proportional to the product of the armature current and the field flux. As a result, dc motor torque can be controlled directly by controlling the armature current as well as the field flux.
Because synchronous motors develop torque only at synchronous speed, the speed of a synchronous motor is simply determined by the electrical frequency of the applied armature excitation. Thus, steady-state speed control is simply a matter of armature frequency control. Torque control is also possible. By transforming the stator quantities into a reference frame rotating synchronously with the rotor (using the dq0 transformation of Appendix C), we found that torque is proportional to the field flux and the component of armature current in space quadrature with the field flux. This is directly analogous to the torque production in a dc motor. Control schemes which adopt this viewpoint are referred to as vector or field-oriented control.
Induction machines operate asynchronously; rotor currents are induced by the relative motion of the rotor with respect to the synchronously rotating stator-produced flux wave. When supplied by a constant-frequency source applied to the armature winding, the motor will operate at a speed somewhat lower than synchronous speed, with the motor speed decreasing as the load torque is increased. As a result, precise speed regulation is not a simple matter, although in most cases the speed will not vary from synchronous speed by an excessive amount.
Analogous to the situation in a synchronous motor, in spite of the fact that the rotor of an induction motor rotates at less than synchronous speed, the interaction between the rotor and stator flux waves is indeed synchronous. As a result, a transformation into a synchronously rotating reference frame results in rotor and stator flux waves which are constant. The torque can then be expressed in terms of the product of the rotor flux linkages and the component of armature current in quadrature with the rotor flux linkages (referred to as the quadrature-axis component of the armature current) in a fashion directly analogous to the field-oriented viewpoint of a synchronous motor. Furthermore, it can be shown that the rotor flux linkages are proportional to the direct-axis component of the armature current, and thus the direct-axis component of armature current behaves much like the field current in a synchronous motor. This field-oriented viewpoint of induction machine control, in combination with the power-electronic and control systems required to implement it, has led to the widespread applicability of induction machines to a wide range of variable-speed applications.
Finally, this chapter ends with a brief discussion of the control of variablereluctance machines. To produce useful torque, these machines typically require relatively complex, nonsinusoidal current waveforms whose shape must be controlled as a function of rotor position. Typically, these waveforms are produced by pulse- idth modulation combined with current feedback using an H-bridge inverter of the type discussed in Chapter 10. The details of these waveforms depend heavily upon the geometry and magnetic properties of the VRM and can vary significantly from motor to motor.
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical Energy Conversion Principles
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
Much of the control flexibility that is now commonly associated with electric machinery comes from the capability of the power electronics that is used to drive these machines. This chapter builds therefore on the discussion of power electronics in Chapter 10.
The starting point is a discussion of dc motors for which it is convenient to subdivide the control techniques into two categories: speed and torque control. The algorithm for speed control in a dc motor is relatively straight forward. With the exception of a correction for voltage drop across the armature resistance, the steadystate speed is determined by the condition that the generated voltage must be equal to the applied armature voltage. Since the generated voltage is proportional to the field flux and motor speed, we see that the steady-state motor speed is proportional to the armature voltage and inversely proportional to the field flux.
An alternative viewpoint is that of torque control. Because the commutator/brush system maintains a constant angular relationship between the field and armature flux, the torque in a dc motor is simply proportional to the product of the armature current and the field flux. As a result, dc motor torque can be controlled directly by controlling the armature current as well as the field flux.
Because synchronous motors develop torque only at synchronous speed, the speed of a synchronous motor is simply determined by the electrical frequency of the applied armature excitation. Thus, steady-state speed control is simply a matter of armature frequency control. Torque control is also possible. By transforming the stator quantities into a reference frame rotating synchronously with the rotor (using the dq0 transformation of Appendix C), we found that torque is proportional to the field flux and the component of armature current in space quadrature with the field flux. This is directly analogous to the torque production in a dc motor. Control schemes which adopt this viewpoint are referred to as vector or field-oriented control.
Induction machines operate asynchronously; rotor currents are induced by the relative motion of the rotor with respect to the synchronously rotating stator-produced flux wave. When supplied by a constant-frequency source applied to the armature winding, the motor will operate at a speed somewhat lower than synchronous speed, with the motor speed decreasing as the load torque is increased. As a result, precise speed regulation is not a simple matter, although in most cases the speed will not vary from synchronous speed by an excessive amount.
Analogous to the situation in a synchronous motor, in spite of the fact that the rotor of an induction motor rotates at less than synchronous speed, the interaction between the rotor and stator flux waves is indeed synchronous. As a result, a transformation into a synchronously rotating reference frame results in rotor and stator flux waves which are constant. The torque can then be expressed in terms of the product of the rotor flux linkages and the component of armature current in quadrature with the rotor flux linkages (referred to as the quadrature-axis component of the armature current) in a fashion directly analogous to the field-oriented viewpoint of a synchronous motor. Furthermore, it can be shown that the rotor flux linkages are proportional to the direct-axis component of the armature current, and thus the direct-axis component of armature current behaves much like the field current in a synchronous motor. This field-oriented viewpoint of induction machine control, in combination with the power-electronic and control systems required to implement it, has led to the widespread applicability of induction machines to a wide range of variable-speed applications.
Finally, this chapter ends with a brief discussion of the control of variablereluctance machines. To produce useful torque, these machines typically require relatively complex, nonsinusoidal current waveforms whose shape must be controlled as a function of rotor position. Typically, these waveforms are produced by pulse- idth modulation combined with current feedback using an H-bridge inverter of the type discussed in Chapter 10. The details of these waveforms depend heavily upon the geometry and magnetic properties of the VRM and can vary significantly from motor to motor.
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical Energy Conversion Principles
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
Introduction to Power Electronics
The goal of this chapter is relatively modest. Our focus has been to introduce some basic principles of power electronics and to illustrate how they can be applied to the design of various power-conditioning circuits that are commonly found in motor drives. Although the discussion in this chapter is neither complete nor extensive, it is intended to provide the background required to support the various discussions of motor control which are presented in this book.
