电子工程师手册（英文版四）：ch066

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 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 ? 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 period- ically varying magnetic field. Thus, emf’s are induced across these wind- ings 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 gener- ating 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 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 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 a 1 a 1 ′ and a 2 a 2 ′. 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 sim- ple two-pole synchronous machine. The rotor body is salient. Current in rotor winding: H20002 into the page, H17018 out of the page. ? 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) struc- tures 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 l 1 = L(q)I F where q is to denote the angular position of the N–S axis (of the rotor field) relative to the surface of aa￠, I F 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, i a , i b , and i c . This flux can be expressed as l 2 = L s i a + L m i b + L m i c , where L s (L m ) is the self (mutual) inductance. Because the rotor is round, L s and L m are not dependent on q, the relative position of the rotor and the winding. Typically, the sum of the stator currents i a + i b + i c is near zero; thus, one can write l 2 = (L s – L m )i a . The total flux seen by winding aa￠ is l = l 1 – l 2 = L(q)I F – (L s – L m )i a , where the minus sign in l 1 – l 2 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 synchro- nous 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. ? 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 w 0 is the angular speed (rad/s) of the rotor in a steady state. This equation can be modeled as a voltage source –E F behind a reactance jX s , as shown in Fig. 66.4; this reactance is usually referred to as synchronous reactance. The resistor R a in the diagram represents the winding resistance, and V t is the voltage measured across the winding. As mentioned, the theory for salient-rotor machines is more com- plicated. In the equation l 2 = L s i a + L m i b + L m i c , the terms L s and L m are now dependent on the (relative) position of the rotor. For example (refer to Fig. 66.1), L s 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 B 2 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 B d , which directly opposes the rotor field, is said to belong to the direct axis; the other component, B q , 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 I a , 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 salient- pole 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 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 their synchronous-speed operation. Intuitively, the impact is more severe for machines closer to the disturbance. When a system is per- turbed, 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 ILL di dt eLL di dt aFsm a Fsm a == = l –(– ) –(– ) D EELLjIEjXI aFsmaFsa =- = (–) –w 0 D FIGURE 66.4Per-phase equivalent cir- cuit of round-rotor synchronous machines. –E F is the internal voltage (phasor form) and V t is the terminal volt- FIGURE 66.5 In the salient-pole the- ory, the stator field (represented by a single vector B 2 ) is decomposed into B d and B q . Note that *B d * > *B q *. ? 2000 by CRC Press LLC 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, 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 d 2 q/dt 2 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 P in ~ driving torque, and P out ~ restoring torque). The term P in , which drives the rotor shaft, can be considered constant in many cases. The term P out , the power sent out to the system, may behave in a very complicated way. Qualitatively, P out 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., e a for phase a, can be obtained by combining the direct-axis and quadrature-axis components e d and e q , 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)l d on the first line, (d/dt)l q 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), X d (s), X q (s) given in an operational form (s denotes the derivative operator). Figure 66.6 gives a general view of the input–output state descrip- tion of machine’s dynamic model, the state variables of which appear in Eqs. (66.1) and (66.2). M d dt D d dt PP 2 2 qq += in out – e d dt d dt ri e d dt d dt ri ddqd qq q =+ =+ ì í ? ? ? ? ? llq llq – – l l dFdd qqq Gsi Xsi Xsi = = ì í ? ? ? ()– () –() FIGURE 66.6A block diagram depicting a qualitative relationship among various electrical and mechanical quantities of a synchronous machine. e a , e b , e c are phase voltages; i a , i b , i c phase currents; i F rotor field current; q relative position of rotor; w deviation of rotor speed from synchro- nous speed; P in mechanical power input. The state variables appear in Eqs. (66.1) and (66.2). ? 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 P in . 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, P in 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 V t drops. The excitation system boosts the internal voltage E F ; this action can increase the voltage V t 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 (armortis- seur) 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] ? 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 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 loss—because 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 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 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 temperature. TABLE 66.1 Typical Synchronous Generator Parameters a Parameter Symbol Round Rotor Salient-Pole Rotor with Damper Windings Synchronous reactance d-axis X d 1.0–2.5 1.0–2.0 q-axis X q 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. ? 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 con- ventional ones (refer to equivalent circuit shown in Fig. 66.4). Recall that the reactance jX s 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 jX s is large. In the superconducting design, the core is basically air; thus, jX s is smaller. The difference is generally a ratio of 5:1 in magnitude. An implication is that, at the same level of output current I a and terminal voltage V t , it requires of the superconducting generator a smaller induced emf E F 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 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. ? 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 V t , 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 applica- tions. 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 wave- form similar to the one in Fig. 66.8, but there are time shifts among these waveforms due to the spatial displacement among the conduc- tor 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 66.7A basic two-pole dc gen- erator. V t is the voltage across the machine terminals. ^# and (# indicate the direc- tion of currents (into or out of the page) that would flow if a closed circuit is made. FIGURE 66.8Open-circuited terminal voltage of the simple dc generator. ? 2000 by CRC Press LLC When the generator is connected to an electrical load, load currents flow through the rotor conductors. Therefore, a magnetic field is set up in addition to that of the permanent magnet. This additional field generally weakens the magnetic flux seen by the rotor conductors. A direct consequence is that the induced emf’s are less than those in an unloaded machine. Similar to the case of ac generators, this phenomenon is referred to as armature reaction, or flux-weakening effect. The use of brushes in the design of dc generators can cause a serious problem in practice. Each time a brush comes into contact with two adjacent copper segments, the corresponding conductors are short-circuited. For a loaded generator, such an event occurs when the currents in these conductors are not zero, resulting in flashover at the brushes. This means that the life span of the brushes can be drastically reduced and that frequent maintenance is needed. A number of design techniques have been developed to mitigate this problem. Mathematical/Circuit Model The (no-load) terminal voltage V t of a dc generator depends on several factors. First, it depends on the construction of the machine (e.g., the number of conductors). Second, the voltage magnitude depends on the magnetic field of the stator: the stronger the field is, the higher the voltage becomes. Third, since the induced emf is proportional to the rate of change of the magnetic flux (Faraday’s law), the terminals have higher voltage with a higher machine speed. One can write V t(no load) = Kln where K is a constant representing the first factor, l is magnetic flux, and n is rotor speed. The foregoing equation provides some insights into the voltage control of dc generators. Among the three terms, it is impractical to modify K, which is determined by the machine design. Changing n over a wide range may not be feasible since this is limited by what drives the rotor. Changing the magnetic flux l can be done if the permanent magnet is replaced by an electromagnet, and this is how the voltage control is done in practice. The control of l is made possible by adjusting the current fed to this electromagnet. Figure 66.9 shows the modified design of the simple dc generator. The stator winding is called the field winding, which produces excitation for the machine. The current in the field winding is adjusted by means of a variable resistor connected in series with this winding. It is also possible to use two field windings in order to have more flexibility in control. The use of field winding(s) on the stator of the dc machine leads to a number of methods to produce the magnetic field. Depending on how the field winding(s) and the rotor winding are connected, one may have FIGURE 66.9A simple two-pole dc generator with a stator winding to produce a magnetic field. Top, main components of the machine; bottom, coupled-circuit representation; the circuit on the left represents the field winding; the induced emf E is controlled by i F . ? 