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foundation of principles that include responsibility to the communities we Principles of electric machines and power electronics / Dr. P.C. Sen, Fellow IEEE. Library of Congress Cataloging-in-Publication Data: Sen, P. C. (Paresh Chandra) . Principles of electric machines and power electronics /. —2nd ed. p. cm. This book deals with the fundamentals of Engineering Drawing to begin with and . The principles Machine D PRINCIPLES OF ELECTRIC CIRCUITS, 9th.

Mohamed E. Ampacity Computations for Transmission George J. Anderson, B. Agrawal, J. Anderson, A. Anderson Hardcover pp Power and Communication Cables:

The transformer transfers power with very high efficiency from one level of voltage to another level. Due to its high efficiency, the power transferred to the secondary is almost the same as the primary. Using a step-up transformer reduces losses in the line, which makes the transmission of power over long distances possible.

Insulation requirements and other practical design problems limit the generated voltage to low values, usually 30 kV.

Thus, step-up transformers are used for transmission of power. At the receiving end of the transmission lines, step-down transfor- mers are used to reduce the voltage to suitable values for distribution or utilization.

The electricity in an electric power system may undergo four or five transformations between generator and consumers. Transmission lines also interconnect neighboring utilities, which allows economic dispatch of power within regions during normal conditions, and the transfer of power between regions during emergencies. High-voltage transmission lines are terminated in substations called high-voltage substations, receiving substations, or primary substations. The function of some substations is to switch circuits in and out of service; these are referred to as switching stations.

At the primary substation, voltage is stepped down to a value more suitable for the next stage toward the load. Very large industrial customers may be served from the transmission system. There is no clear distinction between transmission and subtransmission voltage levels. Typically, the subtransmission voltage level ranges from 69 kV to kV. Some large industrial customers may be served from the subtransmission system.

Capacitor banks and reactor banks for maintaining the transmission-line voltage are usually installed in the substations. The primary distribution lines operate on voltages from 4 kV to Some small industrial customers are served directly by the primary feeders.

The secondary distribution network reduces the voltage for utilization by commercial and residential consumers. Lines and cables not exceeding a few hundred feet in length then deliver power to the individual consumers.

The power for a typical home is derived from a transformer that reduces the primary feeder voltage to V using a three-wire line.

Distribution systems involve both overhead and underground means of power transmission. Power systems loads are divided into industrial, commercial, and residential. Industrial loads are composite loads, and induction motors form a high proportion of these loads. These composite loads are functions of voltage and frequency and form a major part of the system load.

Commercial and residential loads consist largely of lighting, computing, entertainment, heating, and cooking. These loads are independent of frequency and consume negligibly small reactive power.

For reliable and economical operation of the power system, it is necessary to monitor the entire system in a control center. The modem control center of today is called the energy control center ECC. ECCs are equipped with on-line computers performing all signal processing through the remote acquisition system. Computers work in a hierarchical structure to properly coordinate different functional requirements under normal as well as emergency conditions.

Every energy control center contains control consoles, which consist of a visual display unit VDU , keyboard, and light pen. Computers may give alarms as advance warnings to the operators dispatchers when deviation from the normal state occurs. The dispatcher makes decisions and executes them with the aid of a computer. Simulation tools and software packages are implemented for efficient operation and reliable control of the system.

In addition, SCADA, an acronym for "supervisory control and data acquisition," systems are auxiliaries to the energy control center. It is on this basis that Chapter 2 is founded. In Chapter 2, the basic laws, definitions, and relevant phenomena of electromagnetism are discussed. Emphasis is placed on the magnetic circuits of devices treated in this text.

Ampere's circuital law is used to establish engineering-oriented procedures for analyzing the electromagnetic behavior of some simple structures. Properties of ferromagnetic materials, including eddy current and hysterisis loss, are discussed in this chapter. Theoretical foundations of electromechanial energy conversion devices are also treated in Chapter 2.

The object here is the development of a simple but general approach to the energy conversion process in rotating electric machines. The discussion leads to an appreciation of the differences between the various types of electrical machines. Chapter 3, which is titled "Power Electronic Devices and Systems," discusses the characteristics and technical performance characteristics of available solid-state power electronics devices. This is followed with a discussion of the various systems based on the availability of devices.

