# Analytical Modeling of Flux-Switching In-Wheel Motor Using Variable Magnetic Equivalent Circuits

Authors: Y. Tang; J. J. H. Paulides; E. A. Lomonova
Journal: ISRN Automotive Engineering
 (2014)

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## Abstract

Flux-switching motors (FSM) are competitive candidates for in-wheel traction systems. However, the analysis of FSMs presents difficulty due to their complex structure and heavy magnetic saturation. This paper presents a methodology to rapidly construct, adapt, and solve a variable magnetic equivalent circuit of 12-stator-slot 10-rotor-tooth (12/10) FSMs. Following this methodology, a global MEC model is constructed and used to investigate correlations between the radial dimensions and the open-circuit phase flux linkage of the 12/10 FSM. The constructed MEC model is validated with finite element analysis and thus proved to be able to assist designers with the preliminary design of flux-switching motors for different in-wheel traction systems.


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## Body

## 1. Introduction

With the rising concern on environmental issues, hybrid/electric vehicles (HEV/EV) have attracted an increasing interest from public and industry since the end of the 20th century. The fast developing technology of HEV/EV has also introduced a revolutionary traction concept to vehicles designers, namely, in-wheel traction. By putting electric motors in the wheels, the drivetrain is greatly simplified. Mechanical axes can thus be removed, which creates extra space for the cargo and reduces the total weight of the vehicle [1, 2], as shown in Figure 1(a). These advantages of simplicity and freedom make the in-wheel traction a preferable traction mode for vehicle designers.In-wheel traction—(a) an electric truck with large cargo and (b) an in-wheel motor.
(a)
(b)For the design of an in-wheel traction system, there are two possible topologies, namely, direct driving and indirect driving. In a direct-driving system, the electrical motor is directly driving the wheel without a gearbox, as shown in Figure1(b). This direct-driving in-wheel module provides a maximum simplicity for the system design; thus, it is commonly adopted in most existing in-wheel traction systems.However, due to the absence of gearbox, the electrical motor of a direct-driving system needs to provide a high torque. The high torque in-wheel motor increases the wheel mass and consequently reduces the passenger comfort [3]. To solve this problem, an indirect-driving in-wheel module, shown in Figure 2(a), can be adopted, in which the electrical motor is indirectly driving the wheel through a gearbox. By this means, the required torque for the motor is reduced (Figure 2(b)), and the wheel mass is maintained. Different topologies of the in-wheel traction lead to different requirements and constraints for the motor design. Nevertheless, in both topologies, electrical motors need to have a high torque density with certain level of ruggedness.Light-weight in-wheel module—(a) the module structure and (b) torque-speed requirements for electrical motors in direct-driving and indirect-driving in-wheel modules.
(a)
(b)Permanent magnet synchronous motor (PMSM) is advantageous in torque production. However, the performance of PMSM is usually limited due to crucial limits on the thermal and mechanical conditions of the permanent magnets in the rotor. Compared to PMSM, switched reluctance motor (SRM), on the one hand, presents better ruggedness benefitting from the robust rotor structure and, on the other hand, lower torque density resulting from the absence of a second magnetic excitation source [1, 2, 4].Flux-switching motors (FSM), including flux-switching permanent magnet motor (FSPMM) and dc-excited flux-switching motor (DCE-FSM) as shown in Figure3, are preferably considered as they combine the advantages of both SRM and PMSM. By placing the permanent magnets in the stator, FSPMMs gain a similar robust rotor structure as SRMs while preserving comparable torque density as PMSMs due to the flux-concentrating effect [5–7]. Hence, they are a competitive candidate for the application of in-wheel traction where high torque and ruggedness are both concerned.Cross-sections of (a) flux-switching permanent magnet motor (FSPMM) and (b) dc-excited flux-switching motor (DCE-FSM).
(a)
12/10 FSPMM
(b)
12/10 DCE-FSMHowever, the analysis of FSMs presents great difficulty due to their complex structure and heavy magnetic saturation. Numerical methods, such as finite element analysis (FEA), usually require a significant computational time despite of relatively high accuracy. Thus, they are not preferred for preliminary motor design, for example, initial sizing [8, 9]. In contrast, certain analytical methods, such as Fourier analysis, are relatively faster in computing. However, they are less accurate in solving nonlinear magnetic problems resulting from magnetic saturation [10–13]. Magnetic equivalent circuits (MEC) provide a good compromise between computational time and accuracy. Hence, they have been extensively used for analyzing subsystems of electrical motors and actuators which are prone to saturation, for example, the stator and rotor teeth or back iron [14–21].In this paper, 12-stator-slot 10-rotor-tooth (12/10) FSMs selected for various in-wheel traction systems are modeled using nonlinear variable MEC. The methods for constructing and solving the variable MEC of the 12/10 FSM are introduced in Sections2 and 3. Using the constructed variable MEC, correlations between certain design parameters and the open-circuit phase flux linkage of 12/10 FSM are investigated. The results obtained using the MEC method are validated by the FEA results in Section 4.


