ISSN 10683712, Russian Electrical Engineering, 2012, Vol. 83, No. 3, pp. 126–131. © Allerton Press, Inc., 2012.
Original Russian Text © V.A. Dmitrievskii, V.A. Prakht, F.N. Sarapulov, V.A. Klimarev, 2012, published in Elektrotekhnika, 2012, No. 3, pp. 7–13.
A Finite Element Model of Electric Machine
with Flux SwitchingOver for Studying
the Dynamic Operation Modes
V. A. Dmitrievskii, V. A. Prakht, F. N. Sarapulov, and V. A. Klimarev
Received February 23, 2012
Abstract—The article is devoted to modeling the dynamic modes of a flux switching machine using the finite
element method. The proposed model allows one to study highly dynamic transient processes in an electric
machine under arbitrary control algorithms and take into account the geometric features of the machine in
detail, the saturation, and the energy dissipation in magnets and steel.
Keywords: electric machine, flux switchingover, permanent magnet, mathematical modeling.
DOI: 10.3103/S1068371212030042
Electric machines with permanent magnets on a
stator (EMs with PMs on a stator) have attracted con
siderable interest in recent years. In the Englishlan
guage literature [1], such machines are called stator
interior permanent magnet machines, or stator–PM
machines.
The main advantage of stator–PM EMs is the
absence of permanent magnets on the rotating stator.
The magnets do not need to be glued to the rotor or
fixed in a special way. The simpler the rotor fabrica
tion, the cheaper the EM.
The advantage of stator–PM EMs over rotor–PM
EMs is the absence of a centrifugal force acting to
magnets. Paper [2] considers nontrivial methods of
magnet fastening on an EM rotor with high rates of
motion. Simplifying the manufacturing technology
of a rotor of an EM with a PM on the stator leads to
decreasing the EM net cost.
Paper [1] gives a review of existing structures of an
EM with a PM on the stator. One structure is a
machine with flux switching (a fluxswitching
machine (FSM)). Figure 1 shows a schematic sketch
of the structure of such machine [1, 3].
Methods for designing FSMs of the required scale
with specified initial values of power, the speed on the
shaft, and optimal characteristics of EM are still absent.
The development of such techniques is impossible with
out complex theoretical analysis of such EMs.
The present paper describes the dynamic mathe
matical model with using the finiteelement method.
Although the model requires great numerical
resources, it allows one to model highly dynamic tran
sient processes in EMs with arbitrary control algo
rithms and currents (voltages) on the EM and take into
account the geometrical features of the machine in
detail, the saturation, and the energy dissipation in
magnets and steel. Moreover, software based on the
FEMs visualizes the structure and simplifies develop
ment of the assembly documentation for the EM
design.
ASSUMPTIONS FORMING THE BASIS
FOR THE MATHEMATICAL MODEL
The given model is based on traditional assump
tions, including quasistationarity of the electromag
netic field, homogeneity of the media of EMs along
the z axis (the rotation axis of the rotor shaft), orienta
tion of the currents along the z axis, plane parallelism
of the magnet field, and uniform distribution of slot
ting currents over the conducting parts of slots.
Let us consider two variants of the model. In the
first, the magnets are electrically insulated from mag
126
+C
+A
Permanent
magnets
–A
–B
–C
+B
–B
+C
+B
x
y
–C
Rotor
+A
–A
–A
+A
+B
–C
–B
+C
z
+B
–C
+C
–B
–A
Fig. 1.
+A
Stator
A FINITE ELEMENT MODEL OF ELECTRIC MACHINE
netic circuits. In the second, there is an electrical con
tact between magnetic circuits and magnets, which
leads the magnets to the closed electric circuit in the
form of a squirrelcage of the induction motor. Note
that the electric field in the case of insulation of mag
nets from the magnetic circuits has a component of
electric field that is heterogeneous along the z axis in
the transverse plane.
CONSTITUTIVE EQUATION
FOR DIFFERENT MEDIA OF A MOTOR
The constitutive equations in the form of depen
dences of the magnetic field strength on the magnetic
induction H(B) and the z component of the current
density on the z component of the electric field
strength Jz(Ez) for each subdomain have special fea
tures. However, these relationships can be unified to
describe the partial differential equation uniformly for
all subdomains within the calculation domain.
In the reference system related to the medium, the
unified relationship Jz(Ez) showing the absence of the
current in the laminated magnetic circuit and the air,
the uniform current distribution over the conducting
parts of the stator slots, and the linear Ohm’s law for
the vortex current magnets is
e
J z = σE z + J z ,
127
sion approximating the relationship between μ and B
for numerical calculations. The expression from [5]
μ(0)
μ = 1 + ,
2 n
1 + a ( 0.1 + B )
where μ0, a, and n are selected to minimize the differ
ence between the values of μ–1 according to the mag
netization curves for the given electric steel, is applied
upon program implementation of the described
model.
In the magnetic circuits, Br is the vector describing
the steel losses. Assume that the magnet losses in steel
are prescribed by the expression
B⎞ ,
P st = α ⎛ ∂
⎝ ∂t ⎠
2
B.