Here we discuss ac voltage controllers, controlled rectifiers, choppers, and inverters. It is on the basis of the discussion in this chapter that we introduce aspects of adjustable speed drives along with each motor type discussed in subsequent chapters. In Chapter 4 the de machine as generator and motor is considered. The diSCUSSIon of de generators is brief by necessity, but various connections and performance characteristics are given. The discussion of dc motors is more detailed and incorporates such important issues as matching motor to load, stating, speed control, and applications.

Chapter 5 deals with transformers and offers the student an opportunity to appreciate the significance of Faraday's law and principles discussed in Chapter 2 to the development of this important device.

Analysis of the performance of transformers is the central topic of this chapter. The many uses for and connections of transformers are discussed from an application point of view. Chapter 6 is dedicated to the induction motor, which is the versatile workhorse of today's industrial complex.

Our treatment relies on principles discussed in Chapter 2 as well as the treatment of transformers in Chapter 5. Modeling the performance of the induction motor together with important characteristics, is also dealt with.

Motor starting and control issues are also discussed in this chapter. In Chapter 7, the synchronous machine is treated from a systems application point of view. Again, performance characteristics for round-rotor and salient-pole machines are dealt with.

In addition, the applications of synchronous motors are presented in this chapter. A number of fractional-horsepower machines are discussed in Chapter 8 to highlight their applications and performance characteristics. An ample number of drill problems at the end of each chapter are designed to follow the natural development of the text material.

In this chapter we discuss some concepts and terminology used in this text to study electric machines. An electromechanical energy conversion device is essentially a medium of transfer between an input side and an output side, as shown in Fig. In the case of an electric motor, the input is electrical energy drawn from the supply source and the output is mechanical energy supplied to the load, which may be a pump, fan, hoist, or any other mechanical load.

The electric generator is a device that converts mechanical energy supplied by a prime mover such as a steam turbine or a hydroturbine to electrical energy at the output side. Many electromechanical energy conversion devices can act as either generator or motor. The chapter begins with a brief review of the Lorentz force law, Biot-Savart law, and Ampere's circuital law.

Magnetic-circuit concepts are discussed and properties of magnetic materials are covered. Faraday's law, inductance, and energy relations are also treated here. The remainder of the chapter is devoted to the development of the principal relationships that govern the electromechanical energy conversion process. The relationships sought will be valuable in understanding the operation of electric machines studied in subsequent chapters.

Electrical energy Load B Figure 2. If the charges are moving with uniform velocity, a second effect known as magnetism takes place. Thus a magnetic field is associated with moving charges, and as a result, electric currents are sources of magnetic fields.

A magnetic field is identified by a vector B called the magnetic flux density. In the SI system of units, the unit of B is the tesla T. This product is expressed by inserting a cross an x between U and V, thus: The direction of the cross product W is perpendicular to the plane of U and V and follows the right-hand rule, as shown in Fig.

The concept of the cross product of two vectors is useful in formulating magneticfield laws based on the experimental work of Oersted, Ampere, and Biot-Savart.

We start with the Lorentz force law, which deals with the force on a moving charge in a magnetic field. An interpretation of Eq. On the basis of Eq. A pictorial presentation of Eq.

The differential form 2. The force on an entire loop can be obtained by integrating the current element 2.

Another important law in magnetic-field theory is the Biot-Savart law, which is based on Ampere's work showing that electric currents exert forces on each other and that a magnet could be replaced by an equivalent current. The flux density B at a point P, which is at a distance r from q in free space, is given by the Biot-Savart equation: A distribution of charge causes differential magnetic flux density dB due to each incremental charge element dq given by - JJo.

The total magnetic field is obtained by integrating the contributions from all the incremental elements: Lo fldl 4n x a, r2 2. R p R dB into paper 8 A Figure 2. An alternative to this relation is Ampere's circuital law, which we discuss next. To illustrate Ampere's circuital law, let us consider the infinite wire carrying a current I, treated earlier using the Biot-Savart law. We have shown that the field is concentric circles about the wire with a flux density magnitude given by Eq.

As shown in Fig. The incre- The flux density is in the same direction as dl, and thus we have As a result, f cl B I. The closed path need not be circular, and to illustrate this, we consider the path C2 shown in Fig. JC2 B. Thus we have i B. L, 21tR. We will consider path C3 of Fig. Ampere's circuital law asserts that the line integral is zero in this case.

