## 2 Theory on the magnetic circuit of a linear generator

The theory needed to understand the magnetic circuit in a wave power converter is stated in Section 2.1–2.6. The structure of the theory section starts with a wider theoretical view of magnetic properties and then moves towards a deeper understanding, important for the linear generator. Firstly, Maxwell’s equations, which are important for all work with electromagnetism, are stated in Section 2.1. Secondly, the properties of magnetized matter are discussed in Section 2.2. Thirdly, different types of magnetic materials are briefly discussed in Section 2.3, with a focus on ferrimagnetic materials and their hysteresis behaviour. Fourthly, the different types of ferrites called Y30 and Y40 are presented in Section 2.4. Thereafter, the properties of the pole shoes are mentioned in Section 2.5. Finally, the magnetic properties of the linear generator used for wave power conversion is discussed in Section 2.6. All presented theory was used to generate the hypothesis and the work methodology in Section 3, and to understand the given results, presented in Section 4.

### 2.1 Maxwell’s equations

In order to understand the energy conversion in a linear generator, Maxwell’s equations are needed. The vector algebra sufficient for Maxwell’s equations is given in Appendix A. For further reading, Griffith’s book *Introduction to Electrodynamics* [13] is recommended; here it is used as a source of reference.

Maxwell’s equations describe the relations between the vector fields, the charge densities and the current densities. The different fields used in the equations, **D**, **B**, **E** and **H**, have different interpretation. The **E**-field is the electric field produced by stationary charges and the **B**-field is the magnetic flux density generated by moving charges. The **D**-field is the displacement field and the **H**-field is the magnetic field strength. Moreover , ρ* _{f}* is the free charge density and

**J**

*is the free current density. Maxwell’s equations can be stated as shown in Equations (1)–(4),*

_{f}

This way of writing Maxwell’s equations is useful when dealing with electric or magnetic polarized matter, as only the free charge density and the free current density are used in the equations [13]. Equation (1), Gauss’s law, states that the divergence of the displacement field is equal to the free charge density. Equation (2) states that the divergence of the magnetic flux density is zero, which leads to the fact that there are no magnetic monopoles. The magnetic flux **Φ**_{B} through a surface *dA* can be stated as

In the case of a closed surface the magnetic flux is zero, which is equivalent to (2) and the equation

Equation (3) is called Faraday’s law of induction and is an important equation in the understanding of a linear generator. Equation (3) states that a time varying B-field induces an electric field with a direction that opposes the change in the **B**-field. In terms of magnetic flux, Faraday’s law can be written as

which states that a time varying magnetic flux through a closed loop generates an electric field. Equation (4) is called Ampere’s law and states that the sum of the free current density and the time derivative of the displacement field is the same as the curl of the magnetic field strength.

### 2.2 Magnetized matter

More equations are required in order to relate the **D**-field and the **H**-field to the **E**-field and the **B**-field. When a dielectric material is subjected to an electric field, it becomes polarized. This means that the atoms of the material will gain dipole moment and align in the direction of the electric field. The polarization, **P**, is the total dipole moment per unit volume. The relation between the displacement field, the electric field and the polarization is

where ε_{0} is the electric constant [13].

All matter contains atoms with moving charges (electrons); however , the exact description of the movement is beyond the scope of this text. In accordance with Equation 3 (Section 2.1), Ampere’s law, moving charges (electrons) generate magnetic fields. The magnetic field of a small permanent magnet is the same as the magnetic field created by a small current loop [14], as can be seen in Figure 3. Both the permanent magnet and the current loop are called magnetic dipoles, and they have a magnetic moment **m**.

*Figure 3. Magnetic dipoles, with a magnetic moment m, will generate magnetic B-fields. The reference for Figure 3 is [15].* [Figure not shown]

The magnetic fields from the electrons could be cancelled out if the charges move randomly. However , if the material is magnetized, i.e. subjected to an external magnetic field, its magnetic dipoles gain moment and align in the direction of the magnetic field. This can generate steady current loops in the material and a steady magnetic field. The magnetization **M** is the total magnetic dipole moment per unit volume. The magnetization **M** gives a relation between the **H**-field and the **B**-field,

where *μ _{0}* is the permeability of free space. The relation between the magnetization and the magnetic field strength can be determined by the magnetic susceptibility,

*χ*, since

_{m}

If *μ _{r}* is the relative permeability, the relation between the magnetic field strength and the magnetic flux density is

for

according to [16]. Equation 11 is used to set the parameters for the simulation program. Moreover, Lorentz force law describes the force **F** acting on an electric charge **Q** moving with velocity **v** in an electric and magnetic field,

### 2.3 Magnetic material

*Figure 4. The alignment of magnetic dipoles of different magnetic materials in the absence of an external magnetic field. The reference for Figure 4 is [17].* [Figure not shown]

All material can be divided into different groups, depending on how magnetic it is. The different types of magnetic materials are depicted in Figure 4, where the alignment of the magnetic dipoles in the materials, in absence of an external magnetic field, differs between the groups [17]. The magnetic dipoles of a paramagnetic material (A) have random orientation. In contrast , the dipoles of a ferromagnetic material (B) only have magnetic dipoles aligned in the same direction. The antiferromagnetic material (C) has opposite directions of half of the magnetic dipoles, cancelling out the magnetic field. The ferrimagnetic material (D) also has opposite directions on half of the magnetic dipoles, however , the magnetic dipoles in one direction are stronger than in the opposite direction, generating a magnetic field. Ferrite permanent magnets, discussed in this report, are made of ferrimagnetic material. As a result , ferromagnetic and ferrimagnetic materials produce a magnetic field at all times. If the magnetic dipoles in a material are subjected to an external **H**-field, a torque is generated as

Equation 14 shows that a dipole will not experience any torque if the moment of the dipole is in the same direction as the **B**-field. The torque can change the direction of the dipoles and, for example, order the dipoles of a paramagnetic material, which then will produce a magnetic field.

