2. Ferromagnetic core

Magnetic fields are the essential means by which energy is converted from one form to another in motors, generators, and transformers. The most important class of the magnetic materials is the ferromagnetic materials such as iron, cobalt, nickel, and manganese [2].

There are four basic principles which describe how magnetic fields are used [2]:


#### 2.1. The magnetic field

The magnetic field is produced by induced current in Ampere's law:

$$
\oint \mathbf{H}.\mathbf{dl} = I\_{\text{net}} \tag{1}
$$

where Inet produces magnetic field intensity and H and dl are the length integration along a path. If the core is produced from ferromagnetic material (Figure 1), then all the magnetic field produced within the core will remain inside the core. Therefore, the path of integration dl in the Ampere's law is the mean path length lc [2].

The current passing in the path of the integration Inet is NI since the coil of the wire divides the path of integration into N times when the current passes through it:

$$H.l = NI \Rightarrow H = \frac{NI}{l} \tag{2}$$

The magnetic field intensity H is the effort in which a current is applying to establishment of a magnetic field. Strength of the magnetic field depends on the material of core. There is a relationship between the magnetic field intensity, the material magnetic permeability µ, and the magnetic flux produced within the material as shown in Eq. (3):

$$B = \mu H \tag{3}$$

The permeability of free space is called <sup>µ</sup><sup>0</sup> and equal to 4<sup>π</sup> � <sup>10</sup>�<sup>7</sup> H/m, and the relative permeability is the permeability of any other material compared to the free space permeability:

$$
\mu\_r = \frac{\mu}{\mu\_0} \tag{4}
$$

In the core (Figure 1), the magnitude of the flux density is given by

$$B = \mu H = \mu \frac{NI}{l} \tag{5}$$

Therefore, the total flux in a given area is expressed in Eq. (6). This equation reduced if the flux density vector is perpendicular to any plane of area, and if the flux density is constant throughout the area, then to

$$
\oint\_A \phi = B.dA \Rightarrow \phi = B.A = \mu HA = \mu \frac{NI}{l} A \tag{6}
$$

Figure 1. Ferromagnetic core.

The study of an efficient power system starts with understanding the behavior of each component that develops this system. Electric machines used in power systems (generators, motors, and transformers) will be examined through analytical expressions and computer simulation. The importance of simulation is that these components could be studied before it is manufactured; thus, the consequences of changing dimensions and parameters can be assessed.

This simulation will be implemented in an educational tool, going from the basic operation principles, through developing models and equations toward the solution. The graphical user interface of MATLAB allows the students to study and analyze the effect of each parameter in

This chapter will discuss the implementation of ferromagnetic core using graphical user interface taking into consideration the effects of air gap and fringing of a ferromagnetic core. Then, a detailed study of output power and losses with voltage regulation and efficiency of a single- and three-phase transformer will be established. In addition, a special survey will be accomplished concerning the types of DC motors and generators. Finally, this chapter will be concluded by providing an adequate research on the induction machines including their

This chapter presents learning situations going from the theoretical expansion to the graphical

Magnetic fields are the essential means by which energy is converted from one form to another in motors, generators, and transformers. The most important class of the magnetic materials is

1. A wire produces a magnetic field in the area around it when current passes through it.

2. A change in magnetic field, by mutual inductance, induces a voltage in the coil of wire:

3. In the presence of a magnetic field, a current-carrying wire has a force induced on it: this is

4. In the presence of a magnetic field, a moving wire has a voltage induced in it: this is the

∮ H:dl ¼ Inet ð1Þ

order to understand its electric behavior with respect to its electric model.

parametric study, and it will be achieved by a general conclusion of this work.

interpretation. It is a teaching methodology toward the science education.

the ferromagnetic materials such as iron, cobalt, nickel, and manganese [2].

The magnetic field is produced by induced current in Ampere's law:

this is the principle of transformer action.

the principle of motor action.

principle of generator action.

2.1. The magnetic field

There are four basic principles which describe how magnetic fields are used [2]:

2. Ferromagnetic core

154 Science Education - Research and New Technologies

#### 2.2. Magnetic circuits

Magnetic flux is produced when the current in a coil of wire is wrapped around a core. This is similar to a voltage in an electric circuit producing a current flow. Thus, a "magnetic circuit" is defined by equations that are similar to that of an electric circuit. In the design of electric machines and transformers, the magnetic circuit model is used to simplify the complex design process [2].

The voltage or electromotive force drives the current flow in the electric circuit. The magnetomotive force of the magnetic circuit is denoted by where is the magnetomotive force in ampere-turns. In the magnetic circuit, the applied magnetomotive force causes flux (φ) to be produced (Figure 2).