This is confirmed as follows: As a result, " B. The Biot-Savart law given by Eq. For free space B varies linearly with I, and J. We can conclude that for materials that exhibit a linear variation of B with I, all expressions discussed in Section 2. Material can be classified from a B-J-variation point of view into two classes: Nonmagnetic material such as all dielectrics and metals with permeability equal, for all practical purposes, to J.

Magnetic material that belongs to the iron group known as ferromagnetic material. For this class, a given current produces a much larger B field than in free space. Permeability of ferromagnetic material is much higher than that of free space and is nonlinear, since it varies over a wide range with variations in current. Ferromagnetic material can be further categorized into two classes: Soft ferromagnetic material for which a linearization of the B-1 variation in a region is possible.

Hard ferromagnetic material for which it is difficult to give a meaning to the term permeability. Material in this group is suitable for permanent magnets.

The source of B in the case of soft ferromagnetic material can be modeled as being due to the current I. For hard ferromagnetic material, the source of B is a combined effect of current J and material magnetization M, which originates entirely in the medium. In order to separate the two sources of the magnetic B field, the concept of magnetic-field intensity H is introduced.

Equation 2. The B-H characteristic of nonmagnetic materials is shown in Fig. The B-H characteristics of soft ferromagnetic material, often called the magnetization curve, follow the typical pattern displayed in Fig. The permeability of the material in agreement with Eq. The maximum value of Jl occurs at the knee of the B-H characteristic. B H Figure 2. For the electromechanical energy conversion devices treated in this text, a linear approximation to the magnetization curve provides satisfactory answers in the normal region of operation.

The main idea is illustrated in Fig. Within the acceptable range of H values, one can then use the following relation to model the ferromagnetic material: Properties fl. Linear approximation of the magnetization curve for a soft ferromagnetic of magnetic materials are discussed further in the following sections.

For the present, we assume that IJr is constant. If a path encloses the current N times, the right-hand side of Eq. The magnetic flux is confined to paths defined by the structure due to the high permeability of the magnetic material. An elementary example of a magnetic circuit is shown in Fig.

Here, a toroidal ferromagnetic core carrying an N-turn coil is considered. The torus has a cross-sectional area A, an inner radius a, and an outer radius b. The magnitude of the flux density B inside the core is obtained by multiplying H by the permeability of the core material: As a result, we conclude that 2. This relation is depicted in block diagram form Fig. We are already familiar with Ohm's law for de circuits, as interpreted in Fig. Inspection of the figure shows us that an analogy between magnetic variables and electric-circuit variables is possible.

We can thus establish a correspondence between electric voltage V electromotive force and the magnetomotive force fj. As a consequence, we can claim a correspondence between the quantity I! Indeed, we now define the reluctance of the magnetic circuit denoted by 9t as 2. Let us observe that the magnetic reluctance as defined in Eq. The advantages of this analogy include the ability to utilize our circuit analysis tools in dealing with magnetic-field problems.

The development of the magnetic circuit concept in this section relies on the analysis of the toroid of Fig. It is appropriate now to take a look at a simple example.

Example 2. The toroid is wound with a coil of turns and a direct current of 1. Assume that the relative permeability of the toroid is It is necessary to calculate: The reluctance of the circuit 2. The magnetomotive force 3' and magnetic-field intensity H. The flux and flux density inside the toroid. Assuming uniform fields, the magnetic-circuit concept can be applied to structures composed of different materials, such as that shown in Fig.

The path of integration in Ampere's law is shown as the closed contour abcdefgha. J g h Figure 2. The following example deals with a single-loop structure with two materials, the unknown is the MMF, and this problem is the inverse of that given in Example 2. Consider the magnetic structure shown in Fig.

Find the current J in the tum winding to set up a flux density of 1 T in the structure. Assume the relative permeability of the structure material to be Solution The length of the magnetic-material path is denoted by Ii and is found from the geometry of Fig. In Fig. We deduce the equivalent circuit shown in the figure by appealing again to Ampere's circuital law. It is clear that once we have the equivalent circuit, familiar techniques of circuit analysis can be applied.

In the first, the MMF is given and the flux or flux densities values are found in various sections of the structure. The second class involves the inverse problem of finding ty given a flux flux density value in the structure. Assume the following dimensions: Solution We first compute the reluctances involved: An example structure is shown in Fig.

I, 10 em The thickness of the core is 2 cm. Solution We compute the reluctances in the structure by first finding relevant lengths and areas. It should be noted that the structure is nonuniform. The reluctance values are therefore given by 8x 2. The polarity of the sources is determined by the right-hand rule.