2000 by CRC Press LLC shunt excitation, series excitation, etc. Each connection yields a different terminal characteristic. The possible connections and the resulting current–voltage characteristics are given in Table 66.2. [Chapman, 1991] and [Fink and Beaty, 1987] provide more detailed discussions of dc generators. Specifically, [Chapman, 1991] shows how the characteristics are derived for various excitation methods. TABLE 66.2Excitation Methods and Voltage Current Characteristics for DC Generators Excitation Methods Characteristics Separate For low currents, the curve is nearly a straight line. As load current increases, the armature reaction becomes more severe and contributes to the nonlinear drop. Series At no load, there is no field current, and voltage is due to the residual flux of the stator core. The voltage rises rapidly over the range of low currents, but the resistive drop soon becomes dominant. Shunt Voltage buildup depends on the residual flux. The shunt field resistance must be less than a critical value. Compounded There are two field windings. Depending on how they are set up, one may have cumulative if the two fields are additive, differential if the two fields are subtractive. Cumulative: An increase in load current increases the resistive drop, yet creates more flux. At high currents, however, resistive drop becomes dominant. Differential: An increase in load current not only increases the resistive drop, but also reduces the net flux. Voltage drops drastically. i F I L E V t + + – – I L V t Separate i ? I L E V t + + – – I L V t Shunt i a I L i F E V t + + – – I L V t Shunt i a I L i F E V t + + – – I L V t Compounded cumulative dif fer ential ? 2000 by CRC Press LLC FRANK JULIAN SPRAGUE (1857–1934) rank Sprague was a true entrepreneur in the new field of electrical technology. After a brief stint on Thomas Edison’s staff, Sprague went out on his own, founding Sprague Electric Railway and Motor Company in 1884. In 1887, Sprague equipped the first modern trolley railway in the United States. Sprague’s successful construction of a streetcar system for Richmond, Virginia, in 1888 was the beginning of the great electric railway boom. Sprague followed this system with 100 other such systems, both in America and Europe, during the next two years. In less than 15 years, more than F ? 2000 by CRC Press LLC Defining Terms Armature circuit: A winding where the load current is carried. Armature reaction: The phenomenon in which the magnetic field due to currents in the armature circuit counters the effect of the field circuit. Commutation: A mechanical technique in which rectification can be achieved in dc machines. Field circuit: A set of windings that produces a magnetic field so that the electromagnetic induction can take place in electric machines. Revolving fields: A magnetic field created by multiphase currents on spatially displaced windings in rotating machines; the field revolves in the air gap. Swing equation: A nonlinear differential equation describing the rotor dynamics of an ac synchronous machine. Synchronous speed: A characteristic speed of synchronous and induction machines with a revolving field; it is determined by the rotor structure and the line frequency. Related Topics 2.2 Ideal and Practical Sources ? 3.4 Power and Energy ? 104.1 Welding and Bonding 20,000 miles (32,000 km) of electric street rail- way were built. In addition to his work in railroads, Sprague’s diverse talents led to his development of electric elevators, an ac induction smelting furnace, min- iature electric power units for use in small appli- ances, and as a member of the U.S. Naval Consulting Board during World War I, he developed fuses and air and depth bombs. Sprague was awarded the AIEE’s Edison Medal in 1910. (Courtesy of the IEEE Center for the History of Electrical Engineering.) References M. S. Sarma, Synchronous Machines (Their Theory, Stability, and Excitation Systems), New York: Gordon and Breach, 1979. J. R. Bumby, Superconducting Rotating Electrical Machines, New York: Oxford University Press, 1983. S. J. Chapman, Electric Machinery Fundamentals, New York: McGraw-Hill, 1991. G. McPherson, An Introduction to Electrical Machines and Transformers, New York: Wiley, 1981. A. R. Bergen, Power Systems Analysis, Englewood Cliffs, N.J.: Prentice-Hall, 1986. M. A. Laughton and M. G. Say, Eds., Electrical Engineer’s Reference Book, Stoneham, Mass.: Butterworth, 1985. D. G. Fink and H. W. Beaty, Eds., Standard Handbook for Electrical Engineers, New York: McGraw-Hill, 1987. S. S. L. Chang, ed., Fundamentals Handbook of Electrical and Computer Engineering, New York: Wiley, 1982. Further Information Several handbooks, e.g., Electrical Engineer’s Reference Book and Standard Handbook for Electrical Engineers, give more details on the machine design. [Bumby, 1983] covers the subject of superconducting generators. Some textbooks in the area of rotating machines are listed as [Sarma, 1979; Chapman, 1991; McPherson, 1981]. The quarterly journal IEEE Transactions on Energy Conversion covers the field of rotating machinery and power generation. Another IEEE quarterly journal, IEEE Transactions on Power Systems, is devoted to the general aspects of power system engineering and power engineering education. The bimonthly journal Electric Machines and Power Systems, published by Hemisphere Publishing Corpora- tion, covers the broad field of electromechanics, electric machines, and power systems. 66.2 Motors Donald Galler Electric motors are the most commonly used prime mover in industry. The classification of the types of ac and dc motors commonly used in industrial applications is shown in Fig. 66.10. Motor Applications DC Motors Permanent magnet (PM) field motors occupy the low end of the horsepower (hp) range and are commercially available up to about 10 hp. Below 1 hp they are used for servo applications, such as in machine tools, for robotics, and in high-performance computer peripherals. Wound field motors are used above about 10 hp and represent the highest horsepower range of dc motor application. They are commercially available up to several hundred horsepower and are commonly used in traction, hoisting, and other applications where a wide range of speed control is needed. The shunt wound dc motor is commonly found in industrial applications such as grinding and machine tools and in elevator and hoist applications. Compound wound motors have both a series and shunt field component to provide specific torque-speed characteristics. Propulsion motors for transit vehicles are usually compound wound dc motors. AC Motors Single-phase ac motors occupy the low end of the horsepower spectrum and are offered commercially up to about 5 hp. Single-phase synchronous motors are only used below about 1/10 of a horsepower. Typical applications are timing and motion control, where low torque is required at fixed speeds. Single-phase induction motors are used for operating household appliances and machinery from about 1/3 to 5 hp. Polyphase ac motors are primarily three-phase and are by far the largest electric prime mover in all of industry. They are offered in ranges from 5 up to 50,000 hp and account for a large percentage of the total motor industry in the world. In number of units, the three-phase squirrel cage induction motor is the most common. It is commercially available from 1 hp up to several thousand horsepower and can be used on ? 2000 by CRC Press LLC conventional ac power or in conjunction with adjustable speed ac drives. Fans, pumps, and material handling are the most common applications. When the torque-speed characteristics of a conventional ac induction motor need to be modified, the wound rotor induction motor is used. These motors replace the squirrel cage rotor with a wound rotor and slip rings. External resistors are used to adjust the torque-speed characteristics for speed control in such applications as ac cranes, hoists, and elevators. Three-phase synchronous motors can be purchased with PM fields up to about 5 hp and are used for applications such as processing lines and transporting film and sheet materials at precise speeds. In the horsepower range above about 10,000 hp, three-phase synchronous motors with wound fields are used rather than large squirrel cage induction motors. Starting current and other characteristics can be con- trolled by the external field exciter. Three-phase synchronous motors with wound fields are available up to about 50,000 hp. Motor Analysis DC Motor Analysis The separately excited dc motor is the simplest of all dc motors and is the one most commonly found in industrial applications. The equivalent circuit is shown in Fig. 66.11. An adjustable dc voltage V is applied to the motor terminals. This voltage is impressed across the series combination of the armature resis- tance R a and the back emf V a generated by the armature. The field is energized with a separate dc power supply, usually at 300 or 500 V dc. The terminal voltage is given as V = I a R a + V a (66.3) The torque in steady state is T = K t I a F (66.4) FIGURE 66.10Classification of ac and dc motors for industrial applications. FIGURE 66.11Equivalent circuit of separately excited dc motor. ? 