We began with a brief overview of a few of the available solid-state switching devices: diodes, SCRs, IGBTs and MOSFETs, and so on. We showed that, for the purposes of a preliminary analysis, it is quite sufficient to represent these devices as ideal switches. To emphasize the fact that they typically can pass only unidirectional current, we included ideal diodes in series with these switches. The simplest of these devices is the diode, which has only two terminals and is turned ON and OFF simply by the conditions of the external circuit. The remainder have a third terminal which can be used to turn the device ON and, in the case of transistors such as MOSFETS and IGBTs, OFF again.
A typical variable-frequency, variable-voltage motor-drive system can be considered to consist of three sections. The input section rectifies the power-frequency, fixed-voltage ac input and produces a dc voltage or current. The middle section filters the rectifier output, producing a relatively constant dc current or voltage, depending upon the type of drive under consideration. The output inverter section converts the dc to variable-frequency, variable-voltage ac voltages or currents which can be applied to the terminals of a motor.
The simplest inverters we investigated produce stepped voltage or current waveforms whose amplitude is equal to that of the dc source and whose frequency can be controlled by the timing of the inverter switches. To produce a variable-amplitude output waveform, it is necessary to apply additional control to the rectifier stage to vary the amplitude of the dc bus voltage or link current supplied to the inverter.
We also discussed pulse-width-modulated voltage-source inverters. In this type of inverter, the voltage to the load is switched between V0 and -V0 such that the average load voltage is determined by the duty cycle of the switching waveform.
Loads whose time constant is long compared to the switching time of the inverter will act as filters, and the load current will then be determined by the average load voltage. Pulse-width modulated current-source inverters were also discussed briefly. The reader should approach the presentation here with great caution. It is important to recognize that a complete treatment of power electronics and motor drives is typically the topic of a multiple-course sequence of study. Although the basic principles discussed here apply to a wide range of motor drives, there are many details which must be included in the design of practical motor drives. Drive circuitry to turn ON the "switches" (gate drives for SCRs, MOSFETs, IGBTs, etc.) must be carefully designed to provide sufficient drive to fully turn on the devices and to provide the proper switching sequences. The typical inverter includes a controller and a protection system which is quite elaborate. Typically, the design of a specific drive is dominated by the current and voltage ratings of available switches devices. This is especially true in the case of high-power drive systems in which switches must be connected in series and/or parallel to achieve the desired power rating. The reader is referred to references in the bibliography for a much more complete discussion of power electronics and inverter systems than has been presented here.
Motor drives based upon the configurations discussed here can be used to control motor speed and motor torque. In the case of ac machines, the application of power-electronic based motor drives has resulted in performance that was previously available only with dc machines and has led to widespread use of these machines in most applications.
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical Energy Conversion Principles
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
A typical variable-frequency, variable-voltage motor-drive system can be considered to consist of three sections. The input section rectifies the power-frequency, fixed-voltage ac input and produces a dc voltage or current. The middle section filters the rectifier output, producing a relatively constant dc current or voltage, depending upon the type of drive under consideration. The output inverter section converts the dc to variable-frequency, variable-voltage ac voltages or currents which can be applied to the terminals of a motor.
The simplest inverters we investigated produce stepped voltage or current waveforms whose amplitude is equal to that of the dc source and whose frequency can be controlled by the timing of the inverter switches. To produce a variable-amplitude output waveform, it is necessary to apply additional control to the rectifier stage to vary the amplitude of the dc bus voltage or link current supplied to the inverter.
We also discussed pulse-width-modulated voltage-source inverters. In this type of inverter, the voltage to the load is switched between V0 and -V0 such that the average load voltage is determined by the duty cycle of the switching waveform.
Loads whose time constant is long compared to the switching time of the inverter will act as filters, and the load current will then be determined by the average load voltage. Pulse-width modulated current-source inverters were also discussed briefly. The reader should approach the presentation here with great caution. It is important to recognize that a complete treatment of power electronics and motor drives is typically the topic of a multiple-course sequence of study. Although the basic principles discussed here apply to a wide range of motor drives, there are many details which must be included in the design of practical motor drives. Drive circuitry to turn ON the "switches" (gate drives for SCRs, MOSFETs, IGBTs, etc.) must be carefully designed to provide sufficient drive to fully turn on the devices and to provide the proper switching sequences. The typical inverter includes a controller and a protection system which is quite elaborate. Typically, the design of a specific drive is dominated by the current and voltage ratings of available switches devices. This is especially true in the case of high-power drive systems in which switches must be connected in series and/or parallel to achieve the desired power rating. The reader is referred to references in the bibliography for a much more complete discussion of power electronics and inverter systems than has been presented here.
Motor drives based upon the configurations discussed here can be used to control motor speed and motor torque. In the case of ac machines, the application of power-electronic based motor drives has resulted in performance that was previously available only with dc machines and has led to widespread use of these machines in most applications.
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical Energy Conversion Principles
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
Single And Two Phase Motors
One theme of this chapter is a continuation of the induction-machine theory of Chapter 6 and its application to the single-phase induction motor. This theory is expanded by a step-by-step reasoning process from the simple revolving-field theory of the symmetrical polyphase induction motor. The basic concept is the resolution of the statormmf wave into two constant-amplitude traveling waves revolving around the air gap at synchronous speed in opposite directions. If the slip for the forward field is s, then that for the backward field is (2 - s). Each of these component fields produces induction-motor action, just as in a symmetrical polyphase motor. From the viewpoint of the stator, the reflected effects of the rotor can be visualized and expressed quantitatively in terms of simple equivalent circuits. The ease with which the internal reactions can be accounted for in this manner is the essential reason for the usefulness of the double-revolving-field theory.