The first has to do with the value of the reluctance of an air gap relative to that of a path in the magnetic material. In Example 2. We can conclude that the air-gap reluctance is much higher than the reluctance of a path in iron. In this case, the magnetic path can be approximated by a short circuit zero reluctance in the equivalent magnetic circuit.

The second point is that in practice, the magnetic-field lines extend outward somewhat as they cross the air gap, as shown in Fig. This phenomenon is called fringing and is accounted for by taking a larger air-gap cross-sectional area. The common practice is to use an air-gap area made by adding the air-gap length to each of the two dimensions making the cross-sectional area.

As a result, with reference to Fig. The following example illustrates this point. Now, we have from Example 2. We now consider the effects of variations of the magnetic system with time.

A change in the magnetic field is associated with the establishment of an electric field that is manifested as an induced voltage. This basic fact is due to Faraday's experiments and is expressed by Faraday's law of electromagnetic induction.

Each turn encloses or links the total flux and we also note that the total flux links each of the N turns. Inductance is the passive circuit element that is related to the geometry and material properties of the structure.

The unit of inductance is the henry- or weber-turns per ampere. The inductance L is related to the reluctance 9t of the magnetic structure of a single-loop structure, since we have As a result, we get 2. In this case di. Using Faraday's law [Eq. Let us recall the basic relation stating that power p t is the rate of change of energy W t: Part a depicts Eq.

In part b , it is assumed that the iP - tj characteristic is available, and thus Eq. Consider the case of a magnetic structure that experiences a change in state between the time instants 11 and It is clear that the energy per unit volume expended between I. For a linear structure, we can develop these relations further. Using Eq. This information is useful in many ways, as will be seen in this text.

This is generally referred to as a hysteresis characteristic. To illustrate this phenomenon, we use the sequence of portraits of Fig. Assume that the MMF, and hence H, is a slowly varying sinusoidal waveform with period T, as shown in the lower portion graphs of Fig.

We discuss the evolution of the B-H hysteresis loop in the following intervals. The flux density B is observed to decrease along the segment abo Note that ab is higher than oa, and thus for the same value of H we get different value of B. The value of B, is referred to as the residual field, remanence, or retentivity. If we leave the coil unenergized, the core will still be magnetized.

B decreases to zero at point c. The flux density B is negative and increases from d to e. Beyond f, we find that B increases up to a again. Interval V A typical hysteresis loop is shown in Fig. On the same graph, the B-H characteristic for nonmagnetic material is shown to show the relative magnitudes involved. It should be noted that for each maximum value of the ac magnetic-field intensity cycle, there is a steady-state loop, as shown in Fig.

In the figure, the dashed curve connecting the tips of the loops is the de magnetization curve for the material. The distinction between hard and soft magnetic material on the basis of their hysteresis loops is shown in Fig. It is evident from the figure that the coercive force He for the soft magnetic material is much lower than that for a hard magnetic material.

Table 2. In Section 2. The energy supplied by the source in moving from a to b in the graph of Fig. If we continue on from b to d through C, the energy is positive, as H is negative but B is decreasing [see Fig. H Figure 2. The second half of the loop is treated in Fig. Superimposing both halves of the loop, we obtain Fig. This energy is expended in the magnetization-demagnetization process and is dissipated as a heat loss.

Note that the loop is described in one cycle, and as a result, the hysteresis loss per second is equal to the product of the loop area and the frequency f of the waveform applied.

The area of the loop depends on the maximum flux density, and as a result, we say that the power dissipated through hysteresis Ph is given by where k h is a constant, f is the frequency, and B m is the maximum flux density. The exponent n is determined from experimental results and ranges between 1.

TABLE 2. There is another loss, called eddy-current loss, mechanism that arises in connection with the application of time-varying magnetic field. The change in flux will induce voltages in the core material, which will result in currents circulating in the core.

The induced currents tend to establish a flux that opposes the original change imposed by the source. The induced currents, which are essentially the eddy currents, will result in power loss due to heating of the core material. To minimize eddy-current losses, the magnetic core is made of layers of sheet-steel laminations, ideally separated by highly resistive material. It is clear that this effectively results in the actual area of the magnetic material being less than the gross area presented by the stack.