2000 by CRC Press LLC and the generated armature voltage is V = K a wF (66.5) where F is the magnitude of the flux produced by the field winding and is proportional to the field current I f . The torque constant K t and the armature constant K a are numerically equal in a consistent set of units. w is the shaft speed in radians/second. Solving the three equations gives the steady-state speed as (66.6) The input power and output power are P in = I a V (66.7) P out = wT = I a V – I a 2 R a (66.8) The efficiency (neglecting power loss in the field) is (66.9) A simplified torque-speed curve is shown in Fig. 66.12. The torque capability is constant up to the base speed of the motor while the armature and field currents are held constant. The speed is controlled by armature voltage in this range. Operation above base speed is accomplished by reducing the field current. This is called field weakening. The motor operates at constant power in this range, and the torque reduces with increasing speed. FIGURE 66.12Torque-speed capability for the separately excited dc motor. w= VTRK K at a –(/)F F h w = = P P T IV a out in ? 2000 by CRC Press LLC Synchronous Motor Analysis Synchronous motor analysis may be conducted using either a round rotor or salient pole model for the motor. The round rotor model is used in the following discussion. The equivalent circuit is shown in Fig. 66.13. The model consists of two ac voltages V 1 and V 2 connected by an impedance Z = R + jX. Analysis is facilitated by use of the phasor diagram shown in Fig. 66.14. The power delivered through the impedance to the load is P 2 = V 2 I cos f 2 (66.10) where f 2 is the phase angle of I with respect to V 2 . The phasor current (66.11) is expressed in polar form as (66.12) The equations make use of the fact that the three-phase operation is symmetrical and uses a “per-phase” equivalent circuit. This will also be true for the induction motor, which is analyzed in the following section. The real part of I is (66.13) FIGURE 66.14Phasor diagram for the ac synchronous motor (round rotor model). FIGURE 66.13 Per-phase equiva- lent circuit model for the syncho- nous motor (round rotor model). I VV Z = 12 – I VV Z V Z V Z z = DD ° D =- D D 12 12 0 2 d f df f – –– I V Z V Z zz cos cos(– cos 2 fdff= 12 )– (– ) ? 2000 by CRC Press LLC Using Eq. (66.13) in Eq. (66.10) gives (66.14) Letting a = 90° – f z = arctan R/X gives the output power as (66.15) and the input power as (66.16) Usually R is neglected and (66.17) which shows that the power is maximum when d = 90° and is (66.18) The current can be found from Eqs. (66.15) and (66.16) since the only loss occurs in R. Setting I 2 R = P 2 – P 1 (66.19) and solving for I gives (66.20) which is the input line current. The power factor is (66.21) and q = d + f 2 as shown in Fig. 66.14. All the foregoing values are per-phase values. The total input power is P in = 3P 1 (66.22) The mechanical output power is P out = Tw (66.23) = 3 · P 2 P VV Z VR Z z2 cos( – = 12 2 2 2 df)– P VV Z VR Z 2 sin( + = 12 2 2 2 da)– P VV Z VR Z 1 sin( – =+ 12 1 2 2 da) PP VV X 1 sin == 2 12 d P VV X MAX = 12 IPPR=-()/ 21 cos q= P VI 1 1 ? 2000 by CRC Press LLC and the torque is T = 3 · P out /w (66.24) where w is the rotational speed of the motor expressed in radians per second. Synchronous motor operation is determined by the torque angle d and is illustrated in Fig. 66.15 for a typical motor. Input power, output power, and current are shown on a per-unit basis. Torque is not shown but is related to output power only by a constant. Induction Motor Analysis The characteristic algebraic equations for the steady-state power, torque, and efficiency of the ac induction motor are derived from the per-phase equivalent circuit of Fig. 66.16. All voltages and currents are in sinusoidal steady state. The derivation of the equations can be simplified by defining the complex motor impedance as (66.25) By defining the following constants as M 1 = R 1 R 2 2 M 2 = R 2 L m 2 M 3 = L 2 + L m (66.26) M 4 = L 1 + L m M 5 = R 1 M 3 2 + M 2 the terms of Eq. (66.25) become FIGURE 66.15Synchronous motor performance. Zj m =+ a z b z ? 2000 by CRC Press LLC z = R 2 2 + w s 2 M 3 2 (66.27) a = zR 1 + (w m + w s )w s M 2 (66.28) b = (w m + w s ) [zL 1 + L M R 2 2 +w s 2 M 3 L 2 L m ] (66.29) The angular velocity w s is the slip frequency and is defined as follows: w s = w f – w m (66.30) where w f is the frequency applied to the stator and w m = w/N p (66.31) is the rotor angular velocity in terms of an equivalent stator frequency. N p is the number of stator pole pairs. The average mechanical output power of the motor is the power in the resistance R 2 w m /w s and is given as (66.32) where V is the rms line-neutral voltage. Since (66.33) the torque becomes (66.34) The motor efficiency is defined as (66.35) FIGURE 66.16Equivalent circuit of ac induction motor. P VM ms out 2 3 = + zww ab 2 22 T P PN p m = = out out w w T VNM ps = + 3 2 2 2 zw ab 2 h= P P out in ? 2000 by CRC Press LLC where the input power is (66.36) Using Eqs. (66.32) and (66.36), the efficiency becomes (66.37) Typical performance characteristics of the induction motor are shown in Fig. 66.17. Classical analysis represents all the motor expressions in terms of the slip, s, which is defined as (66.38) where w m is the equivalent mechanical frequency of the rotor, w m = w/N p , and w f is the angular velocity of the stator field in radians/second. In this format, the output power is (66.39) FIGURE 66.17Induction motor operating characteristics, fixed voltage, and frequency. P V Z V Z f m f m in cos = = 3 3 2 2 2 q a z ** ** h ww a = ms M 2 s fm f = ww w – PIR s s =× - ( ) 2 2 2 1 ? 2000 by CRC Press LLC ? 2000 by CRC Press LLC AUTOMATIC MOTOR SYNCHRONIZATION CONTROL William P. Lear Patented July 2, 1946 #2,403,098 ear described a system for synchronizing instrumentation throughout an aircraft using DC servo motors instead of mechanical linkages that loaded down the master instrument. Its greater appli- cation came in using it to control and maintain altitude and heading by synchronizing the aircraft’s control surfaces and using the servos to adjust them. This “autopilot” helped reduce pilot fatigue on long flights and was one of the developments that made commercial air practical during the 1950s and beyond. The servo control principles described are still used in automated air and sea navigation today. Lear is perhaps best known for his development of small corporate jet aircraft known as Learjets in the 1960s. He patented the first practical car radio in the 1930s that launched today’s giant Motorola Company. He also developed the eight-track tape system for autos in the 1960s and before his death in 1978, he designed the Lear fan, a high speed propeller aircraft made entirely from composites. (Copyright ? 1995, DewRay Products, Inc. Used with permission.) L The maximum torque, T m , occurs at a slip of (66.40) where X 1 and X 2 are the stator and rotor reactances corresponding to L 1 and L 2 . If R 1 and X are neglected, the torque can be expressed as (66.41) but this expression loses accuracy if s < 0.1 where most practical operation takes place. Another expression, may be used and is useful over the whole slip range. The full equation set of the previous discussion should be used where variable frequency and variable voltage operation is used, such as in adjustable speed drives. These equations are accurate for all regions of motor and generator operation. ACand DC Motor Terms General Terms w: Shaft angular velocity in radians/second P out : Electrical output power P in : Electrical input power h: Efficiency T: Shaft torque DC Motor Terms I a : Armature current I f : Field current V a : Back emf generated by armature V: Motor terminal voltage R a : Armature resistance K t : Torque constant K a : Armature constant F: Field flux AC Induction Motor Terms L 1 : Stator winding inductance R 1 : Stator winding resistance L 2 : Rotor winding inductance R 2 : Rotor winding resistance L M : Magnetizing inductance N p : Number of pole pairs in stator winding w f : Frequency of voltage applied to stator w m : Rotor equivalent mechanical frequency w s : Slip frequency, w s = w f – w m s: Slip s = (w f – w m )/w f T M : Maximum torque s M : Slip at maximum torque AC Synchronous Motor Terms V 1 : Terminal voltage V 2 : Back emf generated by rotor R: Rotor circuit resistance X: Rotor circuit reactance Z: Rotor circuit impedance Z = R + jX d: Torque angle (between V 1 and V 2 ) f 2 : Angle between I and V 2 f Z : Rotor circuit reactance angle f z = tan –1 X/R a:90° – f z q: Power factor angle q = d + f 2 Defining Terms DC motor: A dc motor consists of a stationary active part, usually called the field structure, and a moving active part, usually called the armature. Both the field and armature carry dc. Induction motor: An ac motor in which a primary winding on the stator is connected to the power source and polyphase secondary winding on the rotor carries induced current. S R RXX m = ++ ( ) 2 1 2 12 2 TT ss ss m m m =× + 2 22 TT ss s s sR R sR R m m m m =× ++ ( ) ×++ ( ) ? è ? ? ? è ? ? 