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*
1. Electromechanical equations. Electromechanical equations are to model the effect of input-output imbal ance on the rotor speed (and therefore on the operating frequency ). The rotor of each machine can be described by the so-called swing equation d-e where e denotes the rotor position relative to a certain rotating frame, M the inertia of rotor, and d damping The term de/dt represents the angular velocity and d20/dr is the angular acceleration of the rotor. The preceding differential equation is derived from Newtons law for rotational motions and, in some respects, resembles the namical equation of a swinging pendulum(with Pin -driving torque, and Pout"restoring torque). The term Pin, which drives the rotor shaft, can be considered constant in many cases. The term Pout, the power sent out to the system, may behave in a very complicated way. Qualitatively, Pout tends to increase(respectively, decrease, as the rotor position moves forward(respectively, backward) relative to the synchronous rotating frame However, such a stable operation can take place only when the system is capable of absorbing(respectively, providing) the extra power. In a multimachine system, conflict might arise when various machines compete with each other in sending out more(or sending out less)electrical power; as a result, the stabilizing effect 2. Electromagnetic equations. The(nonlinear)electromagnetic equations are derived from Faraday s law of electromagnetic induction--induced emfs are proportional to the rate of change of the magnetic fluxes. A general form is as follows λ,+λ- dt (66.1) λ,+λ6 d = g(s)if- xd(s)id (s)i The true terminal voltage, e.g., e, for phase a, can be obtained by ombining the direct-axis and quadrature-axis components ea and respectively, which are given in Eq(66.1). On each line of Eq (66.1), P-model>SYSTEM Machin the induced emf is the combination of two sources the first is the rate of change of the flux on the same axis [(d/dt)nd on the first line, (d/dr)%a on the second ] the second comes into effect only when a Igiven by(d/dn)e]. The third term in the voltage equation represents a qualitative relationship among various the ohmic loss associated with the stator winding ectrical and mechanical quantities of a e Equation(66. 2) expresses the fluxes in terms of relevant currents: synchronous machine.e,,6,,e,are phase x is equal to inductance times current, with inductances G(s),X(s), voltages; ia,ib, i phase currents; iF rotor X,(s)given in an operational form(s denotes the derivative operator). field current; 0 relative position of rotor; Figure 66.6 gives a general view of the input-output state descri o deviation of rotor speed from synchro. tion of machine's dynamic model, the state variables of which appear The state variables appear in Eqs. (66.1) nous speed; Pin mechanical power input. in eqs.(66.1)and(662) d(66.2) e 2000 by CRC Press LLC
© 2000 by CRC Press LLC 1. Electromechanical equations. Electromechanical equations are to model the effect of input–output imbalance on the rotor speed (and therefore on the operating frequency). The rotor of each machine can be described by the so-called swing equation, where q denotes the rotor position relative to a certain rotating frame, M the inertia of rotor, and D damping. The term dq/dt represents the angular velocity and d2q/dt2 is the angular acceleration of the rotor. The preceding differential equation is derived from Newton’s law for rotational motions and, in some respects, resembles the dynamical equation of a swinging pendulum (with Pin ~ driving torque, and Pout ~ restoring torque). The term Pin, which drives the rotor shaft, can be considered constant in many cases. The term Pout, the power sent out to the system, may behave in a very complicated way. Qualitatively, Pout tends to increase (respectively, decrease) as the rotor position moves forward (respectively, backward) relative to the synchronous rotating frame. However, such a stable operation can take place only when the system is capable of absorbing (respectively, providing) the extra power. In a multimachine system, conflict might arise when various machines compete with each other in sending out more (or sending out less) electrical power; as a result, the stabilizing effect might be reduced or even lost. 2. Electromagnetic equations. The (nonlinear) electromagnetic equations are derived from Faraday’s law of electromagnetic induction—induced emf’s are proportional to the rate of change of the magnetic fluxes. A general form is as follows: (66.1) where (66.2) The true terminal voltage, e.g., ea for phase a, can be obtained by combining the direct-axis and quadrature-axis components ed and eq, respectively, which are given in Eq. (66.1). On each line of Eq. (66.1), the induced emf is the combination of two sources: the first is the rate of change of the flux on the same axis [(d/dt)ld on