To account for this, a stacking factor is employed for practical circuit calculations. Gross cross-sectional area Typically, lamination thickness ranges from 0. The eddy-current power loss per unit volume can be expressed by the empirical formula This formula shows that the eddy-current power loss per unit volume varies with the square of frequency f, maximum flux density B m , and the lamination thickness II. Of course, K, is the proportionality constant.

The term core loss is used to denote the combination of eddy-current and hysteresis power losses in the material. In practice, manufacturer-supplied data are used to estimate the core loss P; for given frequencies and flux densities for a particular type of material.

Our intention is to develop a model of the process that is practical and easy to follow, and therefore takes a macroscopic approach based on the principle of energy conservation.

The situation is best illustrated using the diagrams of Fig. We assume that an incremental change in electric energy supply, dWe has taken place. This energy flow into the device can be visualized as made up of three components, as shown in Fig.

Part of the energy will be imparted to the magnetic field of the device and will result in an increase in the energy stored in the field, denoted by dWf. A second component of energy will be expended as heat losses dW1oss. The third and most important component is that output energy is made available to the load dWmech. In part b of the figure, the energy flow is shown in a form that is closer to reality by visualizing Ampere's bonne homme making a trip through the machine.

Starting in the stator, ohmic losses will be encountered, followed by field losses and a change in the energy stored in the magnetic field. Having crossed the air gap, our friend witnesses ohmic losses taking place in the rotor windings, and in passing to the shaft, bearing frictional losses are also encountered.

It should be emphasized here that the phenomena dealt with here are distributed in nature, and what we are doing is simply developing an understanding in the form of mathematical expressions called models.

The trip by our Amperean friend can never take place in real life, but is a helpful means of visualizing the process.

We are now ready to write our energy balance equation based on the foregoing arguments. The field energy is a function of two states of the system. This follows since knowledge of A completely specifies i through the A-i characteristic. First let us take the dependence of WI on A and x, and write 2. This is a consequence of Taylor's series for a function of two variables. Comparing Eqs. For rotary motion, we replace x by 0 in the foregoing development to arrive at T.

Our next task, therefore, is to determine the variations of the field energy with A and x for linear motion and that with A and lJ for rotary motion. We thus choose the path OAP shown in Fig. As a result, 2. Find an expression for the force developed by the field. Figure 2. Thus the inductance is obtained using The energy stored in the field is thus given by Eq. Find the current ; t assuming zero initial velocity. L oA ] wN2 Note that the expression of x in terms of Fo, M, w, and t has to be used in conjunction with the last relation.

We encounter this situation in most rotating electromechanical energy conversion devices. The force or torque can be obtained by simple extension of the techniques discussed in the foregoing sections.

We will consider a system with three windings, as shown in Fig. Of course, our discussion can be simply extended to an arbitrary number of windings, n. The differential electric-energy input is 2. If we are dealing with a rotational system, we have 2. Thus dA.. CP ,1. We have 2. Our examples involved use of a linear motion device to illustrate the concepts discussed.

In the preceding development we referred to the case of rotary motion and derived expressions for the torque developed by the electromagnetic field in terms of energy stored. The present section deals with an important class of singly excited electromechanical energy conversion devices commonly referred to as the reluctance motor.

In its simplest form, a reluctance motor consists of a rotor free to rotate on a horizontal axis between the pole pieces of a stationary singly excited magnetic structure, as shown in Fig. This type of motor can be found in clocks and phonograph turntables.

It is assumed that when the rotor is in the vertical position, as shown in Fig. Let us assume that the permeability of the ferromagnetic path is large enough to enable us to neglect its reluctance relative to that of the air gaps. The air-gap length is now gl, and the effective area of the gap is AI. Note that although A 1 is larger than Ao, this is more than compensated for by the fact that g. Let us recall that the inductance L is related to the reluctance 9l by 2. It should be noted that this sinusoidal variation is an approximation of the actual variation.

Design of the pole shapes is based on attempting to achieve this ideal situation. As a typical example we take the system shown in Fig. Here we have a coil on the stator fed by an electric-energy source 1 and a second coil mounted on the rotor and fed by source 2. This is an example that will prove useful as a prototype of many practical machines. Source 1 Source 2 Stationary member Rotor Figure 2. Therefore, the self-inductance L 11 is expressed using Eq.

