Dionysios Aliprantis, Steven Pekarek
©SHUTTERSTOCK.COM/PAUL CRAFT
The electrification of transportation hinges on a number of core technologies, such as power electronics and energy storage. In this article, the spotlight is on electric machines (EMs), a term that encompasses all electromechanical energy conversion devices, i.e., motors and generators. Computational electromagnetics, like the finite element method (FEM), play a vital role in a modern EM design workflow. Such methods have been under development for decades, however, with recent advances in computing, they are now mainstream. One of their key features is that they allow us to search a design space with unprecedented accuracy. But perhaps more importantly in today’s rapidly changing landscape, they are accelerating the pace of innovation, reducing prototyping costs and the time it takes to go from concept to product.
Let us consider, for example, a modern electric vehicle (EV). First, we note that the term EV does not apply only to a passenger car. Today, when we speak of an EV, we may be referring to an electric scooter or bicycle for micromobility, a zero-emissions city bus, a medium-duty box truck for local deliveries, a heavy-duty semitrailer truck for long-haul freight transport, a forklift employed on a factory floor or in a port, a loader used at a construction site, a hauler in a mine, an aircraft or watercraft, and the list goes on, with many products already available in all these markets. The propulsion motors found on board these vehicles are obviously not commercial off-the-shelf devices. For maximum performance and competitiveness in the marketplace, each EM has to be crafted very carefully for the specific requirements of a given vehicle, while accounting for efficiency, weight, size, cost, reliability, and the vehicle’s drive cycle and operating conditions. In addition to the main propulsion motors, we should not forget about all the other actuators that are also being electrified by virtue of having access to an underlying EV power system architecture.
The EM design problem is far from trivial. Even though EMs have been around for more than a century, modeling their behavior remains a challenge, especially when they are connected to power electronic (switched) circuits. This is due to a variety of reasons, but mainly due to their complicated structures as well as the nonlinearities of ferromagnetic materials and our imperfect modeling of their physics (e.g., hysteretic effects). It could be argued that the basic dimensions of an EM can be obtained using relatively simple analytical formulas found in textbooks dating back to the early 20th century. Nevertheless, such designs serve only as an initial point for further fine-tuning. We now rely on sophisticated computer-aided design software that can predict the electromagnetic, mechanical, and thermal behavior of an EM very accurately over its entire operating range. We can thus incorporate every geometric detail, no matter how minute; however, it should be understood that our models are also limited by uncertainties in material parameters and other assumptions.
The FEM is an established and widely used technique in a variety of disciplines. We may think of the FEM as a numerical microscope, allowing us to examine what is transpiring inside EMs over time. For instance, the FEM can be employed to establish the mechanical properties of an EM to ensure the structural integrity of a high-speed rotor, or to predict noise, vibration, and harshness (NVH) characteristics. In what follows, we describe the electromagnetic FEM in more detail, providing a high-level explanation of what it does and how it works in the context of EMs.
Simply speaking, the electromagnetic FEM solves Maxwell’s equations, which are partial differential equations (PDEs) quantifying the variation of the electromagnetic field through space and time. EMs are relatively low-frequency devices, so we can safely ignore electromagnetic radiation effects. We thus have a simpler quasi-magnetostatic problem, where the unknown is the magnetic field throughout a finite region in space surrounding the device. It is possible to formulate the FEM in either two or three dimensions. In two dimensions, it is assumed that nothing changes along the third axis. Luckily, because of how EMs are built, a 2D model often yields a decent approximation of the fields inside the active region of the EM, which is where the ferromagnetic material resides (in contrast to the end winding region, which has to be analyzed separately). A 3D model can be more accurate, but this comes at a significantly higher computational cost.
Physics tells us that there are two sources for the magnetic field. First, we have free current sources due to the flow of electrons inside conductors, such as current in windings or induced eddy currents in conductive regions. Second, we must account for bound current sources; these are more difficult to conceptualize, but they essentially represent the magnetization of matter—and in EMs, they are the dominant source because their effect is what creates the strong magnetic field required to make them function properly. In the FEM, bound currents are not modeled explicitly; rather, their presence is captured indirectly through the magnetization (B–H) characteristics of magnetic materials.