## 2. Variable Magnetic Equivalent Circuits

The airgap magnetic flux distribution of an FSM is highly sensitive to its rotor position due to the double-salient structure of this motor. Hence, the global MEC of the motor needs to be frequently updated in synchronism with rotor position. At certain positions, not only the permeance values but also the circuit topology should be changed. This repetitive adaption usually requires a considerable amount of manual effort and makes it especially difficult to implement the MEC method for motors with low periodicity, in which an elaborate global MEC is required.In this paper, the global MEC of 12/10 FSM is constructed based on a number of MEC modules, including stator MEC modules, rotor MEC modules, and airgap MEC modules. By this means, the requirement of adapting the global MEC is addressed to specific MEC modules. Hence, the involved workload is reduced.


### 2.1. Categorization of  MEC Elements

The MEC method is based on the analogies of the quantity relations in a magnetic field and in a resistive electric circuit [22, 23], as
(1)
Φ
=
ℱ
ℛ
=
ℱ
·
𝒫
,
in which Φ is the flux, ℱ  is the magnetomotive force (mmf), ℛ is the reluctance, and 𝒫 is the permeance.In a MEC, permanent magnets are modeled as mmf sources with reluctances in series or as flux sources with reluctances in parallel, shown in Figure4. In these models, the values of mmf source ℱ
PM and flux source Φ
PM can be calculated as
(2)
ℱ
PM
=
B
r
μ
0
μ
r
l
PM
,
(3)
Φ
PM
=
B
r
A
PM
,
in which B
r, μ
r, l
PM, and A
PM are the remanent magnetic flux density, the relative permeability, the length in the magnetization direction, and the area of the cross-section perpendicular to the magnetization direction, respectively, of the permanent magnet.Magnetic equivalent circuit models of permanent magnets as (a) an mmf source and a reluctance in series; (b) a flux source and a reluctance in parallel.
(a)
(b)Reluctances or their reciprocals permeances are approximately calculated by merging an area of flux paths into flux tubes. These flux tubes are defined as objects that enclose the flux lines with their side face parallel and head face perpendicular to them [22].For the 12/10 FSM, seven types of flux tubes are identified, shown in Figure5. The equally distributed lines in each cross-section indicate the flux paths, and the magnetic properties along the longitudinal dimension of each flux tube are assumed to be homogeneous.Transversal cross-sections of flux tubes and their permeances: (a)𝒫
=
μ
L
θ
/
ln
(
r
2
/
r
1
); (b) 𝒫
=
μ
L
ln
(
r
2
/
r
1
)
/
θ; (c) 𝒫
=
μ
L
x
/
h; (d) 𝒫
=
2
μ
L
·
ln
[
1
+
π
x
/
(
π
r
+
2
h
)
]
/
π; (e) 𝒫
=
μ
L
·
ln
[
1
+
2
π
x
/
(
π
r
1
+
π
r
2
+
2
h
)
]
/
π; (f) 𝒫
=
2
μ
L
x
/
(
π
w
+
2
h
).
(a)
(b)
(c)
(d)
(e)
(f)To simplify the permeance calculation of flux tubes, two coordinate systems are used, namely, Cylindrical and Cartesian. The two types of flux tubes, shown in Figures5(a) and 5(b), are defined in the Cylindrical coordinate system. The other four flux tubes, shown in Figures 5(c)–5(f), are defined in the Cartesian coordinate system.