The relationship that governs the magnetomotive force and flux is given by

$$\mathfrak{S} = NI = \phi \mathfrak{R} \tag{7}$$

The permeance of a magnetic circuit is the reciprocal of its reluctance. Therefore, the relation between magnetomotive force and flux can be expressed as

$$
\phi = \mathfrak{S}P \Rightarrow \phi = \mathfrak{F}\frac{1}{\mathfrak{R}} \tag{8}
$$

It is easier to work with the permeance of a magnetic field than with its reluctance.

The resulting flux and reluctance of a core are shown in Eqs. (9) and (10), respectively:

$$\phi = \mathfrak{F}\frac{\mu A}{l} \tag{9}$$

$$\mathcal{R} = \frac{l}{\mu A} \tag{10}$$

The equivalent reluctance of a number of reluctances in series is just the sum of the individual reluctances:

Figure 2. (a) A simple electric circuit. (b) The magnetic circuit analogue to a transformer core.

Electric Machines: Tool in MATLAB http://dx.doi.org/10.5772/intechopen.68957 157

$$\mathcal{R}\_{eq} = \mathcal{R}\_1 + \mathcal{R}\_2 + \mathcal{R}\_3 + \dots \tag{11}$$

The equivalent reluctance of a number of reluctances in parallel is just the sum of the individual reluctances:

$$\mathcal{R}\_{eq} = \frac{1}{\mathcal{R}\_1} + \frac{1}{\mathcal{R}\_2} + \frac{1}{\mathcal{R}\_3} + \dots \tag{12}$$

The reluctance of each leg of a ferromagnetic core is

$$\mathcal{R}\_{\mathbf{x}} = \frac{l\_{\mathbf{x}}}{\mu\_r \mu\_0 A\_{\mathbf{x}}} \mathbf{A} . t \!/\_{wb} \tag{13}$$

The air-gap reluctance at leg X is

2.2. Magnetic circuits

156 Science Education - Research and New Technologies

produced (Figure 2).

process [2].

reluctances:

Magnetic flux is produced when the current in a coil of wire is wrapped around a core. This is similar to a voltage in an electric circuit producing a current flow. Thus, a "magnetic circuit" is defined by equations that are similar to that of an electric circuit. In the design of electric machines and transformers, the magnetic circuit model is used to simplify the complex design

The voltage or electromotive force drives the current flow in the electric circuit. The magnetomotive force of the magnetic circuit is denoted by where is the magnetomotive force in ampere-turns. In the magnetic circuit, the applied magnetomotive force causes flux (φ) to be

The permeance of a magnetic circuit is the reciprocal of its reluctance. Therefore, the relation

<sup>φ</sup> <sup>¼</sup> <sup>ℑ</sup><sup>P</sup> ) <sup>φ</sup> <sup>¼</sup> <sup>ℑ</sup> <sup>1</sup>

<sup>φ</sup> <sup>¼</sup> <sup>ℑ</sup> <sup>μ</sup><sup>A</sup>

<sup>ℜ</sup> <sup>¼</sup> <sup>l</sup>

The equivalent reluctance of a number of reluctances in series is just the sum of the individual

It is easier to work with the permeance of a magnetic field than with its reluctance.

Figure 2. (a) A simple electric circuit. (b) The magnetic circuit analogue to a transformer core.

The resulting flux and reluctance of a core are shown in Eqs. (9) and (10), respectively:

ℑ ¼ NI ¼ φℜ ð7Þ

<sup>ℜ</sup> <sup>ð</sup>8<sup>Þ</sup>

<sup>l</sup> <sup>ð</sup>9<sup>Þ</sup>

<sup>μ</sup><sup>A</sup> <sup>ð</sup>10<sup>Þ</sup>

The relationship that governs the magnetomotive force and flux is given by

between magnetomotive force and flux can be expressed as

$$\mathcal{R}\_{\text{xa}} = \frac{l\_{\text{xa}}}{\mu\_0 A\_{\text{xa}}} A.t/\_{\text{wb}} \tag{14}$$

The total flux of the ferromagnetic core is

$$
\phi\_{TOT} = \frac{\mathfrak{F}}{\mathfrak{R}\_{eq}} wb \tag{15}
$$

#### 2.3. Implement in MATLAB GUI

When implementing in MATLAB, the user will add certain input which will then be calculated, and the result will be displayed. Below is a block diagram of the system.

The user fills the number of regions with availability of air gap indicating which leg is available and the details for core type such as relative permeability of the material and number of turns with the current (Figure 3). The results of the calculated parameters such as total flux and total reluctance and magnetomotive force of ferromagnetic core are displayed (Figure 4).

Figure 3. Ferromagnetic core GUI block diagram.

Figure 4. Graphical user interface for ferromagnetic core.

Also, the user should add the parameters of the ferromagnetic core such as length, area, air gap, and fringing percentage of each leg of the core; the ferromagnetic core is displayed after entering the inputs. Push buttons are added to load, save data, clear, and quit.