To initiate a finite element analysis (FEA) study, a user has to input the geometry. In a 2D analysis, this corresponds to a cross section of the EM on the plane of interest (i.e., perpendicular to the shaft). In the most elementary fashion, a geometry can be described by two lists: 1) a list of nodes and their coordinates and 2) a list of linear segments interconnecting pairs of nodes. This is similar to how a connect-the-dots game looks like. For convenience, the geometry can also be described through a graphical interface (but the software then creates internally lists of nodes and segments) or even through an application programming interface (which facilitates parametric studies). Curved surfaces can be approximated by polygonal shapes. Subsequently, a user has to define the material type occupying the various regions in space (e.g., air, copper, steel, or permanent magnets) and the electromagnetic parameters of each material (e.g., using a constant magnetic permeability or a B–H curve). After the geometry definition is complete, a user has to specify the variation of the winding currents and the rotor position over time. For example, in a three-phase permanent magnet synchronous machine (PMSM) operating in steady state, the currents can be defined as a balanced set, and the rotor position could be a linear function of time. A separate FEA would then be conducted for each time instant and mechanical position. In more advanced FEM implementations, the variation of the currents and/or rotor position can be determined dynamically by interfacing the FEA with an external circuit and/or mechanical subsystem transient solver. The FEM can also be used for an analysis of linear EMs, where the mechanical degree of freedom is a linear position. Finally, boundary conditions are set, which capture the impact of external magnetic fields (if any). This information is really all that is needed to conduct an FEA study.
What happens under the hood of an FEA program? The first thing that takes place is a partitioning of the geometric domain, which is subdivided into a high number of relatively small finite elements. This action is called meshing. Typically, triangular elements are used for 2D EM FEA because they work fairly well for meshing the common internal features of EMs (e.g., teeth, slots, pockets, or curved surfaces). For a 3D FEA, the finite elements could be, for instance, tetrahedral. There could be tens of thousands of such elements. Commercial FEA software employs sophisticated meshing algorithms, which are intelligent enough to determine which regions need to be meshed with a higher density than others. In general, a decent mesh has more finite elements concentrated in regions where the magnetic field varies rapidly, while a sparser mesh can be used where the field is more constant. (Typically, we avoid meshing the domain with a uniformly small finite-element size because doing so increases the computational burden.) Various mathematical results exist that correlate the size of the finite elements with the error between the FEM solution and the actual solution. An example mesh is shown in Figure 1, depicting an elementary surface-mounted PMSM. To take advantage of the periodicity of the magnetic field, we meshed a single pole (30 mechanical degrees).
Figure 1. FEA mesh and magnetic flux lines for a surface-mounted PMSM. PM: permanent magnet.
Once a mesh has been established, the FEM proceeds with numerical calculations to obtain the magnetic field. We do not describe these details here. The key idea is that the FEM searches among a set of functions (e.g., piecewise linear functions) that are defined on the given mesh. Interestingly, we do not work directly with Maxwell’s PDEs; rather, by using vector calculus identities, we obtain an equivalent variational calculus formulation. The new objective is to identify the function that minimizes an energy-related integral over the domain. So, essentially, we are solving an optimization problem rather than the original PDE.
At the end of the day, the FEM assembles a system of equations where, typically, the unknown is the magnetic vector potential (MVP), denoted as A. These equations become nonlinear if magnetic saturation characteristics of materials are modeled. Specifically, the unknown of the underlying problem is an array of MVP values. If 2D linear triangular elements are used, where a linear variation of the MVP is assumed over each element, the unknown MVP values correspond to potentials at the vertices of the finite elements. The MVP is a physical quantity that is analogous to the electric potential, V or $\varphi$, which is perhaps a more familiar scalar field related to the distribution of electric charge around space. The sources of the MVP are free and bound currents.
Finally, as a postprocessing step, the FEA program will compute various quantities of interest and will generate plots as needed. Typical outputs are the flux linkages of the coils (see Figure 2), electromagnetic torque (see Figure 3), various losses (ohmic and magnetic), and the magnetic forces acting on the inner parts of the device (that inform a subsequent NVH analysis). The visualization aspect is important in its own right because it can help identify issues and possibilities for improving a design. Magnetic flux lines are directly obtained as contour lines of the MVP (see Figure 1). The magnetic flux density (i.e., the B-field) can be readily found as the curl (an operator from vector calculus) of the MVP; when plotted over the domain, it can highlight potential saturation or high-loss regions.
Figure 2. Variation of the a-phase flux linkage (per unit of length in the axial direction) with rotor position. MoM: method of moments.
