Analytical Model for Brushless Double Mechanical Port Flux-Switching Permanent Magnet Machines

Abstract

In this paper, a two-dimensional analytical model is presented for brushless double mechanical port flux-switching permanent magnet machines (BDMPFSPMMs). The BDMPFSPMM is an appropriate candidate to be employed as the electric motor in hybrid electric vehicles (HEVs) due to integrating two rotors and two stators into a compact structure. The radial and tangential components of the magnetic flux density are calculated due to permanent magnets and armature currents in each active region of the machine based on the subdomain method. The effects of the saliency on both the rotor and stator structures as well as their interactions are considered. The approach can be used for the magnetic field calculation in BDMPFSPMMs with any combination of rotor-and stator-pole number. Based on the computed magnetic flux density, electromagnetic torque, self and mutual inductances, flux linkage, induced voltage, local traction and unbalanced magnetic force are obtained. To validate the proposed model, the analytical results are compared with those obtained from the finite element method (FEM).

Full text

IEEE TRANSACTIONS ON MAGNETICS, VOL. 57, NO. 10, OCTOBER 2021 8107713
Analytical Model for Brushless Double Mechanical Port
Flux-Switching Permanent Magnet Machines
E. Shirzad and A. Rahideh
Department of Electrical and Electronics Engineering, Shiraz University of Technology, Shiraz 71557-13876, Iran
In this article, a 2-D analytical model is presented for brushless double mechanical port flux-switching permanent magnet
machines (BDMPFSPMMs). The BDMPFSPMM is an appropriate candidate to be employed as the electric motor in hybrid electric
vehicles (HEVs) due to integrating two rotors and two stators into a compact structure. The radial and tangential components of
the magnetic flux density are calculated due to permanent magnets and armature currents in each active region of the machine
based on the subdomain method. The effects of the saliency on both the rotor and stator structures as well as their interactions
are considered. The approach can be used for the magnetic field calculation in BDMPFSPMMs with any combination of rotor- and
stator-pole numbers. Based on the computed magnetic flux density, electromagnetic torque, self and mutual inductances, flux linkage,
induced voltage, local traction, and unbalanced magnetic force (UMF) are obtained. To validate the proposed model, the analytical
results are compared with those obtained from the finite element method (FEM).
Index Terms—Analytical model, boundary conditions, brushless machines, flux-switching, finite element method (FEM),
subdomain.
NOMENCLATURE
AMagnetic vector potential (V ·s/m).
BMagnetic flux density vector (T).
Brem Permanent magnet residual flux density (T).
BrRadial component of B(T).
BθTangential component of B(T).
FxMagnetic force in x-direction (N).
FyMagnetic force in y-direction (N).
fxLocal traction in x-direction (N/m2).
fyLocal traction in y-direction (N/m2).
HMagnetic field intensity vector (A/m).
JArmature current density vector (A/m2).
MMagnetization vector (A/m).
μ0Free space permeability (H/m).
μrRelative permeability.
μrpm Relative permeability of permanent magnet (PM).
Niss Number of inner stator slots.
Nirs Number of inner rotor slots.
Nipm Number of inner permanent magnets.
Noss Number of outer stator slots.
Nors Number of outer rotor slots.
Nopm Number of outer PMs.
αiCentral angle of ith slot of inner rotor.
βiCentral angle of ith slot of inner stator.
λiCentral angle of ith inner PM.
τiCentral angle of ith outer PM.
σiCentral angle of ith slot of outer rotor.
ψiCentral angle of ith slot of outer stator.
Manuscript received May 3, 2021; revised July 16, 2021; accepted
August 11, 2021. Date of publication August 16, 2021; date of current version
September 20, 2021. Corresponding author: A. Rahideh (e-mail: rahide@
sutech.ac.ir).
Color versions of one or more figures in this article are available at
https://doi.org/10.1109/TMAG.2021.3104938.
Digital Object Identifier 10.1109/TMAG.2021.3104938
Superscripts and subscripts
rRadial direction.
θTangential direction.
zAxial direction.
iry Inner rotor yoke.
irs Inner rotor slot.
ia Inner airgap.
iss Inner stator slot.
isy Inner stator yoke.
ipm Inner PM.
fb Flux barrier.
opm Outer PM.
osy Outer stator yoke.
oss Outer stator slot.
oa Outer airgap.
ors Outer rotor slot.
ory Outer rotor yoke.
I. INTRODUCTION
HYBRID electric vehicles (HEVs) are a configuration
with an internal combustion engine (ICE) and an electric
machine. The electric machine receives the energy stored in
batteries during the motoring mode and transfers energy to
the batteries during the generating mode. Both the electric
motor and the engine engage in providing torque to drive the
vehicle directly according to specific program navigated via a
controllable torque transfer unit. At low speeds or in a heavy