the first line, (d/dt)lq on the second]; the second comes into effect only when a disturbance makes the rotor and stator fields depart from each other [given by (d/dt)q]. The third term in the voltage equation represents the ohmic loss associated with the stator winding. Equation (66.2) expresses the fluxes in terms of relevant currents: flux is equal to inductance times current, with inductances G(s), Xd(s), Xq(s) given in an operational form (s denotes the derivative operator). Figure 66.6 gives a general view of the input–output state description of machine’s dynamic model, the state variables of which appear in Eqs. (66.1) and (66.2). M d dt D d dt P P 2 2 q q + = in – out e d dt d dt ri e d dt d dt ri d d q d q q d q = + = + Ï Ì Ô Ô Ó Ô Ô l l q l l q – – l l d F d d q q q Gsi X s i X s i = = Ï Ì Ô Ó Ô ( ) – ( ) – ( ) FIGURE 66.6 A block diagram depicting a qualitative relationship among various electrical and mechanical quantities of a synchronous machine. ea , eb , ec are phase voltages; ia , ib , ic phase currents; iF rotor field current; q relative position of rotor; w deviation of rotor speed from synchronous speed; Pin mechanical power input. The state variables appear in Eqs. (66.1) and (66.2)
3. Miscellaneous. In addition to the basic components of a synchronous generator (rotor, stator, and thei windings), there are auxiliary devices which help maintain the machine's operation within acceptable limits. Three such devices are mentioned here: governor, damper windings, and excitation control system Governor. This is to control the mechanical power input Pin. The control is via a feedback loop where the speed of the rotor is constantly monitored. For instance, if this speed falls behind the synchronous speed, the input is insufficient and has to be increased. This is done by opening up the valve to increase the steam for turbogenerators or the flow of water through the penstock for hydrogenerators. Governors are mechanical systems and therefore have some significant time lags(many seconds)compared to other lectromagnetic phenomena associated with the machine. If the time duration of interest is short, the effect of governor can be ignored in the study; that is, Pin is treated as a constant. Damper windings(amortisseur windings). These are special conducting bars buried in notches on the rotor surface, and the rotor resembles that of a squirrel-cage-rotor induction machine(see Section 66.2) The damper windings provide an additional stabilizing force for the machine when it is perturbed from an equilibrium. As long as the machine is in a steady state, the stator field rotates at the same speed as the rotor, and no currents are induced in the damper windings. That is, these windings exhibit no effect on a steady-state machine. However, when the speeds of the stator field and the rotor become different (because of a disturbance, currents are induced in the damper windings in such a way as to keep according to Lenz's law, the two speeds from separating Excitation control system. Modern excitation systems are very fast and quite efficient. An excitation ontrol system is a feedback loop that aims at keeping the voltage at machine terminals at a set level. To explain the main feature of the excitation system, it is sufficient to consider Fig. 66.4. Assume that a disturbance occurs in the system, and as a result, the machine's terminal voltage V, drops. The excitation stem boosts the internal voltage EF; this action can increase the voltage V, and also tends to increase the reactive power output. From a system viewpoint, the two controllers of excitation and governor rely on local information(machine terminal voltage and rotor speed). In other words, they are decentralized controls. For large-scale systems, such designs do not always guarantee a desired stable behavior since the effect of interconnection is not taken into account in detail Synchronous Machine Parameters. When a disturbance, such as a short circuit at the machine terminals, takes place, the dynamics of a synchronous machine will be observed before a new steady state is reached. Such a process typically takes a few seconds and can be divided into subprocesses. The damper windings(armortis seur)exhibit their effect only during the first few cycles when the difference in speed between the rotor and ne perturbed stator field is significant. This period is referred to as subtransient. The next and longer period which is between the subtransient and the new steady state, is called transient. Various parameters associated with the subprocesses can be visualized from an equivalent circuit. The d-axis and q-axis(dynamic) equivalent circuits of a synchronous generator consist of resistors, inductors, and voltage sources. In the subtransient period, the equivalent of the damper windings needs to be considered. In the transient period, this equivalent can be ignored. When the new steady state is reached, the