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**COURSE: INTRODUCTION TO ELECTRICAL MACHINES**

Prof Elisete Ternes Pereira, PhD

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**Synopsis Introducing the Basic types of Electric Machines**

A.C. Motors - Induction and Synchronous Motors Ideal and Practical Transformers D.C. Motors and Generators Self and Separately Exited Motors Stepper Motors and their characteristics Assessment of Electric Motors Efficiency Energy losses Motor load analysis Energy efficiency opportunity analysis Improve power quality Rewinding Power factor Speed control

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INTRODUCTION 1

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**Introduction to Electrical Machines**

Essentially all electric energy is generated in a rotating machine: the synchronous generator, and most of it is consumed by: electric motors. In many ways, the world’s entire technology is based on these devices. The study of the behavior of electric machines is based on three fundamental principles: Ampere’s law, Faraday’s law and Newton’s Law.

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**Introduction to Electrical Machines**

Various conﬁgurations result and are classiﬁed generally by the type of electrical system to which the machine is connected: direct current (dc) machines or alternating current (ac) machines.

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**Introduction to Electrical Machines**

Machines with a dc supply are further divided into permanent magnet and wound ﬁeld types, as shown in Figure 4.1.

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**Introduction to Electrical Machines**

The wound motors are further classiﬁed according to the connections used: The ﬁeld and armature may have separate sources (separately excited), they may be connected in parallel (shunt connected), or they may be series (series connected). (figure follows)

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**Introduction to Electrical Machines**

AC machines are usually single-phase or three-phase machines and may be synchronous or asynchronous. See figure next page.

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**Introduction to Electrical Machines**

Several variations are shown in Figure 4.2.

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**1.1 - BASIC ELECTROMAGNETIC LAWS: AMPERE`S LAW AND FARADAY`S LAW**

The two principles that describe the electromagnetic behavior of electric machines are Ampere’s Law and Faraday’s Law. These are two of Maxwell’s equations. Most electric machines operate by attraction or repulsion of electromagnets and/or permanent magnets.

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AMPERE`S LAW Ampere’s law describes the magnetic ﬁeld that can be produced by currents or magnets. In an electric machine, there will always be at least one set of coils with currents. A motor cannot be produced with permanent magnets alone.

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AMPERE`S LAW Ampere’s law states that the line integral of the component of the magnetic ﬁeld along the path of integration is equal to the current enclosed by the path. This is exactly true for static ﬁelds and is a very good approximation for the low-frequency ﬁelds dealt with in electric machines: The right-hand side of the equation represents the current enclosed by the integration path and is called the magnetomotive force (MMF).

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AMPERE`S LAW An example illustrating the determination of the MMF is shown in Figure 4.3, where different integration paths are shown by dotted lines. In electric machines, currents are frequently placed in slots surrounded by ferromagnetic teeth. The MMF corresponding to each path is the total current enclosed by the path. If the slots contain currents that are approximately sinusoidally distributed , then the MMF will be cosinusoidally distributed in space. In this way, the magnetic ﬁeld or ﬂux density in the air gaps of the machine will often have a sinusoidal or cosinusoidal distribution.

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Faraday`s Law & EMF Faraday’s law relates the induced voltage, or electromotive force (EMF), to the time rate of change of the magnetic ﬂux linkage: Magnetic flux Electric circuit (loop of conducting material) For voltage to be induced, there has to be a variation in time between the relative position of the magnetic flux and the electric circuit If the electric circuit is closed and current is allowed to flow, the current will produce a magnetic flux that opposes the increase of the applied flux = Lenz`s law

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Faraday`s Law or where E is the electric ﬁeld and B is the magnetic ﬂux density. This law states that the voltage induced in a loop is equal to the time rate of change of the ﬂux linking the loop. The negative sign indicates that the voltage is induced such that the current would oppose the change in ﬂux linkage. The change in ﬂux linkage can be caused by a change in ﬂux density and/or a change in geometry.

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MAGNETIC FORCE or I I The change in

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**BASIC CONCEPTS - REVIEW**

1. MAGNETIC CIRCUITS

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**CONCEPTS REVIEW MAGNETIC CIRCUITS**

Lets consider first, the most basic ideal circuit: Some relevant parameters: m =mrm0 >> m Ac N lc i

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**CONCEPTS REVIEW MAGNETIC CIRCUITS**

Ampère`s law applied over the typical mean-length (lc) results in: for magnetic circuits The Magnetic Flux can be written as a function of B: for magnetic circuits Substituting we find: The Magnetic Flux Density, B, in terms of the Magnetic Field, H, is:

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**CONCEPTS REVIEW MAGNETIC CIRCUITS**

This found equation: So that: But, Can be written in terms of the `Magnetomotive Force`: Then, for magnetic circuits:

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**CONCEPTS REVIEW MAGNETIC CIRCUITS**

Also, if this equation: is equal to this: Then, the `Magnetic Reluctance` is given by:

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**BASIC CONCEPTS - REVIEW**

2. TRANSFORMERS PRINCIPLES

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**CONCEPTS REVIEW TRANSFORMER BASICS**

Conceptual Schematic – Ideal Transformer Ideal circuit = no loses.