traffic, the electric motor alone transfers power to the vehicle
using energy stored in the batteries. Under the acceleration
condition and during increasing speed, both the ICE and the
motor provide torque to drive the vehicle. In steady state
highway cruising, the ICE alone drives the vehicle. An HEV
cannot be plugged into electric network to charge the battery;
instead, the battery is charged by the regenerative braking
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8107713 IEEE TRANSACTIONS ON MAGNETICS, VOL. 57, NO. 10, OCTOBER 2021
and the ICE. The remaining power provided by the electric
motor can be used for a smaller engine. With respect to stated
configuration, the biggest advantage of HEVs is providing
better fuel economy due to less consumption of fuel [1]–[4].
Flux-switching permanent magnet machines (FSPMMs) have
attracted considerable attention in recent years due to the large
torque capability, sinusoidal back-electromotive force (EMF)
waveforms, high torque (power) density, as well as compact
and robust structure since both the magnets and armature
windings are located in the stator instead of the rotor [5], [19].
There is much interest in double rotor electric machines due
to their versatile configurations and performance characteris-
tics. Double rotor machines show promising applications in
advanced HEV powertrains due to the requirement of dual
electro-mechanical ports in such systems [20]. Integrating
these powertrain systems with double rotor machines not only
brings design freedom in component layout, but also reduces
the number of parts and thus improves compactness [21], [22].
Double mechanical port permanent magnet (DMP-PM) motors
have attracted increasing attention and become one of the
research highlights in the motor field [23]. The four-quadrant
energy transducer (4QT), which is one of the typical DMP-PM
motors, integrates two rotating mechanical ports and two
electric ports into one compound motor frame [24]. Moreover,
the 4QT exhibits potential applications in many fields, such as
modern HEVs and wind power generation systems [25]–[27].
Some studies on FSPMMs are based on finite element method
(FEM) (numerical method) that in [5], a prototype of a high-
power three-phase 12-stator-slot/10-rotor-pole FSPM motor is
designed by FEM for HEVs and in [28], a new flux switch-
ing permanent magnet machine (S-FSPM) with an outer-
rotor configuration is investigated by theoretical analysis and
2-D FEM.
FEM is mainly employed to analyze FSPMMs because of
the doubly-salient airgap characteristics. Despite its proven
accuracy, FEM has high computational burden and has the
limitation for optimal design in which several thousands of
iterations may be required. To solve these problems, some
authors proposed some analytical models. In [29] and [30],
a 2-D analytical method is investigated for FSPM by using
Maxwell equations. Other analytical methods are proposed to
model FSPMMs using magnetic equivalent circuit (MEC) [31],
Fourier analysis methods based on subdomain model [18],
and slot relative permeance calculation [32]. The subdomain
model is more accurate compared to the MEC [31]. In [33],
a simple analytical model is presented for a switched flux
memory machine (SFMM) to provide in-depth insight into
its working mechanism. In [34], a double-stator (DS) hybrid-
excited FSPM is modeled based on permeance and inductance
modeling. In [25], a general analytical subdomain model is
presented for the computation of the magnetic field distribution
in any number of stator slots and rotor poles with and
without electrically excited, permanent magnet-excited, and
hybrid-excited multiphase flux switching. In [29], a MEC
is proposed to be used in the multilevel optimization of
brushless double mechanical port FSPMMs (BDMPFSPMMs).
Analytical subdomain model for magnetic field computation
in segmented permanent magnet switched flux consequent
Fig. 1. Cross section of BDMPFSPMMs in the presence of PM.
pole machine is studied in [35]. In [36], the memory flux
principle is extended to switched flux structures, forming
two newly emerged SFMMs with single-stator (SS) and DS
configurations. In [30], a DS-FSPM is analyzed by frozen
permeability method. The subdomain method provides a good
compromise between the accuracy and computational speed
so that is not only accurate, but also fast compared to other
methods.
The analytical method in this article is based on 2-D solution
of Maxwell equations considered both PM and armature
reaction (AR) effects on the magnetic flux density distribution
of BDMPFSPMMs with single- and double-layer winding
structures. Because of the saliency on both the stator and
rotor structures, the airgap reluctance (or permeance) varies
in terms of the peripheral angle and also the rotor position.
Therefore, the magnetic flux density waveforms versus the
peripheral angle versus time (or the rotor position) vary due to