### 2.2. Stator MEC Modules

A stator MEC module is defined as the MEC model that represents the magnetic flux distribution in a unit section of the stator, that is, the section between central axes of each two neighboring stator slots, as shown in Figure6(a).Stator MEC module of FSM: (a) a stator section; (b) magnetic equivalent circuit for a stator section (𝒫
sl
=
𝒫
sl
1
+
𝒫
sl
2).
(a)
(b)The resulting MEC module is shown in Figure6(b), where the nodes and branches are numbered in Arabic and Roman numerals, respectively.The magnetic source in the stator section is modeled as an mmf sourceℱ
ms with a permeance  𝒫
ms  in series, as shown in Figure 6(a), in which ℱ
ms is calculated using (2), and 𝒫
ms represents the permeance of the permanent magnet or the iron core, accordingly. The flux paths in the stator tooth and stator back iron are modeled as different flux tubes with permeances 𝒫
st and 𝒫
si, respectively. The leakage flux outside the stator is also included and modeled as two parallel flux tubes with permeances 𝒫
sl
1 and 𝒫
sl
2, respectively, as can be seen in Figure 6(a). The types of flux tubes selected from Figure 5 for the permeance calculation are summarized in Table 1.Table 1
Types of flux tubes in the stator and rotor MEC modules.
𝒫
ms
𝒫
st
𝒫
si
𝒫
sl1
𝒫
sl2
𝒫
rt
𝒫
ri
b
a
b
f
e
b
aFor the 12/10 FSM, the MEC of the stator consists of twelve stator MEC modules that are similar to Figure6(b). However, when connecting the circuits, the polarity of the mmf source on branch II should be reversed in each neighboring module. Furthermore, it is worth noting that the actual magnetic flux paths in the stator vary with rotor position. However, to simplify the modeling problem, this variation is neglected as the permeability of the ferromagnetic material is relatively large under nonsaturated conditions.


### 2.3. Rotor MEC Modules

A rotor MEC module is defined as the MEC model that represents the magnetic flux distribution in a unit section of rotor. This rotor section is obtained by dividing the rotor of the FSM into equal sections, each with a rotor tooth in the middle, as shown in Figure7(a).Rotor MEC module of FSM: (a) a rotor section; (b) magnetic equivalent circuit for a rotor section.
(a)
(b)The resulting MEC module is shown in Figure7(b), where the nodes and branches are numbered in Arabic and Roman numerals, respectively. In this module, the flux paths in the rotor tooth and rotor back iron are modeled as different flux tubes with permeances 𝒫
rt and 𝒫
ri, respectively. The types of flux tubes selected from Figure 5 for the permeance calculation are summarized in Table 1.For the 12/10 FSM, the MEC of the rotor consists of ten rotor MEC modules. With the same simplifications as applied to the stator MEC modules, the rotor MEC modules are also assumed to be invariant to rotor position.


### 2.4. Airgap MEC Modules

An airgap MEC module is defined as the MEC model for the magnetic flux distribution in the airgap area around a rotor tooth. The airgap magnetic field is modeled with a certain number of different airgap MEC modules for different rotor positions.To limit the number of module varieties, this paper divides a stator section into eight intervals, shown in Figure8. A different airgap MEC module is used only when the rotor tooth enters a different interval.Figure 8
Interval divisions of a stator section for positioning a rotor tooth.It can be seen from Figure8 that, with respect to the central axis of the stator section, the four intervals on the left, numbered from 1 to 4, are symmetric to the four on the right, numbered from −1 to −4; thus, only four different airgap MEC modules are needed.FEA is performed on the model of 12/10 FSMs to obtain an impression of the magnetic flux distribution in the airgap during the rotor’s rotation, shown in Figure9.FEA results of equipotential contour in part of 12/10 FSM while a rotor tooth travels in (a) interval 1 and (b) interval 2.
(a)
(b)Based on the FEA results, the airgap flux distribution is simplified by grouping the flux paths into a certain number of flux tubes, as shown in Figures10(a), 11(a), 12(a), and 13(a), in which θ
x is the angle that the rotor tooth has traveled in the corresponding interval. The types of flux tubes selected from Figure 5 for the permeance calculation are summarized in Table 2.Table 2
Types of flux tubes in different airgap MEC modules.
𝒫
i
,
j
j
1
2
3
4
5
6
7
8
9
10
11
𝒫
1
,
j
d
e
d
c
d
d
f
d
c
d
—
𝒫
2
,
j
d
f
e
d
c
d
f
f
d
c
d
𝒫
3
,
j
d
f
e
d
c
e
f
d
—
—
—
𝒫
4
,
j
d
f
d
d
c
d
e
f
—
—
—Airgap MEC module 1: (a) flux tubes in the airgap while a rotor tooth travels in interval 1; (b) detailed magnetic equivalent circuit; (c) topology of airgap MEC module 1.
(a)
(b)
(c)Airgap MEC module 2: (a) flux tubes in the airgap while a rotor tooth travels in interval 2; (b) detailed magnetic equivalent circuit; (c) topology of airgap MEC module 2.
(a)
(b)
(c)Airgap MEC module 3: (a) flux tubes in the airgap while a rotor tooth travels in interval 3; (b) detailed magnetic equivalent circuit; (c) topology of airgap MEC module 3.
(a)
(b)
(c)Airgap MEC module 4: (a) flux tubes in the airgap while a rotor tooth travels in interval 4; (b) detailed magnetic equivalent circuit; (c) topology of airgap MEC module 4.
(a)
(b)
(c)Hence, four different airgap MEC modules are built with variable permeances, as shown in Figures10(b), 11(b), 12(b), and 13(b). In these models, permeances are expressed as functions of the angle θ
x using the equations introduced in Figure 5. By integrating the parallel permeances, topologies of the four modules are obtained, as shown in Figures 10(c), 11(c), 12(c), and 13(c).For the 12/10 FSM, the MEC of airgap consists of ten airgap MEC modules selected corresponding to the traveling interval of each rotor tooth.