current in the rotor nding becomes a constant(dc); thus, one can further ignore the equivalent inductance of this winding. This approximate method results in three equivalent circuits, listed in order of complexity: subtransient, transient, and steady state. For each circuit, one can define parameters such as(effective)reactance and time constant For example, the d-axis circuit for the transient period has an effective reactance X' and a time constant Tdo omputed from the R-L circuit) when open circuited. The parameters of a synchronous machine can be mputed from experimental data and are used in numerical studies. Typical values for these parameters are given in Table 66.1 References on synchronous generators are numerous because of the historical importance of these machines in large-scale electric energy production. [Sarma, 1979] includes a derivation of the steady-state and dynamic models, dynamic performance, excitation, and trends in development of large generators. [Chapman, 1991] e 2000 by CRC Press LLC
© 2000 by CRC Press LLC 3. Miscellaneous. In addition to the basic components of a synchronous generator (rotor, stator, and their windings), there are auxiliary devices which help maintain the machine’s operation within acceptable limits. Three such devices are mentioned here: governor, damper windings, and excitation control system. • Governor. This is to control the mechanical power input Pin. The control is via a feedback loop where the speed of the rotor is constantly monitored. For instance, if this speed falls behind the synchronous speed, the input is insufficient and has to be increased. This is done by opening up the valve to increase the steam for turbogenerators or the flow of water through the penstock for hydrogenerators. Governors are mechanical systems and therefore have some significant time lags (many seconds) compared to other electromagnetic phenomena associated with the machine. If the time duration of interest is short, the effect of governor can be ignored in the study; that is, Pin is treated as a constant. • Damper windings (armortisseur windings). These are special conducting bars buried in notches on the rotor surface, and the rotor resembles that of a squirrel-cage-rotor induction machine (see Section 66.2). The damper windings provide an additional stabilizing force for the machine when it is perturbed from an equilibrium. As long as the machine is in a steady state, the stator field rotates at the same speed as the rotor, and no currents are induced in the damper windings. That is, these windings exhibit no effect on a steady-state machine. However, when the speeds of the stator field and the rotor become different (because of a disturbance), currents are induced in the damper windings in such a way as to keep, according to Lenz’s law, the two speeds from separating. • Excitation control system. Modern excitation systems are very fast and quite efficient. An excitation control system is a feedback loop that aims at keeping the voltage at machine terminals at a set level. To explain the main feature of the excitation system, it is sufficient to consider Fig. 66.4. Assume that a disturbance occurs in the system, and as a result, the machine’s terminal voltage Vt drops. The excitation system boosts the internal voltage EF ; this action can increase the voltage Vt and also tends to increase the reactive power output. From a system viewpoint, the two controllers of excitation and governor rely on local information (machine’s terminal voltage and rotor speed). In other words, they are decentralized controls. For large-scale systems, such designs do not always guarantee a desired stable behavior since the effect of interconnection is not taken into account in detail. Synchronous Machine Parameters. When a disturbance, such as a short circuit at the machine terminals, takes place, the dynamics of a synchronous machine will be observed before a new steady state is reached. Such a process typically takes a few seconds and can be divided into subprocesses. The damper windings (armortisseur) exhibit their effect only during the first few cycles when the difference in speed between the rotor and the perturbed stator field is significant. This period is referred to as subtransient. The next and longer period, which is between the subtransient and the new steady state, is called transient. Various parameters associated with the subprocesses can be visualized from an equivalent circuit. The d-axis and q-axis (dynamic) equivalent circuits of a synchronous generator consist of resistors, inductors, and voltage sources. In the subtransient period, the equivalent of the damper windings needs to be considered. In the transient period, this equivalent can be ignored. When the new steady state is reached, the current in the rotor winding becomes a constant (dc); thus, one can further ignore the equivalent inductance of this winding. This approximate