This study proposed an improved analytical method for electromagnetic field analysis of a wound rotor synchronous machine (WRSM) considering the permeability of soft magnetic materials. A simplified analytical model considering the magnetic permeability of each region is presented. The governing equations of each region are derived from Maxwell’s equations and the electromagnetic field theory, and a general solution is derived by using mathematical techniques. The analytical solutions of all domains are derived by calculating the boundary conditions. To validate the proposed analytical method, the radial and circumferential magnetic flux densities are compared with finite element analysis (FEA) results. In addition, electromagnetic performance parameters such as flux linkage, back-electromotive force, and torque are determined using electromagnetic theories. In particular, the magnetic saturation of the soft magnetic material is due to field and armature current, and the superiority of the proposed method is demonstrated by comparing the improved analytical method considering the global saturation of each region with the nonlinear FEA result considering local saturation. The proposed analytical method can be widely used in the initial and optimal design of WRSM because it can consider saturation of the core due to changes in field and armature current.

## I. INTRODUCTION

Similar to a traditional wound rotor synchronous machine (WRSM), a brushless WRSM is a type of electric motor that uses a rotor with field windings that are connected to a variable frequency drive.^{1} Brushless WRSMs transfer electrical energy to the rotor without requiring physical contact. An electronic controller is used to regulate the rotor current, thereby producing a magnetic field.^{2} The primary advantage of a brushless WRSM is its ability to eliminate maintenance issues, such as brush breakage, that are common in relation to conventional WRSMs. Additionally, brushless WRSMs offer precise speed and torque control, which improve energy efficiency. Therefore, brushless WRSMs serve as viable substitutes for the permanent magnet synchronous machine (PMSM), which encounters problems such as resource weaponization and is damaging to the environment. With the advent of brushless WRSM,^{3} the design of WRSM with high power density requires analysis technology that takes into account the saturation characteristics of electrical steel sheets.

In this study, an electromagnetic analysis is conducted using an improved analytical method by considering the finite permeability of the soft magnetic material of a WRSM. As shown in Fig. 1, the subdomain model used for electromagnetic analysis contains a rotor yoke, rotor winding, rotor tooth, stator tooth, stator winding, and stator yoke, as well as an air gap. The general solutions obtained from the tooth and windings of the rotor and those of the stator with discontinuous cycles, as well as the general solutions of the air gap and rotor and stator yoke regions with continuous cycles, are defined using the boundary conditions. By calculating boundary conditions, analytical solutions for each subdomain are derived and used to conduct electromagnetic performance analysis. In particular, by considering the magnetic saturation owing to field and armature current in the soft magnetic materials in a WRSM, an improved analytical method considering global saturation in each region is presented. Finally, the proposed analytical results are validated by comparing them with the finite element analysis (FEA) results. The proposed analytical method, which enables faster analysis than FEA in electromagnetic performance and allows more accurate analysis than conventional analytical methods, can be widely used in the electromagnetic analysis and design stages of WRSM.

## II. ANALYTICAL METHOD CONSIDERING THE PERMEABILITY OF THE SOFT MAGNETIC MATERIAL

### A. Problem description and assumptions

^{4}

**A**

_{z}is the magnetic vector potential,

*μ*

_{0}is the permeability of vacuum, and

**J**is the current density.

Parameters . | Values . | Parameters . | Values . |
---|---|---|---|

r_{01}: Outer radius of the rotor shaft | 5 mm | a: Angle of the rotor tooth | 18° |

r_{02}: Outer radius of the rotor yoke | 18 mm | b: Angle of the rotor slot | 72° |

r_{03}: Outer radius of the rotor | 44.8 mm | c: Angle of the stator tooth | 30° |

r_{04}: Inner radius of the stator | 45.3 mm | d: Angle of the stator winding | 30° |

r_{05}: Inner radius of the stator yoke | 60.3 mm | Q_{r}: Number of rotor slots | 4 |

r_{06}: Outer radius of the stator yoke | 68 mm | Q_{s}: Number of stator slots | 6 |

N_{s}: Number of conductors in the stator slot | 120 | r_{07}: Number of conductors in the rotor slot | 100 |

Parameters . | Values . | Parameters . | Values . |
---|---|---|---|

r_{01}: Outer radius of the rotor shaft | 5 mm | a: Angle of the rotor tooth | 18° |

r_{02}: Outer radius of the rotor yoke | 18 mm | b: Angle of the rotor slot | 72° |

r_{03}: Outer radius of the rotor | 44.8 mm | c: Angle of the stator tooth | 30° |

r_{04}: Inner radius of the stator | 45.3 mm | d: Angle of the stator winding | 30° |

r_{05}: Inner radius of the stator yoke | 60.3 mm | Q_{r}: Number of rotor slots | 4 |

r_{06}: Outer radius of the stator yoke | 68 mm | Q_{s}: Number of stator slots | 6 |

N_{s}: Number of conductors in the stator slot | 120 | r_{07}: Number of conductors in the rotor slot | 100 |

### B. Analytical solutions and boundary conditions

^{5}

**A**

_{z}

^{h}represents a homogeneous solution. Additionally,

*k*

_{n}=

*n*, where

*n*is the order of spatial harmonics and

*A*

_{0},

*B*

_{0},

*A*

_{n},

*B*

_{n},

*C*

_{n}, and

*D*

_{n}represent undetermined coefficients.

^{6,7}

*k*and

*m*are spatial harmonics,

*k*

_{k}=

*k*π/ζ, and

*k*

_{m}=

*m*π/ln(

*r*

_{o}/

*r*

_{i}). Here,

*r*

_{i}and

*r*

_{o}represent the inner and outer diameters of each region, respectively. Furthermore,

*A*

_{k},

*B*

_{k},

*C*

_{k}, and

*D*

_{k}represent undetermined coefficients;

*ζ*represents the width (radian) of each region; and

*θ*

_{1}and

*θ*

_{2}represent the start and end positions of each region, respectively.

