This paper focuses on electromagnetic and loss analyses of an induction machine using the subdomain method. The proposed method derives an analytical solution for each subdomain using electromagnetic theory. On the basis of the derived analytical solution, electromagnetic performances are calculated and copper loss, iron loss, and rotor loss are derived. The validity of the proposed method was verified by comparing the analysis and loss analysis results with the finite element analysis results. In particular, the loss analysis results were input into an equivalent thermal analysis model, thermal analysis was performed, and the results were compared with those measured in the actual model. The reliability of the proposed method was verified through analysis and experimental results.

Induction machines are widely used throughout the industry owing to their advantages such as price competitiveness, wide operating range, and robust construction.1 With the advancement of manufacturing and material technology, the design technology of induction machines is required, and in recent years, the design of high-efficiency induction machines has mainly been studied.2–4 The analytical methods for the design of an induction machine include the loading distribution method5 and the equivalent circuit (EC) method,6 both of which have the disadvantage that the shape and material cannot be considered. An analytical solution for various types of machines has been developed7–9 by employing the subdomain method, which can consider the slotting effect of permanent-magnet machines. In addition, analytical solutions using analytical methods for induction machines have been obtained in previous studies.10,11 However, it is very difficult to calculate and experimentally verify electromagnetic losses, which are a major issue in the design of induction machines. In this paper, the electromagnetic loss analysis of an induction machine is performed using the subdomain method and the temperature characteristics are predicted using thermal EC. On the basis of the derived analytical solutions, EC parameters are calculated and losses due to applied current and slip are analyzed. The proposed method is validated by comparing its results with the finite element (FE) analysis results. The thermal analysis results obtained by substituting the loss analysis results obtained from the proposed and FE analysis methods as heat sources into the EC model were compared with the experimental results. The validity of the proposed method was verified through a comparison of analysis and experimental results.

Figures 1(a) and 1(b) show the prototype and structure of the induction machine. For the subdomain method, characteristic analysis is performed using a simplified analytical model such as that shown in Fig. 1(c) from the 2D FE model shown in Fig. 1(b). A simplified analytical model for electromagnetic analysis is obtained by applying several assumptions.7–11 In Fig. 1(c), θss, θsso, θrso, and θrs are the angular positions of the i-th stator slot opening, i-th stator slot, j-th rotor slot opening, and j-th rotor slot:
θsso=β2+2iπQ,θss=θsso+12βχ,θrso=γ2+2jπB,θrs=θrso+12γδ
(1)

The main parameters of this geometry are defined as given in Table I.

General solutions that can predict the magnetic field distribution using Fourier series and separation of variables are derived in all subdomains.10,11
AzI=A0IJ0λr+B0IY0λr+v=1AvIJvπ/δλr+BvIYvπ/δλr×cosvπδθθrsiz,
(2)
AzII,IV=A0II,IV+B0II,IVlnr+l,m=1Azl,zmII,IVrr2,4kl,m+Bzl,zmII,IVrr3,5kl,m×coskl,mθθrso,ssoiz,
(3)
AzIII=n=1AnIIIrr3n+BnIIIrr4nsinnθ+CnIIIrr3n+DnIIIrr4ncosnθiz,
(4)
AzV=A0V+B0Vlnr14μ0J0Vr2+m=1AmVirr5mπχ+BmVirr6mπχ×cosmπχθθssiz,
(5)
where v, l, n, k, and m are the spatial harmonics orders (a positive integer); λ=jμ0σωsl, kl = /γ, km = /β; j varies from 1 to Qr rotor bars; and δ, γ, β, and χ are the angular widths of the rotor conductor bar, rotor slot opening, stator slot opening, and stator slot, respectively. Further, J/δ(λr) and Y/δ (λr) are the Bessel functions of the first and second kinds, respectively, of order /δ.
From the definition of the magnetic vector potential, the magnetic flux densities of the normal and tangential components are expressed as follows:
Br=1rAθir,Bθ=Aθiθ.
(6)

The proposed analytical model has 10 boundary conditions and a total of 25 equations and 25 undetermined coefficients depending on DC and harmonic terms. A linear matrix system for calculating the boundary conditions is introduced in references.7–11 By calculating the boundary conditions, analytical solutions were derived. The proposed analytical solution and boundary conditions were validated through a comparison with the FE analysis results as shown in Fig. 2.

