This entry includes the Table of Contents, Glossary, References, and Appendix for *Investigation of magnetic circuit of linear generators for wave power plants*.

## Table of contents

Abstract | (2) |

Acknowledgements | (2) |

Nomenclature | (3) |

1. Introduction to renewable and sustainable wave power | (5) |

1.1 Power from the ocean waves | (5) |

1.2 Objective with the project is a more flexible design | (6) |

1.3 Scope of the report and limitations | (7) |

2 Theory on the magnetic circuit of a linear generator | (7) |

2.1 Maxwell’s equations | (7) |

2.2 Magnetized matter | (8) |

2.3 Magnetic material | (10) |

2.4 Ferrites of type Y30 and Y40 | (11) |

2.5 Pole shoes | (12) |

2.6 Linear Generator | (12) |

3 Methodology of project combines theory, simulations and discussions | (14) |

3.1 Hypothesis | (14) |

3.2 Simulations of the linear generator | (15) |

3.2.1 Simulation 1, mixed ferrite permanent magnets | (17) |

3.2.2 Simulation 2, different designed pole shoes | (18) |

3.2.3 Simulation 3, mixed magnets and different pole shoe | (19) |

4 Results for mixed magnets and different pole shoes | (21) |

4.1 Results of Simulation 1, mixed ferrite permanent magnets | (21) |

4.2 Results of Simulation 2, different designed pole shoes | (22) |

4.3 Results of Simulation 3, different magnets and pole shoes | (23) |

5 Analysis on mixed magnets and different pole shoes | (25) |

6 Conclusions of the project and discussion on the possibility to create a more flexible design of the linear generator | (26) |

7 Outlook and future work with a wave power converter with mixed types of magnets and different shaped pole shoes | (27) |

8 References | (29) |

Appendix A | (32) |

## Nomenclature

B |
T | Magnetic flux density |

Br | T | Remanence |

(BH)max | J/m3 | Energy product |

D |
C/m2 | Displacement field |

E |
V/m | Electric field |

ε_{0} |
Electric constant | |

F |
N | Force |

H |
A/m | Magnetic field strength |

Hc | A/m | Coercivity |

Hci | A/m | Intrinsic coercivity |

hpm |
m | Height of permanent magnet |

Ipm |
A | Coil current |

Jf |
A/m2 | Free current density |

M |
A/m | Magnetization |

m |
Am2 | Magnetic moment |

P |
C/m2 | Polarization |

ρf |
C/m3 | Free charge density |

Q | C | Electric charge |

ℜ | A/Wb | Magnetic reluctance |

τ |
Nm | Torque |

v |
m/s | Velocity |

ΦB | Wb | Magnetic flux |

χm |
Magnetic susceptibility | |

μr |
N/A2 | Relative permeability |

μ0 | N/A2 | Permeability of free space |

## 8 References

The references are listed according to the IEEE 2006 style.

[1] International Energy Agency, “Fossil fuel energy consumption (% of total)” 2013. [Online]. Available: http://data.worldbank.org/indicator/EG.USE.COMM.FO.ZS/countries?display=graph. [Accessed 21 November 2013]. [2] C. Curtis, “The White House Blog: President Obama Visits the Argonne National Research Lab to Talk About American Energy Security” 15 March 2013. [Online]. Available: http://www.whitehouse.gov/blog/2013/03/15/president-obama-visits-argonne-national-research-lab-talk-about-american-energy-secu. [Accessed 21 November 2013].[3] “Renewable Energy: What do we want to achieve?” European Commission, [Online]. Available: http://ec.europa.eu/energy/renewables/index_en.htm. [Accessed 21 November 2013].

[4] B. K. Sovacool, “The avian benefits of wind energy: A 2009 update” Renewable Energy, World Renewable Energy Congress — XI, vol. 49, pp. 19-24, 2013.[5] United States Environmental Protection Agency, “Hydroelectricity” 25 September 2013. [Online]. Available: http://www.epa.gov/cleanenergy/energy-and-you/affect/hydro.html. [Accessed 25 November 2013].