2 1 11 2 12 2 12 2 ? 2000 by CRC Press LLC Permanent magnet dc motor: A dc motor in which the field flux is supplied by permanent magnets instead of a wound field. Rotor: The rotating member of a motor including the shaft. It is commonly called the armature on most dc motors. Separately excited dc motor: A dc motor in which the field current is derived from a circuit which is independent of the armature. Squirrel cage induction motor: An induction motor in which the secondary circuit (on the rotor) consists of bars, short-circuited by end rings. This forms a squirrel cage conductor structure which is disposed in slots in the rotor core. Stator: The portion of a motor that includes and supports the stationary active parts. The stator includes the stationary portions of the magnetic circuit and the associated windings and leads. Synchronous motor: An ac motor in which the average speed of normal operation is exactly proportional to the frequency to which it is connected. A synchronous motor generally has rotating field poles which are excited by dc. Wound rotor induction motor: An induction motor in which the secondary circuit consists of a polyphase winding or coils connected through a suitable circuit. When provided with slip rings, the term slip-ring induction motor is used. Related Topics 2.2 Ideal and Practical Sources ? 104.2 Large Drives References P. C. Sen, Thyristor DC Drives, New York: John Wiley, 1981. P. C. Sen, Principles of Electric Machines and Power Electronics, 2nd ed., New York: John Wiley, 1997. G. R. Slemon, Electric Machines and Drives, Reading, Mass.: Addison-Wesley, 1992. I. Boldea and S. A. Nasar, Vector Control of AC Drives, Boca Raton, Fla.: CRC Press, 1992. M. G. Say and E. O. Taylor, Direct Current Machines, 2nd ed., London: Pitman Publishing, 1986. R. H. Engelmann and W. H. Middendorf, Handbook of Electric Motors, New York: Marcel Dekker, 1995. D. W. Novotny and T. A. Lipo, Vector Control and Dynamics of AC Drives, Oxford: Clarendon Press, 1996. Further Information The theory of ac motor drive operation is covered in the collection of papers edited by Bimal K. Bose, Adjustable Speed AC Drive Systems (IEEE, 1981). A good general text is Electric Machinery, by Fitzgerald, Kingsley, and Umans. The analysis of synchronous machines is covered in the book Alternating Current Machines, by M.G. Say (Wiley, 1984). Three-Phase Electrical Machines — Computer Simulation by J. R. Smith (Wiley, 1993) covers computer modeling and simulation techniques. 66.3 Small Electric Motors Elias G. Strangas Introduction Small electrical machines carry a substantial load in residential environments, but also in industrial environ- ments, where they are mostly used to control processes. In order to adapt to the limitations of the power available, the cost requirements, and the widely varying operating requirements, small motors are available in a great variety of designs. Some of the small motors require electronics in order to start and operate, while others can start and run directly connected to the supply line. ? 2000 by CRC Press LLC AC motors that can start directly from the line are mostly of the induction type. Universal motors are also used extensively for small AC powered, handheld tools. They can either run directly from the line or have their speed adjusted through electronics. Stepping motors of many varying designs require electronics to operate. They are used primarily to position a tool or a component and are seldom used to provide steady rotating motion. Besides these motors, permanent magnet AC motors are replacing rapidly both DC and induction motors for accurate speed and position control, but also to decrease size and increase efficiency. They require power and control electronics to start and run. Single Phase Induction Motors To produce rotation, a multi-phase stator winding is often used in an AC motor, supplied from a symmetric and balanced system of currents. The magnetomotive force of these windings interacts with the magnetic field of the rotor (induced or applied) to produce a torque. In three-phase induction motors, the rotor field is created by currents that are induced due to the relative speed of the rotor and the synchronously rotating stator field. In an induction motor that is supplied by a single-phase stator current, it is not as clear how a rotating magnetomotive force can be created and a torque be produced. Two different concepts will be used to generate torque. The first, conceptually simpler design concept, involves the generation of a second current which flows in a second winding of the stator. This auxiliary winding is spatially displaced on the stator. This brings the motor design close to the multi-phase principle. The current in the auxiliary winding has to be out of phase with the current in the main winding, and this is accomplished through the use of increased resistance in it or a capacitor in series with it. A motor can operate in this fashion over its entire speed range. Once the motor is rotating, the second design concept allows that one of the phases, the auxiliary one, be disconnected. The current in the remaining main winding alone produces only a pulsating flux, which can be analyzed as the sum of two rotating fields of equal amplitude but opposite direction. These fields, as seen from the moving rotor, rotate at different speeds, hence inducing in it currents of different frequency and amplitude. If the speed of the rotor is w r , the applied frequency to the stator is f and the number of pole pairs in the motor is p, the frequencies of the currents induced in the rotor are pw r – f and pw r + f. These unequal currents in turn produce unequal torques in the two directions, with a nonzero net torque. The various designs of single-phase induction motors result from the variety of ways that the two phases are generated and by whether the auxiliary phase remains energized after starting. Shaded Pole Motors These motors are simple, reliable, and inefficient. The stator winding is not distributed on the rotor surface, but rather it is concentrated on salient poles. The auxiliary winding, which has to produce flux out of phase with the main winding, is nothing but a hardwired shorted turn around a portion of the main pole as Fig. 66.18. Because of the shorted turn, the flux out of the shaded part of the pole lags behind the flux out of the main pole. The motor always rotates from the main to the shaded pole, and it is not possible to change directions. Shaded pole motors are inefficient and have high starting and running current and low starting torque. They are used where reliability and cost are important, while their small size makes unimportant the overall effect of their disadvantages, e.g., small fans. Their size ranges from 0.002 to 0.1 hp. Resistance Split-Phase Motors These motors have an auxiliary winding which simply has higher resistance than the main winding and is displaced spatially on the stator by about 90°. Both windings are distributed on the stator surface and are connected to the line voltage, but the different time constants between them makes the FIGURE 66.18 A shaded pole motor with tapered poles and mag- netic wedges. (Source: C. G. Veinott and J. E. Martin, Fractional and Subfractional Horsepower Electric Motors, New York: McGraw-Hill, 1986. With permission.) ? 2000 by CRC Press LLC current in the auxiliary winding lead that of the main. This arrangement results in a nonzero, but relatively low starting torque and high starting current. The use of the auxiliary winding is limited only to starting—the motor runs more efficiently without it, as a single phase motor described earlier. A switch, activated by speed (centrifugal) or by stator temperature, disconnects the auxiliary winding shortly after starting. Figure 66.19 represents schematically the connections of this type of motor. These motors represent an improvement in efficiency and starting torque over shaded pole motors, at the expense of increased cost and lower reliability. They are built to larger sizes, but their application is limited by the high starting current. Capacitor Motors Another way to generate a phase angle of current in the auxiliary winding is to include a capacitor in series with it. The capacitor can be disconnected after starting in a capacitor start motor. Their operation is similar to that of the resistance split-phase motor, but they have better starting characteristics and are made as large as 5 hp. Figure 66.20 shows schematically the wiring diagram of the capacitor start motor. To optimize both starting and running, different values of the capacitor are used. One value of the capacitor is calculated to minimize starting current and maximize starting torque, while the other is designed to maximize efficiency at the operating point. A centrifugal switch handles the changeover. Such motors are built for up to 10 hp, and their cost is relatively high because of the switch and two capacitors. Figure 66.21 shows schematically the wiring diagram of the capacitor start and run motor. FIGURE 66.19Connections of a resistive, split-phase motor. FIGURE 66.20Conenctions of a capacitor start motor. ? 