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.

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.

FIG. 1.

Analysis models: (a) two-dimensional finite element model and (b) simplified analytical model.

FIG. 1.

Analysis models: (a) two-dimensional finite element model and (b) simplified analytical model.

Close modal
Figure 1 represents the simplified analytical model of a WRSM based on polar coordinates. Additionally, the specifications of the analysis model are listed in Table I. The following assumptions are necessary to define the simplified analytical model: the end leakage effect is ignored, the width of each region is angle-dependent, the current density distributions in the stator slots are uniform, and the eddy-current effects owing to the conductivity of the iron core material are neglected. Based on these assumptions, the governing equations in each domain can be derived based on Maxwell’s equations and electromagnetic field theories.4 
2Az=0inRegionsI,II,III,V,VI,andVIII2Az=μ0JinRegionsIVandVII
(1)
where Az is the magnetic vector potential, μ0 is the permeability of vacuum, and J is the current density.
TABLE I.

Design parameters of the wound rotor synchronous machine.

ParametersValuesParametersValues
r01: Outer radius of the rotor shaft 5 mm a: Angle of the rotor tooth 18° 
r02: Outer radius of the rotor yoke 18 mm b: Angle of the rotor slot 72° 
r03: Outer radius of the rotor 44.8 mm c: Angle of the stator tooth 30° 
r04: Inner radius of the stator 45.3 mm d: Angle of the stator winding 30° 
r05: Inner radius of the stator yoke 60.3 mm Qr: Number of rotor slots 
r06: Outer radius of the stator yoke 68 mm Qs: Number of stator slots 
Ns: Number of conductors in the stator slot 120 r07: Number of conductors in the rotor slot 100 
ParametersValuesParametersValues
r01: Outer radius of the rotor shaft 5 mm a: Angle of the rotor tooth 18° 
r02: Outer radius of the rotor yoke 18 mm b: Angle of the rotor slot 72° 
r03: Outer radius of the rotor 44.8 mm c: Angle of the stator tooth 30° 
r04: Inner radius of the stator 45.3 mm d: Angle of the stator winding 30° 
r05: Inner radius of the stator yoke 60.3 mm Qr: Number of rotor slots 
r06: Outer radius of the stator yoke 68 mm Qs: Number of stator slots 
Ns: Number of conductors in the stator slot 120 r07: Number of conductors in the rotor slot 100 
The governing equation of each region can be expressed as a general solution in the form of a Fourier series based on a differential equation using polar coordinates. Two types of general solutions exist depending on the presence or absence of boundary conditions in the circumferential direction. First, when the periodic boundary condition is 2π, such as in the shaft, yoke, and air-gap regions, the following homogeneous solution is obtained:5 
Azh=A0+lnrB0+n=1rknAn+rknBncosknθ+rknCn+rknDnsinknθiz,
(2)
where Azh represents a homogeneous solution. Additionally, kn = n, where n is the order of spatial harmonics and A0, B0, An, Bn, Cn, and Dn represent undetermined coefficients.
Second, when the aperiodic boundary condition has the permeability of adjacent regions, such as in the windings and teeth of a rotor and stator, the homogeneous solution obtained is as follows:6,7
Azh=A0+lnrB0+k=1rkkAk+rkkBkcoskkθθ1+m=1sinhkmθθ1sinhkmζAm+sinhkmθθ2sinhkmζBmsinkmlnrriiz,
(3)
where k and m are spatial harmonics, kk = kπ/ζ, and km = mπ/ln(ro/ri). Here, ri and ro represent the inner and outer diameters of each region, respectively. Furthermore, Ak, Bk, Ck, and Dk represent undetermined coefficients; ζ represents the width (radian) of each region; and θ1 and θ2 represent the start and end positions of each region, respectively.
The coil region has nonhomogeneous solutions owing to its current density. The particular solution in the coil domain is as follows:
Azcoil=14μ0J0r2iz,
(4)
where μ0 is the permeability of vacuum, and J0 represents the current density of the slot.
According to electromagnetic theory, the 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:
i)AzIIr2,θ=AzIII,jr2,θ,HθIIr2,θ=HθIII,jr2,θAzIIr2,θ=AzIV,jr2,θ,HθIIr2,θ=HθIV,jr2,θθαj,αj+aθβj,βj+bii)AzVr3,θ=AzIII,jr3,θ,HθVr3,θ=HθIII,jr3,θAzVr3,θ=AzIV,jr3,θ,HθVr3,θ=HθIV,jr3,θθαj,αj+aθβj,βj+biii)AzVr4,θ=AzVI,ir4,θ,HθVr4,θ=HθVI,ir4,θAzVr4,θ=AzV II,ir4,θ,HθVr4,θ=HθV II,ir4,θθγj,γj+cθδj,δj+div)AzV IIIr5,θ=AzVI,ir5,θ,HθV IIIr5,θ=HθVI,ir5,θAzV IIIr5,θ=AzV II,ir5,θ,HθV IIIr5,θ=HθV II,ir5,θθγj,γj+cθδj,δj+d,
(5)
where αj and βj represent the angular position of the j-th slot and tooth of the rotor. Additionally, γj and δj represent the angular position of the j-th slot and tooth of the stator.
To consider the boundary conditions in the coil and tooth areas of the stator and rotor, the boundary conditions according to the circumferential angle can be expressed as follows:6,7
v)AzIV,jr,αj+a=AzIII,jr,βj,AzIV,j+1r,αj+1=AzIII,jr,βj+bHrIV,jr,αj+a=HrIII,jr,βj,HrIV,j+1r,αj+1=HrIII,jr,βj+brr2,r3rr2,r3vi)AzVI,ir,γj+c=AzV II,ir,δi,AzVI,j+1r,γj+1=AzV II,jr,δj+bHrVI,jr,γj+c=HrV II,jr,δi,HrVI,j+1r,γj+1=HrV II,jr,δj+bθr3,r4θr3,r4,
(6)
The derived analytical solution can be expressed as a magnetic flux density in the radial and tangential directions from the definitions of magnetic vector potential and magnetic flux density.
Br=1rAzθir,Bθ=Azriθ,
(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.