## 2.1. Categorization of  MEC Elements

The MEC method is based on the analogies of the quantity relations in a magnetic field and in a resistive electric circuit [22, 23], as
(1)
Φ
=
ℱ
ℛ
=
ℱ
·
𝒫
,
in which Φ is the flux, ℱ  is the magnetomotive force (mmf), ℛ is the reluctance, and 𝒫 is the permeance.In a MEC, permanent magnets are modeled as mmf sources with reluctances in series or as flux sources with reluctances in parallel, shown in Figure4. In these models, the values of mmf source ℱ
PM and flux source Φ
PM can be calculated as
(2)
ℱ
PM
=
B
r
μ
0
μ
r
l
PM
,
(3)
Φ
PM
=
B
r
A
PM
,
in which B
r, μ
r, l
PM, and A
PM are the remanent magnetic flux density, the relative permeability, the length in the magnetization direction, and the area of the cross-section perpendicular to the magnetization direction, respectively, of the permanent magnet.Magnetic equivalent circuit models of permanent magnets as (a) an mmf source and a reluctance in series; (b) a flux source and a reluctance in parallel.
(a)
(b)Reluctances or their reciprocals permeances are approximately calculated by merging an area of flux paths into flux tubes. These flux tubes are defined as objects that enclose the flux lines with their side face parallel and head face perpendicular to them [22].For the 12/10 FSM, seven types of flux tubes are identified, shown in Figure5. The equally distributed lines in each cross-section indicate the flux paths, and the magnetic properties along the longitudinal dimension of each flux tube are assumed to be homogeneous.Transversal cross-sections of flux tubes and their permeances: (a)𝒫
=
μ
L
θ
/
ln
(
r
2
/
r
1
); (b) 𝒫
=
μ
L
ln
(
r
2
/
r
1
)
/
θ; (c) 𝒫
=
μ
L
x
/
h; (d) 𝒫
=
2
μ
L
·
ln
[
1
+
π
x
/
(
π
r
+
2
h
)
]
/
π; (e) 𝒫
=
μ
L
·
ln
[
1
+
2
π
x
/
(
π
r
1
+
π
r
2
+
2
h
)
]
/
π; (f) 𝒫
=
2
μ
L
x
/
(
π
w
+
2
h
).
(a)
(b)
(c)
(d)
(e)
(f)To simplify the permeance calculation of flux tubes, two coordinate systems are used, namely, Cylindrical and Cartesian. The two types of flux tubes, shown in Figures5(a) and 5(b), are defined in the Cylindrical coordinate system. The other four flux tubes, shown in Figures 5(c)–5(f), are defined in the Cartesian coordinate system.


## 2.2. Stator MEC Modules

A stator MEC module is defined as the MEC model that represents the magnetic flux distribution in a unit section of the stator, that is, the section between central axes of each two neighboring stator slots, as shown in Figure6(a).Stator MEC module of FSM: (a) a stator section; (b) magnetic equivalent circuit for a stator section (𝒫
sl
=
𝒫
sl
1
+
𝒫
sl
2).
(a)
(b)The resulting MEC module is shown in Figure6(b), where the nodes and branches are numbered in Arabic and Roman numerals, respectively.The magnetic source in the stator section is modeled as an mmf sourceℱ
ms with a permeance  𝒫
ms  in series, as shown in Figure 6(a), in which ℱ
ms is calculated using (2), and 𝒫
ms represents the permeance of the permanent magnet or the iron core, accordingly. The flux paths in the stator tooth and stator back iron are modeled as different flux tubes with permeances 𝒫
st and 𝒫
si, respectively. The leakage flux outside the stator is also included and modeled as two parallel flux tubes with permeances 𝒫
sl
1 and 𝒫
sl
2, respectively, as can be seen in Figure 6(a). The types of flux tubes selected from Figure 5 for the permeance calculation are summarized in Table 1.Table 1
Types of flux tubes in the stator and rotor MEC modules.
𝒫
ms
𝒫
st
𝒫
si
𝒫
sl1
𝒫
sl2
𝒫
rt
𝒫
ri
b
a
b
f
e
b
aFor the 12/10 FSM, the MEC of the stator consists of twelve stator MEC modules that are similar to Figure6(b). However, when connecting the circuits, the polarity of the mmf source on branch II should be reversed in each neighboring module. Furthermore, it is worth noting that the actual magnetic flux paths in the stator vary with rotor position. However, to simplify the modeling problem, this variation is neglected as the permeability of the ferromagnetic material is relatively large under nonsaturated conditions.