For a single-phase winding, the forward- and backward-component mmf waves are equal, and their amplitude is half the maximum value of the peak of the stationary pulsating mmf produced by the winding. The resolution of the stator mmf into its forward and backward components then leads to the physical concept of the singlephase motor described in Section 9.1 and finally to the quantitative theory developed in Section 9.3 and to the equivalent circuits of Fig. 9.11. In most cases, single-phase induction motors are actually two- hase motors with unsymmetrical windings operated off of a single phase source. Thus to complete our understanding of single-phase induction motors, it is necessary to examine the performance of two-phase motors. Hence, the next step is the application of the double-revolving-field picture to a symmetrical two- hase motor with unbalanced applied voltages, as in Section 9.4.1. This investigation leads to the symmetricalcomponent concept, whereby an unbalanced two-phase system of currents or voltages can be resolved into the sum of two balanced two-phase component systems of opposite phase sequence. Resolution of the currents into symmetrical-component systems is equivalent to resolving the stator-mmf wave into its forward and backward components, and therefore the internal reactions of the rotor for each symmetricalcomponent system are the same as those which we have already investigated. A very similar reasoning process, not considered here, leads to the well-known three-phase symmetrical-component method for treating problems involving unbalanced operation of three-phase rotating machines. The ease with which the rotating machine can be analyzed in terms of revolving-field theory is the chief reason for the usefulness of the symmetrical-component method. Finally, the chapter ends in Section 9.4.2 with the development of an analytical theory for the general case of a two-phase induction motor with unsymmetrical windings. This theory permits us to analyze the operation of single-phase motors running off both their main and auxiliary windings.
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical Energy Conversion Principles
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical Energy Conversion Principles
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
Variable Reluctance Machines and Stepping Motors
Variable-reluctance machines are perhaps the simplest of electrical machines. They consist of a stator with excitation windings and a magnetic rotor with saliency. Torque is produced by the tendency of the salient-pole rotor to align with excited magnetic poles on the stator.
VRMs are synchronous machines in that they produce net torque only when the rotor motion is in some sense synchronous with the applied stator mmf. This synchronous relationship may be complex, with the rotor speed being some specific fraction of the applied electrical frequency as determined not only by the number of stator and rotor poles but also by the number of stator and rotor teeth on these poles.
In fact, in some cases, the rotor will be found to rotate in the direction opposite to the rotation direction of the applied stator mmf. Successful operation of a VRM depends on exciting the stator phase windings in a specific fashion correlated to the instantaneous position of the rotor. Thus, rotor position must be measured, and a controller must be employed to determine the appropriate excitation waveforms and to control the output of the inverter. Typically chopping is required to obtain these waveforms. The net result is that although the VRM is itself a simple device, somewhat complex electronics are typically required to make a complete drive system.The significance of VRMs in engineering applications stems from their low cost, reliability, and controllability. Because their torque depends only on the square of the applied stator currents and not on their direction, these machines can be operated from unidirectional drive systems, reducing the cost of the power electronics. However, it is only recently, with the advent of low-cost, flexible power electronic circuitry and microprocessor-based control systems, that VRMs have begun to see widespread application in systems ranging from traction drives to high-torque, precision position control systems for robotics applications.
Practical experience with VRMs has shown that they have the potential for high reliability. This is due in part to the simplicity of their construction and to the fact that there are no windings on their rotors. In addition, VRM drives can be operated successfully (at a somewhat reduced rating) following the failure of one or more phases, either in the machine or in the inverter. VRMs typically have a large number of stator phases (four or more), and significant output can be achieved even if some of these phases are out of service. Because there is no rotor excitation, there will be no voltage generated in a phase winding which fails open-circuited or current generated in a phase winding which fails short-circuited, and thus the machine can continue to be operated without risk of further damage or additional losses and heating.
Because VRMs can be readily manufactured with a large number of rotor and stator teeth (resulting in large inductance changes for small changes in rotor angle), they can be constructed to produce very large torque per unit volume. There is, however, a trade-off between torque and velocity, and such machines will have a low rotational velocity (consistent with the fact that only so much power can be produced by a given machine frame size). On the opposite extreme, the simple configuration of a VRM rotor and the fact that it contains no windings suggest that it is possible to build extremely rugged VRM rotors. These rotors can withstand high speeds, and motors which operate in excess of 200,000 r/min have been built. Finally, we have seen that saturation plays a large role in VRM performance. As recent advances in power electronic and microelectronic circuitry have brought VRM drive systems into the realm of practicality, so have advances in computerbased analytical techniques for magnetic-field analysis. Use of these techniques now makes it practical to perform optimized designs of VRM drive systems which are competitive with alternative technologies in many applications. Stepping motors are closely related to VRMs in that excitation of each successive phase of the stator results in a specific angular rotation of the rotor. Stepping motors come in a wide variety of designs and configurations. These include variablereluctance, permanent-magnet, and hybrid configurations. The rotor position of a variable-reluctance stepper motor is not uniquely determined by the phase currents since the phase inductances are not unique functions of the rotor angle. The addition of a permanent magnet changes this situation and the rotor position of a permanentmagnet stepper motor is a unique function of the phase currents.
Stepping motors are the electromechanical companions to digital electronics. By proper application of phase currents to the stator windings, these motors can be made to rotate in well-defined steps ranging down to a fraction of a degree per pulse. They are thus essential components of digitally controlled electromechanical systems where a high degree of precision is required. They are found in a wide range of applications including numerically controlled machine tools, in printers and plotters, and in disk drives.
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical Energy Conversion Principles
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
VRMs are synchronous machines in that they produce net torque only when the rotor motion is in some sense synchronous with the applied stator mmf. This synchronous relationship may be complex, with the rotor speed being some specific fraction of the applied electrical frequency as determined not only by the number of stator and rotor poles but also by the number of stator and rotor teeth on these poles.
Practical experience with VRMs has shown that they have the potential for high reliability. This is due in part to the simplicity of their construction and to the fact that there are no windings on their rotors. In addition, VRM drives can be operated successfully (at a somewhat reduced rating) following the failure of one or more phases, either in the machine or in the inverter. VRMs typically have a large number of stator phases (four or more), and significant output can be achieved even if some of these phases are out of service. Because there is no rotor excitation, there will be no voltage generated in a phase winding which fails open-circuited or current generated in a phase winding which fails short-circuited, and thus the machine can continue to be operated without risk of further damage or additional losses and heating.