2000 by CRC Press LLC A permanent split capacitor motor uses the same capacitor throughout the speed range of the motor. Its value requires a compromise between the values of the two-capacitor motors. The result is a motor design optimized for a particular application, e.g., a compressor or a fan. Figure 66.22 shows schematically the wiring diagram of the permanent split capacitor motor. Universal Motors These motors can be supplied from either DC or AC. Their design is essentially similar to a DC motor with series windings. When operated as AC motors, supplied say by a 60 Hz source, the current in the armature and the field windings reverses 120 times per second. As the torque is roughly proportional to both armature and field currents, connecting these windings in series guarantees that the current reverses in both at the same time, retaining the unidirectional torque. Figure 66.23 shows a schematic diagram of the connections of universal motors. They can run at speeds up to 20,000 rpm, thus being very compact for a given horsepower. Their most popular applications include portable drills, food mixers, and fans. Universal motors supplied from AC lend themselves easily to variable speed applications. A potentiometer, placed across the line voltage, controls the firing of a TRIAC thus varying the effective value of the voltage at the motor. FIGURE 66.21Connections of a capacitor-start, capacitor-run motor. FIGURE 66.22Connections of a permanent split capacitor motor. ? 2000 by CRC Press LLC Permanent Magnet AC Motors When compared to induction motors, permanent magnet motors have higher steady state torque for the same size and better efficiency. They carry a polyphase winding in the stator, which can be either rectangular or sinusoidally distributed. The rotor has a steel core, with permanent magnets mounted on it or inset. These magnets can be made from a variety of materials, such as rare earth, ceramic, etc. Figure 66.24 shows a schematic of the cross-section of a motor with surface mounted magnets, and Fig. 66.25 shows a schematic of a motor with inset magnets. The stator windings are supplied by a DC source through power electronic switches that constitute an inverter. Which switches are to be conducting at any time is determined by a controller, which in turn uses as inputs a speed or torque command and a measurement or an estimate of the rotor position. Figure 66.26 shows a schematic of the motor cross-section and of the inverter. FIGURE 66.23Connections of a universal motor. FIGURE 66.24Surface mounted magnets on a Permanent Magnet AC motor. FIGURE 66.25Inset (interior) magnets on a permanent magnet AC motor. r a' c c' Magnet b b' a 2 q b -m g e a' c c' b b' a 2 b - q axis d axis ? 2000 by CRC Press LLC When the stator windings are rectangular and are energized based only on the rotor position, the resulting set of PM motor, inverter, and controller is called a brushless DC motor. The developed torque is proportional to the airgap flux, B g , and the stator current, I s . T = kB g I s Due to the rotor speed, w 0 a voltage, e, (back emf) is induced to the stator windings. e = kB g w 0 Stepping Motors These motors convert a series of power pulses to a corresponding series of equal angular movements. These pulses can be delivered at a variable rate, allowing the accurate positioning of the rotor without feedback. They can develop torque up to 15 Nm and can handle 1500 to 2500 pulses per second. They have zero steady state error in positioning and high torque density. An important characteristic of stepping motors is that when one phase is activated they do not develop a rotating but rather a holding torque, which makes them retain accurately their position, even under load. Stepping motors are conceptually derived either from a variable reluctance motor or from a permanent magnet synchronous motor. FIGURE 66.26 Permanent magnet AC motor and inverter. Rotor Position Signals Controlled direct-current source (a) a bc a' b'c' 1 i 35 a b c 462 u w 0 t ? 2000 by CRC Press LLC One design of stepping motors, based on the doubly salient switched reluctance motor, uses a large number of teeth in the rotor (typically 45) to create saliency, as shown in Fig. 66.27. In this design, when the rotor teeth are aligned in say Phase 1, they are misaligned in Phases 2 and 3. A pulse of current in Phase 2 will cause a rotation so that the alignment will occur at Phase 2. If, instead, a pulse to Phase 3 is given, the rotor will move the same distance in the opposite rotation. The angle corresponding to a pulse is small, typically 3° to 5°, resulting from alternatively exciting one stator phase at a time. A permanent magnet stepping motor uses permanent magnets in the rotor. Figure 66.28 shows the steps in the motion of a four-phase PM stepping motor. Hybrid stepping motors come in a variety of designs. One, shown in Fig. 66.29, consists of two rotors mounted on the same shaft, displaced by one half tooth. The permanent magnet is placed axially between the rotors, and the magnetic flux flows radially at the air gaps, closing through the stator circuit. Torque is created by the interaction of two magnetic fields, that due to the magnets and that due to the stator currents. This design allows a finer step angle control and higher torque, as well as smoother torque during a step. FIGURE 66.27Cross-sectional view of a four-phase variable reluctance motor. Number of rotor teeth 50, step number 200, step angle 1.8°. (Source: Oxford University Press, 1989. With permission.) FIGURE 66.28Steps in the operation of a permanent magnet stepping motor. (Source: T. Kenjo, Stepping Motors and Their Microprocessor Controls, Oxford University Press, 1989. With permission.) ? 2000 by CRC Press LLC Fundamental to the operation of stepping motors is the utilization of power electronic switches, and of a circuit providing the timing and duration of the pulses. A characteristic of a specific stepping motor is the maximum frequency it can operate at starting or running without load. As the frequency of the pulses to a running motor is increased, eventually the motor loses synchronism. The relation between the frictional load torque and maximum pulse frequency is called the pull-out characteristic. References G. R. Slemon, Electrical Machines and Drives, Addison-Wesley, 1992. T. Kenjo, Stepping Motors and Their Microprocessor Controls, Oxford University Press, 1984. R. H. Engelman and W. H. Middendorf, Eds., Handbook of Electric Motors, New York: Marcel Dekker, 1995. R. Miller and M. R. Miller, Fractional Horsepower Electric Motors, Bobs Merrill Co., 1984. G. G. Veinott and J. E. Martin, Fractional and Subfractional Horsepower Electric Motors, New York: McGraw- Hill, 1986. T. J. E. Miller, Brushless Permanent-Magnet and Reluctance Motor Drives, Oxford University Press, 1989. S. A. Nasar, I. Boldea, and L. E. Unnewehr, Permanent Magnet, Reluctance and Self-Synchronous Motors, Boca Raton, Fla.: CRC Press, 1993. Further Information There is an abundance of books and literature on small electrical motors. IEEE Transactions on Industry Appli- cations, Power Electronics, Power Delivery and Industrial Electronics all have articles on the subject. In addition, IEE and other publications and conference records can provide the reader with specific and useful information. Electrical Machines and Drives [Slemon, 1992] is one of the many excellent textbooks on the subject. Stepping Motors and their Microprocessor Controls [Kenjo, 1984] has a thorough discussion of stepping motors, while Fractional and Subfractional Horsepower Electric Motors [Veinott and Martin, 1986] covers small AC and DC motors. Brushless Permanent-Magnet and Reluctance Motor Drives [Miller, 1989] and Permanent Magnet, Reluctance and Self-Synchronous Motors [Nasar et al., 1993] reflect the increased interest in reluctance and brushless DC motors, and provide information on their theory of operation, design and control. Finally, Fractional Horsepower Electric Motors [Miller and Miller, 1984] gives a lot of practical information about the application of small motors. 66.4 Simulation of Electric Machinery Chee-Mun Ong Simulation has been an option when the physical system is too large or expensive to experiment with or simply not available. Today, with powerful simulation packages, simulation is becoming a popular option for conduct- ing studies and for learning, especially when well-established models are available. Modeling refers to the process FIGURE 66.29 Construction of a hybrid stepping motor. (Source: Oxford University Press, 1989. With permission.) ? 