## 2.3. Rotor MEC Modules

A rotor MEC module is defined as the MEC model that represents the magnetic flux distribution in a unit section of rotor. This rotor section is obtained by dividing the rotor of the FSM into equal sections, each with a rotor tooth in the middle, as shown in Figure7(a).Rotor MEC module of FSM: (a) a rotor section; (b) magnetic equivalent circuit for a rotor section.
(a)
(b)The resulting MEC module is shown in Figure7(b), where the nodes and branches are numbered in Arabic and Roman numerals, respectively. In this module, the flux paths in the rotor tooth and rotor back iron are modeled as different flux tubes with permeances 𝒫
rt and 𝒫
ri, respectively. The types of flux tubes selected from Figure 5 for the permeance calculation are summarized in Table 1.For the 12/10 FSM, the MEC of the rotor consists of ten rotor MEC modules. With the same simplifications as applied to the stator MEC modules, the rotor MEC modules are also assumed to be invariant to rotor position.


## 2.4. Airgap MEC Modules

An airgap MEC module is defined as the MEC model for the magnetic flux distribution in the airgap area around a rotor tooth. The airgap magnetic field is modeled with a certain number of different airgap MEC modules for different rotor positions.To limit the number of module varieties, this paper divides a stator section into eight intervals, shown in Figure8. A different airgap MEC module is used only when the rotor tooth enters a different interval.Figure 8
Interval divisions of a stator section for positioning a rotor tooth.It can be seen from Figure8 that, with respect to the central axis of the stator section, the four intervals on the left, numbered from 1 to 4, are symmetric to the four on the right, numbered from −1 to −4; thus, only four different airgap MEC modules are needed.FEA is performed on the model of 12/10 FSMs to obtain an impression of the magnetic flux distribution in the airgap during the rotor’s rotation, shown in Figure9.FEA results of equipotential contour in part of 12/10 FSM while a rotor tooth travels in (a) interval 1 and (b) interval 2.
(a)
(b)Based on the FEA results, the airgap flux distribution is simplified by grouping the flux paths into a certain number of flux tubes, as shown in Figures10(a), 11(a), 12(a), and 13(a), in which θ
x is the angle that the rotor tooth has traveled in the corresponding interval. The types of flux tubes selected from Figure 5 for the permeance calculation are summarized in Table 2.Table 2
Types of flux tubes in different airgap MEC modules.
𝒫
i
,
j
j
1
2
3
4
5
6
7
8
9
10
11
𝒫
1
,
j
d
e
d
c
d
d
f
d
c
d
—
𝒫
2
,
j
d
f
e
d
c
d
f
f
d
c
d
𝒫
3
,
j
d
f
e
d
c
e
f
d
—
—
—
𝒫
4
,
j
d
f
d
d
c
d
e
f
—
—
—Airgap MEC module 1: (a) flux tubes in the airgap while a rotor tooth travels in interval 1; (b) detailed magnetic equivalent circuit; (c) topology of airgap MEC module 1.
(a)
(b)
(c)Airgap MEC module 2: (a) flux tubes in the airgap while a rotor tooth travels in interval 2; (b) detailed magnetic equivalent circuit; (c) topology of airgap MEC module 2.
(a)
(b)
(c)Airgap MEC module 3: (a) flux tubes in the airgap while a rotor tooth travels in interval 3; (b) detailed magnetic equivalent circuit; (c) topology of airgap MEC module 3.
(a)
(b)
(c)Airgap MEC module 4: (a) flux tubes in the airgap while a rotor tooth travels in interval 4; (b) detailed magnetic equivalent circuit; (c) topology of airgap MEC module 4.
(a)
(b)
(c)Hence, four different airgap MEC modules are built with variable permeances, as shown in Figures10(b), 11(b), 12(b), and 13(b). In these models, permeances are expressed as functions of the angle θ
x using the equations introduced in Figure 5. By integrating the parallel permeances, topologies of the four modules are obtained, as shown in Figures 10(c), 11(c), 12(c), and 13(c).For the 12/10 FSM, the MEC of airgap consists of ten airgap MEC modules selected corresponding to the traveling interval of each rotor tooth.


