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Stephen J. Chapman received a B.S. in Electrical Engineering from Louisiana. State University () and an M.S.E. in Electrical Engineering from the Univer-. Electric Machinery Fitzgerald 7th Edition PDF Electric Machinery Fitzgerald 7th Edition Book PDF Extra Link Book Description: This seventh edition of Fitzgerald . Electric machinery. 2. Power electronics. I. Title. TKS44 '— dc Printed in the United States of America. 10 9 8 7 6 5 4 3 2 1.


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Electric Machinery. Sixth Edition. A. E. Fitzgerald. Late Vice President for Academic Affairs and Dean of the Faculty. Northeastern University. Charles Kingsley, Jr. Arthur E. Fitzgerald, Charles Kingsley, Jr., and Stephen D. Umans, Electric Machinery, Sixth Edition, McGraw-Hill, 1. (Three-Phase Circuits) 2. ( Magnetic. Stephen J. Electric machinery fundamentals / Stephen Chapman. - 4th ed. p. em. Includes index. ISBN I. E lectric machinery. I. T itle. T K

Fringing effect, effective Ag increased. With self-excited generators, residual magnetism must be present in the machine iron to get the self-excitation process started. Polyphase Induction Machines 8. Some method of supplying an accurate, low-level representation of these signals to the instrumentation is required. The rotor winding may one of two types. R1 Effect of the exciting current: They cost less, weigh less, require less floor space, and have somewhat higher efficiency.

Part of this energy is dissipated as losses and results in heating of the core. The rest appears as reactive power associated with energy storage in the magnetic field. This reactive power is not dissipated in the core; it is cyclically supplied and absorbed by the excitation source. The first is ohmic I 2 R heating, associated with induced currents in the core material. Eddy currents circulate and oppose changes in flux density in the material. To reduce the effects, magnetic structures are usually built of thin sheets of laminations of the magnetic material.

It is plotted in terms of watts per unit weight as a function of flux density; often a family of curves for different frequencies are given. See Fig. These materials produce significant magnetic flux even in magnetic circuits with air gaps. The second quadrant of a hysteresis loop the magnetization curve is usually employed for analyzing a permanent-magnet material. The significant of remanent magnetization is that it can produce magnetic flux in a magnetic circuit in the absence of external excitation such as winding currents.

Consider Example 1. From 1. A curve of constant B-H product is a hyperbola. In Fig. The static transformer is not an energy conversion device, but an indispensable component in many energy conversion systems. It is a significant component in ac power systems: Electric generation at the most economical generator voltage Power transfer at the most economical transmission voltage Power utilization at the most voltage for the particular utilization device It is widely used in low-power, low-current electronic and control circuits: Matching the impedances of a source and its load for maximum power transfer Isolating one circuit from another Isolating direct current while maintaining ac continuity between two circuits The transformer is one of the simpler devices comprising two or more electric circuits coupled by a common magnetic circuit.

Its analysis involves many of the principles essential to the study of electric machinery. One of these windings, the primary, is connected to an alternating-voltage. An alternating flux will be produced whose magnitude will depend on the primary voltage, the frequency of the applied voltage, and the number of turns. The mutual flux will link the other winding, the secondary, and will induce a voltage in it whose value will depend on the number of secondary turns as well as the magnitude of the mutual flux and the frequency.

By properly proportioning the number of primary and secondary turns, almost any desired voltage ratio, or ratio of transformation, can be obtained.

The essence of transformer action requires only the existence of time-varying mutual flux linking two windings. Iron-core transformer: The magnetic circuit usually consists of a stack of thin laminations. Silicon steel has the desirable properties of low cost, low core loss, and high permeability at high flux densities 1. Silicon-steel laminations 0.

Two common types of construction: Figure 2. Leakage flux links one winding without linking the other. Leakage flux is a small fraction of the total flux. Leakage flux is reduced by subdividing the windings into sections and by placing them as close together as possible. The primary and secondary windings are actually interleaved in practice.