Figure 3. Variation of electromagnetic torque (per unit of axial length) with rotor position. MoM: method of moments.
Although the FEM is perhaps the most well-known approach, there exist alternative numerical approaches to solve for magnetic fields within low-frequency electromagnetic components. These are often placed under the umbrella terms method of moments (MoM) or integral-based methods, of which there are some different flavors. Here we highlight one in particular that we refer to as an MoM, which connects nicely with the earlier description of magnetic materials and with the FEM. Specifically, one can consider that a magnetized material has bound currents that, in addition to the free currents in conductors, act to create magnetic flux. It is these bound currents that provide, for example, the remanent flux that resides in and around a material when there is no externally sourced magnetic field applied. These bound currents also change (respond) when a material is subjected to an externally generated magnetic field. One can show that if a material has uniform permeability, the bound currents reside only on the surface. If the permeability is nonuniform, bound currents also reside within the material volume.
Derivation of the considered MoM starts by asking a basic physics question: What is the value of the magnetic flux density everywhere in a system? It is then recognized, from Ampère’s law, that the flux density at any location can be determined by summing the contributions from all free and bound currents. The flux density within a magnetic material can also be obtained with knowledge of the magnetization and the relative permeability. Finally, there is a straightforward relationship between magnetization and the bound current at the surface of the region. Taken together, these relationships are used to generate an MoM model in which the free currents serve as an input, and the bound currents (or the tangent component of magnetization) on or within magnetic materials are the unknowns.
One may ask, “Where does the discretization take place?” Any region in the system that has a relative permeability greater than one has bound currents and must be discretized. When considering an EM, the stator and rotor iron (and permanent magnets) must be meshed. In the easiest of cases, if the machine is modeled in 2D and any of these materials is assumed to have uniform permeability throughout their respective regions, the mesh consists of line segments that outline the edge of the respective materials. In Figure 4, a line-segment-based MoM mesh is shown for a surface-mounted PMSM. If the materials have nonuniform permeability, which often needs to be represented if a region is saturated, the interior of the material must be meshed. This can be done using the same discretization geometries (e.g., triangles) that are used in an FEA, although other polygons can also be applied.
Figure 4. MoM mesh and magnetic flux lines for a surface-mounted PMSM. PM: permanent magnet.
Looking under the hood of an MoM-based model, one will find the critical postprocessing functions needed to compute winding flux linkage (see Figure 2) and electromagnetic torque (see Figure 3). Similar to the techniques applied in an FEA, flux linkage is determined from the value of the MVP at the winding conductors. In a 2D MoM, one can leverage a common expression for the MVP from filamentary currents in free space to predict the MVP at the stator conductors resulting from all free and bound currents. Because the MoM formulation is based on determining flux everywhere in a system and is structured to solve for bound currents, the forces acting throughout the stator and rotor and the electromagnetic torque are readily predicted by calculating the Lorentz force acting on each element.
One can observe in Figures 2 and 3 that the FEA and MoM responses are close, but they are not exact matches. The differences are attributed to several factors, including that in the MoM model, only a surface mesh was applied and the relative permeability is assumed constant, whereas the FEA model employs a nonlinear B–H curve. In addition, in the MoM-based model, stator slot current is modeled using a single filamentary conductor at the center of the slot. In contrast, within the FEM-based model, the conductors are partitioned by phase into the top and bottom regions of the slot, and the respective current density is modeled across the respective region.
As electrification continues to spread across land-, sea-, and air-based transportation, computational electromagnetics will undoubtedly continue to play a central role. A search of recent literature shows that a key goal is to leverage integrated multiphysics (electromagnetic, thermal, and mechanical) models to enhance machine and system performance. Nevertheless, predicting behavior in space and time (for a true 4D analysis) over different physical domains is still very demanding computationally as well as from a modeling perspective. Artificial intelligence-based techniques, such as deep neural networks that can be trained to learn the EM response surface, and next-generation (e.g., quantum) computing are likely to serve as key tools to enable widespread use of 4D models in multiobjective optimization and real-time control.
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Dionysios Aliprantis (dionysis@purdue.edu) is with Purdue University, West Lafayette, IN 47907 USA.
Steven Pekarek (spekarek@purdue.edu) is with Purdue University, West Lafayette, IN 47907 USA.
Digital Object Identifier 10.1109/MELE.2023.3320510
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