## 3. Construction and Solving Method

### 3.1. Position State Shifting

Periodicity of the 12/10 FSM allows modeling only half of the motor. Hence, the global MEC of this motor with certain rotor position is assembled using six stator MEC modules (S1–S6), five rotor MEC modules (T1–T5), and five airgap MEC modules (A1–A5), shown in Figure14.Global MEC of half of 12/10 FSPMM at a certain rotor position: (a) cross-section in polar coordinate; (b) network topology.
(a)
(b)Each airgap MEC module varies with rotor position. Therefore, this paper proposes a strategy named “position state shifting”. In this strategy, a shifting scheme of rotor position states is allocated to indicate the sequence of the airgap MEC module types for each position. These rotor position states are arrays of interval numbers of the rotor teeth. With certain slot/tooth configurations, for example, 12/10, the shift of rotor position states in an electrical period can be mathematically predicted.Table3 presents the shifting scheme of position states for the considered 12/10 FSM, in which the columns T1–T10 refer to the ten rotor teeth, the rows P1–P8 refer to the position states, and the numbers show the traveling intervals of each rotor tooth. It can be seen that the first eight states (P1–P8) are repeated for different starting-points. This repetition suggests a possibility in reducing the number of varieties of the global MEC.Table 3
Variation of the rotor position state during rotation.
T1
T2
T3
T4
T5
T6
T7
T8
T9
T10
P1
−4
−3
−1
1
3
−4
−3
−1
1
3
P2
−4
−3
−1
2
3
−4
−3
−1
2
3
P3
−4
−2
−1
2
3
−4
−2
−1
2
3
P4
−4
−2
−1
2
4
−4
−2
−1
2
4
P5
−4
−2
1
2
4
−4
−2
1
2
4
P6
−3
−2
1
2
4
−3
−2
1
2
4
P7
−3
−2
1
3
4
−3
−2
1
3
4
P8
−3
−1
1
3
4
−3
−1
1
3
4
−3
−1
1
3
−4
−3
−1
1
3
−4
−3
−1
2
3
−4
−3
−1
2
3
−4
−2
−1
2
3
−4
−2
−1
2
3
−4
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯


### 3.2. Incidence Matrix

The MEC modules can be mathematically described using the incidence matrix method [24], explained as follows.For a circuit withm nodes and n branches, its incidence matrix A is an m
×
n matrix in which
(4)
A
i
j
=
{
1
,
if
branch
j
begins
from
node
i
,
-
1
,
if
branch
j
ends
to
node
i
,
0
,
if
branch
j
is
not
connected
to
node
i
.Variables of this circuit can be further defined in the form of matrix or vector listed as follows:A: incidence matrix (m
×
n matrix);
V: magnetic potential on each node (m
×
1 vector);
U: mmf drop across each branch (n
×
1 vector);
φ: flux through each branch (n
×
1 vector);
E: mmf source in each branch (n
×
1 vector);
Λ: permeance of each branch (n
×
n diagonal matrix);
R: reluctance of each branch (n
×
n diagonal matrix).According to Kirchhoff Circuit Laws, following equations can be derived:(5)
U
=
A
t
·
V
,
A
·
ϕ
=
0
,
U
=
R
·
ϕ
+
E
=
Λ
-
1
·
ϕ
+
E
.Therefore, with givenA, Λ, and E, the magnetic potentials can be solved as
(6)
V
=
(
A
·
Λ
·
A
t
)
-
1
·
(
A
·
Λ
·
E
)
.Hence, using the incidence matrix method, the global MEC is assembled and solved. The magnetic potentials of each node are computed; therefore, the magnetic flux through each flux tube is obtained.In addition, magnetic saturation is taken into account by iteratively solving the global MEC to identify the applicable relative permeability used for certain flux tube.