The core flux is fixed by the applied voltage, and the required exciting current is determined by the magnetic properties of the core; the exciting current must adjust itself so as to produce the mmf required to create the flux demanded by 2.

A curve of the exciting current as a function of time can be found graphically from the ac hysteresis loop as shown in Fig. If the exciting current is analyzed by Fourier-series methods, its is found to consist of a fundamental component and a series of odd harmonics. Core-loss component: Magnetizing current: The peculiarities of the exciting-current waveform usually need not by taken into account, because the exciting current itself is small, especially in large transformers.

Ideal Transformer Fig. Winding resistances are negligible. Leakage flux is assumed negligible. There are no losses in the core.

Only a negligible mmf is required to establish the flux in the core. The impressed voltage, the counter emf, the induced emf, and the terminal voltage: Let a load be connected to the secondary. Impedance transformation properties: Summary for the ideal transformer: Voltages are transformed in the direct ratio of turns. Currents are transformed in the inverse ratio of turns. Impedances are transformed in the direct ratio squared.

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Power and voltamperes are unchanged. Note that the capacitances of the windings will be neglected. Method of the equivalent circuit technique is adopted for analysis. Development of the transformer equivalent circuit Leakage flux: R1 Effect of the exciting current: The actual transformer can be seen to be equivalent to an ideal transformer plus external impedances Refer to the assumptions for an ideal transformer to understand the definitions and meanings of these resistances and reactances.

Two tests serve to determine the parameters of the equivalent circuits of Figs. The high voltage side is usually taken as the primary to which voltage is applied. The circuit parameters referred to the primary can be found as 2.

Note that it is possible to measure R1 and R2 directly by a dc resistance measurement on each winding.

However, no such simple test exists for X l1 and X l2. The test is performed with the secondary open-circuited and rated voltage impressed on the primary. If the transformer is to be used at other than its rated voltage, the test should be done at that voltage. An exciting current of a few percent of full-load current is obtained. The windings of the two-winding transformer are electrically isolated whereas those of the autotransformer are connected directly together.

In the transformer connection, winding ab must be provided with extra insulation. Autotransformer have lower leakage reactances, lower losses, and smaller exciting current and cost less than two-winding transformers when the voltage ration does not differ too greatly from 1: Trsansformers having a primary and multiple secondaries are frequently found in multiple-output dc power supplies.

Distribution transformers used to supply power for domestic purposes usually have two V secondaries connected in series. The three-phase transformer banks used to interconnect two transmission system of different voltages often have a third, or tertiary, set of windings to provide voltage for auxiliary power purposes in substation or to supply a local distribution system.

Static capacitors or synchronous condensers may be connected to the tertiary windings for power factor correction or voltage regulation. The Y- connection is commonly used in stepping down from a high voltage to a medium or low voltage.

The -Y connection is commonly used for stepping up to a high voltage.

Open-delta, or V, connection The Y-Y connection is seldom used because of difficulties with exciting-current phenomenon. Because there is no neutral connection to carry harmonics of the exciting current and harmonic voltages are produced which significantly distort the transformer voltages.

A three-phase bank may consist of one three-phase transformer having all six windings on a common multi-legged core and contained in a single tank. They cost less, weigh less, require less floor space, and have somewhat higher efficiency. It is usually convenient to carry out circuit computations involving three-phase transformer banks under balanced conditions on a single-phase per-phase-Y, line-to-neutral basis.

Some method of supplying an accurate, low-level representation of these signals to the instrumentation is required. Potential Transformer PT and Current Transformer CT , also referred to as Instrumentation Transformer, are designed to approximate the ideal transformer as closely as is practically possible.

The load on an instrumentation transformer is frequently referred to as the burden on that transformer. A potential transformer should ideally accurately measure voltage while appearing as an open circuit to the system under measurement, i.

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An ideal current transformer would accurately measure current while appearing as a short circuit to the system under measurement, i. All pertinent quantities are expressed as decimal fractions of appropriately chose base values.

All the usual computations are then carried out in these per unit values instead of the familiar volts, amperes, ohms, and so on.