method results in three equivalent circuits, listed in order of complexity: subtransient, transient, and steady state. For each circuit, one can define parameters such as (effective) reactance and time constant. For example, the d-axis circuit for the transient period has an effective reactance X ¢ d and a time constant T ¢ do (computed from the R-L circuit) when open circuited. The parameters of a synchronous machine can be computed from experimental data and are used in numerical studies. Typical values for these parameters are given in Table 66.1. References on synchronous generators are numerous because of the historical importance of these machines in large-scale electric energy production. [Sarma, 1979] includes a derivation of the steady-state and dynamic models, dynamic performance, excitation, and trends in development of large generators. [Chapman, 1991]
TABLE 66. 1 Typical Synchronous Generator Parametersa alient-Pole rotor Parameter Symbol Round Rotor Damper windings d-axis q-axis 6-1.2 d-axis 0.2-0.3 0.2-0.45 Subtransient reactance d-axis x 0.1-0.25 0.15-0.25 0.2-0.8 Time constants winding open-circuited Tito 4.5-13 3.0-8.0 Stator winding short-circuited Tt 0.03-0.1 a Reactances are per unit, i.e., normalized quantities. Time constants are in seconds. ource: M.A. Laughton and M.G. Say, eds, Electrical Engineer's Reference Book, Stoneham and [McPherson, 1981] are among the basic sources of reference in electric machinery, where ctical aspects are given. An introductory discussion of power system stability as related to synchronous generators can be found in [Bergen, 1986]. A number of handbooks that include subjects on ac as well as dc generators are also available in [Laughton and Say, 1985; Fink and Beaty, 1987; and Chang, 1982 Superconducting Generators The demand for electricity has increased steadily over the years. To satisfy the increasing demand, there has een a trend in the development of generators with very high power rating. This has been achieved, to a great extent, by improvement in materials and cooling techniques. Cooling is necessary because the loss dissipated as heat poses a serious problem for winding insulation. The progress in machine design based on conventional methods appears to reach a point where further increases in power ratings are becoming difficult. An alternative method involves the use of superconductivity. In a superconducting generator, the field winding is kept at a very low temperature so that it stays super conductive. An obvious advantage to this is that no resistive loss can take place in this winding, and therefore a very large current can flow. A large field current yields a very strong magnetic field, and this means that many issues considered important in the conventional design may no longer be critical. For example, the conventional design makes use of iron core for armature windings to achieve an appropriate level of magnetic flux for these windings; iron cores, however, contribute to heat lossbecause of the effects of hysteresis and eddy cur rents--and therefore require appropriate designs for winding insulation. with the new design, there is no need for iron cores since the magnetic field can be made very strong; the absence of iron allows a simpler winding insulation, thereby accommodating additional armature windings. There is, however, a limit to the field current increase. It is known that superconductivity and diamagnetism are closely related; that is, if a material is in the superconducting state, no magnetic lines of force can enter its lterior. Increasing the current produces more and more magnetic lines of force, and this can continue until the dense magnetic field can penetrate the material. When this happens, the material fails to stay supercon ductive, and therefore resistive loss can take place. In other words, a material can stay superconductive until a certain critical field strength is reached. The critical field strength is dependent on the material and its e 2000 by CRC Press LLC
© 2000 by CRC Press LLC and [McPherson, 1981] are among the basic sources of reference in electric machinery, where many practical aspects are given. An introductory discussion of power system stability as related to synchronous generators can be found in [Bergen, 1986]. A number of handbooks that include subjects on ac as well as dc generators are also available in [Laughton and Say, 1985; Fink and Beaty, 1987; and Chang, 1982]. Superconducting Generators The demand for electricity has increased steadily over the years. To satisfy the increasing demand, there has been a trend in the development of generators with very high power rating. This has been achieved, to a great extent, by improvement in materials and cooling techniques. Cooling is necessary because the loss dissipated as heat poses a serious problem for winding insulation. The progress in machine design based on conventional methods appears to reach a