Because VRMs can be readily manufactured with a large number of rotor and stator teeth (resulting in large inductance changes for small changes in rotor angle), they can be constructed to produce very large torque per unit volume. There is, however, a trade-off between torque and velocity, and such machines will have a low rotational velocity (consistent with the fact that only so much power can be produced by a given machine frame size). On the opposite extreme, the simple configuration of a VRM rotor and the fact that it contains no windings suggest that it is possible to build extremely rugged VRM rotors. These rotors can withstand high speeds, and motors which operate in excess of 200,000 r/min have been built. Finally, we have seen that saturation plays a large role in VRM performance. As recent advances in power electronic and microelectronic circuitry have brought VRM drive systems into the realm of practicality, so have advances in computerbased analytical techniques for magnetic-field analysis. Use of these techniques now makes it practical to perform optimized designs of VRM drive systems which are competitive with alternative technologies in many applications. Stepping motors are closely related to VRMs in that excitation of each successive phase of the stator results in a specific angular rotation of the rotor. Stepping motors come in a wide variety of designs and configurations. These include variablereluctance, permanent-magnet, and hybrid configurations. The rotor position of a variable-reluctance stepper motor is not uniquely determined by the phase currents since the phase inductances are not unique functions of the rotor angle. The addition of a permanent magnet changes this situation and the rotor position of a permanentmagnet stepper motor is a unique function of the phase currents.
Stepping motors are the electromechanical companions to digital electronics. By proper application of phase currents to the stator windings, these motors can be made to rotate in well-defined steps ranging down to a fraction of a degree per pulse. They are thus essential components of digitally controlled electromechanical systems where a high degree of precision is required. They are found in a wide range of applications including numerically controlled machine tools, in printers and plotters, and in disk drives.
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical Energy Conversion Principles
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
DC Machines
This chapter has discussed the significant operating characteristics of dc machines. In general, the outstanding advantage of dc machines lies in their flexibility and versatility. Before the widespread availability of ac motor drives, dc machines were essentially the only choice available for many applications which required a high degree of control. Their principal disadvantages stem from the complexity associated with the armature winding and the commutator/brush system. Not only does this additional complexity increase the cost over competing ac machines, it also increases the need for maintenance and reduces the potential reliability of these machines. Yet the advantages of dc motors remain, and they continue to retain a strong competitive position in both large sizes for industrial applications and in smaller sizes for a wide variety of applications.
Dc generators are a simple solution to the problem of converting mechanical energy to electric energy in dc form, although ac generators feeding rectifier systems are certainly an option which must be considered Among dc generators themselves, separately-excited and cumulatively-compounded, self-excited machines are the most common. Separately-excited generators have the advantage of permitting a wide range of output voltages, whereas self-excited machines may produce unstable voltages at lower output voltages where the field-resistance line becomes essentially tangent to the magnetization curve. Cumulatively-compounded generators may produce a substantially flat voltage characteristic or one which rises with load, whereas shuntor separately-excited generators may produce a drooping voltage characteristic unless external regulating means (such as a series field winding) are added.
Among dc motors, the outstanding characteristics of each type are as follows. The series motor operates with a decidedly drooping speed as load is added, the no-load speed usually being prohibitively high; the torque is proportional to almost the square of the current at low flux levels and to some power between 1 and 2 as saturation increases. The shunt motor at constant field current operates at a slightly drooping but almost constant speed as load is added, the torque being almost proportional to armature current; equally important, however, is the fact that its speed can be controlled over wide ranges by shunt-field control, armature-voltage control, or a combination of both. Depending on the relative strengths of the shunt and series field, the cumulatively-compounded motor is intermediate between the other two and may be given essentially the advantages of one or the other.
In a wide variety of low-power applications in systems which are run from a dc source (automotive applications, portable electronics, etc.), dc machines are the most cost-effective choice. These dc machines are constructed in a wide-range of configurations, and many of them are based upon permanent-magnet excitation. In spite of the wide variety of dc machines which can be found in these various applications, their performance can readily be determined using the models and techniques presented in this chapter.
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical Energy Conversion Principles
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
Among dc motors, the outstanding characteristics of each type are as follows. The series motor operates with a decidedly drooping speed as load is added, the no-load speed usually being prohibitively high; the torque is proportional to almost the square of the current at low flux levels and to some power between 1 and 2 as saturation increases. The shunt motor at constant field current operates at a slightly drooping but almost constant speed as load is added, the torque being almost proportional to armature current; equally important, however, is the fact that its speed can be controlled over wide ranges by shunt-field control, armature-voltage control, or a combination of both. Depending on the relative strengths of the shunt and series field, the cumulatively-compounded motor is intermediate between the other two and may be given essentially the advantages of one or the other.
In a wide variety of low-power applications in systems which are run from a dc source (automotive applications, portable electronics, etc.), dc machines are the most cost-effective choice. These dc machines are constructed in a wide-range of configurations, and many of them are based upon permanent-magnet excitation. In spite of the wide variety of dc machines which can be found in these various applications, their performance can readily be determined using the models and techniques presented in this chapter.
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical Energy Conversion Principles
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
Polyphase Induction Machines
In a polyphase induction motor, slip-frequency currents are induced in the rotor windings as the rotor slips past the synchronously-rotating stator flux wave. These rotor currents, in turn, produce a flux wave which rotates in synchronism with the stator flux wave; torque is produced by the interaction of these two flux waves. For increased load on the motor, the rotor speed decreases, resulting in larger slip, increased induced rotor currents, and greater torque.
Examination of the flux-mmf interactions in a polyphase induction motor shows that, electrically, the machine is a form of transformer. The synchronously-rotating air-gap flux wave in the induction machine is the counterpart of the mutual core flux in the transformer. The rotating field induces emf's of stator frequency in the stator windings and of slip frequency in the rotor windings (for all rotor speeds other than synchronous speed). Thus, the induction machine transforms voltages and at the same time changes frequency. When viewed from the stator, all rotor electrical and magnetic phenomena are transformed to stator frequency. The rotor mmf reacts on the stator windings in the same manner as the mmf of the secondary current in a transformer reacts on the primary. Pursuit of this line of reasoning leads to a singlephase equivalent circuit for polyphase induction machines which closely resemble that of a transformer.