FIG. 2.

Comparison between the flux densities obtained from the analytical and two-dimensional FEA results based on the permeability of the soft magnetic material. (a) Radial flux density of the field coil, (b) tangential flux density of the field coil, (c) radial flux density of the armature coil, and (d) tangential flux density of the armature coil.

FIG. 2.

Comparison between the flux densities obtained from the analytical and two-dimensional FEA results based on the permeability of the soft magnetic material. (a) Radial flux density of the field coil, (b) tangential flux density of the field coil, (c) radial flux density of the armature coil, and (d) tangential flux density of the armature coil.

Close modal

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.

Electromagnetic force characteristics can be calculated as a function of the magnetic flux density in the air gap using the Maxwell stress tensor in the air gap.8 
Tz=r×Fθ=rμ00lstk02πBrgVBθgVrdθdziz.
(8)
Additionally, by applying Stokes’ theorem to the relationship between magnetic flux and magnetic flux density, the magnetic flux linkage can be expressed as a function of the magnetic vector potential.9,10 By applying Faraday’s law using the calculated flux linkage, the induced electromotive force can be calculated as follows:
ϕq=lstkNcAslotγiγi+cr3r4AzV II,jrdrdθ,λaλbλc=Cϕq,
(9)
EaEbEc=ddtλaλbλc,
(10)
where Nc 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.

FIG. 3.

Comparison between the electromagnetic performances according to the permeability of the soft magnetic material in the WRSM based on the analytical and 2D FEA results: (a) flux linkage, (b) back-EMF, and (c) electromagnetic torque.

FIG. 3.

Comparison between the electromagnetic performances according to the permeability of the soft magnetic material in the WRSM based on the analytical and 2D FEA results: (a) flux linkage, (b) back-EMF, and (c) electromagnetic torque.

Close modal

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 (BH) data of the electrical steel sheets, as shown in Fig. 4(b).

FIG. 4.

Nonlinear analytical modeling and analysis results: (a) algorithm for calculating relative permeability, (b) relative permeability of the core, and (c) comparison between the analysis results with those of the analytical and 2D FE analysis.

FIG. 4.

Nonlinear analytical modeling and analysis results: (a) algorithm for calculating relative permeability, (b) relative permeability of the core, and (c) comparison between the analysis results with those of the analytical and 2D FE analysis.

Close modal
The magnetic flux densities of electrical steel are obtained via analytical calculations of the magnetic field, and the relative permeability μr of each region is obtained according to the following equation:11 
ε=μr,initialμr,newμr,new,
(11)
where μr,initial is the initial value of the relative permeability, and μr,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.

FIG. 5.

Analysis results of electromagnetic torque according to field and armature current: (a) conventional analytical method, (b) proposed analytical method, and (c) nonlinear 2D FEM.

FIG. 5.

Analysis results of electromagnetic torque according to field and armature current: (a) conventional analytical method, (b) proposed analytical method, and (c) nonlinear 2D FEM.

Close modal

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.

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.

The authors have no conflicts to disclose.

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).

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

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