Magnetic materials can be characterized by their magnetization curve where an external **H**-field is plotted against the resulting **B**-field or against the magnetization field **M**. For permanently magnetized materials, i.e. ferromagnetic and ferrimagnetic materials, the characteristic shape of the curve is called a hysteresis loop. The hysteresis loops for a ferromagnetic material can be seen in Figure 5.

*Figure 5. A permanently magnetized material is characterized by the hysteresis loop. The material is magnetized by an external H-field until it reaches a saturation level. As the external field is removed, the B-field is lowered to the remanence value. The B-field will be zero as an opposite H-field, called the coercive field, is applied. The intrinsic coercivity and the (BH)max value is shown. The reference for Figure 5 is [11].* [Figure not shown]

The hysteresis loop in Figure 5 shows that a ferromagnetic material can be magnetized by an external magnetic field strength H. The material reaches a saturation level, B_{s}, where a further increase of the external field will not increase the magnetization of the material. If the external field **H** is removed, the material will generate a magnetic field called the remanence, B_{r}. In order to lower the **B**-field to zero, an external **H**-field, called the coercivity H_{c}, in the opposite direction is required [16]. However , as soon as the coercive field is removed, the **B**-field of a permanently magnetized material will return. Moreover , the intrinsic coercivity, H_{ci}, is the point where the magnetization field is zero, also indicating how the matter can be demagnetized. The value (BH)max corresponds to the maximum area under the BH-loop, indicating the operating point [11].

### 2.4 Ferrites of type Y30 and Y40

The permanent magnets used in the translator can be created by different types of magnets; here, ferrite permanent magnets, called ferrites or ceramic magnets, are being discussed. Ferrites consist of iron oxide combined with other materials, for example barium or strontium. Thus , ferrites do not corrode, which is favourable in a wave power converter. Moreover , the cost of the ferrites is relatively low, as stated in Section 1.1.2. There are many different types of ferrites on the market, with different remanence and coercivity, for example. The same ferrite can have different names and the magnet which is called Y30 here can also be called Feroba 2, Fer 2, Ferrite C5 or HF26/18 [10]. In this project, the permanent magnets discussed are mainly Y30 and Y40. In Table 1, some important properties of the two ferrites Y30 and Y40 are shown. Y40 can be considered as a stronger ferrite permanent magnet than Y30. The references used for Table 1 are [7], [10] and [18].

Table 1. Some properties of the ferrite permanent magnets Y30 and Y40.

Values from Table 1 were used to calculate the permeability, since

where the permeability of free space equals 4π10^{-7} N/A². The similarities between a current loop and a magnet, shown in Figure 3, can be used, for example, when simulating the magnetic fields from a magnet. If the magnet is a permanent magnet, the equation

can be used to relate the coil current, * I_{pm}*, equivalent to a permanent magnet of height

*[19]. That is, a permanent magnet of known size and material properties can be modelled with a current.*

**h**_{pm}### 2.5 Pole shoes

Magnetic reluctance, ℜ, describes how well a material can conduct magnetic fields, and is calculated as

for a magnetic circuit with a conductor of the length *l* and area *A*, the permeability of free space *μ _{0}* and the relative permeability

*μ*[11]. A pole shoe is a device made of steel, which has low magnetic reluctance. The purpose of a pole shoe is to lead the magnetic flux in the desired direction. The magnetic reluctance in the pole shoes is lower than the magnetic reluctance in air, and therefore the magnetic flux goes through the pole shoes, rather than through the air [11].

_{r}### 2.6 Linear Generator

Faraday’s law of induction states that a time-varying magnetic flux density induces an electric field in the opposite direction, as discussed in Section 2.1. For a linear generator in a wave power plant, the magnets of the translator move relative to the stator, distanced by a small air gap [7], which can be seen in Figure 6. The moving translator contains permanent magnets that cause a time varying magnetic flux density through the stator steel. The stator consists of copper windings in closed loops through the laminated stator steel.

*Figure 6. A linear generator consists of a moving translator with permanent magnets and a stationary stator with stator steel and copper windings. The reference for Figure 6 is [20].* [Figure not shown]

There is a difference in desired magnetic properties of the permanent magnets on the translator and the stator steel of the stator. A permanent magnet should have a high remanence (hard magnetic material), as it should generate a high magnetic field even though the external magnetic field is removed, and it should have a high coercivity, as a permanent magnet should not lose its magnetization easily [19]. Unlike the permanent magnets, the stator steel should have a low remanence (soft magnetic material) and a low coercivity, as it will generate the time-varying magnetic flux, required in Faraday’s law of induction, in order to induce voltage [19]. The hysteresis loop of a soft magnetic material (a) and a hard magnetic material (b) can be seen in Figure 7.

*Figure 7. A soft magnetic material is characterized by a low remanence and a low coercivity (a). A hard magnetic material is characterized by a high remanence and a high coercivity (b). The reference for Figure 7 is [19].* [Figure not shown]

Worth noting is that the stator steel used in wave power plants is saturated at about 1.8 T, which means that the magnetic field from the permanent magnets should not be higher than this. The wave power converter and the magnetic field in the linear generator are shown in Figure 8, where the upper limit of the magnetic field is at 1.8 T.

*Figure 8. The wave energy conversion system with a closer view on the magnetic field, generated in the linear generator as the buoy moves with the ocean waves. The reference for Figure 8 is [7].* [Figure not shown]