# Electrical Machine Analysis using Finite Elements

*One can find a lot of books about finite element methods, but there is just a couple of them devoted to electrical machines. This book is the most recent one in this area.*

The content of this book can be divided into two parts, with the first 70 pages providing a general introduction about the finite element method for electrical machines, and a second part of 190 pages about how to apply the finite element analysis to different types of electrical machines. The book covers methodologies for two-dimensional finite element analysis.

The first part contains a first section about electromagnetic fields, followed by a section about the basic principles of finite element methods. These principles are then applied to two-dimensional fields in the third section. Here, the mathematical background is laid about how to setup a functional associated with every element of the finite element domain, how to merge it into a functional for the whole domain, how to assign the essential boundary conditions, and how to solve the problem.

The last section of the first part is about the analysis procedure. It discusses how to reduce the field problem to a two-dimensional problem, and discusses different boundary conditions, eg. Dirichlet's condition, Neumann's condition, and periodic or antiperiodic conditions. Finally, Bianchi explains how to extract information from the solution of the field problem, which consists of the knowledge of the magnetic vector potential in every point of the finite element domain: Flux lines, magnetic flux and flux linkage, Joule power loss, magnetic energy and coenergy, and magnetic forces.

"Electrical Machine Analysis using Finite Elements" covers the following types of electrical devices, which together form the second part of the book:

- Cylindrical magnetic devices, such as linear actuators
- Single-phase transformers
- Single-phase variable reactances
- Synchronous generators
- Surface-mounted permanent magnet motors
- Interior permanent magnet and reluctance synchronous motors
- Self-starting single-phase synchronous motors
- Switched reluctance motors
- Three-phase induction motors

Let's have a closer look at the section about "Surface-mounted permanent magnet motors", in order to give you an idea about what this book is mainly about. As all other sections, it starts with an introduction, providing some background and introducing main machine topologies (in this case: Inner and outer rotor, radial and axial flux). Then, Bianchi discusses computation issues at no-load conditions, such as choosing boundary conditions, and computing flux linkage, induced emf, as well as cogging torque. Then, the computation of the torque at load conditions is discussed, both by means of Maxwell's stress tensor, by means of the virtual work principle, and by means of flux linkages and currents. The section also contains a cross reference to another section of the book when discussing the computation of the inductances, which can be derived from the magnetic energy, from the flux linkage, or from the airgap flux density distribution.

The section about "Surface-mounted permanent magnet motors" is concluded with some finite element simulation results. Helpful and illustrative examples are provided all along this book and are a great help in understanding the discussed techniques and methodologies. Some simulation results are also compared with experimental results. In addition, Bianchi provides some code of simple functions and algorithms. This gives the reader an understanding of how to practically implement the discussed theory. As with the rest of the book, these code sniplets are not directly coupled to a specific software program.

### Conclusions

The book "Electrical Machine Analysis Using Finite Elements" has been and still is invaluable for designing the online electric motor design software Emetor. The book provides all necessary tools and equations to setup and solve a finite element simulation, which is sort of a prerequisite. However, the main point of reading this book is that it increased my understanding about necessary mathematical and numerical techniques and methodologies for analyzing the results of the finite element simulations. This book holds answers to most of the questions I'm dealing with on a daily basis. However, it does not include extensive in-depth information about specialized topics, which I think are better off in research publications.

As a conclusion, I fully agree with the publisher, who states: "With step-by-step coverage of the fundamentals and common procedures, this book offers a superior analytical framework that allows you to adapt to any type of electrical machine, to any software platform, and to any specific requirements." I can only recommend you the book "Electrical Machine Analysis using Finite Elements", although I have to say that the language leaves some room for improvements.

### About the author

Nicola Bianchi is a native of Verona, Italy. He received the Laurea and Ph.D. degree in Electrical Engineering from the University of Padova, Italy, in 1991 and 1995 respectively. In 1998, he joined the Department of Electrical Engineering of the same University, as Assistant Professor in Electrotechnics. Since 2005 he is an Associate Professor in Electrical Machines, Converters and Drives at the Electrical Drive Laboratory.

Today, Nicola Bianchi is engaged in research about the design of electrical machines, especially for drives applications. He is author and co-author of several scientific papers on electrical machines and drives, and of two international books on the same subject (one of them is reviewed here). One of his teaching activities deals with the design of electrical machines, where he introduced the finite element analysis approach as part of the course. The book "Electrical Machine Analysis using Finite Elements" originates from the lecture notes of this particular course.

### Unofficial Errata

I have been studying some parts of this book in detail and want to share some errata and helpful remarks with you. Hopefully it helps you to avoid some misunderstandings and problems that I have been running into.

**Page 44:**- Not sure if the area $A_m$ is always getting positive in equation (3.21) as it should. To be on the safe side, consider making the area an absolute value. An alternative representation of the area of the m-th triangular element is: $$A_m = \left|\frac{1}{2} [(x_2-x_1)\cdot(y_3-y_1)-(x_3-x_1)\cdot(y_2-y_1)] \right|$$
**Page 47f:**- The last term (driving force) in the functional of the m-th element $F_m$ in equations (3.33), (3.36), and twice in (3.37) should have a negative sign instead of a positive sign. In such a way, these equations correspond with equations (3.13) and (3.39).
**Page 49f:**- Please consider that the orientation of the local functionals is important. In Figure 3.3, the functional $F_a$ is oriented counterclockwise, while the functional $F_b$ is oriented clockwise. The orientation of the functionals is connected to the direction of the driving forces $t_i$, which is not considered in equations (3.40) and (3.41), respectively.
**Page 58:**- The generic component of the column vector $[T_m]$ in equation (3.82) should read as: $$t_i = \frac{1}{2\mu}\left( B_{res,y}q_i-B_{res,x}r_i\right)$$ The driving force of a permanent magnet is proportional to the residual flux density times the length of the magnet. In a triangular element, however, the average length is only half the length of the maximum length ($r_i$, respectively $q_i$), which explains the missing factor 2 in equation (3.82).

### Alternative reading

Finite Element Analysis of Electrical Machines by **Sheppard J. Salon**, Springer, 1995.

*Electromagnetic Modeling by Finite Element Methods* by **João Pedro A. Bastos, Nelson Sadowski**, CRC Press, 2003.

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