the presence of the saliency on both the rotor and stator parts.
The changes in the magnetic flux density waveforms influence
other quantities such as electromagnetic torque, inductances,
back EMF, etc. Using the subdomain technique, it is possible
to accurately consider the effects and interaction of the rotor
and stator saliency in the modeling process. The contribution
of the proposed analytical model is as follows.
1) The 2-D analytical model based on the Maxwell equa-
tions for double-mechanical port FSPMMs has been
presented for the first time.
2) The electromagnetic quantities such as the torque, induc-
tance, UMF, local traction, flux linkage, and back EMF
for the machine are calculated.
3) The proposed model is able to incorporate the influences
of the inner and outer parts on each other.
The remainder of the article is as follows. In Section II,
the 2-D analytical model and governing equations based on
Maxwell equations are expressed for BDMPFSPMMs and
the assumptions are listed. In Section III, the results of the
analytical model are presented and compared with those of
FEM for a case study. Finally, the article is concluded in
Section VI.
II. PROBLEM FORMULATION
To extract the 2-D analytical model for BDMPFSPMMs,
as shown in Fig. 1, the below procedure is followed.
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SHIRZAD AND RAHIDEH: ANALYTICAL MODEL FOR BDMPFSPMMs 8107713
1) A set of assumptions are made to enable the analytical
solution.
2) The motor geometry is divided into an appropriate
number of subdomains.
3) The Fourier series expansions are derived for the current
density in the winding sub-region and PM magnetization
pattern.
4) The partial differential equations (PDEs) and boundary
conditions are expressed in the subdomains.
5) The general and particular solutions for the sub-regions
are obtained.
6) To obtain the unknown coefficients, boundary conditions
are applied.
7) The analytical results are compared with those of FEM
to validate the proposed model.
A. Assumptions
The following assumptions are considered.
1) End effects are ignored.
2) The magnetic flux density is a 2-D vector with radial and
tangential components. Therefore, the magnetic vector
potential has only z-component.
3) The magnetic flux density vector and the magnetic
vector potential are independent of z.
4) All materials are isotropic.
5) The stator and rotor teeth have no tooth-tip.
6) The rotor/stator iron is infinitely permeability; therefore,
the saturation effects are neglected.
7) The slots and rotor poles have radial sides.
8) Eddy current reaction field is neglected.
B. Subdomains
Based on Fig. 1 and the infinite permeability assumption
of the stator and rotor irons, the active subdomains consist
of the inner rotor slots, inner airgap, inner stator slots, inner
PM, flux barrier, outer PM, outer stator slots, outer airgap,
and outer rotor slots which are, respectively, denoted by irs,
ia, iss, ipm, fb, opm, oss, oa, and ors. Two types of winding
layouts are investigated: the single-layer and the double-layer
according to Figs. 2 and 3, respectively. Therefore, there are
Niss +Noss +Nirs +Nors +Nipm +Nopm +3 regions. The
parameters of the considered machine are illustrated in Fig. 4.
C. Governing Equations
Based on the assumptions given in Section II-A,
the Maxwell equation for a sub-region having both current
density and PMs is expressed as follows:
−∇2Ai
z=μ0μrJi
z+μ0
r∂Mr
∂θ −r∂Mθ
∂r.(1)
It is noted that in the PM sub-regions Jz=0, and in the
winding sub-regions Mr=Mθ=0. For other sub-regions
both Jand Mare zero, as shown in the following expression:
−1
r2
∂2Ai
z
∂θ2−1
r
∂
∂rr∂Ai
z
∂r=0.(2)
Fig. 2. Cross section of the BDMPFSPMM with a single-layer winding.
Fig. 3. Cross section of the BDMPFSPMM with a double-layer winding.
Fig. 4. Geometric parameters of the BDMPFSPMM.
The magnetic flux density components are obtained for each
sub-region by using curl from the magnetic vector potential,
i.e. B=∇×A, and the magnetic field intensity is calculated
by (3) that for PM regions Mis not zero and for the other
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8107713 IEEE TRANSACTIONS ON MAGNETICS, VOL. 57, NO. 10, OCTOBER 2021
regions M=0
H=B
μ0μr
−M
μr
.(3)
D. Boundary Conditions
The magnetic vector potential is continuous at the interface
between two adjacent media. If the interface is source-free,
then the parallel component of the magnetic field intensity
vector on one side of the boundary is equal to that of the
other side. Also, the parallel component of the magnetic field
intensity vector is zero at the interface of those media adjacent
to infinitely permeable domains. Therefore, the boundary
conditions are expressed in Table I.
E. Analytical Solutions of Magnetic Vector Potential
In the ith inner or outer rotor slot sub-domain
(κ={(irs,i), (ors,i)})which has no current and perma-
nent magnet, the general solution can be obtained from the
following equation:
Aκ
z(r,θ)=aκ,0+
∞