point where further increases in power ratings are becoming difficult. An alternative method involves the use of superconductivity. In a superconducting generator, the field winding is kept at a very low temperature so that it stays superconductive. An obvious advantage to this is that no resistive loss can take place in this winding, and therefore a very large current can flow. A large field current yields a very strong magnetic field, and this means that many issues considered important in the conventional design may no longer be critical. For example, the conventional design makes use of iron core for armature windings to achieve an appropriate level of magnetic flux for these windings; iron cores, however, contribute to heat loss—because of the effects of hysteresis and eddy currents—and therefore require appropriate designs for winding insulation. With the new design, there is no need for iron cores since the magnetic field can be made very strong; the absence of iron allows a simpler winding insulation, thereby accommodating additional armature windings. There is, however, a limit to the field current increase. It is known that superconductivity and diamagnetism are closely related; that is, if a material is in the superconducting state, no magnetic lines of force can enter its interior. Increasing the current produces more and more magnetic lines of force, and this can continue until the dense magnetic field can penetrate the material. When this happens, the material fails to stay superconductive, and therefore resistive loss can take place. In other words, a material can stay superconductive until a certain critical field strength is reached. The critical field strength is dependent on the material and its temperature. TABLE 66.1 Typical Synchronous Generator Parametersa Parameter Symbol Round Rotor Salient-Pole Rotor with Damper Windings Synchronous reactance d-axis Xd 1.0–2.5 1.0–2.0 q-axis Xq 1.0–2.5 0.6–1.2 Transient reactance d-axis X¢ d 0.2–0.35 0.2–0.45 q-axis X¢ q 0.5–1.0 0.25–0.8 Subtransient reactance d-axis X² d 0.1–0.25 0.15–0.25 q-axis X² q 0.1–0.25 0.2–0.8 Time constants Transient Stator winding open-circuited T¢ do 4.5–13 3.0–8.0 Stator winding short-circuited T¢ d 1.0–1.5 1.5–2.0 Subtransient Stator winding short-circuited T² d 0.03–0.1 0.03–0.1 a Reactances are per unit, i.e., normalized quantities. Time constants are in seconds. Source: M.A. Laughton and M.G. Say, eds., Electrical Engineer’s Reference Book, Stoneham, Mass.: Butterworth, 1985
A typical superconducting design of an ac generator, as in the conventional design, has the field winding mounted on the rotor and armature winding on the stator. The main differences between the two designs lie in the way cooling is done. The rotor has an inner body which is to support a winding cooled to a very low temperature by means of liquid helium. The liquid helium is fed to the winding along the rotor axis. To maintain the low temperature, thermal insulation is needed, and this can be achieved by means of a vacuum space and a radiation shield. The outer body of the rotor shields the rotors winding from being penetrated by the armature ields so that the superconducting state will not be destroyed. The stator structure is made of nonmagnetic material, which must be mechanically strong. The stator windings(armature)are not superconducting and are typically cooled by water. The immediate surroundings of the machine must be shielded from the strong magnetic fields; this requirement, though not necessary for the machines operation, can be satisfied by the use of a copper or laminated iron screen From a circuit viewpoint, superconducting machines have smaller internal impedance relative to the con- ventional ones(refer to equivalent circuit shown in Fig. 66.4). Recall that the reactance jX, stems from the fact that the armature circuits give rise to a magnetic field that tends to counter the effect of the rotor w the conventional design, such a magnetic field is enhanced because iron core is used for the rotor and stator structures;thus jX, is large. In the superconducting design, the core is basically air; thus, jX, is smaller. The lifference is generally a ratio of 5: 1 in magnitude. An implication is that, at the same level of output current , and terminal voltage V, it requires of the superconducting generator a smaller induced emf EF or, equivalently, a smaller field current It is expected that the use of superconductivity adds another 0.4% to the efficiency of generators. This improvement might seem insignificant(compared to an already achieved figure of 98% by the conventional design) but proves considerable in the long run. It is estimated that given a frame size and weight, a supercon ducting generator's capacity is three times that of a conventional one. However, the new concept has to deal with such practical issues as reliability, availability, and costs before it can be put into large-scale