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**we want to find the voltage induced in the open secondary coil**

CONCEPTS REVIEW TRANSFORMER BASICS FIRST: a voltage source is connected in the primary side and the secondary side is an open circuit; we want to find the voltage induced in the open secondary coil When the primary is energized: current in the primary coil magnetic flux in the core.

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**Flux generated by current 1 in coil 1:**

CONCEPTS REVIEW TRANSFORMER BASICS Flux generated by current 1 in coil 1: By Faraday`s law, the induced voltage is: Since there is no losses, the induced voltage is exactly the same as the applied voltage in coil 1:

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**The flux in coil 2, that was generated by current 1:**

CONCEPTS REVIEW TRANSFORMER BASICS The flux in coil 2, that was generated by current 1: And so, the voltage induced in coil 2 is given by the equation:

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**From the voltage equations:**

CONCEPTS REVIEW TRANSFORMER BASICS From the voltage equations: We get the Transformer Ratio Equation:

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**CONCEPTS REVIEW TRANSFORMER BASICS**

SECOND: there is now a load connected to the secondary coil, so i2(t) can flow. We want to find the new induced voltage. By applying Ampere`s law to the circuit, using the line of average path/length , we get the following expression:

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**When i2(t) equal to zero:**

CONCEPTS REVIEW TRANSFORMER BASICS For this expression: When i2(t) equal to zero: The flux in coil 1, produced by current 1, is: But, and So:

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**We get the expression of the flux in coil 1 produced by current 1:**

CONCEPTS REVIEW TRANSFORMER BASICS Substituting Into: We get the expression of the flux in coil 1 produced by current 1: where L1 is the self inductance of coil 1; in this case given by:

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**For the general case, when i2 0**

CONCEPTS REVIEW TRANSFORMER BASICS For the general case, when i2 0 and:

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**The flux in coil 1, produced by both currents, i1 and i2, is given by:**

CONCEPTS REVIEW TRANSFORMER BASICS The flux in coil 1, produced by both currents, i1 and i2, is given by: The first term in parenthesis is the self-inductance of coil 1, the second term is the mutual inductance between coils 1 and 2; then: In a similar we may find the flux in coil 2:

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**The induced voltages in each coil are, then:**

CONCEPTS REVIEW TRANSFORMER BASICS The induced voltages in each coil are, then: Given that: and

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**CONCEPTS REVIEW TRANSFORMER BASICS**

Another equation very much used in transformers design and analysis is the following: This, however, is not an exact equation and can only be used when the magnetic permeability of the nucleus can be considered infinite. or

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**This relation comes from Ampere`s law, that for this case is:**

CONCEPTS REVIEW TRANSFORMER BASICS This relation comes from Ampere`s law, that for this case is: When we assume a very large r so that H 0. In this case: and The negative sign indicates that the currents produce magnetic fields with opposite polarities.

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**CONCEPTS REVIEW TRANSFORMER BASICS**

Another equation extensively employed in the design of transformers is the following: It is called “the Design Equation” and it encounters many practical usage. To deduce it we assume a sinusoidal voltage applied to the primary side when the secondary is open:

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**With primary voltage: The flux then will be: Such that: Resulting in,**

CONCEPTS REVIEW TRANSFORMER BASICS With primary voltage: The flux then will be: Such that: Resulting in,

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**BASIC CONCEPTS - REVIEW**

2. ELECTROMECHANICAL ENERGY-CONVERSION ENERGY and FORCE

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**Energy storage in a system of current conductors **

CONCEPTS REVIEW ENERGY CONVERSION Energy storage in a system of current conductors Most of the important applications of electromagnetic fields are based in the capacity to store energy. ًًThe instantaneous input power given by the source is: So, the input energy is: In this ideal magnetic circuito the energy must be stored in the system of conductors of current, made of a N turns winding and by currente i.

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**This integral equation gives the total energy stored in the system**

CONCEPTS REVIEW ENERGY CONVERSION Input energy: Faraday`s law: where is the linkage flux This integral equation gives the total energy stored in the system

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**The processes of energizing the winding is seeing in the figure:**

CONCEPTS REVIEW ENERGY CONVERSION The processes of energizing the winding is seeing in the figure: The area above the curve is numerically equal to the Stored Energy. There is no physical correspondence to the area below the curve, but it is called Co-Energy.