From the analytical solution, the EC parameters of the induction machine can be derived. The EC parameters are primary resistance, primary leakage inductance, magnetization inductance, secondary resistance, and secondary leakage inductance.6 The primary resistance can be calculated from the winding specifications, and the magnetizing inductance can be calculated as the synchronous inductance at no load. From the EC method, when current is applied to the primary at a specific slip, the secondary impedance is derived from the flux linkage and synchronous inductance. The real part of the derived impedance represents the secondary resistance, and the imaginary part represents the secondary reactance. Depending on the applied current and slip, the speed-torque characteristics of the induction machine can be calculated. The Maxwell stress tensor was applied to the calculation of torque in the electromagnetic field analysis method.

From the derived analytical solutions, EC parameters were calculated and losses due to applied current and slip were analyzed. The experimental results for temperature were compared by substituting the loss derived from the analytical method and the loss obtained from the FE analysis results into the thermal EC model.

The resistance per phase (Rph) can be calculated as Rph = Nturnρc lc/Ac, where ρc is the electrical resistivity of the conductor, lc is the circumference of the coil, and Ac is the cross-sectional area of the coil given by Ac = πrc2. Here, rc is the radius of the coil. The copper loss of the primary stator can be calculated from the primary current and resistance:12 
Pcopper=3Iph2Rph.
(7)
where Iph represents the rms value of phase current.
The core loss is calculated using the modified Steinmetz equation and Bertotti’s model.13–15 
Pcore=VcoreρsteelAconstkhfBmn+kef2Bm2+kaf1.5Bm1.5,
(8)
where kh, ke, ka, and n are the hysteresis, eddy current, anomalous loss coefficients and the Steinmetz constant, respectively. f and Bm are the operating frequency and the peak flux density in the core. Aconst represents the alternating and the rotating field areas judged from the loci of the time harmonic, ρsteel is the specific gravity of steel, and Vcore is the effective volume of the core.

In Eq. (8), the magnetic flux density in the core area is required to calculate the core loss of the induction machine. However, the proposed analytical method cannot directly predict the magnetic flux density of the core because it assumes the permeability of the core is infinite. Using the search coil, the magnetic flux density of each area of the stator shoe, stator teeth, and stator yoke and the rotor shoe, rotor teeth, and rotor yoke are predicted. The calculation of core loss considering the rotating and alternating magnetic fields is expressed in Ref. 16.

The eddy current densities induced in the rotor bar are given in the complex domain as17 
Jerr,θ=jωrmσAzvIr,θ,
(9)
where ωrm is the frequency of eddy-currents in the rotor bars, σ is the electrical conductivity of the rotor bars.
Further, the current flowing through the rotor bar is calculated as
Ier=r1r2θrsθrs+δjerr,θrdrdθ.
(10)
The rotor loss can be calculated from the relationship between the current flowing through the rotor bar and the resistance of the rotor bar:
Protor=BIer2Rrotor.
(11)
where B represents the number of rotor bars.

The torque calculation and loss analysis results obtained using the proposed analytical method were confirmed to be in good agreement with the FE results as shown in Fig. 3.

For the prediction of convection heat transfer and correctly modeling internal interface resistance, a 3D lumped circuit model is used to calculate the steady-state and transient thermal characteristics of an induction machine. Firstly, EC modeling for thermal analysis is derived based on information such as the overall structure, size, and material of the induction machine. Secondly, the cooling method was selected as natural convection under room temperature (25 °C) conditions. Third, the gap of each component of the induction machine in thermal EC was selected considering manufacturing tolerances. Convection occurring on all external surfaces of the motor was calculated using a validated natural convective heat transfer correlation formula. The thermal resistances affecting the conduction heat transfer paths were calculated from the dimensions and thermal conductivity of each component. Figure 4(a) shows the arrangement of thermal resistors. Depending on the heat transfer method, it can be classified into conduction, convection, and radiation thermal resistance. Thermal resistance, radiation resistance, convection resistance, and interface resistance were calculated according to the theory and corrected through experiments.18,19 Finally, the losses derived through the proposed analysis and FE analysis were substituted as a heat source in the thermal EC model to obtain the thermal characteristics under the operating conditions of the induction machine.

The thermal EC model was derived using Ansys Motor-CAD and can be expressed as thermal EC modeling in the radial and axial directions as shown in Figs. 4(b) and 4(c), respectively. Based on thermal EC modeling, analysis of steady-state and transient thermal resistance, power flow, and temperature data of nodes is performed.

The experimental setup was implemented using a 0.4-kW, 1710-rpm industrial induction machine model and a hysteresis brake load (Magtrol Co., Ltd.). Three thermocouples were attached to the induction machine. Two of these thermocouples were mounted on each of the stator teeth and end-turns inside the machine. The machine’s internal and ambient temperatures were measured using a thermograph (DP10, Yokogawa). Figure 5 shows the temperature at the stator value and end-turn obtained when the induction machine was operated at 1750 rpm and 1.1 Nm. The good agreement between the experimental and temperature analysis results confirmed that the proposed loss analysis and temperature calculation processes were valid.