operation [Bumby, 1983] provides more details on superconducting electric machines with issues such as design, performance, and application of such machines. Induction Generators Conceptually, a three-phase induction machine is similar to a synchronous machine, but the former has a much rotor circuit. a typical design of the rotor is the squirrel-cage structure, where conducting bars are led in the rotor body and shorted out at the ends. When a set of three-phase currents(waveforms of mplitude, displaced in time by one-third of a period) is applied to the stator winding, a rotating magnetic field is produced. ( See the discussion of a revolving magnetic field for synchronous generators in the section Principle of Operation". Currents are therefore induced in the bars, and their resulting magnetic field interacts with the stator field to make the rotor rotate in the same direction In this case. the machine acts as a motor since, in order for the rotor to rotate, energy is drawn from the electric power source. When the machine acts as a motor, its rotor can never achieve the same speed as the rotating field (this is the synchronous speed)for that would imply no induced currents in the rotor bars. If an external mechanical torque is applied to the rotor to drive it beyond the synchronous speed, however, then electric energy is pumped to the power grid, and the machine will act as a generato An advantage of induction generators is their simplicity(no separate field circuit) and flexibility in speed. These features make induction machines attractive for applications such as windmills a disadvantage of induction generators is that they are highly inductive. Because the current and voltage have very large phase shifts, delivering a moderate amount of power requires an unnecessarily high current on ne power line. This current can be reduced by connecting capacitors at the terminals of the machine. Capacitors have negative reactance; thus, the machine's inductive reactance can be compensated. Such a scheme is known as capacitive compensation. It is ideal to have a compensation in which the capacitor and equivalent inductor completely cancel the effect of each other. In windmill applications, for example, this faces a great challenge because the varying speed of the rotor(as a result of wind speed) implies a varying equivalent inductor Fortunately, strategies for ideal compensation have been designed and put to commercial use. e 2000 by CRC Press LLC
© 2000 by CRC Press LLC A typical superconducting design of an ac generator, as in the conventional design, has the field winding mounted on the rotor and armature winding on the stator. The main differences between the two designs lie in the way cooling is done. The rotor has an inner body which is to support a winding cooled to a very low temperature by means of liquid helium. The liquid helium is fed to the winding along the rotor axis. To maintain the low temperature, thermal insulation is needed, and this can be achieved by means of a vacuum space and a radiation shield. The outer body of the rotor shields the rotor’s winding from being penetrated by the armature fields so that the superconducting state will not be destroyed. The stator structure is made of nonmagnetic material, which must be mechanically strong. The stator windings (armature) are not superconducting and are typically cooled by water. The immediate surroundings of the machine must be shielded from the strong magnetic fields; this requirement, though not necessary for the machine’s operation, can be satisfied by the use of a copper or laminated iron screen. From a circuit viewpoint, superconducting machines have smaller internal impedance relative to the conventional ones (refer to equivalent circuit shown in Fig. 66.4). Recall that the reactance jXs stems from the fact that the armature circuits give rise to a magnetic field that tends to counter the effect of the rotor winding. In the conventional design, such a magnetic field is enhanced because iron core is used for the rotor and stator structures; thus jXs is large. In the superconducting design, the core is basically air; thus, jXs is smaller. The difference is generally a ratio of 5:1 in magnitude. An implication is that, at the same level of output current Ia and terminal voltage Vt, it requires of the superconducting generator a smaller induced emf EF or, equivalently, a smaller field current. It is expected that the use of superconductivity adds another 0.4% to the efficiency of generators. This improvement might seem insignificant (compared to an already achieved figure of 98% by the conventional design) but proves considerable in the long run. It is estimated that given a frame size and weight, a superconducting generator’s capacity is three times that of a conventional one. However, the new concept has to deal with such practical issues as reliability, availability, and costs before it can