For applications requiting a substantially constant speed without excessively severe starting conditions, the squirrel-cage motor usually is unrivaled because of its ruggedness, simplicity, and relatively low cost. Its only disadvantage is its relatively low power factor (about 0.85 to 0.90 at full load for four-pole, 60-Hz motors and considerably lower at light loads and for lower-speed motors). The low power factor is a consequence of the fact that all the excitation must be supplied by lagging reactive power taken from the ac source. One of the salient facts affecting induction-motor applications is that the slip at which maximum torque occurs can be controlled by varying the rotor resistance. A high rotor resistance gives optimum starting conditions but poor running performance. A low rotor resistance, however, may result in unsatisfactory starting conditions. However, the design of a squirrel-cage motor is, therefore, quite likely to be a compromise. Marked improvement in the starting performance with relatively little sacrifice in running performance can be built into a squirrel-cage motor by using a deep-bar or double-cage rotor whose effective resistance increases with slip. A wound-rotor motor can be used for very severe starting conditions or when speed control by rotor resistance is required. Variable-frequency solid-state motor drives lend considerable flexibility to the application of induction motors in variable-speed applications. These issues are discussed in Chapter 11.
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical Energy Conversion Principles
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
Examination of the flux-mmf interactions in a polyphase induction motor shows that, electrically, the machine is a form of transformer. The synchronously-rotating air-gap flux wave in the induction machine is the counterpart of the mutual core flux in the transformer. The rotating field induces emf's of stator frequency in the stator windings and of slip frequency in the rotor windings (for all rotor speeds other than synchronous speed). Thus, the induction machine transforms voltages and at the same time changes frequency. When viewed from the stator, all rotor electrical and magnetic phenomena are transformed to stator frequency. The rotor mmf reacts on the stator windings in the same manner as the mmf of the secondary current in a transformer reacts on the primary. Pursuit of this line of reasoning leads to a singlephase equivalent circuit for polyphase induction machines which closely resemble that of a transformer.
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical Energy Conversion Principles
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
Synchronous Machines
We have seen that a simple set of tests can be used to determine the significant parameters of a synchronous machine including the synchronous reactance Xs or Xd. Two such tests are an open-circuit test, in which the machine terminal voltage is measured as a function of field current, and a short-circuit test, in which the armature is short-circuited and the short-circuit armature current is measured as a function of field current. These test methods are a variation of a testing technique applicable not only to synchronous machines but also to any electrical system whose behavior can be approximated by a linear equivalent circuit to which Thevenin's theorem applies. From a Thevenin-theorem viewpoint, an open-circuit test gives the intemal voltage, and a short-circuit test gives information regarding the internal impedance. From the more specific viewpoint of electromechanical machinery, an open-circuit test gives information regarding excitation requirements, core losses, and (for rotating machines) friction and windage losses; a short-circuit test gives information regarding the magnetic reactions of the load current, leakage impedances, and losses associated with the load current such as I2R and stray load losses. The only real complication arises from the effects of magnetic nonlinearity, effects which can be taken into account approximately by considering the machine to be equivalent to an unsaturated one whose magnetization curve is the straight line Op of Fig. 5.9 and whose synchronous reactance is empirically adjusted for saturation as in Eq. 5.29. In many cases, synchronous machines are operated in conjunction with an external system which can be represented as a constant- requency, constant-voltage source known as an infinite bus. Under these conditions, the synchronous speed is
determined by the frequency of the infinite bus, and the machine output power is proportional to the product of the bus voltage, the machine internal voltage (which is, in tum, proportional to the field excitation), and the sine of the phase angle between them (the power angle), and it is inversely proportional to the net reactance between them.
While the real power at the machine terminals is determined by the shaft power input to the machine (if it is acting as a generator) or the shaft load (if it is a motor), varying the field excitation varies the reactive power. For low values of field current, the machine will absorb reactive power from the system and the power angle will be large. Increasing the field current will reduce the reactive power absorbed by the machine as well as the power angle. At some value of field current, the machine power factor will be unity and any further increase in field current will cause the machine to supply reactive power to the system.
Once brought up to synchronous speed, synchronous motors can be operated quite efficiently when connected to a constant-frequency source. However, as we have seen, a synchronous motor develops torque only at synchronous speed and hence has no starting torque. To make a synchronous motor self-starting, a squirrel-cage winding, called an amortisseur or damper winding, can be inserted in the rotor pole faces,
as shown in Fig. 5.31. The rotor then comes up almost to synchronous speed by induction-motor action with the field winding unexcited. If the load and inertia are not too great, the motor will pull into synchronism when the field winding is energized from a dc source.
Altematively, as we will see in Chapter 11, synchronous motors can be operated from polyphase variable-frequency drive systems. In this case they can be easily started and operated quite flexibly. Small permanent-magnet synchronous machines operated under such conditions are frequently referred to as brushless motors or brushless-dc motors, both because of the similarity of their speed-torque characteristics to those of dc motors and because of the fact that one can view these motors as inside-out dc motors, with the commutation of the stator windings produced electronically by the drive electronics.
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical Energy Conversion Principles
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
Sunday, December 19, 2010
Introduction to Rotating Machines
This chapter presents a brief and elementary description of three basic types of rotating machines: synchronous, induction, and dc machines. In all of them the basic principles are essentially the same. Voltages are generated by the relative motion of a magnetic field with respect to a winding, and torques are produced by the interaction of the magnetic fields of the stator and rotor windings. The characteristics of the various machine types are determined by the methods of connection and excitation of the windings, but the basic principles are essentially similar.
The basic analytical tools for studying rotating machines are expressions for the generated voltages and for the electromechanical torque. Taken together, they express the coupling between the electric and mechanical systems. To develop a reasonably quantitative theory without the confusion arising from too much detail, we have made several simplifying approximations. In the study of ac machines we have assumed sinusoidal time variations of voltages and currents and sinusoidal space waves of air-gap flux density and mmf. On examination of the mmf produced by distributed ac windings we found that the space-fundamental component is the most important. On the other hand, in dc machines the armature-winding mmf is more nearly a sawtooth wave. For our preliminary study in this chapter, however, we have assumed sinusoidal mmf distributions for both ac and dc machines. We examine this assumption more thoroughly for dc machines in Chapter 7. Faraday's law results in Eq. 4.50 for the rms voltage generated in an ac machine winding or Eq. 4.53 for the average voltage generated between brushes in a dc machine.