2000 by CRC Press LLC of analysis and synthesis to determine a suitable mathematical description that captures the relevant dynamical characteristics and simulation to the techniques of setting up and experimenting with the model. Models of three-phase synchronous and induction machines for studying electromechanical and low-fre- quency electrical transients are well established because of the importance of generator and load behavior in stability and fault studies. Electric machines, however, do interact with other connected components over a wide range of frequencies, from fractions of Hertz for electromechanical phenomena to millions of Hertz for electromagnetic phenomena. Reduced models suitable for limited frequency ranges are often preferred over complex models because of the relative ease in usage — as in determining the values of model parameter and in implementing a simulation. In practice, reduced models that portray essential behavior over a limited frequency range are obtained by making judicious approximations. Hence, one has to be aware of the assumptions and limitations when deciding on the level of modeling details of other components in the simulation and when interpreting the simulation results. Basics in Modeling Most machine models for electromechanical transient studies are derived from a lumped-parameter circuit representation of the machine’s windings. Such lumped-parameter circuit representations are adequate for low- frequency electromechanical phenomena. They are suited for dynamical studies, often times to determine the machine’s performance and control behavior or to learn about the nature of interactions from electromechanical oscillations. Studies of interactions occurring at higher frequencies, such as surge or traveling waves studies, may require a distributed-parameter circuit representation of the machine windings. A lumped-parameter model for dynamical studies typically will include the voltage equations of the windings, derived using a coupled circuit approach, and an expression for the developed electromagnetic torque. The latter is obtained from an expression of the developed electromagnetic power by considering the input power expression and allowing for losses and magnetic energy storage terms. The expression for the developed electromagnetic torque is obtained by dividing that for developed electromagnetic power by the rotor mechan- ical speed. The rotor speed, in turn, is determined by an equation of the rotor’s dynamics that equates the rotor’s inertia torque to its acceleration torque. For example, in a reduced order model of a separately excited dc machine that ignores the details of commutation action and only portrays the average values of voltage, current, and power, the armature winding can be represented as an equivalent winding whose axis is determined by the position of the commutator brushes. The induced voltage in the armature, E a , due to field flux can be expressed as k a ωφ, k a being a machine constant; ω, the rotor speed; and φ the flux per pole. When armature reaction is ignored, φ will be the flux produced by the field winding. (See Fig. 66.30). Using motoring convention, the voltage equations of the armature and field windings with axes that are at right angles to each other can be expressed as (66.42) where V a is the terminal voltage of the armature winding, R a its effective resistance including brush drops, L aq its inductance, v f the applied field voltage, R f and L f , the field circuit resistance and inductance. For motoring, positive I a will flow into the positive terminal of V a , as power flows from the external voltage source into the armature winding. Like the physical device, the model is capable of motoring and generating, and the transition from one mode to the other will take place naturally. FIGURE 66.30 dc machine. VERIL I dt vRiL i t aaaaaq a ffff f =+ + =+ d V d d ? 2000 by CRC Press LLC Equating the acceleration torque of the rotor to its inertia torque, one obtains: (66.43) where T em is the electromagnetic torque developed by the machine; T loss , the equivalent torque representing friction and windage and stray load losses; and T mech , the externally applied mechanical torque on the shaft in the direction of rotation. As shown in Fig. 66.30, T em is positive for motoring and negative for generating; T mech is negative for motoring and positive for generating. Like the derivation of E a , the developed torque, T em , can be shown to be equal to k a φI a by considering first the total power flow into the windings, that is, (66.44) Summing the input powers to both windings, dropping the resistive losses and magnetic energy storage terms, and equating the remaining term to developed power, one will obtain the following relationships from which an expression of the developed torque can be written. T em ω m = P em = E a I a W (66.45) Figure 66.31 shows the flowchart for a simple dc machine simulation. The required inputs are V a , v f , and T mech . Solving the windings’ voltage equations with the required inputs, we can obtain the winding currents, I a and i f . The magnetizing curve block con- tains open-circuit test data of the machine to translate i f to k a φ or the ratio of the open-circuit armature voltage to some fixed speed, that is E a /ω mo . The sim- ulation yields the output of the two winding currents, the field flux, the developed torque, and the rotor speed. Modular Approach Simulation of larger systems consisting of electric machines can be assembled directly from basic equations of the individual components and their connections. On a higher level of integration, it will be more convenient and advantageous to utilize templates of subsystems to construct the full system in a modular manner. Subsystem templates once verified can be reused with confidence for studies that are within the scope of the models implemented. The tasks of constructing and debugging a simulation using the modular approach can be much easier than building the same simulation from elementary representations. Proper considerations to matching inputs to outputs of the connected templates are required when using templates. Take, for example, a template of the above dc motor simulation. Such a template will require inputs of V a , v f , and T mech to produce outputs of I a , i f , flux, and rotor speed. On the mechanical side, the motor template has to be interfaced to the simulation of the mechanical prime mover or load for its remaining input of T mech . In the case of a simple load, the load torque, T mech , could be constant or some simple function of rotor speed, as shown in Fig. 66.32. On the electrical side, the motor template has to be interfaced to the templates of the power supplies to the armature and field windings for its inputs of V a and v f . These voltages can come from TT T J t em loss mech m – += ? d d Nm ω VI EI RI LI t vi Ri Li t aa aa aa aq a ff ff ff =++ () =+ () 2 2 2 2 2 2 d d W d d FIGURE 66.31 dc machine simulation flowchart. ? 2000 by CRC Press LLC the simulations of the power supply circuits if they provide outputs of these voltages. If not, as in the case where the templates of the power supply circuits also require voltages V a and v f as their inputs, the intercon- nection of the motor and power supply circuits templates will require an interface module with current as input and voltage as output as shown in Fig. 66.32(b). In practice, the interface module can be of physical or fictitious origin — the latter essentially a convenient but acceptable approximation. Referring again to Fig. 66.32 of the power supply connected to the motor, examples of an interface module of physical origin would be where shunt filtering capacitors or bleeding resistors are actually present at the terminals of the motor windings. Written with current as input and voltage as output, the equations for simulating the shunt capacitor and the shunt bleeding resistor are (66.46) where i is the net current flowing into the branch in both cases. But if the actual system does not have elements with equations that can be written to accept current as input and voltage as output, shunt R and C of fictitious origin can be inserted at the cost of some loss in accuracy to fulfill the necessary interface requirement. For good accuracy, the current in the introduced fictitious branch element should be kept very small relative to the currents of the actual branches by using very large R or very small C. In practice, loop instability in analog computation and numerical stiffness in digital computation will determine the lower bound on how small an error can be attained. In spite of the small error introduced, this technique can be very useful, as evident from trying to use the above dc motor simulation to simulate an open- circuit operation of the motor. While it is possible to reformulate the equations to handle the open-circuit condition, the equations as given with voltage input and current output can be used for the open-circuit condition if one is willing to accept a small inaccuracy of introducing a very large resistor to approximate the open-circuit. On the other hand, the simulation of short-circuit operation with the above model can be easily implemented using a V a of zero. Mathematical Transformations Mathematical transformations are used in the analysis and simulation of three-phase systems, mostly to decouple variables. For example, the transformations to symmetrical components or αβ0 components are used FIGURE 66.32 Interface for dc motor simulation. V C it V Ri== ∫ 1 d ? 