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## A General Outline of Induction Motors

**The 3ph induction motor structure:**

**Stator**

The stator is formed from thin plates with a thickness of (0.35 – 0.5 mm) of Ferromagnetic iron [1, p48]. These plates are painted before being assembled with insulating materials to reduce Eddy currents losses. The inner part of the stator has a cylindrical shape filled with insulated copper inductors that form the stator windings. These windings form a balanced three-phase circuit connected in either star or delta formation.

Figure 2.1 Induction motor structure

[Source:https://en.wikipedia.org/wiki/Electric_motor#/media/File:Rotterdam_Ahoy_Europort_2011_(14).JPG]

**Rotor**

This part is also formed from thin plates of Ferromagnetic iron assembled together to form a cylindrical shape which has slots on its outer surface.

As in the stator, the rotor coils can also be a three-phase circuit or a multi-phase circuit. The endings of the phases of the rotor coils are connected in star formation and fixed on the axis of the machine. Rotors are sorted in terms of the type of its coils into two types:

**B.1. Squirrel cage Rotor**

In this case an uninsulated aluminum or copper bar is found in each slot where both endings are shorted with rings made of the same material as the bars .The bars and the rings form a square cage shape, which is opened at its end .Most induction motors – especially small and medium capacity ones -are designed this way, where the square cage represents a multi-phase coil connected in star formation at which the number of phases and slots are equal. This kind of rotors is shown in figure 2.2 and in figure 2.3 (C).

[Source: httpsen.wikipedia.orgwikiSquirrel-cage_rotor#mediaFileSquirrel_cage.jpg]

**B.2. Slip Ring Rotor:**

In this case, isolated copper inductors are fixed in the slots connected in series as 3ph groups forming a stable circuit.

The endings are normally connected in star formation where the star point is fixed to the rotor’s body or to the axis of rotation.

On the other hand, the beginnings of the inductors are assembled with isolated copper slip rings that are installed to the rotor’s axis and shorted through external resistances (starting resistances).

The rotor’s windings are formed to have a number of magnetic poles equal to the number of poles in the stator. This kind of rotors is shown in figure 2.3 (A).

**C. Air-Gap:**

The rotor is set apart from the stator by an air-gap which is design to be as small as possible (0.4-0.5 mm) to reduce the leakage flux and the no-load current, enhancing the power factor and the efficiency. In case of large capacities this air-gap gets larger (few mm).

### Basic Theory of Three Phase Induction Motors:

When the stator windings are connected to a three-phase balanced network, a rotating magnetic field is generated -due to current passage through its coils- in a speed that corresponds to the electrical source frequency where the source is connected to the stator.

If the rotating magnetic field creates a pair of poles (N, S) on the stator surface this corresponds to one cycle of the alternative current AC, in this case the number of pole pairs (P) equals (1) [1, p 88].

1 = 1/60 (1)

If the stator has a number of pole pairs (P>1), then the rotating magnetic field frequency will be larger by a factor of (P).This means that the axis of the rotating magnetic field will revolve 360 degrees for every (P) number of the AC cycle as shown in the equations below [1, p 88]:

1 = . 1/60 (2)

Then:

1 = 60. 1 (3)

The rotating magnetic field passes through the stator and the rotor windings inducing electromotive forces (E1, E2). Since the rotor windings are connected to form a closed circuit, a current (I2) passes through those windings.

As a result to the interplay between I2 and the rotating magnetic field, mechanical forces ( = • • ) and electromagnetic torques comes into existence.

These forces and torques revolve the motor in a speed n (rpm), which has the same direction of the magnetic field created by the passage of the electrical current through the stator windings.

The values of E2 and I2 and their frequency of altering depend on the speed of the intersection between the rotating magnetic field and the rotor windings. The speed (n) is less than (n1) due to the losses in the rotating magnetic field spent to overcome the various losses in the machine.

The frequency of (E1, E2) depends on the deference between the synchronous speed (n1) and the motor speed (n) which is called the slip (S) as shown in the equation bellow [1, p89] :

% = 1− . 100% = 1− . 100% = Ω1−Ω . 100% (4)

Where:

• (Ω1) is the mechanical angular speed of the magnetic field rotation and is given by [1, p 90]: Ω1 = 2. . 1 (5)

• ( 1) is the electrical angular speed of the magnetic field rotation and is given by [1, p 90]: = Ω . = 2. . 1 . = 2. . (6) 1 1 60 1

• ( ) is the mechanical angular speed of the motor and is given by [1, p 90]: Ω = 2. . 60 (7)

• ( ) is the electrical angular speed of the motor. Therefore [1, p91]: =(1− ). 1 (8) =(1− ). 1 (9)

The rotor does not revolve synchronically with the magnetic field ( ≠ 1) and this is where the name « Asynchronous motors » comes from. This type of motors is also called « induction motors » because the current that passes through the rotor is created in an inductive way that is not given from an external source.