The parameter values typically fall in a reasonably narrow numerical range when expressed in a per-unit system based upon their rating. When transformer equivalent-circuit parameters are converted to their per-unit values, the ideal transformer turns ratio becomes 1: Actual quantities: For a single-phase system: In typical usage, values of VAbase and Vbase are chosen first; values of I base and all other quantities in 2.

The value of VAbase must be the same over the entire system under analysis. When a transformer is encountered, the values of Vbase differ on each side and should be chosen in the same ratio as the turns ratio of the transformer. The per-unit ideal transformer will have a unity turns ratio and hence can be eliminated.

Usually the rated or nominal voltages of the respective sides are chosen. The procedure for performing system analyses in per-unit is summarized as follows: The physics behind each type of device is the same and, in a crude sense, they can each be considered to be simply scaled versions of the same basic device. When normalized to their own rating, the effect of the scaling is eliminated and the result is a set of per-unit parameter values which is quite similar over the whole size range of that device.

Manufacturers often supply device parameters in per unit on the device base. When performing a system analysis, it may be necessary to convert the supplied per-unit parameter values to per-unit values on the base chosen for the analysis.

Relations for base values: Three-phase problems can thus be solved in per unit as if they were single-phase problems. Emphasis is placed on the analysis of systems that use magnetic fields as the conversion medium.

The concepts and techniques can be applied to a wide range of engineering situations involving electromechanical energy conversion. Based on the energy method, we are to develop expressions for forces and torques in magnetic-field-based electromechanical systems.

Figure 3.

Unlike the case in Example 3. Forces act directly on the magnetic material of these devices which are constructed of rigid, nondeforming structures. The performance of these devices is typically determined by the net force, or torque, acting on the moving component. It is rarely necessary to calculate the details of the internal force distribution. In a motor, the stator magnetic field rotates ahead of that of the rotor, pulling on it and performing work.

For a generator, the rotor does the work on the stator. A lossless magnetic-energy-storage system with two terminals The electric terminal has two terminal variables: The mechanical terminal has two terminal variables: The interaction between the electric and mechanical terminals, i.

Equations 3. The ability to identify a lossless-energy-storage system is the essence of the energy method. This is done mathematically as part of the modeling process. For the lossless magnetic-energy-storage system of Fig. It is through this reaction voltage that the external electric circuit supplies power to the coupling magnetic field and hence to the mechanical output terminals. Combining 3. This field acts as the energy-conversion medium, and its energy is the reservoir between the electric and mechanical system.

The predominant energy storage occurs in the air gap, and the properties of the magnetic circuit are determined by the dimensions of the air gap. Since the magnetic energy storage system is lossless, it is a conservative system. Therefore, 3. Equation 3. The selection of energy or coenergy as the state function is purely a matter of convenience.

Consider the relay in Fig. Assume the relay armature is at position x so that the device operating at point a in Fig. In a singly-excited device, the force acts to increase the inductance by pulling on members so as to reduce the reluctance of the magnetic path linking the winding. Measurement systems: Energy conversion devices: A simple system with two electrical terminals and one mechanical terminal: Three independent variables: That is, the coenergy function is a relatively simple function of displacement.

The use of a coenergy function of the terminal currents simplifies the determination of torque or force. Systems with more than two electrical terminals are handled in analogous fashion. In rotating machines, voltages are generated in windings or groups of coils by rotating these windings mechanically through a magnetic field, by mechanically rotating a magnetic field past the winding, or by designing the magnetic circuit so that the reluctance varies with rotation of the rotor.

The flux linking a specific coil is changed cyclically, and a time-varying voltage is generated. Electromagnetic energy conversion occurs when changes in the flux linkage result from mechanical motion. A set of such coils connected together is typically referred to as an armature winding, a winding or a set of windings carrying ac currents. In ac machines such as synchronous or induction machines, the armature winding is typically on the stator. In dc machines, the field winding is found on the stator.

In synchronous machines, the field winding is found on the rotor. Permanent magnets can be used in the place of field windings. In most rotating machines, the stator and rotor are made of electrical steel, and the windings are installed in slots on these structures.

