This paper presents an analytical method for analyzing the electromagnetic performance of a superconducting (SC) machine according to the 10 MW shielding configuration and the presence or absence of an armature core. To establish the SC machine design process, an analytical method considers shielding conditions and armature core type is included. The presented analytical method derives the governing equations and general solutions for each region using Maxwell’s equations and electromagnetic field theory. Furthermore, it calculates the analytical solutions by applying appropriate boundary conditions. Additionally, the performance of actively and passively shielded, with the same size and shielding performance of SC machine is compared. The electromagnetic performance obtained through the analytical method is compared with finite element analysis to validate its accuracy. The accuracy of the presented analytical method can be utilized in various design analyzes, including initial design and optimal design.

Research is being conducted on superconducting (SC) machines that reduce the weight and volume by enhancing specific power.1–7 Notably, EcoSwing’s partial SC machine prototype achieved a remarkable 24% reduction in weight.8–10 However, the SC coil has the capability to transmit a current density ∼100 times higher than that of copper, resulting in a strong magnetic flux density of up to 4 T.11,12 Such high magnetic fields can have adverse effects on the human body, prompting the international commission on non-ionizing radiation protection to generally recommend limiting exposure to less than 0.4 T.13 In SC machines, passive shielding is generally used to shield leakage flux using an iron core. However, it has the disadvantage of increasing weight. Consequently, research has been conducted to achieve comparable shielding performance by using SC coils for shielding, thereby reducing weight and enhancing specific power.14,15

Therefore, this paper presents an analytical method for electromagnetic analysis of SC machines according to the materials of the armature core and shielding structures. The shielding structures are designed to satisfy the 0.27 T, which is 67.5% of 0.4 T, the magnetic flux density that has a harmful effect on the human body at a specific radius of the SC machine. In the proposed analytical method to consider the armature core, the permeability of the air-core is the same as that of vacuum, and the permeability of the electrical steel-core is calculated using an iterative method based on electromagnetic analysis. The feasibility of the analytical method presented herein is validated through finite element (FE) analysis of electromagnetic performance.

Figures 1(a) and 1(b) show the configurations of actively and passively shielded SC machines according to materials of an armature core. Both the field coil and shield coil are SC coils and are manufactured using RE(Gd)BCO. The field coil and shield coil are 72 mm apart in the radial direction, and the number of turns per layer is 2026 turns and 500 turns, respectively. The proposed analytical models are represented by four models according to the permeability and current modeling of region II and region VII, as shown in Fig. 1(c).

FIG. 1.

Structure of SC machine according to armature core materials with (a) actively shielded and (b) passively shielded, and (c) analytical model.

FIG. 1.

Structure of SC machine according to armature core materials with (a) actively shielded and (b) passively shielded, and (c) analytical model.

Close modal

Detailed analytical modeling and assumptions are defined based on previously published papers.16  Table I shows the design parameters of the SC machine with the presence or absence of an armature core and actively shielded and passively shielded based on the same output power condition.

TABLE I.

Key design parameters of the superconducting machine.

ParametersSymbolValuesParametersSymbolValues
Number of harmonics N 100 Axial length lstk 1000 mm 
Number of poles P Rotating speed Nrpm 500 rpm 
Number of turns for field coil Nmf 8104 Number of turns for shield coil Nsh 1800 
Field current Imf 100 Adc Shield current Ish 90/195 Adc 
Number of turns for armature coila,b,c,d Nac 10/7/13/7 Number of parallel branchesa,b,c,d Npb 2/2/4/4 
Armature current Iac 930 Arms Number of slot per pole per phase Nsp 
Field coil aperture Amf 392 mm Shield coil aperture Ash 520 mm 
Field coil width wmf 259 mm Shield coil width wsh 80 mm 
Field coil height hmf 24 mm Shield coil height hsh 9 mm 
Armature coil height hac 50 mm Ratio of coil width to slot pitch αc 0.9 
Inner air-gap hgi 16 mm Outer air-gap hgo 86 mm 
ParametersSymbolValuesParametersSymbolValues
Number of harmonics N 100 Axial length lstk 1000 mm 
Number of poles P Rotating speed Nrpm 500 rpm 
Number of turns for field coil Nmf 8104 Number of turns for shield coil Nsh 1800 
Field current Imf 100 Adc Shield current Ish 90/195 Adc 
Number of turns for armature coila,b,c,d Nac 10/7/13/7 Number of parallel branchesa,b,c,d Npb 2/2/4/4 
Armature current Iac 930 Arms Number of slot per pole per phase Nsp 
Field coil aperture Amf 392 mm Shield coil aperture Ash 520 mm 
Field coil width wmf 259 mm Shield coil width wsh 80 mm 
Field coil height hmf 24 mm Shield coil height hsh 9 mm 
Armature coil height hac 50 mm Ratio of coil width to slot pitch αc 0.9 
Inner air-gap hgi 16 mm Outer air-gap hgo 86 mm 
a