be put into large-scale operation. [Bumby, 1983] provides more details on superconducting electric machines with issues such as design, performance, and application of such machines. Induction Generators Conceptually, a three-phase induction machine is similar to a synchronous machine, but the former has a much simpler rotor circuit. A typical design of the rotor is the squirrel-cage structure, where conducting bars are embedded in the rotor body and shorted out at the ends. When a set of three-phase currents (waveforms of equal amplitude, displaced in time by one-third of a period) is applied to the stator winding, a rotating magnetic field is produced. (See the discussion of a revolving magnetic field for synchronous generators in the section “Principle of Operation”.) Currents are therefore induced in the bars, and their resulting magnetic field interacts with the stator field to make the rotor rotate in the same direction. In this case, the machine acts as a motor since, in order for the rotor to rotate, energy is drawn from the electric power source. When the machine acts as a motor, its rotor can never achieve the same speed as the rotating field (this is the synchronous speed) for that would imply no induced currents in the rotor bars. If an external mechanical torque is applied to the rotor to drive it beyond the synchronous speed, however, then electric energy is pumped to the power grid, and the machine will act as a generator. An advantage of induction generators is their simplicity (no separate field circuit) and flexibility in speed. These features make induction machines attractive for applications such as windmills. A disadvantage of induction generators is that they are highly inductive. Because the current and voltage have very large phase shifts, delivering a moderate amount of power requires an unnecessarily high current on the power line. This current can be reduced by connecting capacitors at the terminals of the machine. Capacitors have negative reactance; thus, the machine’s inductive reactance can be compensated. Such a scheme is known as capacitive compensation. It is ideal to have a compensation in which the capacitor and equivalent inductor completely cancel the effect of each other. In windmill applications, for example, this faces a great challenge because the varying speed of the rotor (as a result of wind speed) implies a varying equivalent inductor. Fortunately, strategies for ideal compensation have been designed and put to commercial use
In [Chapman, 1991], an analysis of induction generators and the effect of capacitive compensation on machine's performance are given. DC Generators To obtain dc electricity, one may prefer an available ac source with an electronic rectifier circuit. Another possibility is to generate dc electricity directly. Although the latter method is becoming obsolete, it is still important to understand how a dc generator works. This section provides a brief discussion of the basic issues associated with dc generators Principle of Operation As in the case of ac generators, a basic design will be used to explain the essential ideas behind the operation of dc generators. Figure 66.7 is a schematic diagram showing an end of a simple dc machine The stator of the simple machine is a permanent magnet with two poles labeled N and S. The rotor is a lindrical body and has two (insulated) conductors embedded in its surface. At one end of the rotor, as illustrated in Fig. 66.7, the two conductors are connected to a pair of copper segments; these semicircular gments, shown in the diagram, are mounted on the shaft of the rotor. Hence, they rotate together with the rotor. At the other end of the rotor, the two conductors are joined to form a coil. Assume that an external torque is applied to the shaft so that the rotor rotates at a certain speed. The rotor ding formed by the two conductors experiences a periodically varying magnetic field, and hence an emf is nduced across the winding. Note that this voltage periodically alternates in sign, and thus, the situation ron pte ly the sam se as t herne neouitertid in ec de he tors. cema r hts machin a t ad a dosibpeew te whe of copper segments and brushes. According to Fig 66.7, each copper segment comes into contact with one brush half of the time during each rotor revolution. The 事N placement of the(stationary) brushes guarantees that one brush always has positive potential relative to the other. For the chose direction of rotation, the brush with higher potential is the one directly beneath the N-pole. ( Should the rotor rotate in the reverse direction, the opposite is true. )Thus, the brushes can serve as the terminals of the dc source In electric machinery, the rectifying action of the copper segments and brushes is referred to as commutation, and the machine is called a commutating machine A qualitative sketch of V, the voltage across terminals of FIGURE 66.7 A basic unloaded simple dc generator, as a function of time is given in erator. V, is the voltage across the Fig. 66.8. Note that this voltage is not a constant. A unidirectional terminals.