The stator and rotor structures are typically built from thin laminations of electrical steel, insulated from each other, to reduce eddy-current losses. In synchronous machines, rotor-winding currents are supplied directly from the stationary frame through a rotating contact. In induction machines, rotor currents are induced in the rotor windings by a combination of the time-variation of the stator currents and the motion of the rotor relative to the stator.

Synchronous Machines Fig. The armature winding is on the stator, and the field winding is on the rotor. The field winding is excited by direct current conducted to it by means of stationary carbon brushes that contact rotating slip rings or collector rings.

It is advantages to have the single, low-power field winding on the rotor while having the high-power, typically multiple-phase, armature winding on the stator. Conductors forming these coil sides are connected in series by end connections. The rotor is turned at a constant speed by a source of mechanical power connected to its shaft.

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Flux paths are shown schematically by dashed lines. Assume a sinusoidal distribution of magnetic flux in the air gap of the machine in Fig. The radial distribution of air-gap flux density B is shown in Fig. As the rotor rotates, the flux —linkages of the armature winding change with time and the resulting coil voltage will be sinusoidal in time as shown in Fig 4. The frequency in cycles per second Hz is the same as the speed of the rotor in revolutions in second rps.

A two-pole synchronous machine must revolve at rpm to produce a Hz voltage. Figure 4. A great many synchronous machines have more than two poles. Fig 4. The field coils are connected so that the poles are of alternate polarity. There are two complete wavelengths, or cycles, in the flux distribution around the periphery, as shown in Fig.

The generated voltage goes through two complete cycles per revolution of the rotor. The frequency in Hz is thus twice the speed in rps. When a machine has more than two poles, it is convenient to concentrate on a single pair of poles and to express angles in electrical degrees or electrical radians rather than in physical units.

One pair of poles equals electrical degrees or 2 electrical radians. The rotors shown in Figs. The field winding is a two-pole distributed winding; the coil sides are distributed in multiple slots around the rotor periphery and arranged to produce an approximately sinusoidal distribution of radial air-gap flux. Most power systems in the world operate at frequencies of either 50 or 60 Hz. A salient-pole construction is characteristic of hydroelectric generators because hydraulic turbines operate at relatively low speeds, and hence a relatively large number of poles is required to produce the desired frequency.

Steam turbines and gas turbines operate best at relatively high speeds, and turbine- driven alternators or turbine generators are commonly two- or four-pole cylindrical- rotor machines. With very few exceptions, synchronous generators are three-phase machines. A simplified schematic view of a three-phase, two-pole machine with one coil per phase is shown in Fig. Note that a minimum of two sets of coils must be used.

In an elementary multipole machine, the minimum number of coils sets is given by one half the number of poles. Then the coils of the three phases may then be either Y- or -connected. The electromechanical torque is the mechanism through which a synchronous generator converts mechanical to electric energy. When a synchronous generator supplies electric power to a load, the armature current creates a magnetic flux wave in the air gap that rotates at synchronous speed.

This flux reacts with the flux created by the field current, and an electromechanical torque results from the tendency of these two magnetic fields to align. In a generator this torque opposes rotation, and mechanical torque must be applied from the prime mover to sustain rotation.

The counterpart of the synchronous generator is the synchronous motor. Ac current supplied to the armature winding on the stator, and dc excitation is supplied to the field winding on the rotor. The magnetic field produced by the armature currents rotates at synchronous speed. To produce a steady electromechanical torque, the magnetic fields of the stator and rotor must be constant in amplitude and stationary with respect to each other. In a motor the electromechanical torque is in the direction of rotation and balances the opposing torque required to drive the mechanical load.

Note that the flux produced by currents in the armature of a synchronous motor rotates ahead of that produced by the field, thus pulling on the field and hence on the rotor and doing work.

This is the opposite of the situation in a synchronous generator, where the field does work as its flux pulls on that of the armature, which is lagging behind. Induction Machines Alternating currents are applied directly to the stator windings. Rotors currents are then produced by induction, i. Alternating currents flow in the rotor windings of an induction machine, in contrast to a synchronous machine in which a field winding on the rotor is excited with dc current.