This paper presents analytical predictions and experimental verification of the magnetic field distribution, eddy current, EC parameters, and temperature characteristic analysis of an induction machine. The proposed analytical method considers the slotting effect of the stator and rotor and derives a general solution from the Laplace and Poisson equations depending on the presence or absence of sources and the Helmholtz equation considering the time-varying magnetic field. From the derived analytical solutions, EC parameters are calculated and losses due to applied current and slip are analyzed. The validity of the proposed method is verified by comparing the proposed analysis results with the FE analysis results. The thermal analysis results obtained by substituting the loss analysis results obtained from the proposed and FE analysis methods as heat sources into the EC model were compared with the experimental results. The proposed method was validated through a comparison of analysis and experimental results.

This research was supported by Korea Institute of Marine Science and Technology Promotion (KIMST) funded by the Ministry of Oceans and Fisheries (Grant No. 1525013774/PMS5470). This work was also supported by a “Development of basis technologies in eco-friendly ship fuel reliability and safety evaluation (Grant No. 1525014866/PES4740)” program funded by Korea Research Institute of Ships and Ocean Engineering.

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: Validation (equal); Visualization (equal). Soyoung Sung: Project administration (equal); Resources (equal). Jung-Hyung Park: Project administration (equal); Resources (equal). Han-Wook Cho: Conceptualization (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – review & editing (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.
K.
Rajashekara
,
IEEE Trans. Ind. Appl.
30
,
897
(
1994
).
2.
V. T.
Buyukdegirmenci
,
A. M.
Bazzi
, and
P. T.
Krein
,
IEEE Trans. Ind. Appl.
50
,
395
(
2014
).
3.
Z.
Yang
,
F.
Shang
,
I. P.
Brown
, and
M.
Krishnamurthy
,
IEEE Trans. Transp. Electrif.
1
,
245
(
2015
).
4.
G.
Pellegrino
,
A.
Vagati
,
B.
Boazzo
, and
P.
Guglielmi
,
IEEE Trans. Ind. Appl.
48
,
2322
(
2012
).
5.
T. A.
Lipo
,
Introduction to AC Machine Design
(
University of Wisconsin
,
2004
).
6.
P.
Alger
,
Induction Machines, Their Behavior and Uses
(
Gordon and Breach
,
New York
,
1995
).
7.
T.
Lubin
,
S.
Mezani
, and
A.
Rezzoug
,
IEEE Trans. Magn.
47
,
479
(
2011
).
8.
B.
Hannon
,
P.
Sergeant
, and
L.
Dupre
,
IEEE Trans. Magn.
50
,
8101410
(
2014
).
9.
K.
Boughrara
,
R.
Ibtiouen
, and
T.
Lubin
,
IEEE Trans. Magn.
48
,
2121
(
2012
).
10.
K.
Boughrara
,
F.
Dubas
, and
R.
Ibtiouen
,
IEEE Trans. Magn.
50
,
7028214
(
2014
).
11.
K.
Boughrara
,
N.
Takorabet
,
R.
Ibtiouen
,
O.
Touhami
, and
F.
Dubas
,
IEEE Trans. Magn.
51
,
8200317
(
2015
).
12.
M.-D.
Nguyen
,
J.-H.
Woo
,
H.-S.
Shin
,
Y.-K.
Lee
,
H.-K.
Lee
,
K.-H.
Shin
,
A.-T.
Phung
, and
J.-Y.
Choi
,
AIP Adv.
13
,
025140
(
2023
).
13.
G.
Novak
,
J.
Kokošar
,
M.
Bricelj
,
M.
Bizjak
,
D. S.
Petrovič
, and
A.
Nagode
,
IEEE Trans. Magn.
53
,
2001805
(
2017
).
14.
C. P.
Steinmetz
,
Proc. IEEE
72
,
197
221
(
1984
).
15.
G.
Bertotti
,
IEEE Trans. Magn.
24
,
621
630
(
1988
).
16.
K. H.
Shin
,
K.
Hong
,
H. W.
Cho
, and
J.-Y.
Choi
,
IEEE Trans. Appl. Supercond.
28
,
5205005
(
2018
).
17.
P.-D.
Pfister
,
X.
Yin
, and
Y.
Fang
,
IEEE Trans. Magn.
52
,
8103013
(
2016
).
18.
K. H.
Seong
,
J. H.
Wang
,
J. Y.
Shim
, and
H. W.
Cho
,
IEEE Trans. Magn.
50
,
358
(
2014
).
19.
See www.motor-design.com for Motor-CAD.