The torque is then 2. From a broad geometric configuration point of view, such machines can be classified as being either of the salient-pole type or round-rotor smooth air gap classes. We now discuss the salient-pole type, as this class is a simple extension of the discussion of the preceding section. In a salient-pole machine, one member the rotor in our discussion has protruding or salient poles, and thus the air gap between stator and rotor is not uniform, as shown in Fig. It is clear that the results of Section 2.

Subscript 1 is replaced by s to represent stator quantities, and subscript 2 is replaced by r to represent rotor quantities. Thus we rewrite Eq. The developed torque given by Eq.

We define the primary or main torque TI by 2. The primary or main torque T 1, expressed by Eq. Examining Eq. It thus follows that as a condition for the nonzero average of T 1, we must satisfy one of the following: Our discussion led to developing conditions for producing a torque with a nonzero average value under sinusoidal excitation to the rotor and stator coils.

A round-rotor machine is a special case of salient-pole machine where the air gap between the stator and rotor is relatively uniform. The term smooth air gap is an idealization of the situation illustrated in Fig.

Therefore, for the machine of Fig. We have concluded that for an average value of T1 to exist, one of the conditions of Eq. Although these terms are of zero average value, they can cause speed pulsations and vibrations that may be harmful to the machine's operation and life. The alternating torques can be eliminated by adding additional windings to the stator and rotor, as discussed presently. It is clear that this is an extension of the machine of Fig. Our analysis of this machine requires first setting up the inductances required.

This can be done best using vector terminology. We can write for this four-winding system: The lower right-hand comer block corresponds to phase b, and we note that we assume that phase a and phase b coils are similar in this discussion. What remains to be discussed is the upper right comer block which is seen to be equal to the transponse of the lower left comer block , representing the mutual interaction between phase a and phase b.

The mutual terms L as bs and L ar br are zero, since phases a and bare at right angles. The term L as br is equal to -Mo sin J by the use of projections of flux, and similarly, L ar bs is equal to Mo sin 8.

The field energy is the same as given by Eq. The torque is obtained in the usual manner. Let us now assume that the terminal currents are given by the balanced, two-phase current sources Is coswst 2.

Condition 2. The machine in Fig. Equations 2.

We have, by Eq. The left sides are sinusoidal voltages of equal magnitude but are 90 degrees apart in phase. The rotor currents will have a frequency of co, - Wm , which satisfies condition 2. The induction machine is the subject of detailed study in Chapter 6. For now let us emphasize that currents induced in the rotor have a frequency of co, - rom and that average torque can be produced.

An important point to consider is the convention, adopted for assigning polarities in schematic diagrams, which is discussed presently. Consider the bar magnet of Fig. The magnetic flux lines are shown as closed loops oriented from the south pole to the north pole within the magnetic material.

The situation with a two-pole stator is explained in Fig. First consider Fig. According to our convention, the flux lines are oriented away from the south pole toward the north pole within the magnetic material not in air gaps. The flux lines are oriented in accordance with the right-hand rule, and we conclude that the north and south pole orientations are as shown in the figure. Consider now the situation illustrated in Fig. An extension of the prior arguments concerning a two-pole machine results from the combination of the stator and rotor of Fig.

It is clear that any arbitrary even number of poles can be achieved by placing the coils of a given phase in symmetry around the periphery of the stator and rotor of a given machine. The number of poles is simply the number encountered in one round-trip around the periphery of the air gap. It is necessary for successful operation of the machine to have the same number of poles on the stator and rotor.

We note here that our treatment of the electric machines was focused on two-pole configurations. As an example, relations 2. The torque T 1 under the sinusoidal excitation conditions 2. A time-saving and intuitively appealing concept in dealing with P-pole machines is that of electrical degrees. Consider the first condition of Eq. Assume that the relative permeability of the core material is and that the outside radius of the toroid is 0.

The area of cross section is circular, with diameter 0. Find the flux in the core. Problem 2. N Figure 2. Assume that the current is 40 A and that the core is nonmagnetic. It is required to calculate: Find the current in the tum winding to set up a flux density of 0.

Use the concept of reluctance in your solution. For the magnetic structure shown in Fig. Assume that the relative Problem 2. Use the concepts of magnetic circuit and reluctance in your solution. The coil current is 0. The two rotors are made of a material with relative permeability J. Lr' while the core material has a permeability of J.

With the rotors absent, the flux is found to be 0. Find the relative permeabilities J. Find the length LOa Problem 2. Draw an equivalent magnetic circuit for the system. Assume that the relative permeability of the core is Calculate the flux in the right-hand air gap for an MMF in the coil of At.