The induction machine may be regarded as a generalized transformer in which electric power is transformed between rotor and stator together with a change of frequency and a flow of mechanical power.

The induction motor is the most common of all motors. The induction machine is seldom used as a generator. In recent years it has been found to be well suited for wind-power applications. It may also be used as a frequency changer. In the induction motor, the stator windings are essentially the same as those of a synchronous machine. The rotor windings are electrically short-circuited. The rotor windings frequently have no external connections.

Currents are induced by transformer action from the stator winding. Squirrel-cage induction motor: The armature flux in the induction motor leads that of the rotor and produces an electromechanical torque. The rotor does not rotate synchronously. An ideal current transformer would accurately measure current while appearing as a short circuit to the system under measurement, i. All pertinent quantities are expressed as decimal fractions of appropriately chose base values.

All the usual computations are then carried out in these per unit values instead of the familiar volts, amperes, ohms, and so on. The parameter values typically fall in a reasonably narrow numerical range when expressed in a per-unit system based upon their rating. When transformer equivalent-circuit parameters are converted to their per-unit values, the ideal transformer turns ratio becomes 1: Actual quantities: For a single-phase system: In typical usage, values of VAbase and Vbase are chosen first; values of I base and all other quantities in 2.

The value of VAbase must be the same over the entire system under analysis. When a transformer is encountered, the values of Vbase differ on each side and should be chosen in the same ratio as the turns ratio of the transformer. The per-unit ideal transformer will have a unity turns ratio and hence can be eliminated.

Usually the rated or nominal voltages of the respective sides are chosen. The procedure for performing system analyses in per-unit is summarized as follows: The physics behind each type of device is the same and, in a crude sense, they can each be considered to be simply scaled versions of the same basic device. When normalized to their own rating, the effect of the scaling is eliminated and the result is a set of per-unit parameter values which is quite similar over the whole size range of that device.

Manufacturers often supply device parameters in per unit on the device base. When performing a system analysis, it may be necessary to convert the supplied per-unit parameter values to per-unit values on the base chosen for the analysis.

Relations for base values: Three-phase problems can thus be solved in per unit as if they were single-phase problems. Emphasis is placed on the analysis of systems that use magnetic fields as the conversion medium.

The concepts and techniques can be applied to a wide range of engineering situations involving electromechanical energy conversion.

Based on the energy method, we are to develop expressions for forces and torques in magnetic-field-based electromechanical systems. Figure 3. Unlike the case in Example 3. Forces act directly on the magnetic material of these devices which are constructed of rigid, nondeforming structures. The performance of these devices is typically determined by the net force, or torque, acting on the moving component. It is rarely necessary to calculate the details of the internal force distribution.

In a motor, the stator magnetic field rotates ahead of that of the rotor, pulling on it and performing work. For a generator, the rotor does the work on the stator. A lossless magnetic-energy-storage system with two terminals The electric terminal has two terminal variables: The mechanical terminal has two terminal variables: The interaction between the electric and mechanical terminals, i.

Equations 3. The ability to identify a lossless-energy-storage system is the essence of the energy method. This is done mathematically as part of the modeling process. For the lossless magnetic-energy-storage system of Fig. It is through this reaction voltage that the external electric circuit supplies power to the coupling magnetic field and hence to the mechanical output terminals. Combining 3. This field acts as the energy-conversion medium, and its energy is the reservoir between the electric and mechanical system.

The predominant energy storage occurs in the air gap, and the properties of the magnetic circuit are determined by the dimensions of the air gap. Since the magnetic energy storage system is lossless, it is a conservative system.

Therefore, 3. Equation 3. The selection of energy or coenergy as the state function is purely a matter of convenience. Consider the relay in Fig.

Assume the relay armature is at position x so that the device operating at point a in Fig. In a singly-excited device, the force acts to increase the inductance by pulling on members so as to reduce the reluctance of the magnetic path linking the winding. Measurement systems: Energy conversion devices: A simple system with two electrical terminals and one mechanical terminal: Three independent variables: That is, the coenergy function is a relatively simple function of displacement.