[6] United Nations, “UN Documents : Our Common Future, Chapter 2: Towards Sustainable Development” in A/42/427. Our Common Future: Report of the World Commission on Environment and Development, 1987.

[7] B. Ekergård, Full Scale Applications of Permanent Magnet Electromagnetic Energy Converters: From Nd2Fe14B to Ferrite, Uppsala: Acta Universitatis Upsaliensis, 2013.[8] Division for Electricity: The Ångström Laboratory, “Wave Power Project – Lysekil” 9 April 2013. [Online]. Available: http://www.el.angstrom.uu.se/forskningsprojekt/WavePower/Lysekilsprojektet_E.html. [Accessed 9 December 2013].

[9] J. Emsley, “Neodymium” in Nature’s Building Blocks: An A-Z Guide to the Elements, Oxford, Oxford University Press, 2011, pp. 337-340.

[10] E-Magnets UK, “Introduction to Ferrite Magnets” [Online]. Available: http://www.ferrite-info.com/information.aspx. [Accessed 27 November 2013].

[11] J. Karlsson and O. Söderström, “Review of Magnetic Materials Along With a Study of the Magnetic Stability and Solidity of Y40” Uppsala University, Uppsala, 2012.[12] ABB Coporate Research, Ace 2.2 User Manual, The ABB Common Platform for 2D Field Analysis and Simulation, Västerås: Asea Brown Boveri, 1993.

[13] D. J. Griffiths, “Introduction to electrodynamics” Pearson Education, San Fransisco, 2008.[14] N. A. Spaldin, Magnetic materials: fundamentals and applications, 2. ed., Cambridge: Cambridge University Press, 2011.

[15] Wikipedia, “Magnetic Dipole” 7 November 2013. [Online]. Available: http://en.wikipedia.org/wiki/Magnetic_dipole. [Accessed 15 November 2013].

[16] O. Danielsson, Wave Energy Conversion: Linear Synchronous Permanent Magnet Generator, Uppsala: Universitatis Upsaliensis, 2006.

[17] M. Divine and M. Schilling, “The Attraction of Medical Magnets” July 2013. [Online]. Available: http://www.onlinetmd.com/tmd0713-medical-magnets-attraction.aspx. [Accessed 15 November 2013].

[18] S. Eriksson and H. Bernhoff, “Rotor Design for PM Generators Reflecting the Unstable Neodymium Price” 2012.

[19] S. Eriksson, Direct Driven Generators for Vertical Axis Wind Turbines, Uppsala: Acta Universitatis Upsaliensis, 2008.

[20] S. Lindroth, Buoy and Generator Interaction with Ocean Waves: Studies of a Wave Energy Conversion System, Uppsala: Acta Universitatis Upsaliensis, 2011.

## Appendix A

Vector Analysis

In order to understand Maxwell’s equations, vector analysis is needed. The vector operator called *del* is used in Maxwell’s equations and can be written as

The del-operator can act upon a scalar, S, to form the gradient, ∇S. Moreover, the del-operator can act upon a vector, **v**. When the del-operator acts with the dot product, the divergence, ∇·**v**, will be formed. When the del-operator acts with the cross product, the curl, ∇×**v**, will be formed. The use of the del-operator can be interpreted by geometrical meanings. The gradient ∇S has direction and magnitude. The direction of the gradient is the direction where the function S increases the most, and the magnitude of the gradient is the rate of change in this direction. The divergence, ∇·**v**, is related to how the vector v diverges from a reference point, whereas the curl, ∇×**v**, is related to how much the vector field is curled around a reference point. The fundamental theorem of calculus,

states that the integral of a derivative, calculated from the boundaries *a* to *b*, is the difference of the function values at these boundaries. A fundamental theorem can be stated for the divergence of a vector field **v**,

and this theorem is known as Gauss’s theorem. The flux of the vector field **v** from a reference point inside the volume is the same as the amount of the field flowing out of the surface. A fundamental theorem for the curl of a vector field **v** can be stated as Stokes’ theorem,

which can be interpreted with the fundamental theorem of calculus. That is, the sum of all curls over a specified surface S is the same as the total flow around the closed loop of the edge of the surface. The vector analysis used for calculations with electric and magnetic fields can be further studied in [13].