The flux density in the cast-iron portion is 0. This corresponds to an MMF of At and a flux of 0. Assume that the permeability of cast iron is 4 x and that the permeability of sheet steel is 4. Find the flux in the air gap Problem 2. Assume that the rotor and stator iron have relative permeabilities of and , respectively. The outside and inside diameters of the stator yoke are 1 m and 0. The rotor length is 0. The air gaps are each 1 em long. The axial length of the machine is 1 m.

Find the required MMF to set up a flux of 0. The air-gap length is 2 mm. Find the flux in the air gap given that Problem 2. Cross section of air gap Problem 2. Find the inductance of the coil of Problem 2. Find the energy stored in the air gap and the magnetic core. Find the change in core losses. Show that the minimum current required to lift a slab of mass M is given by. Show that the dynamics of the system are described by dA.

Neglect core reluctance. The coil has turns. Find the force in terms of x and time. Calculate the average force in terms of x. Assume that x is fixed and find the necessary voltage applied to the coil terminals, given that its resistance is 1 n.

The angle 0 is the rotor angular displacement from the stator coil axis. Find the rotor current in the steady state and the torque developed. The advent of power electronics revolutionized the concept of power control for power conversion and for control of electrical-motor drives. Power electronics may be defined as the applications of solid-state electronics for the control and conversion of electric power. It is based primarily on the switching of the power semiconductor devices.

With the development of power semiconductor technology, the power-handling capabilities and the switching speed of the power devices have improved tremendously.

Power electronics are used in a great variety of high-power products, including heat controls, light controls, motor controls, power supplies, vehicle propulsion systems, and high-voltage direct-current HVDe systems. It is difficult to draw the boundaries for the applications of power electronics; especially with the present trends in the development of power devices and microprocessors, the upper limit is undefined.

Since then, there has been much progress in the power semiconductor devices. Until , the conventional thyristors had been exclusively used for power control in industrial applications. These can be divided broadly into four types: The thyristors can be subdivided into seven types: Static induction transistors SIT are also commercially available.

A diode has two terminals: Schottky diodes have low onstate voltage and very small recovery time, typically nanoseconds. The leakage current increases with the voltage rating, and their ratings are limited to V, A. A diode conducts when its anode voltage is higher than that of the cathode, and the forward voltage drop of a power diode is very low, typically 0.

If the cathode voltage is higher than its anode voltage, a diode is said to be in a blocking mode. There are three types of power diodes: General-purpose diodes are available up to V, A, and the rating of fast-recovery diodes can go up to V, A.

The reverse recovery time varies between 0. The fast-recovery diodes are used for high-frequency switching of power converters. Forward bias is a term describing the application of an external voltage to the anode terminal higher than that at the cathode i. For low-power ac applications, TRIACs are widely used in all types of simple heat controls, light controls, motor controls, and ac switches.

A TRIAC is similar to two thyristors connected in inverse parallel and having only one gate terminal. GTOs and SITHs are essentially self-turned-offthyristors, which are turned on by applying a short positive pulse to the gates and are turned off by the application of a short negative pulse to the gate. They do not require commutation circuits. SITHs are applied for medium-power converters with a frequency of several hundred kilohertz and beyond the frequency range of GTOs.

High-power bipolar transistors are employed in power converters at a frequency below 10 kHz and are applied in power ratings up to kWand V. A bipolar transistor has three terminals: It is operated as a switch in the common-emitter configuration.

As long as the base of an NPN transistor is at a higher potential than the emitter and the base current is sufficiently large to drive the transistor in the saturation region, the transistor remains on, provided that the collector-to-emitter junction is properly biased.

The forward drop of a conducting transistor is in the range of 0. If the base drive voltage is withdrawn, the transistor remains in the nonconduction or off mode. Power MOSFETs are used in high-speed power converters and are available for relatively low power rating in the range of V, 50 A at a frequency range of several tens of kilohertz. Table 3. Note that thyristor technology is superior to transistors in blocking voltages above 2. The required output is obtained by varying the conduction time of these switching devices..

Figure 3. Once a thyristor is conducting, the gate signal of either positive or negative magnitude has no effect on the conduction property, and this is shown in Fig. When a power semiconductor device is in a normal conduction mode, there is a small voltage drop across the device.