SC machine with active shielding and air-core.

b

SC machine with passive shielding and air-core.

c

SC machine with active shielding and steel-core.

d

SC machine with passive shielding and steel-core.

The governing equations of each region are derived from electromagnetic theories and Maxwell’s equations, and the general solution expressed in Fourier series using separation of variables can be expressed as17,18
AznSD=n=1AnSDrrlnp+BnSDrrhnp+WnsSDsinnpθ+CnSDrrlnp+DnSDrrhnp+WncSDcosnpθ
(1)
where superscript SD represents each region, and rl and rh represent the lower and upper boundaries of each region. AnSD, BnSD, CnSD, and DnSD are unknown coefficients. WnsSD and WncSD are particular solutions according to current density distribution, respectively. The undetermined coefficients of general solutions are substituted into the boundary conditions of each model to derive an analytical solution.17–19 

Table II shows the particular solution and relative permeability conditions according to actively and passively shielded and the presence or absence of the armature core. For the steel-core armature model, the relative permeability of each region was determined from an iterative method.20 

TABLE II.

Particular solutions and relative permeabilities in regions II and VII based on the structure of SC machines.

Type of SC machinesRegion IIRegion VII
Active shielding and air-core μr = 1 Wns=μ0r2Js/np24,Wnc=μ0r2Jc/np24 
Passive shielding and air-core  Wns = 0, Wnc = 0 
Active shielding and steel-core  Wns=μ0r2Js/np24,Wnc=μ0r2Jc/np24 
  
   
   
   
   
   
   
   
   
   
   
   
Passive shielding and steel-core  Wns = 0, Wnc = 0 
Type of SC machinesRegion IIRegion VII
Active shielding and air-core μr = 1 Wns=μ0r2Js/np24,Wnc=μ0r2Jc/np24 
Passive shielding and air-core  Wns = 0, Wnc = 0 
Active shielding and steel-core  Wns=μ0r2Js/np24,Wnc=μ0r2Jc/np24 
  
   
   
   
   
   
   
   
   
   
   
   
Passive shielding and steel-core  Wns = 0, Wnc = 0 

Figures 2(a)2(d) shows the magnetic flux density distribution and mesh distribution of the SC machine according to armature core materials and shielding. The mesh of the FE analysis is analyzed at ∼597 000 nodes, and the computation time takes ∼40 min (2400 s) for each model. The proposed analytical method calculates up to the 100th harmonic order, and the computation time takes 4.38 s.

FIG. 2.

Flux density distribution and 2-D mesh model of SC machine: air-cored SC machine with (a) actively shielded and (b) passively shielded, steel-cored SC machine with (c) actively shielded and (d) passively shielded.

FIG. 2.

Flux density distribution and 2-D mesh model of SC machine: air-cored SC machine with (a) actively shielded and (b) passively shielded, steel-cored SC machine with (c) actively shielded and (d) passively shielded.

Close modal

Figure 3 compares the magnetic flux density derived from the proposed analytical method and FE analysis. From the analysis results, the air gap flux density increases significantly when steel-core and passive shielding are applied to SC machine. However, SC machine with steel-cored armature and passive shielding causes disadvantages in terms of weight and loss.

FIG. 3.

Comparison of the flux density at the air gap with analytical results and FE results: air-cored SC machine with (a) actively shielded and (b) passively shielded, steel-cored SC machine with (c) actively shielded and (d) passively shielded.

FIG. 3.

Comparison of the flux density at the air gap with analytical results and FE results: air-cored SC machine with (a) actively shielded and (b) passively shielded, steel-cored SC machine with (c) actively shielded and (d) passively shielded.