2000 by CRC Press LLC in the analysis of unbalanced three-phase network analysis. Transformations to decouple variables, to facilitate the solution of difficult equations with time-varying coefficients, or to refer variables from one frame of reference to another are employed in the analysis and simulation of three-phase ac machines. For example, in the analysis and simulation of a three-phase synchronous machine with a salient pole rotor, transformation of stator quantities onto a frame of reference attached to the asymmetrical rotor results in a much simpler model. In this rotor reference model, the inductances are not dependent on rotor position and, in steady-state operation, the stator voltages and currents are not time varying. Park’s transformation decouples and rotates the stator variables of a synchronous machine onto a dq reference frame that is fixed to the rotor. The positive d-axis of the dq frame is aligned with the magnetic axis of the field winding, that of the positive q-axis is ahead in the direction of rotation or lead the positive d-axis by π/2. Defined in this manner, the internal excitation voltage given by E f = ωL af i f is in the direction of the positive q-axis. Park’s original dq0 transfor- mation [1929] was expressed in terms of the angle, θ d , between the rotor’s d-axis and the axis of the stator’s a-phase winding. The so-called qd0 transformation in more recent publications is Park’s transformation expressed in terms of the angle, θ q , between the rotor’s q-axis and the axis of the stator’s a-phase winding. The row order of components in the qd0 transformation matrix from top of bottom is q-, d-, and then 0. As evident from Fig. 66.33, the angle θ q is equal to θ d + π/2. The transformation from abc variables to a qd0 reference frame is accomplished with [f qd0 ] = [T qd0 (θ q )][f abc ] (66.47) where f can be voltage, current, or flux, and (66.48) Transforming back from qd0 to abc is accomplished by premultiplying both sides of Eq. (66.47) with the inverse (66.49) The angle θ q can be determined from (66.50) FIGURE 66.33 qd0 transformation. T qd q qq q qq q0 2 3 2 3 2 3 2 3 2 3 1 2 1 2 1 2 θ θθ π θ π θθ π θ π () [] = ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? cos cos – cos sin sin – sin T qd q qq qq qq 0 1 1 2 3 2 3 1 2 3 2 3 1 θ θθ θ π θ π θ π θ π () [] = ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? – cos sin cos – sin – cos sin θωθ d t q ttdt () = () + () ∫ 0 0 rad ? 2000 by CRC Press LLC where ω(t) is the rotational speed of the qd reference frame and θ q (0), the initial value of θ q at t = 0. In the case of the rotor’s qd reference frame, ω(t) is equal to the rotor’s speed in electrical radians (per second), that is, ω(t) = ω r (t). Figure 66.34 shows a circuit representation of an idealized synchronous machine with damper windings, kd and kq, and field winding, f, on the rotor. The equivalent circuit representation and equations of the machine in its own rotor qd0 reference frame and in motor convention are shown in Fig. 66.35 and Table 66.3, respectively. Base Quantities Oftentimes, the machine equations are expressed in terms of the flux linkages per second, ψ, and reactances, x, instead of λ and L. These are related simply by the base or rated value of angular frequency, ω b , that is, ψ = ω b λ and x = ω b L (66.51) FIGURE 66.34 Circuit representation of idealized synchronous machine. FIGURE 66.35 Equivalent qd0 circuits of synchronous machine. ? 2000 by CRC Press LLC where ω b = 2πf rated electrical radian per second, f rated being the rated frequency in Hertz of the machine. When dealing with complex waveforms, it is logical to use peak rather than the rms value as the base value. The base quantities with peak rather rms value of a P-pole, three-phase induction machines with rated line-to-line rms voltage, V rated , and rated volt-ampere, S rated , area as follows: Simulation of Synchronous Machines Table 66.4 shows the main steps in a simulation of a three-phase synchronous machine with T mech , v f , and abc stator voltages as input. The rotor’s speed and angle, ω r (t) and δ, are determined by the rotor’s equation of motion. (66.52) The developed torque, T em , is positive for motoring operation and negative for generating operation. The rotor angle, δ, is defined as the angle of the q r axis of the rotor with respect to the q e axis of the synchronously rotating reference frame, that is, (66.53) Since the synchronous speed, ω e , is a constant, (66.54) TABLE 66.3 qd0 Model of Synchronous Machine Voltage equations Flux linkage equations Torque equation vri d dt d dt vri d dt vri d dt d dt vri d dt vri d dt vri d dt qsq q d r fff f dsd d q r kd kd kd kd skqk kq =+ + ′ = ′′+ =+ ′ = ′′+ ′ =+ ′ = ′′+ ′ λ λ θ λ λ λ θλ λ λ – 00 0 λλ qqqmqkq f md mdkdff d d d md f md kd kd md d md f kdkd kd ls kq mq q kqkq kq Li Li Li Li Li Li Li Li Li Li L i Li L i L i =+′′=+′ + ′′ =+′ + ′′=+′ + ′′ = ′ =+′′ 00 T P ii em d q q d = () 3 22 λλ–N–m base voltage, base volt-ampere, base current, base impedance, base torque, base speed, VV SS ISV ZVI TS P b rated b rated b b b bbb bbbm bm b ===23 2 3 2ωωω TT T J dt dt em mech damp rm += () – ω N–m δθ θ ωωθθ ttt tdt re t rere () = () () = () + () () ∫ – –– elect. rad 0 00 dt dt dt dt re r ωω ω() = () – ? 2000 by CRC Press LLC Using (2/P) ω r (t) in place of ω rm (t) and Eq. (66.54) to replace dω r (t)/dt, Eq. (66.52) can be rewritten in terms of the slip speed: (66.55) The angles θ r (t) and θ e (t) are the respective angles of the q r and q e axes of the rotor and synchronously rotating reference frames measured with respect to the stationary axis of the a-phase stator winding. Note that δ is the angle between the q r axis of the rotor and the q e axis of the reference synchronously rotating frame. For multi-machine systems, the rotor angles of the machines could all be referred to a common synchronously rotating reference frame at some bus or to the q r axis of the rotor of a chosen reference machine. A flowchart showing the main blocks for the above simulation is given in Fig. 66.36. As shown, the input voltages and output currents are in abc phase quantities. For some studies, the representation of the supply TABLE 66.4 Simulation of Synchronous Machine Transform input stator abc voltages to the qd reference frame attached to the rotor using [v qd0 ] = [T qd0 (θ r )][v abc ] where θ r (t) = ∫ t 0 ω r (t)dt + θ r (0). The currents or flux linkages of the cut set of three inductors in both the q- and d-axis circuits of Fig. 66.35, are not independent. Using the winding flux linkages per second as states, the mutual flux linkages per second are expressed as where Solve winding flux linkages using the following integral form of the winding voltage equations: where E f = x md , and Determine qd0 winding currents from winding flux linkages. Transform qd0 currents to abc using [i abc ] = [T –1 qd0 (θ r )][i qd0 ]. ψω ψψ ψω ψψ ψ mq b mq q kq MQ q ls kq lkq md b md d kd f MD d ls kd lkd f lf Lii x xx Lii i x xx x =+′ () =+ ′ ′ ? ? ? ? ? ? =+′ + ′ () =+ ′ ′ + ′ ′ ? ? ? ? ? ? 111111111 xxxxxxxxx MQ mq lkq ls MD md lkd lf ls =+ ′ +=+ ′ + ′ + ψω ω ω ψψψ ψ ω ψψ ψω ω ω ψψψ ψ ω ψψ ψω qbq r b d s ls mq q kq bkq lkq mq kq dbd r b q s ls md d kd bkd lkd md kd v r x dt r x dt v r x dt r x dt =+ ′ = ′ ′ ′ =++ ′ = ′ ′ ′ = ( ) ? ? ? ? ? ? ( ) ( ) ? ? ? ? ? ? ( ) ∫∫ –– – –– 0 b s ls f bf md f md lf md f v r x dt r x E x x dt 00 ω ψψ ? ? ? ? ? ? ( ) ? ? ? ? ? ? ′ = ′ + ′ ′ ν′ f f τ ψψψψψ ψ q ls q mq d ls d md ls f lf f md kd lkd kd md kq lkq kq mq xi xi xi xi x i x i =+ =+ = ′ = ′′+ ′ = ′′+ ′ = ′′+ 00 i x i x i x i x i x i x q qmq ls d dmd ls ls kd kd md lkd kq kq mq lkq f fmd lf === ′ = ′ ′ ′ = ′ ′ ′ = ′ ? ′ ψψ ψψ ψ ψψ ψψ ψψ – – – – 0 0 ωω re t em mech damp t P J TT Tdt () =+ () ∫ –– 2 0 elect. rad/s ? 