n=1
aκ,nf(n,r)cosnπ
cθ−θκ+c
2 (30)
where κis a symbol to show the ith inner or outer rotor
slot sub-domain, cis the width of each region, and θκis the
central angle of each rotor slot, moreover, f(n,r)is obtained
from (31) using the boundary conditions
f(n,r)=r−nπ
c+rnπ
c/R2nπ
c(31)
where Ris the radius of the rotor slot region in which
(∂ Ai
z/∂r)=0. In the ith inner or outer stator slot sub-domain
(ι={(iss,i), (oss,i)}), the general solution can be obtained
from the following equation:
Aι
z(r,θ)=aι,0+bι,0ln(r)−1
4μ0Jιr2
+
∞

n=1aι,nr−nπ
c+bι,nrnπ
ccosnπ
cθ−θι+c
2
(32)
where
Jι(θ)=Jι,0+
∞

n=1Jι,ncosnπ
cθ−θι+c
2 (33)
and Jι,0=(Jι,1d+Jι,2d)/c,Jι,n=(2/nπ)sin(nπd/c)
(Jι,1+(Jι,2)(−1)n).
Jι,1,Jι,2are, respectively, the right and left side current
densities in slot ιfor double-layer winding and dis the width
of the winding region.
For the ith inner or outer permanent magnet sub-domain
(ς={(ipm,i), (opm,i)}), the general solution of the mag-
netic vector potential is obtained from the following equation:
Aς
z(r,θ)=aς,0+bς,0ln(r)−μ0Mςr
+
∞

n=1aς,nr−nπ
c+bς,nrnπ
ccos nπ
cθ−θς+c
2
(34)
Mς=(−1)iBrem/μ0.(35)
TAB L E I
BOUNDARY CONDITIONS BETWEEN THE REGIONS
For the inner or outer airgap sub-domain or flux barrier
(χ={(ia), (oa), (fb)}), the general solution is obtained
from the following equation:
Aχ
z(r,θ)=
∞