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**If the system is LINEAR the ENERGY is EQUAL to the CO-ENERGY**

CONCEPTS REVIEW ENERGY CONVERSION If the system is LINEAR the ENERGY is EQUAL to the CO-ENERGY In linear systems: Linear System

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**If the system is LINEAR the ENERGY is EQUAL to the CO-ENERGY**

CONCEPTS REVIEW ENERGY CONVERSION If the system is LINEAR the ENERGY is EQUAL to the CO-ENERGY In linear systems: Non-Linear System Linear System

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**If the system is LINEAR the ENERGY is EQUAL to the CO-ENERGY**

CONCEPTS REVIEW ENERGY CONVERSION If the system is LINEAR the ENERGY is EQUAL to the CO-ENERGY It can also be shown that the Energy per Volume Unit is: Linear System

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**This is how the interaction occurs.**

CONCEPTS REVIEW ENERGY CONVERSION Force Lets now consider a magneto-mechanic arrangement, to see the exchange of energy between the magnetic field and the mechanic system, and how the magnetic force can be derived: When the current flows in the coil the magnetic flux will produce a force on the iron-magnetic core pulling it to the coil nucleus. This is how the interaction occurs.

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**Since: , a small energy variation is given by:**

CONCEPTS REVIEW ENERGY CONVERTION The force and the magnetic flux are depended of current and position, that is: (i,x), Fm(i,x) Or, it is equally true to state that, the force and the current are depended of flux and position, that is: i( ,x), Fm( ,x) The law of energy conservation requires that any variation in the magnetic energy stored in the magnetic circuit should be balanced, either by a variation in the input energy from the voltage source or by a variation of energy in the mechanical system; the following equation describes this requirement: Since: , a small energy variation is given by:

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**By substitution we arrive in the following equation:**

CONCEPTS REVIEW ENERGY CONVERTION By substitution we arrive in the following equation: The magnetic force can, then, be found, as a function of current (i) and position (x or ) - by the equation:

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**or or CONCEPTS REVIEW MAGNETIC FORCE**

The magnetic force can, then, be found, as a function of current (i) and position (x or ) - by the equation: In lienar systems: ; for this case we can write the equations: or for rotating systems or for rotating systems

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**or or CONCEPTS REVIEW MAGNETIC FORCE**

Alternatively, we can obtain the force as a function of flux () and position (x or ), when and x/ are chosen as independent variables: And for linear systems: or or

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**CONCEPTS REVIEW STEP MOTOR**

Now lets consider a machine with six poles in the stator (armature), arranged in three groups (phases) a-aa, b-bb, c-cc. Coils are wounded for the three phases but, for clarity's sake only the coils in phase a-aa are indicated. The rotor in this example has four poles, as shown. The idea is to review qualitatively the behavior of this system after the energizing of each phase sequentially

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**CONCEPTS REVIEW STEP MOTOR**

When coil a-aa is energized, the rotor searches for a position of minimum reluctance, corresponding, in this case, to the alignment of the rotor in the position: a-aa with I-II Then, the current in coil a-aa is interrupted and coil b-bb is energized The position of minimum reluctance now is reached when b-bb is aligned with m- mm. So, the rotor moves clockwise by an angle of: 90º - 60º = 30º

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**CONCEPTS REVIEW STEP MOTOR**

As the windings become energized sequentially, one at each time, following from a-aa b-bb c-cc, the rotor moves clockwise in steps of 30º . This is a very useful and widely employed machine, known as the “Step Motor”. If the windings are energized in the sequence a-aa c-cc b-bb, the rotor will turn anticlockwise. The speed of the rotation is determined by the rate the current is switched from one winding to the next.

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**CONCEPTS REVIEW STEP MOTOR**

Consider now the case where windings a-aa and b-bb are energized simultaneously. The position of minimum reluctance is not reached by the alignment of a-aa with I-II or by aligning b-bb with m-mm. In fact, the rotor will stop in a position of partial alignment between poles a-I and b-m. This corresponds to a 15º rotation Step motors may be easily electronically controlled . They may be operated at low speeds and admit acceleration without difficulty.

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**CONCEPTS REVIEW MOTORS**

Consider now the case where windings are present in both, the stator and the rotor part of the machine This is a more practical case. The energy stored in such systems can be described by the equation: 1 and 2 are the total linkage flux in coils 1 and 2. The linkage flux in coil 1 is partialy due to curren i1 and partialy to currente i2 :

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**Similarly, the linkage flux in coil 2 is given by:**

CONCEPTS REVIEW MOTORS The linkage flux in coil 1 is partialy due to current i1 and partialy to currente i2 and is given by: Where L1 is the self inductance of coil 1 and M is the mutual inductance Similarly, the linkage flux in coil 2 is given by:

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**Similarly, the linkage flux in coil 2 is given by:**

CONCEPTS REVIEW MOTORS So that, Similarly, the linkage flux in coil 2 is given by:

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**and or CONCEPTS REVIEW MOTORS Substituting:**

in this previously given equation: We obtain: or In a linear system , so the torque in the rotor is obtained:

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**Prof. Elisete Ternes Pereira, 2010**

The presented developments (equations+ideas) are useful in the study of the behavior of electrical machines, and are used in the study of electromechanical energy conversion. Prof. Elisete Ternes Pereira, 2010

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