*μ*

_{0}is the permeability of vacuum, and

*J*

_{0}represents the current density of the slot.

*θ*direction of magnetic field intensity where line current density does not exist and the magnetic vector potential in the

*z*direction are continuous in two adjacent subdomains.

^{4}The undetermined coefficients of the general solutions of each domain can be obtained by substituting the boundary conditions, which can be expressed as follows based on the radius used to derive the undetermined coefficients:

*α*

_{j}and

*β*

_{j}represent the angular position of the

*j*-th slot and tooth of the rotor. Additionally,

*γ*

_{j}and

*δ*

_{j}represent the angular position of the

*j*-th slot and tooth of the stator.

^{6,7}

To validate the proposed analytical method, the radial and circumferential flux densities are compared with the FEA results, as shown in Fig. 2. The reliability of the general solutions and boundary conditions of the proposed method was verified using the obtained flux densities.

### C. Calculation of electromagnetic performances

The analytical solutions derived from the general solutions and boundary conditions can be applied to electromagnetic performance analysis such as magnetic flux density, flux linkage, induced electromotive force, and torque based on the electromagnetic field theory.

^{8}

^{9,10}By applying Faraday’s law using the calculated flux linkage, the induced electromotive force can be calculated as follows:

*N*

_{c}represents the number of conductors in the slot and

*C*is the connecting matrix representing the distribution of armature windings.

The electromagnetic performance results such as flux linkage, back-electromotive force (EMF), and torque are obtained using electromagnetic theories, as shown in Fig. 3. The proposed method was verified by comparing the results obtained with the two-dimensional (2D) FEA results. The flux linkage, back-EMF, and torque calculated based on the analytical solutions are in good agreement with the 2D FEA results according to the change in relative permeability.

## III. NONLINEAR ANALYTICAL CALCULATION USING ITERATION BASED ON THE PERMEABILITY OF THE CORE

Because the stator and rotor are constructed from electrical steel sheets with nonlinear-magnetic-field characteristics, the electromagnetic analysis of the WRSM must be conducted by considering saturation. Therefore, in this study, the relative permeabilities of the stator and rotor are calculated as shown in Fig. 4(a). First, the flux density of each analysis region is calculated based on the initial permeability. Second, to consider the global saturation effect of electrical steel, the permeability is derived from the magnetic flux density vs relative permeability (*B*–*H*) data of the electrical steel sheets, as shown in Fig. 4(b).

*μ*

_{r}of each region is obtained according to the following equation:

^{11}

*μ*

_{r,initial}is the initial value of the relative permeability, and

*μ*

_{r,new}represents the relative permeability corresponding to the calculated magnetic flux density.

After converging and satisfying the specified error rate, the relative permeability of each subdomain was computed. Nonlinear electromagnetic analysis, which considers relative permeability, applies a calculation algorithm to each step, resulting in the induced voltage analysis, as depicted in Fig. 4(c), for determining the changes in the field current.

Figure 5 shows the torque analysis results of WRSM according to field and armature current. For torque analysis, the conventional analytical method took 3 s, the proposed analytical method took 8 s, and the nonlinear FEM took 68 s. Although the conventional analytical method provides the fastest analysis, the proposed analytical method also allows for faster analysis compared to nonlinear FEM. In addition, the accuracy of electromagnetic analysis has an error of ∼3% in both the existing and proposed analytical methods when compared to nonlinear FEM under the condition that saturation does not occur. However, compared to nonlinear FEM, the proposed analytical method had an error within 3% at the torque where maximum saturation occurred, while the conventional analytical method had an error of about 241%. The comparison of the analysis results revealed that the proposed analytical method agrees well with the nonlinear FEA results.

## IV. CONCLUSION

This paper presents an improved analytical method that considers the relative permeability of the stator and rotor regions of a WRSM. A WRSM, which has a complex structure, is represented by a simplified analytical model constructed using electromagnetic assumptions. Using the electromagnetic field theory and differential equations, the general solutions of the Laplace and Poisson equations of all the subdomains calculated based on boundary conditions are expressed in the Fourier-series form. To calculate the undetermined coefficients of the general solution, the analytical solution is determined by applying boundary conditions along the radial and circumferential directions. Electromagnetic performance is analyzed based on the derived analysis solution, and the proposed analytical method is verified by comparing the results obtained with the FEA results. In particular, the magnetic saturation of the soft magnetic material in the WRSM is caused by field and armature current, and the superiority of the proposed method is demonstrated by comparing the improved analytical method considering the global saturation of each region with the nonlinear FEA results considering local saturation. The proposed analytical method considering the magnetic permeability of the iron core ensures fast analysis and high accuracy in electromagnetic performances; therefore, this method can be used for several applications, such as electromagnetic field analysis, initial design, and optimal design of WRSMs.

## ACKNOWLEDGMENTS

This research was supported by Korea Institute of Marine Science and Technology Promotion (KIMST) funded by the Ministry of Oceans and Fisheries (1525013494/PMS5490). This work was also supported by a “Development of basis technologies in eco-friendly ship fuel reliability and safety evaluation (1525014866/PES4740)” program funded by Korea Research Institute of Ships and Ocean Engineering.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Kyeong-Tae Yu**: Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal). **Hwi-Rang Ban**: Data curation (equal). **Ju-Hyeong Lee**: Investigation (equal). **Jang-Young Choi**: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal). **Soyoung Sung**: Project administration (equal); Resources (equal). **Jung-Hyung Park**: Project administration (equal); Resources (equal). **Kyung-Hun Shin**: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding authors upon reasonable request.