## 3.1. Position State Shifting

Periodicity of the 12/10 FSM allows modeling only half of the motor. Hence, the global MEC of this motor with certain rotor position is assembled using six stator MEC modules (S1–S6), five rotor MEC modules (T1–T5), and five airgap MEC modules (A1–A5), shown in Figure14.Global MEC of half of 12/10 FSPMM at a certain rotor position: (a) cross-section in polar coordinate; (b) network topology.
(a)
(b)Each airgap MEC module varies with rotor position. Therefore, this paper proposes a strategy named “position state shifting”. In this strategy, a shifting scheme of rotor position states is allocated to indicate the sequence of the airgap MEC module types for each position. These rotor position states are arrays of interval numbers of the rotor teeth. With certain slot/tooth configurations, for example, 12/10, the shift of rotor position states in an electrical period can be mathematically predicted.Table3 presents the shifting scheme of position states for the considered 12/10 FSM, in which the columns T1–T10 refer to the ten rotor teeth, the rows P1–P8 refer to the position states, and the numbers show the traveling intervals of each rotor tooth. It can be seen that the first eight states (P1–P8) are repeated for different starting-points. This repetition suggests a possibility in reducing the number of varieties of the global MEC.Table 3
Variation of the rotor position state during rotation.
T1
T2
T3
T4
T5
T6
T7
T8
T9
T10
P1
−4
−3
−1
1
3
−4
−3
−1
1
3
P2
−4
−3
−1
2
3
−4
−3
−1
2
3
P3
−4
−2
−1
2
3
−4
−2
−1
2
3
P4
−4
−2
−1
2
4
−4
−2
−1
2
4
P5
−4
−2
1
2
4
−4
−2
1
2
4
P6
−3
−2
1
2
4
−3
−2
1
2
4
P7
−3
−2
1
3
4
−3
−2
1
3
4
P8
−3
−1
1
3
4
−3
−1
1
3
4
−3
−1
1
3
−4
−3
−1
1
3
−4
−3
−1
2
3
−4
−3
−1
2
3
−4
−2
−1
2
3
−4
−2
−1
2
3
−4
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯


## 3.2. Incidence Matrix

The MEC modules can be mathematically described using the incidence matrix method [24], explained as follows.For a circuit withm nodes and n branches, its incidence matrix A is an m
×
n matrix in which
(4)
A
i
j
=
{
1
,
if
branch
j
begins
from
node
i
,
-
1
,
if
branch
j
ends
to
node
i
,
0
,
if
branch
j
is
not
connected
to
node
i
.Variables of this circuit can be further defined in the form of matrix or vector listed as follows:A: incidence matrix (m
×
n matrix);
V: magnetic potential on each node (m
×
1 vector);
U: mmf drop across each branch (n
×
1 vector);
φ: flux through each branch (n
×
1 vector);
E: mmf source in each branch (n
×
1 vector);
Λ: permeance of each branch (n
×
n diagonal matrix);
R: reluctance of each branch (n
×
n diagonal matrix).According to Kirchhoff Circuit Laws, following equations can be derived:(5)
U
=
A
t
·
V
,
A
·
ϕ
=
0
,
U
=
R
·
ϕ
+
E
=
Λ
-
1
·
ϕ
+
E
.Therefore, with givenA, Λ, and E, the magnetic potentials can be solved as
(6)
V
=
(
A
·
Λ
·
A
t
)
-
1
·
(
A
·
Λ
·
E
)
.Hence, using the incidence matrix method, the global MEC is assembled and solved. The magnetic potentials of each node are computed; therefore, the magnetic flux through each flux tube is obtained.In addition, magnetic saturation is taken into account by iteratively solving the global MEC to identify the applicable relative permeability used for certain flux tube.


## 4. Validation with Finite Element Analysis

The nonlinear MEC model of 12/10 FSMs is constructed and used to investigate the correlation between the open-circuit phase flux linkage resulting and certain design parameters such as the motor size and the airgap length. To validate the accuracy of this model, the results of open-circuit phase flux linkage are compared to corresponding FEA results. To simplify the comparison, the number of turns is assumed to ben
c
=
1. Properties of steel laminations and permanent magnets adopted for the modeling are shown in Table 4.Table 4
Magnetic properties.
Description
Symbol
Value
Unit
Steel lamination type
M400-50A
—
—
PM remanent flux density
B
r
1.2
T
PM relative permeability
µ
r
1.05
—


### 4.1. Motor Size

To research the robustness of the method, three sets of geometric parameters, noted as Geom1, Geom2, and Geom3 in Figure15 and Table 5, are implemented into the MEC and FEA models of 12/10 FSPMM. In Figure 16, the MEC results show a good agreement with the FEA results.Table 5
Geometric parameters of three flux-switching PM motors.
Description
Symbol
Geom1
Geom2
Geom3
Unit
Stack length
L
120
120
120
mm
Stator outer diameter
D
SO
90
180
270
mm
Stator back iron height
h
si
3.6
7.2
10.8
mm
Stator slot depth
h
s
14
28
42
mm
Rotor outer diameter
D
RO
54
108
162
mm
Rotor back iron height
h
ri
10
20
30
mm
Rotor tooth height
h
rt
10
20
30
mm
Airgap length
δ
1
1
1
mmFigure 15
Cross-sections of three FSPMM models in various motor sizes.Figure 16
Open-circuit phase flux linkages of 12/10 flux-switching permanent magnet motor (FSPMM) in various sizes, obtained using magnetic equivalent circuits (MEC) and finite element analysis (FEA).


