Topology Optimization and AI-based Design of Power Electronic and Electrical Devices: Principles and Methods

By Hajime Igarashi

Topology Optimization and AI-based Design of Power Electronic and Electrical Devices: Principles and Methods provides an essential foundation in the emergent design methodology as it moves towards commercial development in such electrical devices as traction motors for electric motors, transformers, inductors, reactors and power electronics circuits. Opening with an introduction to electromagnetism and computational electromagnetics for optimal design, the work outlines principles and foundations in finite element methods and illustrates numerical techniques useful for finite element analysis. It summarizes the foundations of deterministic and stochastic optimization methods, including genetic algorithm, particle swarm optimization and simulated annealing, alongside representative algorithms. The work goes on to discuss parameter optimization and topology optimization of electrical devices alongside current implementations including magnetic shields, 2D and 3D models of electric motors, and wireless power transfer devices. The work concludes with a lengthy exposition of AI-based design methods, including surrogate models for optimization, deep neural networks, and automatic design methods using Monte-Carlo tree searches for electrical devices and circuits.

• Assists researchers and design engineers in applying emergent topology design optimization to power electronics and electrical device design, supported by step-by-step methods, heuristic derivation, and pseudocodes

• Proposes unique formulations of AI-based design for electrical devices using Monte Carlo tree search and other machine learning methods

• Is richly accompanied by detailed numerical examples and repletes with computational support materials in algorithms and explanatory formulae

• Includes access to pedagogical videos on topics including the evolutionary process of topology optimization, the distribution of genetic algorithms, and CMA-ES

• Language: English
• Publisher: Academic Press
• Release date: Jan 15, 2024
• ISBN: 9780323996754

Hajime Igarashi

Hajime Igarashi has been a Professor at the Graduate School of Information Science and Technology, Hokkaido University (Japan), since 2004. He was a guest researcher at the Technical University of Berlin from 1995 to 1997. His research areas are computational electromagnetism, design optimization, and energy harvesting. Igarashi is a vice president of the International COMPUMAG Society, a fellow of Institute of Electrical Engineers in Japan (IEEJ), a member of the Institute of Electronics, Information and Communication Engineers (IEICE), and a member of IEEE. He was the recipient of the Ministry of Education, Culture, Sports, Science and Technology Award and the Outstanding Technical Paper Award by IEEJ in 2016. He has authored or coauthored more than 300 international journal papers.

Book preview

1: Equations of electromagnetic field

Abstract

This chapter provides the basics of electromagnetism which are necessary for the next chapters. This chapter starts with introduction of Maxwell equations and derives the conservation laws from them. Next sections derive the governing equations for electrostatic, magnetostatic, magneto-quasistatic, and electro-quasistatic fields, as well as electromagnetic waves. The last section describes the boundary conditions which are important for electromagnetic field analysis.

Keywords

Maxwell equations; Faraday's law; Ampere's law; conservation law; Maxwell stress tensor; quasistatic approximation; boundary condition

This chapter describes the basics of electromagnetism which are necessary for the next chapters. This chapter starts with Maxwell equations and derives the conservation laws from them. Sections 1.3.1, 1.4, and 1.5 derive the governing equations for electrostatic, magnetostatic, magneto-quasistatic, and electro-quasistatic fields, as well as electromagnetic waves. Finally, Sect. 1.6 describes the boundary conditions which are important for electromagnetic field analysis.

1.1 Maxwell equations

The electromagnetic field is governed by Maxwell equations

(1.1a)

(1.1b)

(1.1c)

(1.1d)

Eqs. (1.1a)–(1.1c) denote Ampere's law, Faraday's law, and Gauss's law, respectively. Eq. (1.1d) states that there is no divergence in the magnetic flux density, i.e., the flux lines are always closed.

The current density in Eq. (1.1a) consists of the conduction current , the magnetization current , and the displacement current as follows:

(1.2)

In analysis using finite element method (FEM), an imposed (external) current is often included in . Also, the charge density of (1.1c) consists of the true charge density ρ and the polarized charge density as follows:

(1.3)

Density ρ is composed of charged particles such as free electrons and ions while represents the apparent charge density caused by the dielectric polarization. Since the latter represents the charge polarization, it cannot be taken out to an external system in contrast to the former.