The use of a coenergy function of the terminal currents simplifies the determination of torque or force. Systems with more than two electrical terminals are handled in analogous fashion. In rotating machines, voltages are generated in windings or groups of coils by rotating these windings mechanically through a magnetic field, by mechanically rotating a magnetic field past the winding, or by designing the magnetic circuit so that the reluctance varies with rotation of the rotor.

The flux linking a specific coil is changed cyclically, and a time-varying voltage is generated. Electromagnetic energy conversion occurs when changes in the flux linkage result from mechanical motion.

A set of such coils connected together is typically referred to as an armature winding, a winding or a set of windings carrying ac currents. In ac machines such as synchronous or induction machines, the armature winding is typically on the stator. In dc machines, the field winding is found on the stator. In synchronous machines, the field winding is found on the rotor. Permanent magnets can be used in the place of field windings.

In most rotating machines, the stator and rotor are made of electrical steel, and the windings are installed in slots on these structures.

The stator and rotor structures are typically built from thin laminations of electrical steel, insulated from each other, to reduce eddy-current losses. In synchronous machines, rotor-winding currents are supplied directly from the stationary frame through a rotating contact.

In induction machines, rotor currents are induced in the rotor windings by a combination of the time-variation of the stator currents and the motion of the rotor relative to the stator. Synchronous Machines Fig. The armature winding is on the stator, and the field winding is on the rotor.

The field winding is excited by direct current conducted to it by means of stationary carbon brushes that contact rotating slip rings or collector rings. It is advantages to have the single, low-power field winding on the rotor while having the high-power, typically multiple-phase, armature winding on the stator. Conductors forming these coil sides are connected in series by end connections.

The rotor is turned at a constant speed by a source of mechanical power connected to its shaft. Flux paths are shown schematically by dashed lines. Assume a sinusoidal distribution of magnetic flux in the air gap of the machine in Fig.

The radial distribution of air-gap flux density B is shown in Fig. As the rotor rotates, the flux —linkages of the armature winding change with time and the resulting coil voltage will be sinusoidal in time as shown in Fig 4. The frequency in cycles per second Hz is the same as the speed of the rotor in revolutions in second rps. A two-pole synchronous machine must revolve at rpm to produce a Hz voltage.

Figure 4. A great many synchronous machines have more than two poles. Fig 4. The field coils are connected so that the poles are of alternate polarity. There are two complete wavelengths, or cycles, in the flux distribution around the periphery, as shown in Fig. The generated voltage goes through two complete cycles per revolution of the rotor.

The frequency in Hz is thus twice the speed in rps. When a machine has more than two poles, it is convenient to concentrate on a single pair of poles and to express angles in electrical degrees or electrical radians rather than in physical units. One pair of poles equals electrical degrees or 2 electrical radians. The rotors shown in Figs. The field winding is a two-pole distributed winding; the coil sides are distributed in multiple slots around the rotor periphery and arranged to produce an approximately sinusoidal distribution of radial air-gap flux.

Most power systems in the world operate at frequencies of either 50 or 60 Hz. A salient-pole construction is characteristic of hydroelectric generators because hydraulic turbines operate at relatively low speeds, and hence a relatively large number of poles is required to produce the desired frequency. Steam turbines and gas turbines operate best at relatively high speeds, and turbine- driven alternators or turbine generators are commonly two- or four-pole cylindrical- rotor machines.

With very few exceptions, synchronous generators are three-phase machines. A simplified schematic view of a three-phase, two-pole machine with one coil per phase is shown in Fig.

Note that a minimum of two sets of coils must be used. In an elementary multipole machine, the minimum number of coils sets is given by one half the number of poles. Then the coils of the three phases may then be either Y- or -connected. The electromechanical torque is the mechanism through which a synchronous generator converts mechanical to electric energy. When a synchronous generator supplies electric power to a load, the armature current creates a magnetic flux wave in the air gap that rotates at synchronous speed.