### 4.2. Airgap Length

The airgap length of an electrical motor is an important parameter for the motor design as it directly influences the airgap flux density and the saturation level in the rotor and stator iron. Therefore, the flexibility of the MEC model for different airgap lengths is researched. In this respect, Figure17 illustrates the open-circuit phase flux-linkage results of the 12/10 FSPMM (Geom1) with different values of the airgap length δ. It can be seen that the MEC results are close to the FEA results, although the later presents an improved sensitivity to the airgap length variation. This difference suggests room for improving the accuracy of the MEC model, which can be achieved by refining the MEC modules while preserving the structure of the global MEC model.Open-circuit phase flux linkages of 12/10 flux-switching permanent magnet motor (FSPMM) with the airgap length varying from 0.5 mm to 1.5 mm, obtained using magnetic equivalent circuits (MEC) and finite element analysis (FEA): (a) waveforms of the phase flux linkage; (b) root-mean-square (rms) values of the phase flux linkage.
(a)
(b)


## 4.1. Motor Size

To research the robustness of the method, three sets of geometric parameters, noted as Geom1, Geom2, and Geom3 in Figure15 and Table 5, are implemented into the MEC and FEA models of 12/10 FSPMM. In Figure 16, the MEC results show a good agreement with the FEA results.Table 5
Geometric parameters of three flux-switching PM motors.
Description
Symbol
Geom1
Geom2
Geom3
Unit
Stack length
L
120
120
120
mm
Stator outer diameter
D
SO
90
180
270
mm
Stator back iron height
h
si
3.6
7.2
10.8
mm
Stator slot depth
h
s
14
28
42
mm
Rotor outer diameter
D
RO
54
108
162
mm
Rotor back iron height
h
ri
10
20
30
mm
Rotor tooth height
h
rt
10
20
30
mm
Airgap length
δ
1
1
1
mmFigure 15
Cross-sections of three FSPMM models in various motor sizes.Figure 16
Open-circuit phase flux linkages of 12/10 flux-switching permanent magnet motor (FSPMM) in various sizes, obtained using magnetic equivalent circuits (MEC) and finite element analysis (FEA).


## 4.2. Airgap Length

The airgap length of an electrical motor is an important parameter for the motor design as it directly influences the airgap flux density and the saturation level in the rotor and stator iron. Therefore, the flexibility of the MEC model for different airgap lengths is researched. In this respect, Figure17 illustrates the open-circuit phase flux-linkage results of the 12/10 FSPMM (Geom1) with different values of the airgap length δ. It can be seen that the MEC results are close to the FEA results, although the later presents an improved sensitivity to the airgap length variation. This difference suggests room for improving the accuracy of the MEC model, which can be achieved by refining the MEC modules while preserving the structure of the global MEC model.Open-circuit phase flux linkages of 12/10 flux-switching permanent magnet motor (FSPMM) with the airgap length varying from 0.5 mm to 1.5 mm, obtained using magnetic equivalent circuits (MEC) and finite element analysis (FEA): (a) waveforms of the phase flux linkage; (b) root-mean-square (rms) values of the phase flux linkage.
(a)
(b)


## 5. Conclusions

Flux-switching motors (FSM) combine the advantages of synchronous motors and switched reluctance motors; thus, they are competitive candidates for in-wheel traction systems. However, the analysis of FSMs presents difficulty due to their complex structure and heavy magnetic saturation.The magnetic equivalent circuit (MEC) method is recommended as it provides a good compromise between computational time and accuracy. However, constructing a global MEC for FSMs that varies in synchronism with the rotor position is an elaborate work due to their double-salient structure and low periodicity.This paper presents a methodology to rapidly construct, adapt, and solve the global MEC of 12-stator-slot 10-rotor-tooth (12/10) FSMs. With this methodology, MEC models of different parts of FSM are defined, namely, stator MEC module, rotor MEC module, and airgap MEC module, in which the airgap MEC models are adaptive corresponding to the rotor position. Using the incidence matrix method, the global MEC is assembled with the MEC modules and iteratively solved to take into account the magnetic saturation.Using this methodology, the variable global MEC of 12/10 FSM is constructed and used to investigate the correlations between certain design parameters and the open-circuit phase flux linakge. The results of the MEC model show a good agreement with the corresponding FEA results; thus, the accuracy of this methodology is validated. Therefore, this model is proved to be able to assist designers with the preliminary design of flux-switching motors for different in-wheel traction systems.



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*Source: 530260-2014-01-01.xml*