In the output voltage waveforms in Fig. One can classify power semiconductor switching devices on the following basis: Uncontrolled turn-on and -off e. Controlled turn-on and uncontrolled turnoff e. Controlled tum-on and -off characteristics e. A Vo Yo 0 8 Thyristor switch v. T 4. The diodes are also used in rectifier circuits. Practical diodes differ from the ideal characteristics and have certain limitations. The power diodes are similar to pn-junction signal diodes.

However, power diodes have larger power-, voltage-, and current-handling capabilities than that of ordinary signal diodes. The frequency response or switching speed is low compared to signal diodes. When the anode potential is positive with respect to the cathode, the diode is said to beforward biased and the diode conducts.

There is a relatively small forward voltage drop across a conducting diode, and the magnitude of this drop depends on the manufacturing process and temperature. When the cathode potential is positive with respect to the anode, the diode is said to be reverse biased. Under reverse-biased conditions, a small reverse current also known as leakage current in the range of micro- or milliampere flows and this leakage current increases slowly in magnitude with the reverse voltage until the avalanche or zener voltage is reached.

In a forward-biased junction diode, the current is due to the net effect of majority and minority carriers. The minority carriers require a certain time to recombine with opposite charges and to be neutralized. This time is called the reverse recovery time of the diode. The soft recovery type is more common. The reverse recovery time is denoted as t rr and is measured from the initial zero crossing of the two components, ta and tb. The ratio Iblta is known as the softness factor SF.

The storage charge depends on the forward diode current, IF. The peak reverse recovery current, IRR , reverse charge, QRR, and the softness factor are all of interest in circuit design. If a diode is reverse biased, a leakage current flows due to the minority carriers. Then applying forward voltage would force the diode to carry current in the forward direction. However, it requires a certain time, known asforward recovery or tumon time, before all the majority carriers over the whole junction can contribute to the current flow.

If the rate of rise of the forward current is high and the forward current is concentrated in a small area of the junction, the diode may fail. The forward o In many applications, the effects of reverse recovery time are insignificant, and cheap diodes can be used.

Depending on the recovery characteristics and manufacturing techniques, power diodes can be classified into three categories.

The characteristics and practical limitations of each type restrict their applications. Standard or general-purpose diodes 2. Fast-recovery diodes 3. Schottky diodes General-purpose Diodes General-purpose rectifier diodes are characterized by relatively high reverse recovery time, typically 25 J.

These diodes have current ratings starting at less than I A. Fast-recovery Diodes Fast-recovery diodes have short recovery time, typically less than 5 J.

They are used in chopper and inverter circuits, where the speed of recovery is often important. These diodes have current ratings starting from less than I A to hundreds of amperes, with voltage ratings from 50 V to around 3 kV. Schottky Diodes In a Schottky diode the charge storage problem of a pnjunction can be eliminated or minimized. This is achieved by setting up a "barrier potential" with a contact between a metal and a semiconductor. A layer of metal is deposited on a thin epitaxial layer of n-type silicon.

The potential barrier simulates the behavior of a pn-junction. However, the rectifying action depends on the majority carriers only, and as a result there is no excess of minority carriers to recombine. The recovery effect is caused only by the self-capacitance of the semiconductor junction.

The recovered charge of a Schottky diode is much lower than that of an equivalent pn-junction diode. Since it is due only to the junction capacitance, it is largely independent of the reverse dildt. A Schottky diode has a relatively low forward voltage drop.

The leakage current of a Schottky diode is higher than that of a pn-junction diode. A Schottky diode with relatively low conduction voltage has relatively high leakage current, and vice versa.

Therefore, its maximum allowable voltage is limited to V. The current rating of Schottky diodes varies from 1 A to A. The Schottky diodes are ideal for high-current and low-voltage de power supplies. However, they are also used in low-current power supplies to achieve improved efficiency.

The designer needs to determine the correct semiconductor diode to meet given specifications. These parameters are nonlinear and depend on a number of factors. The manufacturer normally provides characteristic curves for important parameters, in the form of a data sheet. Maximum Average Forward Current,IF Av is the maximum allowable value of average forward current at a specified temperature.

This is related to the heating effect due to ;2R dissipation and is limited due to the thermal stress on the device.

This is normally specified for a half-sinusoidal waveform. A repetition is permissible only after expiration of a minimum interval to reduce the junction temperature to the allowable range. Common terms and phrases. Analog electronic circuits by u a bakshi a p godse pdfasset View or edit your browsing history.

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