On examination of the mmf wave of a three-phase winding, we found that balanced three-phase currents produce a constant-amplitude air-gap magnetic field rotating at synchronous speed, as shown in Fig. 4.31 and Eq. 4.39. The importance of this fact cannot be overstated, for it means that it is possible to operate such machines, either as motors or generators, under conditions of constant torque (and hence constant electrical power as is discussed in Appendix A), eliminating the double-frequency, time-varying torque inherently associated with single-phase machines. For example, imagine a multimegawatt single-phase 60-Hz generator subjected to multimegawatt instantaneous power pulsation at 120 Hz! The discovery of rotating fields led to the invention of the simple, rugged, reliable, self-starting polyphase induction motor, which is analyzed in Chapter 6. (A single-phase induction motor will not start; it needs an auxiliary starting winding, as shown in Chapter 9.)
In single-phase machines, or in polyphase machines operating under unbalanced conditions, the backward-rotating component of the armature mmf wave induces currents and losses in the rotor structure. Thus, the operation of polyphase machines under balanced conditions not only eliminates the second-harmonic component of generated torque, it also eliminates a significant source of rotor loss and rotor heating.
It was the invention of polyphase machines operating under balanced conditions that made possible the design and construction of large synchronous generators with ratings as large as 1000 MW.
Having assumed sinusoidally-distributed magnetic fields in the air gap, we then derived expressions for the magnetic torque. The simple physical picture for torque production is that of two magnets, one on the stator and one on the rotor, as shown schematically in Fig. 4.35a. The torque acts in the direction to align the magnets.
To get a reasonably close quantitative analysis without being hindered by details, we assumed a smooth air gap and neglected the reluctance of the magnetic paths in the iron parts, with a mental note that this assumption may not be valid in all situations and a more detailed model may be required.
In Section 4.7 we derived expressions for the magnetic torque from two viewpoints, both based on the fundamental principles of Chapter 3. The first viewpoint regards the machine as a set of magnetically-coupled circuits with inductances which depend on the angular position of the rotor, as in Section 4.7.1. The second regards the machine from the viewpoint of the magnetic fields in the air gap, as in Section 4.7.2.
It is shown that the torque can be expressed as the product of the stator field, the rotor field, and the sine of the angle between their magnetic axes, as in Eq. 4.73 or any of the forms derived from Eq. 4.73. The two viewpoints are complementary, and ability to reason in terms of both is helpful in reaching an understanding of how machines work.
This chapter has been concerned with basic principles underlying rotatingmachine theory. By itself it is obviously incomplete. Many questions remain unanswered.
How do we apply these principles to the determination of the characteristics of synchronous, induction, and dc machines? What are some of the practical problems that arise from the use of iron, copper, and insulation in physical machines? What are some of the economic and engineering considerations affecting rotating-machine applications? What are the physical factors limiting the conditions under which a machine can operate successfully? Appendix D discusses some of these problems.
Taken together, Chapter 4 along with Appendix D serve as an introduction to the more detailed treatments of rotating machines in the following chapters.
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical Energy Conversion Principles
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
The basic analytical tools for studying rotating machines are expressions for the generated voltages and for the electromechanical torque. Taken together, they express the coupling between the electric and mechanical systems. To develop a reasonably quantitative theory without the confusion arising from too much detail, we have made several simplifying approximations. In the study of ac machines we have assumed sinusoidal time variations of voltages and currents and sinusoidal space waves of air-gap flux density and mmf. On examination of the mmf produced by distributed ac windings we found that the space-fundamental component is the most important. On the other hand, in dc machines the armature-winding mmf is more nearly a sawtooth wave. For our preliminary study in this chapter, however, we have assumed sinusoidal mmf distributions for both ac and dc machines. We examine this assumption more thoroughly for dc machines in Chapter 7. Faraday's law results in Eq. 4.50 for the rms voltage generated in an ac machine winding or Eq. 4.53 for the average voltage generated between brushes in a dc machine.
On examination of the mmf wave of a three-phase winding, we found that balanced three-phase currents produce a constant-amplitude air-gap magnetic field rotating at synchronous speed, as shown in Fig. 4.31 and Eq. 4.39. The importance of this fact cannot be overstated, for it means that it is possible to operate such machines, either as motors or generators, under conditions of constant torque (and hence constant electrical power as is discussed in Appendix A), eliminating the double-frequency, time-varying torque inherently associated with single-phase machines. For example, imagine a multimegawatt single-phase 60-Hz generator subjected to multimegawatt instantaneous power pulsation at 120 Hz! The discovery of rotating fields led to the invention of the simple, rugged, reliable, self-starting polyphase induction motor, which is analyzed in Chapter 6. (A single-phase induction motor will not start; it needs an auxiliary starting winding, as shown in Chapter 9.)
In single-phase machines, or in polyphase machines operating under unbalanced conditions, the backward-rotating component of the armature mmf wave induces currents and losses in the rotor structure. Thus, the operation of polyphase machines under balanced conditions not only eliminates the second-harmonic component of generated torque, it also eliminates a significant source of rotor loss and rotor heating.
It was the invention of polyphase machines operating under balanced conditions that made possible the design and construction of large synchronous generators with ratings as large as 1000 MW.
Having assumed sinusoidally-distributed magnetic fields in the air gap, we then derived expressions for the magnetic torque. The simple physical picture for torque production is that of two magnets, one on the stator and one on the rotor, as shown schematically in Fig. 4.35a. The torque acts in the direction to align the magnets.
To get a reasonably close quantitative analysis without being hindered by details, we assumed a smooth air gap and neglected the reluctance of the magnetic paths in the iron parts, with a mental note that this assumption may not be valid in all situations and a more detailed model may be required.
In Section 4.7 we derived expressions for the magnetic torque from two viewpoints, both based on the fundamental principles of Chapter 3. The first viewpoint regards the machine as a set of magnetically-coupled circuits with inductances which depend on the angular position of the rotor, as in Section 4.7.1. The second regards the machine from the viewpoint of the magnetic fields in the air gap, as in Section 4.7.2.
It is shown that the torque can be expressed as the product of the stator field, the rotor field, and the sine of the angle between their magnetic axes, as in Eq. 4.73 or any of the forms derived from Eq. 4.73. The two viewpoints are complementary, and ability to reason in terms of both is helpful in reaching an understanding of how machines work.
This chapter has been concerned with basic principles underlying rotatingmachine theory. By itself it is obviously incomplete. Many questions remain unanswered.