This flux reacts with the flux created by the field current, and an electromechanical torque results from the tendency of these two magnetic fields to align. In a generator this torque opposes rotation, and mechanical torque must be applied from the prime mover to sustain rotation. The counterpart of the synchronous generator is the synchronous motor. Ac current supplied to the armature winding on the stator, and dc excitation is supplied to the field winding on the rotor. The magnetic field produced by the armature currents rotates at synchronous speed.

To produce a steady electromechanical torque, the magnetic fields of the stator and rotor must be constant in amplitude and stationary with respect to each other. In a motor the electromechanical torque is in the direction of rotation and balances the opposing torque required to drive the mechanical load. Note that the flux produced by currents in the armature of a synchronous motor rotates ahead of that produced by the field, thus pulling on the field and hence on the rotor and doing work.

This is the opposite of the situation in a synchronous generator, where the field does work as its flux pulls on that of the armature, which is lagging behind. Induction Machines Alternating currents are applied directly to the stator windings.

Rotors currents are then produced by induction, i. Alternating currents flow in the rotor windings of an induction machine, in contrast to a synchronous machine in which a field winding on the rotor is excited with dc current.

The induction machine may be regarded as a generalized transformer in which electric power is transformed between rotor and stator together with a change of frequency and a flow of mechanical power.

The induction motor is the most common of all motors. The induction machine is seldom used as a generator. In recent years it has been found to be well suited for wind-power applications. It may also be used as a frequency changer. In the induction motor, the stator windings are essentially the same as those of a synchronous machine.

The rotor windings are electrically short-circuited. The rotor windings frequently have no external connections. Currents are induced by transformer action from the stator winding. Squirrel-cage induction motor: The armature flux in the induction motor leads that of the rotor and produces an electromechanical torque.

The rotor does not rotate synchronously. It is the slipping of the rotor with respect to the synchronous armature flux that gives rise to the induced rotor currents and hence the torque. Induction motors operate at speeds less than the synchronous mechanical speed. A typical speed-torque characteristic for an induction motor is shown in Fig. The armature winding is on the rotor with current conducted from it by means of carbon brushes. The field winding is on the stator and is excited by direct current.

An elementary two-pole dc generator is shown in Fig. The air-gap flux distribution usually approximates a flat-topped wave, rather than the sine wave found in ac machines, and is shown in Fig. Rotation of the coil generates a coil voltage which is a time function having the same waveform as the spatial flux-density distribution. The voltage induced in an individual armature coil is an alternating voltage and rectification is produced mechanically by means of a commutator. Stationary carbon brushes held against the commutator surface connect the winding to the external armature terminal.

The need for commutation is the reason why the armature windings are placed on the rotor. The commutator provides full-wave rectification, and the voltage waveform between brushes is shown in Fig.

It is the interaction of the two flux distributions created by the direct currents in the field and the armature windings that creates an electromechanical torque. If the machine is acting as a motor, the torque acts in the direction of the rotation. The individual coils are interconnected so that the result is a magnetic field having the same number of poles as the field winding.

Consider Fig. Full-pitch coil: In the design of ac machines, serious efforts are made to distribute the coils making up the windings so as to minimize the higher-order harmonic components. The rectangular air-gap mmf wave of the concentrated two-pole, full-pitch coil of Fig. The windings of the three phases are identical and are located with their magnetic axes degrees apart. The winding is arranged in two layers, each full-pitch coil of N c turns having one side in the top of a slot and the other coil side in the bottom of a slot a pole pitch away.

It can be seen that the distributed winding produces a closer approximation to a sinusoidal mmf wave than the concentrated coil of Fig. The modified form of 4. The application of three-phase currents will produce a rotating mmf wave. Rotor windings are often distributed in slots to reduce the effects of space harmonics.

As shown in Fig. The armature coil connections are such that the armature winding produces a magnetic field whose axis is vertical and thus is perpendicular to the axis of the field winding. As the armature rotates, the magnetic field of the armature remains vertical due to commutator action and a continuous unidirectional torque results. The mmf wave is illustrated and analyzed in Fig.

