Liu, C.C., Vu, K.T., Yu, Y, Galler, D, Strange, E.G., Ong, Chee-Mun"Electrical Machines The electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRc Press llc. 2000
Liu, C.C., Vu, K.T., Yu, Y., Galler, D., Strange, E.G., Ong, Chee-Mun “Electrical Machines” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
66 Chen-Ching Liu Electrical machines Khoi Tien vu ABB Transmission Technical Yixin Yu 66.1 Generators AC Generators DC Generators Donald galler 66.2 Motors Motor Applications Motor Analysis massachusetts Institute 66.3 Small electric motors Single phase Induction Motors. Universal Motors. Permanent Elias G. Strangas Magnet AC Motors. Stepping Motors 66.4 Simulation of Electric Machinery Basics in Modeling. Modular Approach. Mathematical Chee- Mun One g Transformations. Base Quantities. Simulation of Synchronous Purdue University Machines. Three- Phase Induction Machines 66.1 Generators Chen-Ching Liu, Khoi Tien Vu, and Yixin Yu Electric generators are devices that convert energy from a mechanical form to an electrical form. This process, known as electromechanical energy conversion, involves magnetic fields that act as an intermediate medium There are two types of generators: alternating current(ac)and direct current(dc). This section explains how these devices work and how they are modeled in analytical or numerical studies The input to the machine can be derived from a number of energy sources. For example, in the generation of large-scale electric power, coal can produce steam that drives the shaft of the machine. Typically, for such a thermal process, only about 1/3 of the raw energy (i.e, from coal) is converted into mechanical energy. The final step of the energy conversion is quite efficient, with an efficiency close to 100%. The generators operation is based on Faraday's law of electromagnetic induction. In brief, if a coil winding) is linked to a varying magnetic field, then an electromotive force, or voltage, emf, is induced across the coil. Thus, generators have two essential parts: one creates a magnetic field, and the other where the emfs induced. The magnetic field is typically generated by electromagnets(thus, the field intensity can be adjusted for control purposes), whose windings are referred to as field windings or field circuits. The coils where the mfs are induced are called armature windings or armature circuits. One of these two components is stationary (stator), and the other is a rotational part(rotor) driven by an external torque. Conceptually, it is immaterial which of the two components is to rotate because, in either case, the armature circuits always "see "a varying magnetic field. However, practical considerations lead to the common design that for ac generators, the field windings are mounted on the rotor and the armature windings on the stator. In contrast, for dc generators, the field windings are on the stator and armature on the rotor. AC Generators day,most electric power is produced by synchronous generators. Synchronous generators rotate at a constant d ,called synchronous speed. This speed is dictated by the operating frequency of the system and the machine structure. There are also ac generators that do not necessarily rotate at a fixed speed such as those c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 66 Electrical Machines 66.1 Generators AC Generators • DC Generators 66.2 Motors Motor Applications • Motor Analysis 66.3 Small Electric Motors Single Phase Induction Motors • Universal Motors • Permanent Magnet AC Motors • Stepping Motors 66.4 Simulation of Electric Machinery Basics in Modeling • Modular Approach • Mathematical Transformations • Base Quantities • Simulation of Synchronous Machines • Three-Phase Induction Machines 66.1 Generators Chen-Ching Liu, Khoi Tien Vu, and Yixin Yu Electric generators are devices that convert energy from a mechanical form to an electrical form. This process, known as electromechanical energy conversion, involves magnetic fields that act as an intermediate medium. There are two types of generators: alternating current (ac) and direct current (dc). This section explains how these devices work and how they are modeled in analytical or numerical studies. The input to the machine can be derived from a number of energy sources. For example, in the generation of large-scale electric power, coal can produce steam that drives the shaft of the machine. Typically, for such a thermal process, only about 1/3 of the raw energy (i.e., from coal) is converted into mechanical energy. The final step of the energy conversion is quite efficient, with an efficiency close to 100%. The generator’s operation is based on Faraday’s law of electromagnetic induction. In brief, if a coil (or winding) is linked to a varying magnetic field, then an electromotive force, or voltage, emf, is induced across the coil. Thus, generators have two essential parts: one creates a magnetic field, and the other where the emf’s are induced. The magnetic field is typically generated by electromagnets (thus, the field intensity can be adjusted for control purposes), whose windings are referred to as field windings or field circuits. The coils where the emf’s are induced are called armature windings or armature circuits. One of these two components is stationary (stator), and the other is a rotational part (rotor) driven by an external torque. Conceptually, it is immaterial which of the two components is to rotate because, in either case, the armature circuits always “see” a varying magnetic field. However, practical considerations lead to the common design that for ac generators, the field windings are mounted on the rotor and the armature windings on the stator. In contrast, for dc generators, the field windings are on the stator and armature on the rotor. AC Generators Today, most electric power is produced by synchronous generators. Synchronous generators rotate at a constant speed, called synchronous speed. This speed is dictated by the operating frequency of the system and the machine structure. There are also ac generators that do not necessarily rotate at a fixed speed such as those Chen-Ching Liu University of Washington Khoi Tien Vu ABB Transmission Technical Institute Yixin Yu Tianjing University Donald Galler Massachusetts Institute of Technology Elias G. Strangas Michigan State University Chee-Mun Ong Purdue University
found in windmills(induction generators); these generators, however, account for only a very small percentage of today's generated power. Synchronous Generators Principle of Operation. For an illustration of the steady-state operation, refer to Fig. 66. 1 which shows a cross section of an ac machine. The rotor onsists of a winding wrapped around a steel body. a dc current is made to flow in the rotor winding(or field winding), and this results in a magnetic field (rotor field). When the rotor is made to rotate at a constant d,the three stationary windings ad, bb, and cc experience a period- ically varying magnetic field. Thus, emfs are induced across these wind ings in accordance with Faradays law. These emf's are ac and periodic ach period corresponds to one revolution of the rotor. Thus, for 60-Hz electricity, the rotor of Fig. 66.1 has to rotate at 3600 revolutions per il minute(rpm); this is the synchronous speed of the given machine. Because the windings aa, bb, and cc are displaced equally in space from each FIGURE 66.1 Cross section of a sim- other(by 120 degrees), their emf waveforms are displaced in time by 1/3 ple two-pole synchronous machine. of a period. In other words, the machine of Fig. 66. 1 is capable of gener- The rotor body is salient. Current in ting three-phase electricity. This machine has two poles since its rotor rotor winding o into the page, O out ield resembles that of a bar magnet with a north pole and a south pole. When the stator windings are connected to an external (electrical)system to form a closed circuit, the steady-state currents in these windings are also periodic. These currents create magnetic lds of their own. Each of these fields is pulsating with time because the associated current is ac; however, the combination of the three fields is a revolving field. This revolving field arises from the space displacements of the dings and the phase differences of their currents. This combined magnetic field has two poles and rotates at the same speed and direction as the rotor. In summary, for a loaded synchronous(ac) generator operating in teady state, there are two fields rotating at the same speed: one is due to the rotor winding and the other due to the stator windings. It is important to observe that the armature circuits are in fact exposed to two rotating fields, one of which, the armature field, is caused by and in fact tends to counter the effect of the other, the rotor field. The result is that the induced emf in the armature can be reduced when compared with an unloaded machine (ie, open-circuited stator windings). This phenomenon is referred to as armature reaction. It is possible to build a machine with p poles, where p= 4, 6, 8,..(even numbers). For example, the cross- sectional view of a four-pole machine is given in Fig. 66.2. For the specified direction of the(dc)current in the rotor windings, the rotor field has two pairs of north and south poles arranged as shown. The emf induced in a stator winding completes one period for every pair of north and south poles sweeping by; thus, each revolution of the rotor corresponds to two periods of the stator emfs. If the machine is to operate at 60 Hz then the rotor needs to rotate at 1800 rpm. In general, a p-pole machine operating at 60 Hz has a rotor speed of 3600/(/2)rpm. That is, the lower the number of poles is, the higher the rotor speed has to be. In practice, the number of poles dictated by the mechanical system(prime mover)that drives the rotor. Steam turbines operate best at a high speed; thus, two-or four-pole machines are suitable Machines driven by hydro turbines usually have more poles. Usually, the stator windings are arranged so that the resulting armature field has the same number of pole as the rotor field. In practice, there are many possible ways to arrange these windings; the essential idea, howeve can be understood via the simple arrangement shown in Fig. 66. 2. Each phase consists of a pair of windings (thus occupies four slots on the stator structure), e.g., those for phase a are labeled a, a, and a, 42. Geometry suggests that, at any time instant, equal emf s are induced across the windings of the same phase. If the individual windings are connected in series as shown in Fig. 66.2, their emf s add up to form the phase voltage. Mathematical/Circuit Models. There are various models for synchronous machines, depending on how much detail one needs in an analysis. In the simplest model, the machine is equivalent to a constant voltage source in series with an impedance. In more complex models, numerous nonlinear differential equations are involved. Steady-state model. When a machine is in a steady state, the model requires no differential equations. The representation, however, depends on the rotor structure: whether the rotor is cylindrical (round) or salient. c 2000 by CRC Press LLC
© 2000 by CRC Press LLC found in windmills (induction generators); these generators, however, account for only a very small percentage of today’s generated power. Synchronous Generators Principle of Operation. For an illustration of the steady-state operation, refer to Fig. 66.1 which shows a cross section of an ac machine. The rotor consists of a winding wrapped around a steel body. A dc current is made to flow in the rotor winding (or field winding), and this results in a magnetic field (rotor field). When the rotor is made to rotate at a constant speed, the three stationary windings aa′, bb′, and cc′ experience a periodically varying magnetic field. Thus, emf’s are induced across these windings in accordance with Faraday’s law. These emf’s are ac and periodic; each period corresponds to one revolution of the rotor. Thus, for 60-Hz electricity, the rotor of Fig. 66.1 has to rotate at 3600 revolutions per minute (rpm); this is the synchronous speed of the given machine. Because the windings aa′, bb′, and cc′ are displaced equally in space from each other (by 120 degrees), their emf waveforms are displaced in time by 1/3 of a period. In other words, the machine of Fig. 66.1 is capable of generating three-phase electricity. This machine has two poles since its rotor field resembles that of a bar magnet with a north pole and a south pole. When the stator windings are connected to an external (electrical) system to form a closed circuit, the steady-state currents in these windings are also periodic. These currents create magnetic fields of their own. Each of these fields is pulsating with time because the associated current is ac; however, the combination of the three fields is a revolving field. This revolving field arises from the space displacements of the windings and the phase differences of their currents. This combined magnetic field has two poles and rotates at the same speed and direction as the rotor. In summary, for a loaded synchronous (ac) generator operating in a steady state, there are two fields rotating at the same speed: one is due to the rotor winding and the other due to the stator windings. It is important to observe that the armature circuits are in fact exposed to two rotating fields, one of which, the armature field, is caused by and in fact tends to counter the effect of the other, the rotor field. The result is that the induced emf in the armature can be reduced when compared with an unloaded machine (i.e., open-circuited stator windings). This phenomenon is referred to as armature reaction. It is possible to build a machine with p poles, where p = 4, 6, 8, . . . (even numbers). For example, the crosssectional view of a four-pole machine is given in Fig. 66.2. For the specified direction of the (dc) current in the rotor windings, the rotor field has two pairs of north and south poles arranged as shown. The emf induced in a stator winding completes one period for every pair of north and south poles sweeping by; thus, each revolution of the rotor corresponds to two periods of the stator emf’s. If the machine is to operate at 60 Hz then the rotor needs to rotate at 1800 rpm. In general, a p-pole machine operating at 60 Hz has a rotor speed of 3600/(p/2) rpm. That is, the lower the number of poles is, the higher the rotor speed has to be. In practice, the number of poles is dictated by the mechanical system (prime mover) that drives the rotor. Steam turbines operate best at a high speed; thus, two- or four-pole machines are suitable. Machines driven by hydro turbines usually have more poles. Usually, the stator windings are arranged so that the resulting armature field has the same number of poles as the rotor field. In practice, there are many possible ways to arrange these windings; the essential idea, however, can be understood via the simple arrangement shown in Fig. 66.2. Each phase consists of a pair of windings (thus occupies four slots on the stator structure), e.g., those for phase a are labeled a1a1′ and a2a2′. Geometry suggests that, at any time instant, equal emf’s are induced across the windings of the same phase. If the individual windings are connected in series as shown in Fig. 66.2, their emf’s add up to form the phase voltage. Mathematical/Circuit Models. There are various models for synchronous machines, depending on how much detail one needs in an analysis. In the simplest model, the machine is equivalent to a constant voltage source in series with an impedance. In more complex models, numerous nonlinear differential equations are involved. Steady-state model. When a machine is in a steady state, the model requires no differential equations. The representation, however, depends on the rotor structure: whether the rotor is cylindrical (round) or salient. FIGURE 66.1 Cross section of a simple two-pole synchronous machine. The rotor body is salient. Current in rotor winding: into the page, out of the page.
N 十 FIGURE 66.2 Left, cross section of a four-pole synchro- FIGURE 66. 3 Cross section of a nous machine. Rotor has a salient pole structure. righ chrono schematic diagram for phase a windings The rotors depicted in Figs. 66. 1 and 66.2 are salient since the poles are protruding from the shaft. Such structures are mechanically weak, since at a high speed(3600 rpm and 1800 rpm, respectively) the centrifugal force becomes a serious problem. Practically, for high-speed turbines, round-rotor(or cylindrical-rotor)struc tures are preferred. The cross section of a two-pole, round-rotor machine is depicted in Fig. 663. Fror ractical viewpoint, salient rotors are easier to build because each pole and its winding can be manufacture separately and then mounted on the rotor shaft. For round rotors, slots need to be reserved in the rotor where the windings can be placed The mathematical model for round-rotor machines is much simpler than that for salient-rotor ones. Th stems from the fact that the rotor body has a permeability much higher than that of the air In a steady state, he stator field and the rotor body are at a standstill relative to each other. (They rotate at the same speed discussed earlier. )If the rotor is salient, it is easier to establish the magnetic flux lines along the direction of the rotor body(when viewed from the cross section). Therefore, for the same set of stator currents, different positions of the rotor alter the stator field in different ways; this implies that the induced emfs are different. If the rotor is round, then the relative position of the rotor structure does not affect the stator field. Hence, the associated mathematical model is simplified. In the following, the steady-state models of the round-rotor and salient-rotor generators are explained. Refer to Fig 66.3 which shows a two-pole round-rotor machine. without loss of generality, one can select phase a(i.e, winding ad) for the development of a mathematical model of the machine. As mentioned previously, the(armature or stator) winding of phase a is exposed to two magnetic fields: rotor field and stator fie 1. Rotor field. Its flux as seen by winding ad varies with the rotor position; the flux linkage is largest when the N-S axis is perpendicular to the winding surface and minimum(zero) when this axis aligns with the surface. Thus, one can express the flux due to the rotor field as seen by winding ad as M,=l(e)I where 0 is to denote the angular position of the n-S axis (of the rotor field) relative to the surface of aa, IF is the rotor current(a dc current, and L is a periodic function of 8. 2. Stator field. Its flux as seen by winding aa is a combination of three individual fields which are due to currents in the stator windings, ia, ib, and ie. This flux can be expressed as M2=L, i+ Imi+ lmie, where the self (mutual) inductance. Because the rotor is round, L, and Lm are not dependent on 8, the relative position of the rotor and the winding. Typically, the sum of the stator currents i+ i,+ i is near zero; thus, one can write n2=(L-Lm)i The total flux seen by winding ad is n =L(O)IF-(L-Lm), where the minus sign in n-n,is due to the fact that the stator field opposes ptor field. The induced emf across the winding ad is dN/dt, the time derivative of n: e 2000 by CRC Press LLC
© 2000 by CRC Press LLC The rotors depicted in Figs. 66.1 and 66.2 are salient since the poles are protruding from the shaft. Such structures are mechanically weak, since at a high speed (3600 rpm and 1800 rpm, respectively) the centrifugal force becomes a serious problem. Practically, for high-speed turbines, round-rotor (or cylindrical-rotor) structures are preferred. The cross section of a two-pole, round-rotor machine is depicted in Fig. 66.3. From a practical viewpoint, salient rotors are easier to build because each pole and its winding can be manufactured separately and then mounted on the rotor shaft. For round rotors, slots need to be reserved in the rotor where the windings can be placed. The mathematical model for round-rotor machines is much simpler than that for salient-rotor ones. This stems from the fact that the rotor body has a permeability much higher than that of the air. In a steady state, the stator field and the rotor body are at a standstill relative to each other. (They rotate at the same speed as discussed earlier.) If the rotor is salient, it is easier to establish the magnetic flux lines along the direction of the rotor body (when viewed from the cross section). Therefore, for the same set of stator currents, different positions of the rotor alter the stator field in different ways; this implies that the induced emf’s are different. If the rotor is round, then the relative position of the rotor structure does not affect the stator field. Hence, the associated mathematical model is simplified. In the following, the steady-state models of the round-rotor and salient-rotor generators are explained. Refer to Fig. 66.3 which shows a two-pole round-rotor machine. Without loss of generality, one can select phase a (i.e., winding aa¢) for the development of a mathematical model of the machine. As mentioned previously, the (armature or stator) winding of phase a is exposed to two magnetic fields: rotor field and stator field. 1. Rotor field. Its flux as seen by winding aa¢ varies with the rotor position; the flux linkage is largest when the N–S axis is perpendicular to the winding surface and minimum (zero) when this axis aligns with the surface. Thus, one can express the flux due to the rotor field as seen by winding aa¢ as l1 = L(q)IF where q is to denote the angular position of the N–S axis (of the rotor field) relative to the surface of aa¢, IF is the rotor current (a dc current), and L is a periodic function of q. 2. Stator field. Its flux as seen by winding aa¢ is a combination of three individual fields which are due to currents in the stator windings, ia, ib, and ic. This flux can be expressed as l2 = Ls ia + Lmib + Lmic, where Ls (Lm) is the self (mutual) inductance. Because the rotor is round, Ls and Lm are not dependent on q, the relative position of the rotor and the winding. Typically, the sum of the stator currents ia + ib + ic is near zero; thus, one can write l2 = (Ls – Lm)i a. The total flux seen by winding aa¢ is l = l1 – l2 = L(q)IF – (Ls – Lm)ia, where the minus sign in l1 – l2 is due to the fact that the stator field opposes the rotor field. The induced emf across the winding aa¢ is dl/dt, the time derivative of l: FIGURE 66.2 Left, cross section of a four-pole synchronous machine. Rotor has a salient pole structure. Right, schematic diagram for phase a windings. FIGURE 66.3 Cross section of a two-pole round-rotor synchronous machine
The time-varying quantities are normally sinusoidal, and for practical purposes, can be represented by phasors. Thus the above expression becomes E=E-(L2-Ln)j0og。全E-jxF where Oo is the angular speed (rad/s)of the rotor in a steady state. This equation can be modeled as a voltage burce-Er behind a reactance jX, as shown in Fig. 66.4; this reactance is usually referred to as synchronot reactance. The resistor Ra in the diagram represents the winding resistance, and V, is the voltage measured acro am6m=4++,mmm→人。 are now dependent on the(relative) position of the rotor For example (refer to Fig. 66.1), L, is maximum when the rotor is in a vertical position and minimum when the rotor is 90 away. ea In the derivation of the mathematical/ circuit model for salient-rotor machines, the stator field B, can be resolved into two components when the rotor is viewed from a cross section, one component aligns along the rotor and the other is perpendicular to the rotor(Fig. 66.5) The component Ba, which directly opposes the rotor field, is said to FIGURE 66.4 Per-phase equivalent cir- elong to the direct axis, the other component, Be is weaker and machines -E is the internal voltage belongs to the quadrature axis. The model for a salient-rotor machine consists of two circuits, direct-axis circuit and quadrature-axis circuit, (phasor form) and V, is the terminal vol each similar to Fig. 66.4. Any quantity of interest, such as Ia, the current inding ad, is made up of The round-rotor machine can be viewed as a special case of the salient pole theory where the corresponding parameters of the d-axis and q-axis circuits are equal B Dynamic models. When a power system is in a steady state (ie operated at an equilibrium), the electrical output of each generator is equal to the power applied to the rotor shaft. Various losses have beer a neglected without affecting the essential ideas provided in this discus- sion)Disturbances occur frequently in power systems, however Examples of disturbances are load changes, short circuits, and equip- ment outages. A disturbance results in a mismatch between the power input and output of generators, and therefore the rotors depart from IHITHmTHILIH their synchronous-speed operation. Intuitively, the impact is more FIGURE 66.5 In the salient-pole the- severe for machines closer to the disturbance. When a system is per- ory, the stator field(represented by a turbed, there are several possibilities for its subsequent behavior. If the single vector B, is decomposed into B, disturbance is small, the machines may soon reach a new steady speed, and B. Note that [ BI>IB which is close to or identical to their synchronous speed, in which case the system is said to be stable. It may also happen that some machines speed up while others slow down. In a more complicated situation, a rotor may oscillate about its synchronous speed. This results in an unstable case An unstable situation can result in abnormal changes in system frequency and voltage and, unless properly controlled, may lead to damage to machines (e.g, broken shafts). To study these phenomena, dynamic models are required. Details of a dynamic model depend on a number of factors such as location of disturbance and time duration of interest. An overview of dynamic generator models is given here. In essence, there are two aspects that need be modeled: electromechanical and electromagnetic e 2000 by CRC Press LLC
© 2000 by CRC Press LLC The time-varying quantities are normally sinusoidal, and for practical purposes, can be represented by phasors. Thus the above expression becomes: where w0 is the angular speed (rad/s) of the rotor in a steady state. This equation can be modeled as a voltage source–EF behind a reactance jXs , as shown in Fig. 66.4; this reactance is usually referred to as synchronous reactance. The resistor Ra in the diagram represents the winding resistance, and Vt is the voltage measured across the winding. As mentioned, the theory for salient-rotor machines is more complicated. In the equation l2 = Ls ia + Lmib + Lmic, the terms Ls and Lm are now dependent on the (relative) position of the rotor. For example (refer to Fig. 66.1), Ls is maximum when the rotor is in a vertical position and minimum when the rotor is 90° away. In the derivation of the mathematical/circuit model for salient-rotor machines, the stator field B2 can be resolved into two components; when the rotor is viewed from a cross section, one component aligns along the rotor and the other is perpendicular to the rotor (Fig. 66.5). The component Bd , which directly opposes the rotor field, is said to belong to the direct axis; the other component, Bq, is weaker and belongs to the quadrature axis. The model for a salient-rotor machine consists of two circuits, direct-axis circuit and quadrature-axis circuit, each similar to Fig. 66.4.Any quantity of interest, such as Ia, the current in winding aa¢, is made up of two components, one from each circuit. The round-rotor machine can be viewed as a special case of the salientpole theory where the corresponding parameters of the d-axis and q-axis circuits are equal. Dynamic models. When a power system is in a steady state (i.e., operated at an equilibrium), the electrical output of each generator is equal to the power applied to the rotor shaft.(Various losses have been neglected without affecting the essential ideas provided in this discussion.) Disturbances occur frequently in power systems, however. Examples of disturbances are load changes, short circuits, and equipment outages. A disturbance results in a mismatch between the power input and output of generators, and therefore the rotors depart from their synchronous-speed operation. Intuitively, the impact is more severe for machines closer to the disturbance. When a system is perturbed, there are several possibilities for its subsequent behavior. If the disturbance is small, the machines may soon reach a new steady speed, which is close to or identical to their synchronous speed, in which case the system is said to be stable. It may also happen that some machines speed up while others slow down. In a more complicated situation, a rotor may oscillate about its synchronous speed. This results in an unstable case. An unstable situation can result in abnormal changes in system frequency and voltage and, unless properly controlled, may lead to damage to machines (e.g., broken shafts). To study these phenomena, dynamic models are required. Details of a dynamic model depend on a number of factors such as location of disturbance and time duration of interest. An overview of dynamic generator models is given here. In essence, there are two aspects that need be modeled: electromechanical and electromagnetic. e d dt dL dt I L L di dt e L L di dt a F s m a F s m a = = = l – ( – ) – ( – ) D E E L L j I E jX I a = - F s m a = F s a ( – ) w0 D – FIGURE 66.4 Per-phase equivalent circuit of round-rotor synchronous machines. –EF is the internal voltage (phasor form) and Vt is the terminal voltFIGURE 66.5 In the salient-pole theory, the stator field (represented by a single vector B2 ) is decomposed into Bd and Bq . Note that *Bd* > *Bq*