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