m=1aχ,1,mrm+aχ,2,mr−mcos(mθ)
+bχ,1,mrm+bχ,2,mr−msin(mθ).(36)
Finally, unknown coefficients for each region are obtained
based on Table I and expressions in the Appendix.
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SHIRZAD AND RAHIDEH: ANALYTICAL MODEL FOR BDMPFSPMMs 8107713
TAB L E I I
SPECIFICATIONS OF THE BDMPFSPMM
III. RESULTS
To show the efficacy of the derived analytical magnetic field
expressions, a BDMPFSPMM with the specifications listed
in Table II is selected as the case study. Rotor pole/stator
pole for both outer and inner parts is 10/12. The following
three scenarios are implemented to validate the analytical
method: 1) only PMs are active and the windings are open-
circuited; 2) only single-layer armature windings are carrying
currents and PMs are nullified; and 3) only double-layer
armature windings are carrying currents and PMs are nullified.
Thereafter some quantities of the machine such as the self and
mutual inductances, electromagnetic torque, magnetic forces,
magnetic local traction, and induced voltage are analytically
calculated and compared with those of FEM. The proposed
model can be used to accurately analyze the influence and
interference of the inner part on the outer part and vice versa.
For example, the results of the magnetic flux density in the
inner airgap can be obtained due to only the excitation of the
outer part.
A. Magnetic Flux Density Due to Only PMs
In this subsection, the results of the radial and tangential
components of the magnetic flux density vector are analyti-
cally and numerically obtained for the BDMPFSPMM with
Fig. 5. Inner airgap flux density due to PMs (r=37.5 mm). (a) Radial
component. (b) Tangential component.
Fig. 6. Outer airgap flux density due to PMs (r=93 mm). (a) Radial
component. (b) Tangential component.
the specifications listed in Table II and compared with each
other.
The radial and tangential components of the magnetic flux
density in the inner airgap in the presence of both the inner
and outer PMs are shown in Fig. 5. The radial and tangential
components of the magnetic flux density in the outer airgap
in the presence of both the inner and outer PMs are depicted
in Fig. 6. Fig. 7 shows the radial and tangential components
of the magnetic flux density in the flux barrier region in the
presence of both the inner and outer PMs. The analytical and
numerical results are in good agreement, which confirm the
accuracy of the proposed analytical model. The waveform of
the magnetic flux density components in the airgap at each
instance mainly depends on the airgap reluctance and the
excitation distribution. In the FSPMMs, the feature of double
saliency leads to the irregular waveform of the magnetic flux
density components due to the high variation of the reluctance
at each peripheral angle. The maximum points are usually
related to the alignment position of the stator and rotor teeth.
B. Magnetic Flux Density Due to Only Single-Layer Winding
Currents and Single-Layer Winding Currents and PMs
Fig. 8 shows the radial and tangential components of the
magnetic flux density in the inner airgap in the presence
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8107713 IEEE TRANSACTIONS ON MAGNETICS, VOL. 57, NO. 10, OCTOBER 2021
Fig. 7. Flux barrier flux density due to PMs (r=70 mm). (a) Radial
component. (b) Tangential component.
Fig. 8. Inner airgap flux density due to the currents of the single-layer
windings (r=37.5 mm and t=0). (a) Radial component. (b) Tangential
component.
Fig. 9. Outer airgap flux density due to the currents of the single-layer
windings (r=93 mm and t=0). (a) Radial component. (b) Tangential
component.
of both the inner and outer winding currents. Fig. 9 shows
the radial and tangential components of the magnetic flux
density in the outer airgap. Fig. 10 shows the radial and
tangential components of the magnetic flux density in the
flux barrier region in the presence of both the inner and outer
single-layer winding currents. The results of the flux density
Fig. 10. Flux barrier flux density due to the currents of the single-layer
windings (r=70 mm and t=0). (a) Radial component. (b) Tangential
component.
Fig. 11. Inner airgap flux density due to the currents of the single-layer
windings and PMs (r=37.5 mm and t=0). (a) Radial component.
(b) Tangential component.
in the presence of both of the single-layer winding currents
and PMs are shown in Figs. 11–13. The results for single layer
layout are illustrated at the initial time and for the maximum
current density of 6 A/mm2. The results of the analytical model
and FEM are compared with each other.
C. Magnetic Flux Density Due to Only Double-Layer Winding
Currents and Double-Layer Winding Currents and PMs
Similar results for the double-layer windings are shown
in Figs. 14–19.
D. Electromagnetic Torque
The electromagnetic torque is developed due to the pres-
ence of both the winding currents and PMs. The following
expression is used to calculate the electromagnetic torque:
T(t)=π
−π
LsR2
c
μ0
BrBθdθ(37)
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SHIRZAD AND RAHIDEH: ANALYTICAL MODEL FOR BDMPFSPMMs 8107713
Fig. 12. Outer flux density due to the currents of the single-layer windings
and PMs (r=93 mm and t=0). (a) Radial component. (b) Tangential
component.
Fig. 13. Flux barrier flux density due to the currents of the single-
layer windings and PMs (r=70 mm and t=0). (a) Radial component.
(b) Tangential component.
Fig. 14. Inner airgap flux density due to the currents of the double-layer
windings (r=37.5 mm and t=0). (a) Radial component. (b) Tangential
component.
where Ls,Rc,Br,andBθare, respectively, the machine stack
length, the airgap radius (for the inner part it is 37.5 mm
and for the outer part it is 93 mm), the radial and tangential
Fig. 15. Outer airgap flux density due to the currents of the double-layer
windings (r=93 mm and t=0). (a) Radial component. (b) Tangential
component.
Fig. 16. Flux barrier flux density due to the currents of the double-layer
windings (r=70 mm and t=0). (a) Radial component. (b) Tangential
component.
Fig. 17. Inner airgap flux density due to the currents of the double-
layer windings and PMs (r=37.5 mm and t=0). (a) Radial component.
(b) Tangential component.
components of the magnetic flux density in the airgap due to
both PMs, and winding currents.
The electromagnetic torque waveforms of the inner and
outer parts are shown in Fig. 20. As mentioned before, one
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8107713 IEEE TRANSACTIONS ON MAGNETICS, VOL. 57, NO. 10, OCTOBER 2021