⑧#and⊙# indicate th current can flow when a resistor is connected across the terminals of that would flow if a closed circuit is mad the machine The pulsating voltage waveform generated by the simple dc machine usually cannot meet the requirement of practical applica- tions. An improvement can be made with more pairs of conductors. These conductors are placed in slots that are made equidistant on the ptor surface. Each pair of conductors can generate a voltage wave these waveforms due to the spatial displacement among the cond g 0 form similar to the one in Fig. 66.8, but there are time shifts amo tor pairs. For instance, when an individual voltage is minimum FIGURE 66.8 Open-circuited terminal (zero),other voltages are not. If these voltage waveforms are added, voltage of the simple dc gene the result is a near constant voltage waveform. This improvement of the dc waveform requires many pairs of the copper segments and a pair of brushes e 2000 by CRC Press LLC
© 2000 by CRC Press LLC In [Chapman, 1991], an analysis of induction generators and the effect of capacitive compensation on machine’s performance are given. DC Generators To obtain dc electricity, one may prefer an available ac source with an electronic rectifier circuit. Another possibility is to generate dc electricity directly. Although the latter method is becoming obsolete, it is still important to understand how a dc generator works. This section provides a brief discussion of the basic issues associated with dc generators. Principle of Operation As in the case of ac generators, a basic design will be used to explain the essential ideas behind the operation of dc generators. Figure 66.7 is a schematic diagram showing an end of a simple dc machine. The stator of the simple machine is a permanent magnet with two poles labeled N and S. The rotor is a cylindrical body and has two (insulated) conductors embedded in its surface. At one end of the rotor, as illustrated in Fig. 66.7, the two conductors are connected to a pair of copper segments; these semicircular segments, shown in the diagram, are mounted on the shaft of the rotor. Hence, they rotate together with the rotor. At the other end of the rotor, the two conductors are joined to form a coil. Assume that an external torque is applied to the shaft so that the rotor rotates at a certain speed. The rotor winding formed by the two conductors experiences a periodically varying magnetic field, and hence an emf is induced across the winding. Note that this voltage periodically alternates in sign, and thus, the situation is conceptually the same as the one encountered in ac generators. To make the machine act as a dc source, viewed from the terminals, some form of rectification needs be introduced. This function is made possible with the use of copper segments and brushes. According to Fig. 66.7, each copper segment comes into contact with one brush half of the time during each rotor revolution. The placement of the (stationary) brushes guarantees that one brush always has positive potential relative to the other. For the chosen direction of rotation, the brush with higher potential is the one directly beneath the N-pole. (Should the rotor rotate in the reverse direction, the opposite is true.) Thus, the brushes can serve as the terminals of the dc source. In electric machinery, the rectifying action of the copper segments and brushes is referred to as commutation, and the machine is called a commutating machine. A qualitative sketch of Vt , the voltage across terminals of an unloaded simple dc generator, as a function of time is given in Fig. 66.8. Note that this voltage is not a constant. A unidirectional current can flow when a resistor is connected across the terminals of the machine. The pulsating voltage waveform generated by the simple dc machine usually cannot meet the requirement of practical applications. An improvement can be made with more pairs of conductors. These conductors are placed in slots that are made equidistant on the rotor surface. Each pair of conductors can generate a voltage waveform similar to the one in Fig. 66.8, but there are time shifts among these waveforms due to the spatial displacement among the conductor pairs. For instance, when an individual voltage is minimum (zero), other voltages are not. If these voltage waveforms are added, the result is a near constant voltage waveform. This improvement of the dc waveform requires many pairs of the copper segments and a pair of brushes. FIGURE 66.7 A basic two-pole dc generator. Vt is the voltage across the machine terminals. ^# and (# indicate the direction of currents (into or out of the page) that would flow if a closed circuit is made. FIGURE 66.8 Open-circuited terminal voltage of the simple dc generator