In order to calculate ( 2) for (E2), (I2) and a speed ( 1 − ) it is found that [1, p92]:

= ( 1− ). (10) 2 60

But: = 1− (11) 1

Therefore:

1 = . 1 (12)

The electromotive forces (emf) induced in the stator and rotor winding are given by the equations [1, p 95]:

1 =4.44∙ 1∙ ℎ1∙ ∙ 1 (13)

2 = 4.44 ∙ 2 ∙ ℎ2 ∙ ∙ 2 (14)

Where ( 2 ) is the emf induced in the rotor windings when revolving with a slip (S) and is also given by the equation [1, p 95]:

2 =4.44∙ 2∙ ℎ2∙ ∙ 2=4.44∙ ∙ 1∙ ℎ2∙ ∙ 2 = ∙ 2 (15)

The values of the slip (S) are determined by the mechanical load applied to the rotation axis. It increases proportionally with the load (the speed decreases).

But if the motor would revolve in the synchronous speed ( = 1) and in the same direction ( = 0), the magnetic field would not have crossed the rotor winding, therefore no emf would have been induced in the rotor and no current would have passed in the rotor windings. Thus, neither mechanical forces nor rotating torques would have come into existence.

As a conclusion, the induction motor should revolve in a speed less than the synchronous speed in order for it to work as a motor, therefore its domain is limited by the following conditions [1, p 96]:

1> >0 (16)

1> >0 (17)

#### The general theory of electrical machines:

This theory is based on representing the real machine with a corresponding idealistic machine in a way where all the physical effects are similar. This idealistic machine is symmetrical (has two poles and two phases) and has two pairs of identical windings on perpendicular axes in both the stator and the rotor. The windings are fed by two AC currents that are shifted by (90°) from each other according to time as shown in Figure (3.1) The idealistic machine is treated as a two poles machine because the magnetic flux distribution is repeated after each pair of poles, no matter how many poles there are in the real machine [1, p 15].

In the idealistic machine, the electrical and the geometrical axes are identical. This property simplifies detecting the rotor’s position with reference to the stator during transient state. This transient state has a very complicated physical effect, which makes it almost impossible to mathematically study the machine without applying some theories and assumptions. The complexity is caused by the nonlinearity of the magnetization curve and the elements of the machine, and their dependence to the currents passing through the windings. Another factor of the complexity is the nonsinusoidal curve of the windings electromagnetic force which alters according to the working system of the machine.

Considering the previous obstacles, it is found that the mathematical study would lead to nonlinear equations that are hard to analyze. Therefore, the basic factors are approximated whereas subsidiary factors are totally neglected. Applying these steps would lead to an analyzable idealistic machine which has the following specifications [1, p 19]:

• The air-gap is steady.

• The winding are distributed on the circumference of the rotor and the stator in a way that ensures a steady distribution for the current, and a sinusoidal distribution for the electromotive force.

• The magnetic circuit is unsaturated.

• The absolute resemblance between the stator and the rotor windings.

• The windings leakage flux is independent from the rotor location.

• The reluctance of the stator and rotor should be neglected.

In all electrical machines, studies in [1, p 21] show that the magnetic fields in both of the rotor and the stator are static with respect to each other. Furthermore, this is a crucial condition of power conversion.

When running electrical machines, a few phenomena occur and it is required to represent these phenomena as mathematical equations in order to study the machine. These equations depend on each other, but one can, anyway, consider them to belong to one of three groups which describe [1, p 22]:

• The windings voltages.

• The torques on the axis of the machine.

• The mechanical motion.

In order for these equations to be written in a standard way, it is important to consider the following [1, p 23]:

• The positive direction of the current passing through the windings is from the windings ending towards its beginning.

• The electromotive force has the same positive direction as the current.

• The positive direction of the magnetic flux is the direction of the flux coming out of the rotor.

• The positive direction of the winding axes and the electromagnetic force is the same positive direction of the current.

• The positive direction of the machine revolution, the electromotive force and the angles calculation is counter clockwise.

To be able to study the mathematical model, it is customary to choose one of the following rectangular coordinates that are commonly used in this case [1, p 24]:

1. The coordinates system of (α) and (β) which is fixed on the stator.

2. The coordinates system of (d) and (q) which is fixed on the rotor and revolves with it in the same speed.

3. The coordinates system of (u) and (v) which is fixed on optional coordinates and revolves in an optional speed as well.

The power conversion process in electrical machines does not depend on choosing the coordinates system. However, it is preferred to use the (α) and (β) coordinates system to study power conversion equations, because the voltage equations in the stator have a minimum number of terms. Furthermore, the observer is static in reference to the stator and the frequency of the network voltage in this coordinates system.

**Table of contents :**

Abstract

Acknowledgment

**1. Introduction **

**2. A General Outline of Induction Motors**

2,1. The 3ph induction motor structure

2.2. Basic Theory of Three Phase Induction Motors

**3. The general theory of electrical machines **

**4. Simulating and starting the 3ph induction motor in Matlab**

4.A. The induction motor model

4. B. Simulating starting curves in No-load \ Load cases

4. C. Direct starting method

4. C.1. Direct starting curves in the No-load case

4. C.2 Direct starting curves in the 75% of nominal load case

4. C.3 Direct starting curves in the nominal load case.

**5. Soft staring method **

5.1 Soft starting curves in the nominal load case

5.2 Soft starting curves in the 75% of nominal load case

5.3 Soft starting curves in the 50% of nominal load case

5.4 Soft starting curves in the 25% of nominal load case

**6. Conclusions and discussion… **

**References**

Appendix 1

Notes

Matlab code

Glossary of symbols

Appendix 2