How do we apply these principles to the determination of the characteristics of synchronous, induction, and dc machines? What are some of the practical problems that arise from the use of iron, copper, and insulation in physical machines? What are some of the economic and engineering considerations affecting rotating-machine applications? What are the physical factors limiting the conditions under which a machine can operate successfully? Appendix D discusses some of these problems.
Taken together, Chapter 4 along with Appendix D serve as an introduction to the more detailed treatments of rotating machines in the following chapters.
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical Energy Conversion Principles
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
Electromechanical Energy Conversion Principles
In electromechanical systems, energy is stored in magnetic and electric fields. When the energy in the field is influenced by the configuration of the mechanical parts constituting the boundaries of the field, mechanical forces are created which tend to move the mechanical elements so that energy is transmitted from the field to the mechanical system.
Singly excited magnetic systems are considered first in Section 3.3. By removing electric and mechanical loss elements from the electromechanical-energy-conversion system (and incorporating them as loss elements in the external electrical and mechanical systems), the energy conversion device can be modeled as a conservative system. Its energy then becomes a state function, determined by its state variables i and x or (teta) . Section 3.4 derives expressions for determining the force and torque as the negative of partial derivative of the energy with respect to the displacement, taken while holding the flux-linkage )~ constant. In Section 3.5 the state function coenergy, with state variables i and x or 0, is introduced. The force and torque are then shown to be given by the partial derivative of the coenergy with respect to displacement, taken while holding the current i constant. These concepts are extended in Section 3.6 to include systems with multiple windings. Section 3.7 further extends the development to include systems in which permanent magnets are included among the sources of the magnetic energy storage.
Energy conversion devices operate between electric and mechanical systems. Their behavior is described by differential equations which include the coupling terms between the systems, as discussed in Section 3.8. These equations are usually nonlinear and can be solved by numerical methods if necessary. As discussed in
Section 3.9, in some cases approximations can be made to simplify the equations.
For example, in many cases, linearized analyses can provide useful insight, both with respect to device design and performance. This chapter has been concerned with basic principles applying broadly to the electromechanical-energy-conversion process, with emphasis on magnetic-field systems.
Basically, rotating machines and linear-motion transducers work in the same way. The remainder of this text is devoted almost entirely to rotating machines. Rotating machines typically include multiple windings and may include permanent magnets. Their performance can be analyzed by using the techniques and principles developed in this chapter.
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical Energy Conversion Principles
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
Singly excited magnetic systems are considered first in Section 3.3. By removing electric and mechanical loss elements from the electromechanical-energy-conversion system (and incorporating them as loss elements in the external electrical and mechanical systems), the energy conversion device can be modeled as a conservative system. Its energy then becomes a state function, determined by its state variables i and x or (teta) . Section 3.4 derives expressions for determining the force and torque as the negative of partial derivative of the energy with respect to the displacement, taken while holding the flux-linkage )~ constant. In Section 3.5 the state function coenergy, with state variables i and x or 0, is introduced. The force and torque are then shown to be given by the partial derivative of the coenergy with respect to displacement, taken while holding the current i constant. These concepts are extended in Section 3.6 to include systems with multiple windings. Section 3.7 further extends the development to include systems in which permanent magnets are included among the sources of the magnetic energy storage.
Energy conversion devices operate between electric and mechanical systems. Their behavior is described by differential equations which include the coupling terms between the systems, as discussed in Section 3.8. These equations are usually nonlinear and can be solved by numerical methods if necessary. As discussed in
Section 3.9, in some cases approximations can be made to simplify the equations.
For example, in many cases, linearized analyses can provide useful insight, both with respect to device design and performance. This chapter has been concerned with basic principles applying broadly to the electromechanical-energy-conversion process, with emphasis on magnetic-field systems.
Basically, rotating machines and linear-motion transducers work in the same way. The remainder of this text is devoted almost entirely to rotating machines. Rotating machines typically include multiple windings and may include permanent magnets. Their performance can be analyzed by using the techniques and principles developed in this chapter.
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical Energy Conversion Principles
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
Saturday, December 11, 2010
Transformers
Although not an electromechanical device, the transformer is a common and indispensable component of ac systems where it is used to transform voltages, currents, and impedances to appropriate levels for optimal use. For the purposes of our study of electromechanical systems, transformers serve as valuable examples of the analysis techniques which must be employed. They offer us opportunities to investigate the properties of magnetic circuits, including the concepts of mmf, magnetizing current, and magnetizing, mutual, and leakage fluxes and their associated inductances.
In both transformers and rotating machines, a magnetic field is created by the combined action of the currents in the windings. In an iron-core transformer, most of this flux is confined to the core and links all the windings. This resultant mutual flux induces voltages in the windings proportional to their number of turns and is responsible for the voltage-changing property of a transformer. In rotating machines, the situation is similar, although there is an air gap which separates the rotating and stationary components of the machine. Directly analogous to the manner in which transformer core flux links the various windings on a transformer core, the mutual flux in rotating machines crosses the air gap, linking the windings on the rotor and stator. As in a transformer, the mutual flux induces voltages in these windings proportional to the number of turns and the time rate of change of the flux. A significant difference between transformers and rotating machines is that in rotating machines there is relative motion between the windings on the rotor and stator.
This relative motion produces an additional component of the time rate of change of the various winding flux linkages. As will be discussed in Chapter 3, the resultant voltage component, known as the speed voltage, is characteristic of the process of electromechanical energy conversion. In a static transformer, however, the time variation of flux linkages is caused simply by the time variation of winding currents; no mechanical motion is involved, and no electromechanical energy conversion takes place. The resultant core flux in a transformer induces a counter emf in the primary which, together with the primary resistance and leakage-reactance voltage drops, must balance the applied voltage. Since the resistance and leakage-reactance voltage drops usually are small, the counter emf must approximately equal the applied voltage and the core flux must adjust itself accordingly. Exactly similar phenomena must take place in the armature windings of an ac motor; the resultant air-gap flux wave must adjust itself to generate a counter emf approximately equal to the applied voltage. In both transformers and rotating machines, the net mmf of all the currents must accordingly adjust itself to create the resultant flux required by this voltage balance. In any ac electromagnetic device in which the resistance and leakage-reactance voltage drops are small, the resultant flux is very nearly determined by the applied voltage and frequency, and the currents must adjust themselves accordingly to produce the mmf required to create this flux.