Fig. 18. Outer airgap flux density due to the currents of the double-
layer windings and PMs (r=93 mm and t=0). (a) Radial component.
(b) Tangential component.
Fig. 19. Flux barrier flux density due to the currents of the double-
layer windings and PMs (r=70 mm and t=0). (a) Radial component.
(b) Tangential component.
Fig. 20. Electromagnetic torque waveform for the machine with double-layer
windings. (a) Inner part. (b) Outer part.
of the assumptions of the analytical model is to neglect the
saturation effect. As depicted in Fig. 21, the analytically
calculated electromagnetic torque is compared with those of
Fig. 21. Average electromagnetic torque versus current density for the
machine with double-layer windings. (a) Inner part. (b) Outer part.
Fig. 22. Flux linkage of phase A for double-layer windings. (a) Inner part.
(b). Outer part.
FEM considering the saturation effects to show the efficacy of
the proposed analytical model.
E. Induced Voltage
The magnetic flux is calculated according to the following
equation:
φ(t)=B·dS(38)
where φis the magnetic flux, Bis the flux density vector,
and dSis the vector surface of the integration for winding
slots of each phase. Using the proposed analytical model,
the magnetic flux density due to armature currents and the
magnetic flux density due to PMs can be computed separately
or in combined. Therefore, to calculate the no-load back EMF,
it is possible to set the armature currents to zero and find
the magnetic flux linked to the winding due to only the PMs
and using (39), the no-load back EMF is obtained. Similar
procedure can be applied to find the under-load back EMF,
if the magnetic flux density is due to both PMs and armature
currents. Fig. 22 depicts comparison between analytic and
FEM for flux linkage either inner part or outer part for
phase A.
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SHIRZAD AND RAHIDEH: ANALYTICAL MODEL FOR BDMPFSPMMs 8107713
Fig. 23. Induced voltage of phase A for double-layer windings. (a) Inner
part. (b) Outer part.
The back EMF induced in coil jcan be calculated according
to Faraday’s law as follows:
Ej=Ndφ/dt (39)
where Nis the number of turns of each coil and ϕis the
airgap magnetic flux. Fig. 23 depicts comparison between the
analytical and FEM results of the induced voltage of phase A.
F. Inductance
Equation (38) can be used to calculate the magnetic flux
linked to each coil due to only the armature currents. To this
end, the integral constants are also calculated when only coil j
is carrying current without the effect of PMs. Having assumed
the series connections of all coils in each phase, the self-
inductance of phase kis calculated using the summation
of the flux linked with the coils of phase kdue to solely
the corresponding coil current of phase kdivided by the
corresponding coil current of phase k. It is mathematically
expressed for the self-inductance as follows:
Lkk =
j,j∈k
λj,j
Ij
(40)
and for the mutual inductance it is as follows:
Lkk=
j∈k,j∈k
λj,j
Ij
(41)
where λis Nmultiplied by φ.Fig.24showstheselfand
mutual inductances of the inner and outer parts of the machine,
respectively.
G. Magnetic Local Traction and Force
The radial and tangential components of the local traction
exerted on the rotor surface can be obtained by Maxwell stress
tensor as follows:
fr=B2
r−B2
θ
2μ0
(42)
fθ=BrBθ
μ0
.(43)
Fig. 24. Self and mutual inductances for double-layer windings. (a) Inner
part. (b) Outer part.
Fig. 25. UMF exerted on (a) inner rotor and (b) outer rotor for double-layer
windings.
Fig. 26. UMF exerted on (a) inner rotor and (b) outer rotor for single-layer
windings.
By transforming these local tractions to the Cartesian plane,
the following expressions are obtained:
fx=frcosθ−fθsinθ(44)
fy=frsinθ+fθcosθ. (45)
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8107713 IEEE TRANSACTIONS ON MAGNETICS, VOL. 57, NO. 10, OCTOBER 2021
Fig. 27. Local traction for inner rotor for double-layer windings (t=0).
(a) Horizontal component. (b) Vertical component.
Fig. 28. Local traction for outer rotor for double-layer winding (t=0).
(a) Horizontal component. (b) Vertical component.
Fig. 29. Cross section of 12s/11p BDMPFSPMM.
The magnetic force components are then calculated as
follows:
Fx(t)=Lsπ
−π
fxrdθ(46)
Fy(t)=Lsπ
−π
fyrdθ. (47)
Fig. 30. (a) Radial and (b) tangential components of force for inner part
(t=0).
Fig. 31. (a) Radial and (b) tangential components of force for outer part
(t=0).
Fig. 32. UMF for inner and outer part in 12s/11p BDMPFSPMM.
The unbalanced magnetic force (UMF) exerted on the rotors
are depicted in Figs. 25 and 26 for the inner and outer parts in
double-layer winding layout. The UMF is almost zero for the
case study with double-layer or single-layer windings because
of the symmetrical structure of the machine. The xand y
components of the local traction on the inner and outer rotors
are shown in Figs. 27 and 28.
Because of the symmetrical structure of the BDMPFSPMM
with 12s/10p, UMF for the single and double-layer windings is
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SHIRZAD AND RAHIDEH: ANALYTICAL MODEL FOR BDMPFSPMMs 8107713
almost zero. In order to validate the efficacy of the proposed
model in UMF calculation, a BDMPFSPMM with 12s/11p
and double layer winding for both the inner and outer parts,
as shown in Fig. 29, is examined. Using relations (42)–(47),
UMF, radial, and tangential forces for the BDMPFSPMM with
12s/11p are calculated to observe the effect of an asymmetric
structure on the forces exerted on the rotors. Figs. 30 and 31
show the force components for the inner and outer parts of the
BDMPFSPMM with 12s/11p. The UMF of the BDMPFSPMM
with 12s/11p is shown in Fig. 32.
IV. CONCLUSION
In this article, a general formulation of the analytical
model has been proposed for predicting the magnetic field
distribution and quantities for BDMPFSPMM. Two types of
single- and double-layer windings have been studied. The
magnetic flux density components due to permanent magnet
and armature currents have been used to compute the elec-
tromagnetic quantities such as electromagnetic torque, back
EMF, inductances, local traction, and UMs. Analytical results
are in good agreement with those obtained by FEM.
APPENDIX
The expression requires for calculating the integral coeffi-
cients are as follows:
airs,0=1
wirs αi+wirs/2
αi−wirs/2
Aia
z(Rirs,θ)dθ(48)
airs,nf(n,Rirs)
=2
wirs αi+wirs/2
αi−wirs/2
Aia
z(Rirs,θ)cosnπ
wirs θ−αi
+wirs
2dθ
(49)
m−bia,1,mRm−1
irs +bia,1,mR−m−1
irs 
=
Nirs