The right-hand sides of Eqs. (1.1a) and (1.1c) contain terms such as magnetization current and polarization vector that depend on the material characteristics. These terms are transferred to the left-hand side to introduce the flux density D and magnetic field H defined by

(1.4)

(1.5)

Now, the source terms on the right-hand sides of Eqs. (1.1a) and (1.1c) are the true charge density, conduction current, and displacement current, and they are expressed by

(1.6)

(1.7)

As the externally applied magnetic field H increases, the magnetization M of the magnetic material is expected to increase. Similarly, the polarization P of the dielectric is expected to increase with E. Then, assuming the relation and , Eqs. (1.4) and (1.5) are rewritten as

(1.8)

(1.9)

where , μ is the permeability, is the relative permeability, , ε is the permittivity, and is the relative permittivity. In electromagnetic field analysis, magnetic reluctance is often used instead of μ. Equations representing the relationship between dual quantities E and D, B and H, such as in Eqs. (1.4), (1.5) and (1.8), (1.9), are called constitutive relations.

Note that there exists a remanent magnetization in ferromagnetic materials such as iron even when H is set to zero (i.e., when ). When considering the remanent magnetization, we have to use (1.5) instead of (1.9) because is not valid. The same applies to permanent magnets (PMs). PMs will be discussed in Sect. 2.1.

Maxwell and constitutive equations considered in this book are summarized below:

(1.10a)

(1.10b)

(1.10c)

(1.10d)

(1.10e)

(1.10f)

(1.10g)

Maxwell equations (1.10a)–(1.10d) can be expressed in integral forms as:

(1.11a)

(1.11b)

(1.11c)

(1.11d)

Next, let us derive Ohm's law. We consider a cylindrical conductor of area ΔS and length l. The resistance R of this conductor is proportional to l and inversely proportional to ΔS, so that where σ is conductivity, which is an intrinsic quantity of a material independent of its shape (σ is the inverse of the resistivity ρ). For copper, σ is S/m, for rubber (an insulator) it is S/m, and for semiconductors it is somewhere in between. Various electrical devices are made by taking advantage of such a large difference in σ.

Suppose that when a voltage is applied to both ends of the above cylinder, a current flows. Substituting these into Ohm's law, , we obtain . In vector form, Ohm's law can be expressed as follows:

(1.12)

The Joule loss (power) [W] due to resistance can be expressed as

(1.13)

1.2 Conservation laws

From Maxwell equations, we can derive the laws of conservation of charge, energy, and momentum, which will be described below.

1.2.1 Conservation of electric charge

Taking divergence of Ampere's law (1.10a), we get the charge conservation law given by

(1.14)

where Eq. (1.10c) is used. Eq. (1.14) is also called the current continuity equation. Integrating Eq. (1.14) in volume V, we obtain

(1.15)

where represents the charge included in V. This implies that the amount of charge, or current, entering V per unit time through the boundary surface ∂V is equal to the amount of increase per unit time of the charge in V.¹ If the time variation of the charge is negligible, is approximately valid. This approximation is employed in Sect. 1.4.

1.2.2 Conservation of energy

Taking the inner product of Ampere's law (1.10a) and E, and Faraday's law (1.10b) and H, and subtracting these sides, we obtain

(1.16)

where we used the vector identity

(1.17)

Assuming that ε and μ are constant, the integral of Eq. (1.16) in volume V gives

(1.18)

Eq. (1.18) represents energy conservation, which states that the energy entering V per unit time through the boundary surface ∂V, i.e., the surface integral of Poynting vector , is partly consumed by the Joule loss in V and the rest is stored as electrical and magnetic energy.

Assuming that the electromagnetic field varies sinusoidally as

we can express Eq. (1.18) in phasor form. From the expression

(1.19)

the average power of a Pointing vector period T can be expressed as follows:

(1.20)

Eq. (1.20) represents the average power per unit area entering the system. Evaluating the average power density on the right-hand side of Eq. (1.18) in the same way, we obtain

(1.21)

When permittivity ε and magnetic permeability μ are real, E and D as well as H and B are in phase, the second and third terms on the right-hand side of Eq. (1.21) vanish. In other words, the balance of electric and magnetic energy stored in the system is zero. The power entering the system is consumed by the Joule losses in the first term. To express dielectric, eddy current, and hysteresis losses, ε and μ are sometimes set to be complex numbers. In this case, the second and third terms on the right-hand side of Eq. (1.21) do not vanish and become

(1.22a)

(1.22b)

We will discuss the complex permeability in detail in Sects. 2.5.5 and 4.1.

1.2.3 Conservation of momentum

Taking the vector product of Eq. (1.10a) and B, and (1.10b) and D, respectively, and adding them, we obtain

(1.23)

Assuming that Eqs. (1.8) and (1.9) hold, and expanding the x component of the first term on the left-hand side of Eq. (1.23), we obtain

(1.24)

The y and z components are expressed similarly. Thus, we obtain

(1.25)

where in the first term on the right-hand side denotes Maxwell stress tensor defined by

(1.26)

The components of can be expressed by

(1.27)

where .

The third term on the right-hand side of Eq. (1.25) becomes zero due to Eq. (1.10d). Similarly, the following equation is obtained for the electric field:

(1.28)

From Eq. (1.10c), we can see that the third term on the right-hand side of Eq. (1.28) represents the force acting on the charge. The components of are obtained by replacing and in .

Using the Maxwell stress tensor defined by

(1.29)

Eq. (1.23) can be expressed in the form of law of momentum conservation as

(1.30)

where and represent mechanical and electromagnetic momentums, respectively, the latter of which is defined by

(1.31)

This represents momentum propagating through space as an electromagnetic field. The effect of electromagnetic momentum can be seen, for example, in the tail of a comet due to electromagnetic waves (light pressure) from the Sun.² The time variation of is the electromagnetic force acting on the object and can be expressed as

(1.32)

The first term on the right-hand side of Eq. (1.32) is the Coulomb force acting on a charge and the second term is the Lorentz force acting on a current. The third and fourth terms represent the forces acting on dielectric and magnetic materials, respectively. If we have to consider the energy for deformation of material, the corresponding terms have to be added to the right-hand side of Eq. (1.32) [1].

Since the time variation of the momentum is equal to the force, in Eq. (1.30) is interpreted as the force per unit area acting on the matter and electromagnetic field in V through ∂V. In the case of a static or quasistatic magnetic field, can be neglected. The electromagnetic force acting on the material in V can then be computed from

(1.33)

In FE analysis, the electromagnetic force is computed from Eq. (1.33), which will be described in Sect. 3.4.

In the above, we derived the Maxwell stress tensor T in Cartesian coordinate system, but how can the components of T be expressed in cylindrical or spherical coordinate systems? As it turns out, regardless of the coordinates, the components can be expressed as in Eq. (1.29). This property is called covariance. For example, can be expressed in cylindrical coordinates as follows:

(1.34)

The proof is given in Appendix A.

1.3 Static fields

1.3.1 Electrostatic field

When there is no time variation of the electromagnetic field ( ), the right-hand side of Faraday's law (1.10b) becomes zero. In this case, from Eqs. (1.10b), (1.10c), and (1.10e), we obtain the following governing equations for the electrostatic field:

(1.35a)

(1.35b)

(1.35c)

Since Eq. (1.35a) holds in all spaces, it can be expressed by scalar potential (electrostatic potential) as follows:

(1.36)

From Eqs. (1.35b), (1.35c), and (1.36), we obtain the Poisson equation for φ given by

(1.37)

In the absence of a true charge ( ), Eq. (1.37) reduces to the Laplace equation

(1.38)

In electrostatic field analysis, the Poisson or Laplace equation is solved.

Assume that the charge ρ distributes in a finite volume V in infinite space. It is also assumed that there is no spatial variation in permittivity. In this case, Eq. (1.37) reduces to

(1.39)

where

and

is a Laplacian. To solve Eq. (1.39), we employ the fundamental solution³ to the Laplace equation given by

(1.40)

which satisfies

(1.41)

The solution (1.40) can be derived by Fourier transformation of Eq. (1.41) in space.

In Eq. (1.41), δ denotes the Dirac delta function which satisfies

(1.42a)

(1.42b)

Multiplying the fundamental solution by both sides of Eq. (1.39) and integrating by parts, we obtain

(1.43)

Furthermore, if we apply −∇ to both sides of Eq. (1.43) and note that the derivative is taken with respect to the coordinate x, we obtain Coulomb's law

(1.44)

Eq. (1.44) represents the superposition of electric fields generated by a small volume of electric charge , as shown in Fig. 1.1a.

Figure 1.1 Fields generated by distributed electric charge and current.

As described above, assuming a uniform infinite space, the solution of the Poisson equation (1.37) can be obtained and the electric field can be expressed analytically. However, when a dielectric exists in a region, it is generally difficult to represent the entire electric field analytically, except in the case of a dielectric with a simple shape such as a sphere or a cylinder. Even if there is no dielectric, it is generally difficult to express analytically the solution (Green's function) of Eq. (1.41) that satisfies the boundary conditions in a finite domain. In such general cases, a field computation method such as FEM, described in Sect. 3, and moment method is required to solve it.

1.3.2 Magnetostatic field

The static magnetic field is described by Eq. (1.10a) with and Eqs. (1.10d) and (1.10f), that is,

(1.45)

(1.46)

(1.47)

When considering magnetic hysteresis, Eq. (1.10g) is used instead of Eq. (1.47). Since Eq. (1.46) is valid in the whole space, the magnetic flux density B can be expressed by using vector potential A from the Poincaré lemma as follows:

(1.48)

Since the rotation of the gradient of a scalar function is zero, i.e., , transforming A to gives the same B (gauge invariance). In order to uniquely determine A, it is necessary to impose (fix) a gauge. However, in magnetostatic field analysis using FEM, we usually solve the equations without (explicitly) imposing a gauge (see Sect. 3.2.5).

From Eqs. (1.45), (1.47), and (1.48), we obtain the following differential equation for A:

(1.49)

In magnetostatic field analysis, Eq. (1.49) is usually solved, and B is obtained from Eq. (1.48). As in the case of the electrostatic field, to solve Eq. (1.49), we consider infinite space with uniform permeability and assume that the current is distributed in a finite volume V. Moreover, we impose the Coulomb gauge given by

(1.50)

Then, Eq. (1.49) reduces to the vector Poisson equation given by

(1.51)

where we used the vector identity

(1.52)

Note that is the vector Laplacian, which coincides with the Laplacian in Cartesian coordinates but differs from the Laplacian in curvilinear coordinates such as polar coordinates. Eq. (1.51) can be solved by using the fundamental solution of the Laplace equation (1.40), as in the case of the Poisson equation for an electrostatic field, resulting in the following equation:

(1.53)

Taking the rotation of Eq. (1.53), we obtain the Biot–Savart law

(1.54)

Eq. (1.54) shows the superposition of magnetic flux densities generated by , as shown in Fig. 1.1b. If the current exists in a uniform infinite space, B can be obtained from Eq. (1.54). However, when magnetic materials such as transformers and motors exist, the magnetic permeability changes accross the core surface. And when magnetic saturation of magnetic materials is considered, it is necessary to treat μ as a function of the magnetic field and space. In such general cases, a numerical field analysis method such as FEM and the moment method is required to solve Eq. (1.49).

When considering a system with PMs, does not hold in PMs, and it is necessary to consider as the constitutive relation. Thus, we consider

(1.55)

where in PMs. If there are materials other than PMs that have magnetic hysteresis, we also consider Eq. (1.55) there. The modeling of PMs is described in Sect. 2.1, while the mathematical model of magnetic hysteresis is discussed in Sect. 2.5.6, and its FE analysis is discussed in Sect. 3.1.1.

1.4 Quasistatic fields

1.4.1 Magneto-quasistatic field

If permittivity and magnetic permeability are constant, the system is linear and the principle of superposition is valid. Under these conditions, when the source of the electromagnetic field (charge and current) varies sinusoidally with an angular frequency ω, that is, when the time variation obeys , the time variation of the electromagnetic field also obeys . In this case, the electromagnetic field can be expressed in phasor forms as

(1.56)

where and represent the phasors, also called complex amplitudes.⁴ The Maxwell equations (1.10a) and (1.10b) are then expressed in terms of phasors as follows:

(1.57)

(1.58)

Using Ohm's law (1.12) and the constitutive relation (1.10e), we can express the right-hand side of Eq. (1.57) as , where denotes complex permittivity defined by

(1.59)

The dielectric loss factor, also called loss tangent, defined by

is used to describe the loss characteristics of dielectric materials.

The current consists of the displacement and conduction currents. The values of the dimensionless quantity , which represents the effect of the conduction current, are summarized in Table 1.1 for various cases. When is sufficiently smaller than 1, the displacement current can be neglected. Such an approximation is referred to as magneto-quasistatic approximation. The governing equation under this approximation is given by

(1.60)

(1.61)

(1.62)

(1.63)

Taking divergence of Eq. (1.60), we have

(1.64)

Eq. (1.64) expresses the law of current continuity (charge conservation law) under the magneto-quasistatic approximation. Expressing the magnetic flux density as from Eq. (1.62) and substituting it into Faraday's law (1.61), we