Close modal

Figure 4 shows the flux linkage, back-EMF, synchronous inductance, and torque calculated from the proposed analytical method and FE analysis, respectively. The analysis models used in this study were designed by selecting the number of armature turns and parallel branches to achieve same output power. From comparison of analysis results, the proposed analytical method has high reliability in electromagnetic analysis of SC machines.

FIG. 4.

Electromagnetic performances: (a) flux linkage, (b) back-EMF, (c) synchronous inductances, and (d) torque.

FIG. 4.

Electromagnetic performances: (a) flux linkage, (b) back-EMF, (c) synchronous inductances, and (d) torque.

Close modal

Figure 5(a) shows the leakage density of the active shielded SC machine with air-cored armature according to the shielding current (Ish) and the number of turns of the shielding coil (Nsh). When Ish is 400 A and Nsh is 2000 turns, the leakage flux decreases significantly (0.062 T), and when the shielding current increases, the leakage flux due to the shielding coil increases. Figure 5(b) shows the leakage density of actively shielded SC machines with steel-cored armature according to the shielding current and the number of turns of the shielding coil.

FIG. 5.

Comparison of leakage magnetic flux density according to shielding current and shielding conditions: actively shielded SC machine with (a) air-core and (b) steel-core, and passively shielded SC machine with (c) air-core and (d) steel-core.

FIG. 5.

Comparison of leakage magnetic flux density according to shielding current and shielding conditions: actively shielded SC machine with (a) air-core and (b) steel-core, and passively shielded SC machine with (c) air-core and (d) steel-core.

Close modal

Figures 5(c) and 5(d) show the shielding performance of a passively shielded SC machine with air-cored and steel-cored armature, focusing on the thickness of the passive shield (Tps) and inner radius of passive shield (R06). In the case of passively shielded SC machines, the shielding effect is mainly determined by the thickness of the shielding core, so a thicker shielding core directly improves the shielding performance. From the analysis results, shielding conditions with the same shielding performance were derived. The shielding current of an actively shielded SC machine that satisfies the requirements for the outermost leakage flux is 90 A for an air-core and 195 A for a steel-core. In addition, the thickness of the core of the passively shielded SC machine is 90 mm for the air-core and 310 mm for the steel-core.

A quantitative comparison of the four models was performed at the same output power of 10 MW. Figure 6(a) compares the weight of electrical components of actively and passively shielded SC machines with and without an armature core. Notably, the actively shielded SC machine with air-cored configuration exhibits the lightest weight. A comprehensive comparison of electromagnetic performances is presented in Fig. 6(b). The y-axis represents the ratio of each model to its maximum value of performance, and each quantitative value is displayed on the graph.

FIG. 6.

Comparison of weight and performances: (a) weight of electrical components and (b) electromagnetic performances.

FIG. 6.

Comparison of weight and performances: (a) weight of electrical components and (b) electromagnetic performances.

Close modal

To compare weight and specific power, actively shielded and passively shielded SC machines are compared under conditions of the same size and shielding performance. As a result, it was confirmed that the actively shielded SC machine has the advantage of lighter weight than the passively shielded SC machine, regardless of the presence or absence of a core.

In air-core, the weight was reduced by 60%, and the specific power was also improved by 154%. In steel-core, the weight was reduced by 54%, and the specific power was also improved by 103%.

This paper presents an analytical method for conducting electromagnetic analysis on 10 MW class SC machines with both actively and passively shielded considering the presence or absence of an armature core. Furthermore, to validate comparative analysis with FE analysis results. The comparison between the FE analysis results and the results of presented analytical method show an error of less than 3%. Actively shielded SC machines yielded a substantial weight reduction of 60% in the air-core and 54% in the steel-core, accompanied by a significant improvement in specific power, with an increase of 154% in the air-core and 103% in the steel-core. It was also demonstrated that the specific power of actively shielded SC machine surpassed that of passive shielded SC machine. These results, derived through an analytical method that actively shielded SC machine can deliver equivalent shielding performance to passively shielded while offering a significant weight advantage. The proposed analytical method can be used for electromagnetic analysis, initial design, and optimal design of various SC machines.