In a transformer, the secondary current is determined by the voltage induced in the secondary, the secondary leakage impedance, and the electric load. In an induction motor, the secondary (rotor) current is determined by the voltage induced in the secondary, the secondary leakage impedance, and the mechanical load on its shaft. Essentially the same phenomena take place in the primary winding of the transformer and in the armature (stator) windings of induction and synchronous motors. In all three, the primary, or armature, current must adjust itself so that the combined mmf of all currents creates the flux required by the applied voltage. In addition to the useful mutual fluxes, in both transformers and rotating machines there are leakage fluxes which link individual windings without linking others. Although the detailed picture of the leakage fluxes in rotating machines is more complicated than that in transformers, their effects are essentially the same. In both, the leakage fluxes induce voltages in ac windings which are accounted for as leakagereactance voltage drops. In both, the reluctances of the leakage-flux paths are dominanted by that of a path through air, and hence the leakage fluxes are nearly linearly proportional to the currents producing them. The leakage reactances therefore are often assumed to be constant, independent of the degree of saturation of the main magnetic circuit.
Further examples of the basic similarities between transformers and rotating machines can be cited. Except for friction and windage, the losses in transformers and rotating machines are essentially the same. Tests for determining the losses and equivalent circuit parameters are similar: an open-circuit, or no-load, test gives information regarding the excitation requirements and core losses (along with friction and windage losses in rotating machines), while a short-circuit test together with dc resistance measurements gives information regarding leakage reactances and winding resistances. Modeling of the effects of magnetic saturation is another example:
In both transformers and ac rotating machines, the leakage reactances are usually assumed to be unaffected by saturation, and the saturation of the main magnetic circuit is assumed to be determined by the resultant mutual or air-gap flux.
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical-Energy-ConversionPrinciples
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
In both transformers and rotating machines, a magnetic field is created by the combined action of the currents in the windings. In an iron-core transformer, most of this flux is confined to the core and links all the windings. This resultant mutual flux induces voltages in the windings proportional to their number of turns and is responsible for the voltage-changing property of a transformer. In rotating machines, the situation is similar, although there is an air gap which separates the rotating and stationary components of the machine. Directly analogous to the manner in which transformer core flux links the various windings on a transformer core, the mutual flux in rotating machines crosses the air gap, linking the windings on the rotor and stator. As in a transformer, the mutual flux induces voltages in these windings proportional to the number of turns and the time rate of change of the flux. A significant difference between transformers and rotating machines is that in rotating machines there is relative motion between the windings on the rotor and stator.
This relative motion produces an additional component of the time rate of change of the various winding flux linkages. As will be discussed in Chapter 3, the resultant voltage component, known as the speed voltage, is characteristic of the process of electromechanical energy conversion. In a static transformer, however, the time variation of flux linkages is caused simply by the time variation of winding currents; no mechanical motion is involved, and no electromechanical energy conversion takes place. The resultant core flux in a transformer induces a counter emf in the primary which, together with the primary resistance and leakage-reactance voltage drops, must balance the applied voltage. Since the resistance and leakage-reactance voltage drops usually are small, the counter emf must approximately equal the applied voltage and the core flux must adjust itself accordingly. Exactly similar phenomena must take place in the armature windings of an ac motor; the resultant air-gap flux wave must adjust itself to generate a counter emf approximately equal to the applied voltage. In both transformers and rotating machines, the net mmf of all the currents must accordingly adjust itself to create the resultant flux required by this voltage balance. In any ac electromagnetic device in which the resistance and leakage-reactance voltage drops are small, the resultant flux is very nearly determined by the applied voltage and frequency, and the currents must adjust themselves accordingly to produce the mmf required to create this flux.
In a transformer, the secondary current is determined by the voltage induced in the secondary, the secondary leakage impedance, and the electric load. In an induction motor, the secondary (rotor) current is determined by the voltage induced in the secondary, the secondary leakage impedance, and the mechanical load on its shaft. Essentially the same phenomena take place in the primary winding of the transformer and in the armature (stator) windings of induction and synchronous motors. In all three, the primary, or armature, current must adjust itself so that the combined mmf of all currents creates the flux required by the applied voltage. In addition to the useful mutual fluxes, in both transformers and rotating machines there are leakage fluxes which link individual windings without linking others. Although the detailed picture of the leakage fluxes in rotating machines is more complicated than that in transformers, their effects are essentially the same. In both, the leakage fluxes induce voltages in ac windings which are accounted for as leakagereactance voltage drops. In both, the reluctances of the leakage-flux paths are dominanted by that of a path through air, and hence the leakage fluxes are nearly linearly proportional to the currents producing them. The leakage reactances therefore are often assumed to be constant, independent of the degree of saturation of the main magnetic circuit.
Further examples of the basic similarities between transformers and rotating machines can be cited. Except for friction and windage, the losses in transformers and rotating machines are essentially the same. Tests for determining the losses and equivalent circuit parameters are similar: an open-circuit, or no-load, test gives information regarding the excitation requirements and core losses (along with friction and windage losses in rotating machines), while a short-circuit test together with dc resistance measurements gives information regarding leakage reactances and winding resistances. Modeling of the effects of magnetic saturation is another example:
In both transformers and ac rotating machines, the leakage reactances are usually assumed to be unaffected by saturation, and the saturation of the main magnetic circuit is assumed to be determined by the resultant mutual or air-gap flux.
1 Magnetic Circuits and Magnetic Materials
2 Transformers
3 Electromechanical-Energy-ConversionPrinciples
4 Introduction to Rotating Machines
5 Synchronous Machines
6 Polyphase Induction Machines
7 DC Machines
8 Variable-Reluctance Machines and Stepping Motors
9 Single- and Two-Phase Motors
10 Introduction to Power Electronics
11 Speed and Torque Control
Appendix A Three phase circuits
Appendix B Voltages, Magnetic Fields, and Inductances of Distributed AC Windings
Appendix C The dq0 Transformation
Appendix D Engineering Aspects of Practical Electric Machine Performance and Operation
Appendix E Table of Constants and Conversion
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