2000 by CRC Press LLC network connected to the machine may not be in phase variables. For example, in linearized analysis, and also in transient stability of power systems, the network representation is usually in a synchronously rotating reference frame. In linearized analysis, the small-signal representation of the system is obtained by making small perturbations about an operating point. When the machines are in their respective rotor qd0 reference frames and the power network is in a synchronously rotating qd0 reference frame, the qd0 variables of the network and machines are in steady-state; thus linearized analysis about an operating point can be performed. In transient stability, the main interest is the stability of the system after some large disturbances. The models employed are to portray the transient behavior of the power flows in the network and the electromechanical response of the machines. When dealing with large networks, the fast electromagnetic transients of the network are usually ignored and a static network representation is used. At each new time step of the dynamic simulation, an update of the network condition can be obtained by solving the phasor equations of the static network along with power or current injections from the machines. Because the phasor quantities of the network can be expressed as qd components of a synchronously rotating qd0 reference [Ong, 1998], the exchange of voltage variables between network and machine at the bus will require a rotational transformation given below (66.56) where δ is the rotor angle of the q r of the machines qd0 rotor reference frame measured with respect to the q e axis of the network’s synchronously rotating reference frame. The above transformation is also applicable to the exchange of current variables between network and machine. Other synchronous machine models, besides that given in Table 66.3, are used in power system analysis. Typically, when the network is large and the phenomenon of interest is somewhat localized in nature, machines further away from the action can be represented by simpler models to save computation time. On the other hand, certain phenomena may require an even more sophisticated model than that given in Table 66.3. Canay [1993] described refinements in both the rotor circuit representation and the method of parameter determi- nation to obtain a closer fit of the rotor variables. For studying shaft torsion, the damper circuit representation should not be ignored. In transient stability studies, machines beyond the first two neighborhoods of the disturbance can be represented by progressively simpler models with distance from the disturbance. Ignoring just the pψ q and pψ d terms and also setting ω = ω e in the stator equations will yield a so-called subtransient model of two orders less than the above given model. Further simplification by setting pψ′ kq and pψ′ kd to zero or omitting the damper winding equations will yield a so-called transient model of another two orders less. Finally, setting pψ′ f to zero and holding field flux linkage constant yields the constant field flux linkage model. Significant savings in computing time can also be made by neglecting the subtransient and transient saliency of the machine. When rotor saliency is ignored, the effective stator impedances along the rotor’s q r and d r axes are equal. In other words, the stator impedance in the synchronously rotating reference frame of the network will not be a function of rotor angle. Because its value need not be updated with the rotor angle at each time step of the dynamic simulation, the constant stator impedance of this model can be absorbed into the network’s admittance or impedance representation. FIGURE 66.36 Block diagram of synchronous machine simulation. v v v v v v q d q e d e 00 0 0 001 ? ? ? ? ? ? ? ? ? ? = ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? cos sin sin cos δδ δδ ? 2000 by CRC Press LLC Three-Phase Induction Machines Figure 66.37 shows a circuit representation of a symmetrical three-phase induction machine with uniform airgap. The axes of the qd0 reference frames are assumed to be rotating at an arbitrary angular speed of ω. The angles θ(t) and θ r (t), in electrical radians, can be determined from (66.57) where θ(0) and θ r (0) are their respective initial values at time t = 0. As before, the voltage equations of the stator and rotor windings can be written using the coupled circuit approach. Corresponding voltage equations in the arbitrary qd0 reference frame can be obtained by applying the transformation T qd0 (θ) to the stator variables and the transformation T qd0 (θ – θ r ) to the rotor variables. The equations of a symmetrical induction machine in the arbitrary reference frame in terms of the flux linkages per second and reactances are summarized in Table 66.5. Seldom is there a need to simulate an induction machine in the arbitrary rotating reference frame. Induction machine loads are often simulated on the network’s synchronously rotating reference frame in power system studies. However, in transient studies of adjustable speed drives, it is usually more convenient to simulate the induction machine and its converter on a stationary reference frame. Equations of the machine in the stationary and synchronously rotating reference frames can be obtained by setting the speed of the arbitrary reference frame, ω, to zero and ω e , respectively. Often the stator windings are connected to the supply by a three-wire connection, as shown in Fig. 66.38. With a three-wire connection, the stator zero-sequence current, i 0s , or (i as + i bs + i cs )/3, is zero by physical constraint, irrespective of whether the phase currents are balanced or not. The phase currents could be unbalanced, as in single-phasing operation. The stator neutral is free-floating. Its voltage, v sg , measured with respect to some ground point g, need not be zero. Where the applied voltages are non-sinusoidal, as in the case when the supply is from a bridge inverter, v sg is not zero. In general, the input stator phase voltages, v ag , v bg , and v cg , for the simulation of the induction machine can be established from the following relationships: v as = v ag – v sg v bs = v bg – v sg v cs = v cg – v sg (66.58) When point s is solidly connected to point, g, v sg will be zero. Otherwise, if R sg and L sg are the resistance and inductance of the connection between points s and g, v sg can be determined from FIGURE 66.37 Circuit representation of induction machine. θωθθ ωθt t dt t dt t r t rr () = () + () () =+ () ∫∫ 00 00 ? 2000 by CRC Press LLC (66.59) Where the stator windings’ neutral is free-floating, v sg can be determined from an open-circuit approximation of the form shown in Eq. (66.46). TABLE 66.5 Model of induction machine in arbitrary qdo Voltage equations Flux linkage equations Torque equation FIGURE 66.38 Three-wire power supply connection. v p ri v p ri v p ri v p ri v p r qs b qs b ds s qs qr b qr r b dr r qr ds b ds b qs s ds dr b dr r b qr r dr s b s =++ ′ = ′ + ?? ? ? ? ? ? ′ + ′′ =?+ ′ = ′ ? ?? ? ? ? ? ? ′ + ′′ =+ ω ψ ω ω ψ ω ψ ωω ω ψ ω ψ ω ω ψ ω ψ ωω ω ψ ω ψ 00sss r b rrr iv p ri 0000 ′ = ′ + ′′ ω ψ ψ ψ ψ ψ ψ ψ qs ds s qr dr r ls m m ls m m ls mlrm mlrm lr xx x xx x x xxx xxx x 0 0 00 00 0000 0 00 00 0000 000 ′ ′ ′ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = + + ′ + ′ + ′ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ′ ′ ′ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? i i i i i i qs ds s qr dr r 0 0 T P ii ii P ii P ii P xii ii em rb ds qs qs ds r b dr qr qr dr b qr dr dr qr b ds qs qs ds b mdrqs qrds =? () + ? ′′? ′′ () ? ? ? ? ? ? ? = ′′? ′′ () =? () = ′ ? ′ () 3 22 3 22 3 22 3 22 ω ω ω ψψ ωω ω ψψ ω ψψ ω ψψ ω Nm vRii L d dt iii RL d dt i sg sg as bs cs sg as bs cs sg sg s =++ () +++ () =+ ? ? ? ? ? ? 3 0 ? 2000 by CRC Press LLC Defining Terms Model of an electric machine: Differential algebraic equations describing the dynamic behavior of the electric machine. Dynamic simulation: Setting up of a model capable of portraying certain dynamic behavior of the real device and performing experiments on the model. Rotating reference frame: A rotating qd plane. For example, a synchronously rotating reference frame is a qd plane that is rotating at synchronous speed as defined by the fundamental excitation frequency. References Canay, I. M. (1993) Modelling of Alternating-Current Machines Having Multiple Rotor Circuits, IEEE Trans. on Energy Conversion, Vol. 8, No. 2, June 1993, pp. 280–296. Demerdash, N. A. O. and Alhamadi M. A. (1995) Three-Dimensional Finite Element Analysis of Permanent Magnet Brushless DC Motor Drives – Status of the State of Art, IEEE Trans. on Industrial Electronics, Vol. 43, No. 2, April 1995, pp. 268–275. Ong, C. M. (1998) Dynamic Simulation of Electric Machinery, Prentice-Hall PTR, New Jersey. Park, R. H. (1929) Two-Reaction Theory of Synchronous Machines Generalized Method of Analysis. Part I, A.I.E.E. Transactions, Vol. 48, 1929, pp. 716–727. Preston, T. W., Reece, A. B. J., and Sangha, P. C. (1988) Induction Motor Analysis by Time-Stepping Techniques, IEEE Trans. on Magnetics, Vol. 24, No. 1, Jan. 1998, pp. 471–474. Rahman, M. A. and Little, T. A. (1984) Dynamic Performance Analysis of Permanent Magnet Synchronous Magnet Motors, IEEE Trans. on Power Apparatus and Systems, Vol. 103, No. 6, June 1984, pp. 1277–1282. Salon, S. J. (1995) Finite Element Analysis of Electrical Machines, Kluwer Academic Publishers, Boston. Shen, J. (1995) Computational Electromagnetics Using Boundary Element: Advances in Modeling Eddy Currents, Computational Mechanics Publication, Southampton, UK. Further Information The above chapter section has briefly described some of the techniques of the coupled-circuit approach and qd0 transformation in modeling, and the treatment of interface and floating neutral conditions in implementing a simulation. For more information on modeling and implementation of machine simulations, see [Ong, 1998]. Some techniques of modeling permanent magnet machines are described in [Rahman and Little, 1984; Ong, 1998]. Problems concerning effects of local saturation, anisotropic magnetic properties, and eddy-currents in machines require detailed modeling of the field region. Two- and three-dimensional models of the field region can be solved using finite-element [Salon, 1995] and boundary-element [Shen, 1995] techniques. Although the field models are not as amenable as the circuit models for use in large system studies, they have been successfully integrated with lumped circuit element models in dynamic simulations [Demerdash and Alhamadi, 1995; Preston et al., 1988]. ? 2000 by CRC Press LLC