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2021R1G1A1013741). This research was also supported by Korea Institute of Marine Science and Technology Promotion (KIMST) funded by the Ministry of Oceans and Fisheries (RS-2023-00254688).

The authors have no conflicts to disclose.

Hwi-Rang Ban: Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal). Kyeong-Tae Yu: 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). Kyong-Hwan Kim: Project administration (equal); Resources (equal). Han-Wook Cho: 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.

1.
A. B.
Abrahamsen
et al,
IEEE Trans. Appl. Supercond.
19
,
1678
(
2009
).
2.
T.-K.
Hoang
,
L.
Quéval
,
L.
Vido
, and
D.-Q.
Nguyen
,
IEEE Trans. Appl. Supercond.
32
,
5202606
(
2022
).
3.
D.
Lee
,
T.
Balachandran
,
H.-W.
Cho
, and
K.
Haran
,
IEEE Trans. Magn.
55
,
8106506
(
2019
).
4.
Y.
Liang
,
M. D.
Rotaru
, and
J. K.
Sykulski
,
IEEE Trans. Appl. Supercond.
23
,
5202805
(
2013
).
5.
J.
Wang
,
R.
Qu
,
Y.
Liu
,
J.
He
,
Z.
Zhu
, and
H.
Fang
,
IEEE Trans. Appl. Supercond.
25
,
5201806
(
2014
).
6.
T.
Balachandran
,
A.
Yoon
,
D.
Lee
,
J.
Xiao
, and
K. S.
Haran
,
IEEE Trans. Magn.
58
,
8700805
(
2021
).
7.
R.
Qu
,
Y.
Liu
, and
J.
Wang
,
IEEE Trans. Appl. Supercond.
23
,
5201108
(
2013
).
8.
X.
Song
et al,
IEEE Trans. Energy Convers.
35
,
757
(
2019
).
9.
X.
Song
et al,
IEEE Trans. Energy Convers.
35
,
1697
(
2020
).
10.
X.
Song
,
Y.
Wang
,
D.
Liu
,
S.
Wang
, and
D.
Liang
,
IEEE Trans. Appl. Supercond.
29
,
5202605
(
2019
).
11.
Y.
Guan
,
Z.
Zhu
,
Z.
Azar
,
A. S.
Thomas
,
F.
Vedreño-Santos
,
G.-J.
Li
, and
M.
Odavic
,
IEEE Trans. Appl. Supercond.
27
,
5204211
(
2017
).
12.
K.
Zhang
,
X.
Huang
,
L.
Wu
,
Y.
Fang
, and
W.
Cao
,
IEEE Trans. Appl. Supercond.
29
,
5201205
(
2019
).
13.
International Commission on Non-Ionizing Radiation Protection (ICNIRP)
,
Health Phys.
96
,
504
(
2009
).
14.
H.-W.
Cho
,
T.-K.
Bang
,
J.-I.
Lee
,
K.-H.
Shin
,
H.-S.
Lee
,
J.-S.
Hur
, and
K. S.
Haran
,
IEEE Trans. Appl. Supercond.
32
,
5202505
(
2022
).
15.
K. S.
Haran
,
D.
Loder
,
T. O.
Deppen
, and
L.
Zheng
,
IEEE Trans. Appl. Supercond.
26
,
5202508
(
2016
).
16.
K.-H.
Shin
,
T.-K.
Bang
,
J.-Y.
Choi
,
H.-W.
Cho
, and
K. S.
Haran
,
AIP Adv.
11
,
025306
(
2021
).
17.
H.-W.
Cho
and
K. S.
Haran
,
IEEE Trans. Appl. Supercond.
28
,
5206808
(
2018
).
18.
K. H.
Shin
,
T. K.
Bang
,
H. W.
Cho
,
K. H.
Kim
,
K.
Hong
, and
J. Y.
Choi
,
IEEE Trans. Magn.
55
,
7501805
(
2019
).
19.
G.
Malé
,
T.
Lubin
,
S.
Mezani
, and
J.
Lévêque
,
Supercond. Sci. Technol.
24
,
035014
(
2011
).
20.
M.
Shen
,
P.-D.
Pfister
,
C.
Tang
, and
Y.
Fang
,
IEEE Trans. Magn.
57
,
7400105
(
2020
).