i=1
μ0
παi+wirs/2
αi−wirs/2
Hirs,i
θ(Rirs,θ)sin(mθ)dθ(50)
m−aia,1,mRm−1
irs +aia,1,mR−m−1
irs 
=
Nirs

i=1
μ0
παi+wirs/2
αi−wirs/2
Hirs,i
θ(Rirs,θ)cos(mθ)dθ(51)
aιss,0+bιss,0ln(Riss)−1
4μ0Jiss,0R2
iss
=1
wiss βi+wiss/2
βi−wiss/2
Aia
z(Riss,θ)dθ(52)
2
nπsin nπd
wiss −1
4μ0R2
issJiss,1+Jiss,2(−1)n
×aiss,nR
−nπ
wiss
iss +biss,nR
nπ
wiss
iss 
=2
wiss βi+wiss/2
βi−wiss/2
Aia
z(Riss,θ)cosnπ
wiss θ−βi
+wiss
2dθ
(53)
n−bia,1,mRm−1
iss +bia,1,mR−m−1
iss 
=
Niss

i=1
μ0
πβi+wiss/2
βi−wiss/2
Hiss,i
θ(Riss,θ)sin(mθ)dθ
+
Nipm

i=1
μ0
πλi+wipm/2
λi−wipm/2
Hipm,i
θ(Riss,θ)sin(mθ)dθ(54)
m−aia,1,mRm−1
iss +aia,1,mR−m−1
iss 
=
Niss

i=1
μ0
πβi+wiss/2
βi−ss/2
Hiss,i
θ(Riss,θ)cos(mθ)dθ
+
Nipm

i=1
μ0
πλi+wipm/2
λi−wipm/2
Hipm,i
θ(Riss,θ)cos(mθ)dθ(55)
aipm,0+bipm,0ln(Rirs )−μ0Mipm Rirs
=1
wipm λi+wipm/2
λi−wipm/2
Aia
z(Riss,θ)dθ(56)
aipm,nR
−nπ
wipm
iss +bipm,nR
nπ
wipm
iss 
=2
wipm λi+wipm/2
λi−wipm/2
Aia
z(Riss,θ)cosnπ
wipm θ−λi
+wipm
2dθ
(57)
aors,0=1
wors σi+wors /2
σi−wors /2
Aoa
z(Rors,θ)dθ(58)
aors,nf(n,Rors)
=2
wors σi+wors
2
σi−wors
2
Aoa
z(Rors,θ)cosnπ
wors θ−σi
+wors
2dθ
(59)
m−boa,1,mRm−1
ors +boa,1,mR−m−1
ors 
=
Nors