DC machines often have a magnetic structure with more than two poles. The machine is shown in laid-out form in Fig. The investigations of both ac and dc machines are based on the assumption of sinusoidal spatial distribution of mmf.

Results from examining a two-pole machine can immediately be extrapolated to a multipole machine. Detailed analysis of the magnetic field distributions requires complete solutions of the field problem.

Field coils excited; no current in armature coils. General Electric Company. Note that from Eq. The air-gap mmf of a single-phase winding exicted by a source of ac current can be resolved into rotating traveling waves.

This decomposition is shown graphically in Fig. In a three-phase machine, the windings of the individual phases are displaced from each other by electrical degrees in space around the air-gap circumference as shown in Fig.

Under balanced three-phase conditions, the excitation currents Fig. It is the interaction of this magnetic flux wave with that of the rotor which produces torque. Constant torque is produced when rotor-produced magnetic flux rotates in synchronism with that of the stator. As time passes, the resultant mmf wave retains its sinusoidal form and amplitude but rotates progressively around the air gap.

The net result is an mmf wave of constant amplitude rotating at uniform angular velocity. Practice Problem 4. A synchronous machine is an ac machine whose speed under steady-state conditions is proportional to the frequency of the current in its armature.

The rotor, along with the magnetic field created by the dc field current on the rotor, rotates at the same speed as, or in synchronism with, the rotating magnetic field produced by the armature currents, and a steady torque results.

Armature winding: Field winding: Cylindrical rotor: Salient-pole rotor: Acting as a voltage source: Frequency determined by the speed of its mechanical drive or prime mover. The amplitude of the generated voltage is proportional to the frequency and the field current.

When a synchronous generator is connected to a large interconnected system containing many other synchronous generators, the voltage and frequency at its armature terminals are substantially fixed by the system. It is often useful, when studying the behavior of an individual generator or group of generators, to represent the remainder of the system as a constant-frequency, constant-voltage source, commonly referred to as an infinite bus.

Analysis of a synchronous machine connected to an infinite bus. Torque equation: In a generator, the prime-mover torque acts in the direction of rotation of the rotor, and the electromechanical torque opposes rotation. The rotor mmf wave leads the resultant air-gap flux. In a motor, the electromechanical torque is in the direction of rotation, in opposition to the retarding torque of the mechanical load on the shaft.

Torque-angle curve: Figure 5. Any further increase in prime-mover torque cannot be balanced by a corresponding increase in synchronous electromechanical torque, with the result that synchronism will no longer be maintained and the rotor will speed up. Single-phase, line-to-neutral equivalent circuits for a three-phase machine operating under balanced, three-phase conditions. Both the external system and the machine itself can be represented as an impedance in series with a voltage source.

Note that E1 and E 2 are the line-to-neutral voltages. RT The analysis begins with the development of single-phase equivalent circuits. The general form is suggested by the similarity of an induction machine to a transformer.

The equivalent circuits can be used to study the electromechanical characteristics of an induction machine as well as the loading presented by the machine on its supply source. The stator winding is excited from a balanced polyphase source and produces a magnetic field in the air gap rotating at synchronous speed. The rotor winding may one of two types.

A wound rotor is built with a polyphase winding similar to, and wound with the same number of poles as, the stator. The rotor terminals are available external to the motor. A squirrel-cage rotor has a winding consisting of conductor bars embedded in slots in the rotor iron and short-circuited at each end buy conducting end rings.

It is the most commonly used type of motor in sizes ranging from fractional horsepower on up. The difference between synchronous speed and the rotor speed is commonly referred to as the slip of the rotor.

The rotor currents produce an air-gap flux wave that rotates at synchronous speed and in synchronism with that produced by the stator currents.

With the rotor revolving in the same direction of rotation as the stator field, the rotor currents produce a rotating flux wave rotating at sns with respect to the rotor in the forward direction. Such torque is called an asynchronous torque. The factors influencing the shape of this curve can be appreciated in terms of the torque equation. Figure 6.

Under normal running conditions the slip is small: