With the recent advancements in computer performance and computational technology, numerical electromagnetic field analysis has become an indispensable technique in the field of electrical and electronic engineering. However, further speed improvements are needed to perform larger-scale and more accurate analyses. Therefore, I am engaged in theoretical research on numerical electromagnetic field analysis techniques based on the mathematical properties of the governing equations of electromagnetic fields, methods for speeding up these techniques, and the development of optimal design methods utilizing these techniques.
Fast eddy current analysis of Litz wires using integral equations
Hybrid optimization of topology and parameters of electromagnetic devices using evolutionary computation
Study of the mathematical properties of Cauer circuit derivation algorithms
Design of efficient preprocessing for large-scale linear systems arising from Maxwell's equations
A New Computational Method for High-Speed and High-Precision Analysis of Eddy Currents in Litz Wires
High-frequency currents are used in motors, power supplies, and wireless power transfer devices. These devices often utilize a special type of wire called "Litz wire." Litz wire is a wire with a structure of many thin wires bundled and twisted together, developed to reduce power loss at high frequencies.
However, within Litz wire, unwanted currents called "eddy currents" are generated by the magnetic field created by the current, which then converts into heat, causing energy loss (proximity effect). Because the wires interact with each other in a complex way, accurately calculating this phenomenon is extremely difficult, and conventionally, large-scale numerical calculations using supercomputers (finite element method) were necessary.
This research proposes a new computational method to efficiently solve this problem. The key is to model the Litz wire not as a "thin three-dimensional structure," but as a "single wire running through space." It is known that eddy currents generated in wires by alternating current can be expressed as having magnetic properties (magnetization). This property is utilized to calculate the magnetization distribution along the wire using a one-dimensional integral equation, instead of finely dividing the wire's cross-section. This significantly reduces the computational freedom, enabling much faster analysis compared to conventional methods.
The accuracy of this method was verified using coils consisting of 100 wires and coils with magnetic cores. Compared to the conventional finite element method, the calculation results for power loss (heat generation) showed good agreement within 3-5% error, confirming the reliability of the new method. Furthermore, this study also investigated the effect of Litz wire structure on loss. Comparing wires simply arranged parallel to each other, bundled and twisted, and further twisted into a rope-like structure, it was shown that Litz wires with a twisted structure exhibited lower power loss. This is because twisting the wires averages out the magnetic field bias, suppressing eddy currents.
This research opens the way to accurately evaluate power loss in a short time, even for complex wire structures like Litz wires. This is a significant achievement that will lead to more advanced and faster design of highly efficient motors, power supplies, electric vehicles, and renewable energy equipment.
A new optimization technology that simultaneously designs the "shape" and "structure" to maximize the performance of permanent magnet motors.
Permanent magnet motors (PM motors), used in electric vehicles and industrial robots, perform best when their torque output and rotational smoothness are paramount. These performance characteristics are largely determined by the shape of the magnets inside the motor and the cavity structure called a "magnetic flux barrier" within the iron core. However, optimally designing these components manually is extremely difficult, making computer-aided design (optimization) crucial.
Traditional design methods have broadly followed two approaches. One is "parameter optimization (PO)," which adjusts the width, angle, and position of the magnets as numerical parameters, offering the advantage of maintaining a manufacturable shape while making adjustments. The other is "topology optimization (TO)," which allows for the free adjustment of material presence or absence, a powerful method for automatically generating complex cavity structures like magnetic flux barriers. However, neither method alone could simultaneously optimize both the magnets and the magnetic flux barrier.
This research proposes a new hybrid optimization method that combines these two approaches. The magnet shape is represented by PO, and the arrangement of the magnetic flux barrier is freely changed by TO. For the Topology Element (TO), we use a mathematical model called a "Normalized Gaussian Network (NGnet)" to represent the material distribution (iron or air) as a continuous function. This allows the shape of the magnetic flux barrier to be treated as a number of numerical parameters, enabling simultaneous optimization of both the Topology Element (PO) and TO using a genetic algorithm (GA). Furthermore, this research introduces a technique that automatically regenerates the mesh (computational partitioning) to match the motor shape after each optimization. This allows for efficient and highly accurate electromagnetic field analysis while avoiding jagged, unnatural shapes.
This method was applied to the rotor design of an IPM (Internal Magnet) motor. By simultaneously optimizing the magnet curvature and the shape of the magnetic flux barrier, we obtained a design with higher average torque and lower torque ripple (rotational unevenness) compared to conventional topology optimization alone. In particular, the rotor obtained using this new method effectively utilizes a magnetic attraction effect called "reluctance torque," resulting in an increase in overall torque. Furthermore, optimization was performed on practical structures such as U-shaped and V-shaped magnets, demonstrating that it is possible to achieve both high torque and manufacturability.
This research establishes a groundbreaking design technology that simultaneously optimizes both the "magnet shape" and the "iron core structure" inside a motor. It is expected to significantly contribute to the high performance of next-generation motors used in electric vehicles and energy-saving equipment.
Cauer circuit-based model reduction technique that converts electromagnetic field simulations into "high-speed equivalent circuits".
Inside electrical equipment such as electric motors, transformers, and inductors, eddy currents are generated by the magnetic field created by the electric current, causing energy loss (heat generation). Accurately evaluating this phenomenon requires electromagnetic field analysis using the finite element method (FEM). However, in real-world design, this requires repeated calculations involving hundreds of thousands to millions of degrees of freedom, sometimes taking tens of hours.
To address this problem, our research group has developed a model reduction technique based on the Cauer ladder network (CLN). CLN is a method of representing phenomena occurring in electromagnetic fields as a "ladder-shaped equivalent circuit" consisting of a resistor and an inductor in series. This circuit has the same impedance characteristics as the original electromagnetic field model, but can be analyzed with far fewer degrees of freedom.
In our 2020 paper, we focused on the fact that the electromagnetic field equations obtained by FEM are mathematically a "linear system with a symmetric positive definite matrix," and combined this with the Krylov subspace method (Lanczos method). This led to the development of a new algorithm that can directly calculate the resistance and inductance of a Cauer circuit from the governing equations of the electromagnetic field. This method has a theoretical background where the circuit's impedance is expressed as a "continued fraction," automatically yielding a stable and physically meaningful equivalent circuit.
While Cauer circuits become more accurate with increasing the number of stages (the number of ladder steps), it was previously unclear where to stop. In a 2022 paper, we developed a method for guaranteed error estimation to address this problem.
By utilizing the mathematical properties based on the energy of the electromagnetic field (magnetic and electric field energy), we demonstrated that a quantity that provides an upper bound on "how far the current Cauer circuit is from the exact solution" can be directly calculated from the circuit's final inductor and resistance. This error estimator has been applied to analytical and FEM models of copper foil, and its ability to reliably suppress actual errors from above has been numerically verified. This makes it possible to automatically determine the "minimum number of circuit stages that satisfy the required accuracy."
An electromagnetically separated numerical method dramatically speeds up eddy current analysis connected to external circuits.
In the design of electric motors and power supply equipment, accurately predicting losses (heat generation) due to eddy currents in coils and conductors is essential. Especially in equipment connected to high-frequency power supplies such as inverters and PWM power supplies, complex currents flow within the coils and conductors, significantly impacting efficiency and reliability. Analyzing such problems requires coupled analysis of electromagnetic field simulations using the finite element method (FEM) and external circuits.
However, in eddy current problems involving the coupling of FEM and external circuits, the systems of linear equations used in the calculations become extremely difficult to solve. Conventional iterative methods with incomplete Cholesky decomposition (IC) often require thousands or even tens of thousands of iterations to converge. This is because the "potential (scalar potential)" component, derived from the equation representing the continuity of current, generates a mathematically extremely unfavorable matrix. This problem is particularly pronounced with long conductors, fine meshes, and high-frequency conditions.
In this study, we identified the essence of this problem as the mixing of "components representing magnetic fields" and "components representing the continuity of current" within the same matrix, and proposed a new preprocessing method that separates and treats them based on their physical meaning. This is electromagnetic decoupling (EMD).
In the A–φ (magnetic vector potential and electric scalar potential) formulation of FEM, the unknowns are broadly divided into two types. One is the vector potential representing the magnetic field, and the other is the scalar potential governing the current flow, as well as the circuit voltage and current. In the EMD method, these are mathematically separated into block structures, and a computationally lightweight IC preprocessing is applied to the magnetic field portion, and a "stronger" preprocessing method such as AMG (multiple grid method), domain decomposition method, or direct method is applied to the portion governing the continuity of the current. Since the degrees of freedom of the scalar potential are far fewer than those of the magnetic field, even if computationally expensive preprocessing is used here, the overall computation time does not increase significantly. This allows us to target and remove only the cause of the poor convergence. Furthermore, this method has the characteristic of being very compatible with parallelization based on physical structure. When a coil is made up of multiple independent conductors, the current continuity equations for each conductor are independent of each other. EMD utilizes this structure to apply scalar potential preprocessing in parallel to each conductor. This enables "physical structure-based parallelization" in addition to conventional numerical parallelization, significantly improving scalability for large-scale problems.
Experiments tested practical and demanding problems, including a 21-turn coil, a high-frequency model with a fine skin-effect mesh, and a model with 18 independent conductors. As a result, the number of iterations required for conventional IC preprocessing, which exceeded 10,000, was reduced to a few hundred with EMD, and computation time was sped up by up to approximately 28 times.
This research addresses the industrially critical yet computationally difficult problem of eddy current analysis coupled with external circuits. It identifies the numerical inefficiencies from both physical and mathematical perspectives and provides a practical algorithm to directly solve them. This technology holds great significance as a foundational technology for designing complex circuits and electromagnetic fields in devices such as electric vehicle motors, power supply coils, transformers, and wireless power transfer equipment, enabling faster and more accurate design.