i=1
μ0
πσi−wors/2
σi−wors/2
Hors,i
θ(Rors,θ)sin(nθ)dθ(60)
n−aoa,1,nRn−1
ors +aoa,1,nR−n−1
ors 
=
Nors

i=1
μ0
πσi+wors/2
σi−wors/2
Hors,i
θ(Rors,θ)cos(nθ)dθ(61)
aoss,0+boss,0ln(Ross)−1
4μ0J0,oss R2
oss
=1
woss ψi+woss/2
ψi−woss/2
Aoa
z(Ross,θ)dθ(62)
2
nπsinnπd
woss −1
4μ0R2
issJoss,1+Joss,2(−1)n
×aoss,nR
−nπ
woss
oss +boss,nR
nπ
woss
oss 
=2
woss ψi+woss
2
ψi−woss
2
Aoa
z(Ross,θ)cosnπ
wiss θ−ψi
+woss
2dθ
(63)
m−boa,1,mRm−1
oss +boa,1,mR−m−1
oss 
=
Noss

i=1
μ0
πψi+woss/2
ψi−woss/2
Hoss,i
θ(Ross,θ)sin(mθ)dθ
+
Nopm

i=1
μ0
πτi+wopm/2
τi−wopm/2
Hopm,i
θ(Ross,θ)sin(mθ)dθ(64)
m−aoa,1,mRm−1
iss +aoa,1,mR−m−1
iss 
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8107713 IEEE TRANSACTIONS ON MAGNETICS, VOL. 57, NO. 10, OCTOBER 2021
=
Noss

i=1
μ0
πψi+woss/2
ψi−woss/2
Hoss,i
θ(Ross,θ)cos(mθ)dθ
+
Nopm

i=1
μ0
πτi+wopm/2
τi−wopm/2
Hopm,i
θ(Ross,θ)cos(mθ)dθ(65)
aopm,0+bopm,0ln(Rors )−μ0Mopm Rors
=1
wopm τi+wopm/2
τi−wopm/2
Aoa
z(Ross,θ)dθ(66)
aopm,nr
−nπ
wopm +bopm,nr
nπ
wopm 
=2
wopm τi+wopm
2
τi−wopm
2
Aoa
z(Ross,θ)cosnπ
wopm θ−τi
+wopm
2dθ
(67)
m−bfb,1,mRm−1
ipm +bfb,1,mR−m−1
ipm 
=
Nipm

i=1
μ0
πλi+wipm
2
λi−wipm
2
Hipm,i
θRipm ,θsin(mθ)dθ(68)
m−afb,1,mRm−1
ipm +afb,1,mR−m−1
ipm 
=
Nipm

i=1
μ0
πλi+wipm/2
λi−wipm/2
Hipm,i
θRipm ,θcos(mθ)dθ(69)
aipm,0+bipm,0lnRipm −μ0Mipm Ripm
=1
wipm λi+wipm/2
λi−wipm/2
Afb
zRipm,θdθ(70)
aipm,nR
−nπ
wipm
ipm,n+bipm,nR
nπ
wipm
ipm 
=2
wipm λi+wipm/2
λi−wipm/2
Afb
zRipm,θcosnπ
wipm θ−λi
+wipm
2dθ
(71)
m−bfb,1,mRm−1
opm +bfb,1,mR−m−1
opm 
=
Nopm

i=1
μ0
πτi+wopm/2
τi−wopm/2
Hopm,i
θRopm,θsin(mθ)dθ(72)
m−afb,1,mRm−1
opm +afb,1,mR−m−1
opm 
=
Nopm

i=1
μ0
πτi+wopm/2
τi−wopm/2
Hopm,i
θRopm,θcos(mθ)dθ(73)
aopm,0+bopm,0lnRopm −μ0Mopm Ropm
=1
wopm τi+wopm/2
τi−wopm/2
Afb
zRopm,θdθ(74)
aopm,nR
−nπ
wopm
opm,n+bopm,nR
nπ
wopm
opm 
=2
wopm τi+wipm/2
τi−wipm/2
Afb
z(Ropm,θ)cosnπ
wopm θ−λi
+wopm
2dθ.
(75)
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