⚡ electrical engineering

Design of E-Core Double Mechanical Rotor Double Stator Flux Switching Permanent Magnet Synchronous Machine by Analytical Model

This paper presents an analytical design method using a sub-region approach for a novel E-Core Double Mechanical Rotor Double Stator Flux Switching Permanent Magnet Synchronous Machine to mitigate high cogging torque and saturation issues, with results validated against finite element analysis.

Original authors: E . Shirzad

Published 2026-09-10

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: E . Shirzad

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quest to power the vehicles of tomorrow, engineers are constantly searching for motors that are smaller, stronger, and more efficient than the ones we use today. A key player in this search is a type of electric motor known as the flux-switching permanent magnet machine. Imagine a motor where the heavy, heat-generating copper wires and the powerful magnets are all packed into the stationary outer shell, rather than spinning inside. This design makes the motor incredibly robust and easy to cool, which is a major advantage for hybrid electric vehicles that need to handle high power without overheating. However, these machines have a persistent flaw: they tend to produce a rough, jerky vibration called cogging torque, which happens because the magnets and the metal teeth of the motor pull on each other unevenly as they spin. This vibration creates noise and can wear out the machine. To solve this, researchers have been experimenting with different shapes for the motor's core, moving away from simple designs toward more complex structures that can smooth out these forces.

A researcher has now taken a significant step forward by designing a highly complex version of this motor and creating a new, faster way to predict how it will behave. Instead of relying solely on slow, computer-heavy simulations that take hours to run for every tiny change in shape, they developed a mathematical model that can calculate the motor's performance almost instantly. This new machine is a double-layered wonder, featuring two separate spinning rotors and two stationary stators arranged like a sandwich, all built around a specific "E-shaped" core structure. The goal was to combine the high power of these motors with a design that eliminates the rough vibration, making them suitable for the demanding environment of a hybrid car.

The researcher began by breaking the motor down into a series of distinct zones, treating the magnetic fields in each area like a puzzle piece that fits into the next. They mapped out the inner and outer rotors, the slots where the wires sit, and the permanent magnets, creating a complete mathematical picture of how electricity and magnetism interact within the device. By solving a set of equations that describe how magnetic fields flow through these different zones, they could predict the strength of the magnetic pull at every point inside the motor. This analytical approach allowed them to see exactly how the magnetic forces would behave under load, including how the motor would react when the magnets and the electric currents were working together. To ensure their math was correct, they compared their rapid calculations against a detailed, time-consuming computer simulation known as finite element analysis, which is the industry standard for accuracy. The results showed that their fast method was remarkably close to the slow, heavy simulation, proving that their new model could be trusted for design work.

With a reliable model in hand, the researcher turned their attention to the motor's most stubborn problem: the cogging torque. They systematically tested how changing the physical dimensions of the motor would affect this vibration. They varied the width of the slots, the size of the magnets, and the number of teeth on the rotors, watching closely to see which changes made the motor run smoother. They discovered that the number of teeth on the rotors was the most critical factor. By adjusting the count of these teeth, they could dramatically reduce the jerky pulling forces. Specifically, they found that using eleven teeth on both the inner and outer rotors created a configuration where the magnetic pulls canceled each other out much more effectively than with other numbers. This simple change in the count of the teeth reduced the vibration on the inner part of the motor by about eighty-four percent and nearly eliminated it on the outer part, dropping the force from over three newton-meters to a tiny fraction of that.

The final design, optimized with these eleven teeth and carefully chosen slot widths, proved to be a success. The researcher confirmed that the motor could handle the intense electrical currents needed for a vehicle without the metal parts becoming magnetically "saturated," a state where the motor loses its efficiency and overheats. The magnetic fields stayed within safe limits, reaching a maximum of about 1.42 Tesla in the inner parts and 1.38 Tesla in the outer parts, well below the breaking point of the steel used. The motor also generated a steady, strong twisting force, or torque, capable of driving a vehicle, with the outer section producing about 5.2 newton-meters and the inner section about 2.8 newton-meters. While the complex shape of the motor did create some uneven magnetic forces that push and pull on the shaft, these forces were predictable and could be managed with proper engineering.

This work demonstrates that it is possible to design these advanced, double-rotor motors with a high degree of precision without waiting days for computer simulations to finish. The new mathematical model offers a practical tool for engineers to quickly explore different shapes and find the best configuration for a specific job. By proving that a simple change in the number of teeth can almost completely remove the rough vibration, the study provides a clear path forward for building quieter, smoother, and more powerful motors for the next generation of hybrid vehicles. The approach balances speed and accuracy, showing that with the right mathematical framework, complex machines can be understood and refined with surprising efficiency.

Technical Summary: Design of E-Core Double Mechanical Rotor Double Stator Flux Switching Permanent Magnet Synchronous Machine by Analytical Model

Problem Statement
Flux Switching Permanent Magnet Machines (FSPMMs) are recognized for their high power density and suitability for hybrid electric vehicle (HEV) applications, particularly due to their non-overlapping windings and mechanically robust stator-mounted magnets. However, conventional FSPMMs (specifically U-core structures) face significant challenges, including high cogging torque resulting from a large number of permanent magnets and a tendency to enter magnetic saturation rapidly under load. Furthermore, advanced HEV powertrains increasingly require dual mechanical ports to accommodate separate electro-mechanical interfaces, necessitating complex machine topologies. The paper addresses the need for a design methodology that simultaneously mitigates saturation and cogging torque while accommodating a double-rotor, double-stator configuration, without relying solely on time-consuming numerical simulations for every geometric iteration.

Methodology
The paper proposes a two-dimensional analytical model based on the sub-domain method to design and analyze an E-Core Double Mechanical Rotor Double Stator Flux Switching Permanent Magnet Synchronous Machine (ECDMRDSFSPMSM).

The modeling process follows a structured seven-step approach:

Assumptions: The model assumes 2-D behavior (ignoring end effects), infinite permeability for stator and rotor back-iron (linear analysis), isotropic materials, and the absence of tooth tips and eddy current reaction fields.
Sub-domain Definition: The machine geometry is divided into active regions, including inner/outer rotor slots, air gaps, stator slots, permanent magnet (PM) regions, and flux barriers.
Governing Equations: Maxwell's equations are utilized to formulate the magnetic vector potential ( $A$ ) in each sub-domain. The model accounts for both the magnetization vector ( $M$ ) in PM regions and current density ( $J$ ) in winding regions.
Fourier Series Representation: Current and magnetization patterns are expressed using Fourier series to solve the partial differential equations within each domain.
Boundary Conditions: Continuity of the magnetic vector potential and the tangential component of magnetic field intensity is enforced at the interfaces between sub-domains.
Coefficient Calculation: Unknown coefficients are determined by solving a system of linear equations derived from the boundary conditions using the orthogonality of trigonometric functions.
Validation: The analytical results are validated against Finite Element Analysis (FEA) simulations to ensure accuracy in predicting magnetic flux density, cogging torque, and electromagnetic torque.

Key Contributions

Novel Topology: The paper introduces an E-core structure for a dual-rotor, dual-stator FSPMM. This configuration aims to provide larger slot areas for windings compared to U-core designs, thereby reducing saturation risks and fabrication costs while maintaining high torque density.
Analytical Framework: A comprehensive analytical model is developed that simultaneously considers the effects of permanent magnets and armature reaction (double-layer windings) on the magnetic flux density distribution.
Design Optimization: A sensitivity analysis is conducted to evaluate the impact of key geometric parameters (e.g., rotor tooth count, slot widths, PM widths, and yoke radii) on cogging torque. The study identifies the number of rotor teeth as the most critical parameter for minimizing cogging torque.
Performance Prediction: The model successfully predicts radial and tangential flux densities, cogging torque, electromagnetic torque, and Unbalanced Magnetic Force (UMF) for both the inner and outer sections of the machine.

Results
The study presents a specific design case with 12 stator slots and 10 rotor teeth (initially), later optimized to 11 rotor teeth for both inner and outer rotors.

Flux Density: Under rated load conditions (6 A/mm² current density and 0.4 T residual flux density), the maximum flux density in the stator and rotor yokes reaches approximately 1.42 T and 1.38 T, respectively. These values remain below the saturation limit of 1.7 T, confirming the design avoids saturation.
Validation Accuracy: The analytical model shows strong agreement with FEA results. The maximum deviation for flux density is approximately 6.2%. For cogging torque and electromagnetic torque, the errors between the analytical model and FEA are within 2.8%–5.0% and 1.0%–1.7%, respectively.
Cogging Torque Reduction: The sensitivity analysis revealed that increasing the number of rotor teeth from 8 to 11 significantly reduced cogging torque. The optimized design (11 teeth) achieved a peak cogging torque of 0.1264 N·m for the inner part and 0.0190 N·m for the outer part (for a 100 mm stack length), representing reductions of approximately 84% and 99.4% compared to an 8-tooth configuration.
Torque and Forces: The optimized machine produces an average electromagnetic torque of approximately 2.8 N·m (inner) and 5.2 N·m (outer). Due to the E-core asymmetry, Unbalanced Magnetic Forces (UMF) were observed, with peak values reaching approximately 8,000 N (inner) and 20,000 N (outer) in the radial direction, which the author notes must be considered for mechanical bearing design.

Significance and Claims
The paper claims that the proposed sub-domain analytical model serves as a fast and efficient tool for the initial design and parametric exploration of complex FSPMMs, particularly for light-duty applications. By demonstrating that the analytical approach can replace time-consuming numerical simulations for early-stage design, the author argues that their method offers a practical compromise between simplicity and accuracy. The study concludes that the E-core double mechanical rotor topology, when optimized using this analytical model, effectively addresses the traditional drawbacks of FSPMMs (saturation and cogging torque) while providing the dual-port capability required for advanced HEV powertrains. The author explicitly states the model is designed under linear assumptions (infinite permeability) and is intended for initial design exploration rather than final high-fidelity saturation analysis.

More like this