B = – αμμ 0 ∂
∂t
r
e
Jz
where
is the density of the uniformly distributed
slot current, which is uniquely specified by the phase
currents; σ is the electrical conduction.
n
⎛ f⎞
⎝ 50⎠
α = ρP ,
2
( 2πf )
(6)
is used in the present paper. This provides coincidence
between the fundamental harmonic of (4) and the
given expression for calculating the steel losses [4]:
n
2
P st = ρP ⎛ f⎞ B ,
⎝ 50⎠
r
B–B
H = ,
μμ 0
(2)
where μ is the magnetic permeability; Br is the vector
taking the nonzero value in the domains of magnets
and magnetic circuits. In this case, the physical mean
ing of the vector Br is different in these domains.
For all nonmagnetic media (air and the conducting
domains of the slot), we assume that μ = 1 and Br = 0.
For magnets NdFeB, μ = 1, while Br is the vector of
residual induction.
For the magnetic circuits, the magnetic permeabil
ity is stated by the main curve magnetization curve μ =
B/(μ0H) [4] and depends on the magnetic induction
module. There is a need to have an analytical expres
RUSSIAN ELECTRICAL ENGINEERING
(5)
The expression
e
The relationship H(B) can be written in the form
(4)
where α is the proportionality coefficient defined
based on empirical formulas.
As is known, the volume density of the work per
unit time over the change in the magnetic induction is
r
∂B
B ∂B
B ∂B
H , or, taking (2) into account, – .
μμ 0 ∂t
∂t
μμ 0 ∂t
r
The summand containing B needs to coincide with
(4). Then,
(1)
It is taken that Jz = 0 or σ = 0, J z = 0, for the air
and magnetic circuits. To take the losses due to vortex
current in magnets into account, σ is given according
e
to the reference data, J z = 0. For the conducting parts
of the slot, σ = 0.
(3)
Vol. 83
(7)
where ρ is the steel density; P are the losses per 1 kg of
steel at the frequency 50 Hz and actual value of induc
tion 1 T; f is the frequency; and n is the parameter,
which is n = 1, 3 for the majority of dynamo steels.
GAGE SELECTION
It is known that the plane parallel magnetic field
can be described by a vector potential having only one
zindependent component Az,
No. 3
∂A z
B x = ;
∂y
2012
∂A z
B y = – .
∂x
(8)
128
DMITRIEVSKII et al.
Since the conduction of the magnets is described
by the σ coefficient, the electric field strength for them
∂A
is E = – – ∇ϕ, where ϕ is the scalar potential.
∂t
The requirement to the parallelism of currents of
∂ϕ ∂ϕ
the z axis is met if = = 0 within the media with
∂x
∂y
the electric conduction described by σ. This means
that the z component of the potential electric field
E 0 = –∂ϕ/∂z within the magnet does not depend on x
and y and
∂A
0
E z = – z + E .
∂t
(10)
If the magnets are electrically insulated from the
stator and have a dielectric coating, E0 is different for
each magnet. Thus, the electromagnetic field in the
chosen gage is specified by the field variable Az and
0
The coordinate system related to domain II is not
inertial. However, the Maxwell equations in quasista
tionary approximation are invariant with respect to the
transition into the rotating coordinate system.
EQUATIONS FOR THE VARIABLES
SPECIFYING THE ELECTRIC FIELD
Equations (1) and (2) are matched by the equation
for the field variable Az:
∂A
e
0
∂ ⎛ 1 ∂A z⎞
σ ⎛ – z + E ⎞ + J z = – ⎝ ∂t
⎠
∂x ⎝ μμ 0 ∂x ⎠
r
r
B
By
∂ 1 ∂A
∂ – ⎛ z⎞ – + ∂ x.
∂y ⎝ μμ 0 ∂y ⎠ ∂x μμ 0 ∂y μμ 0
(12)
0
additional ordinary variables E 1 , …, E Nm as values for
each magnet (Nm is the number of magnets).
If the magnets are not insulated, E 0 is the same for
all magnets, E 0 = 0, and the electromagnetic field is
thus specified by only the field variable Az.
MODELING THE ROTATION
OF THE ROTOR ABOUT THE STATOR
When the solids move relative to each other, the
calculation domain can be divided into subdomains
where the timedependent joint condition is specified
at the common boundary. This technique for modeling
forward motion in a linear induction motor is consid
ered in [6].
For modeling of the rotation, the calculation
domain is divided into two domains by circle passing
through the center of the nonmagnetic gap: domain I
includes the area of the stator and half of the air gap;
domain II includes the area of rotor and the other half
of the air gap. Each domain is considered in the asso
ciated reference system.
Joining of the component of vector of magnetic
potential Az was fulfilled at the common boundary of
domains I and II. The independent degrees of freedom
at the boundary of domains are the values of Az in
domain I. These values in the point (x, y) prescribe the
value of Az in the point on the boundary of domain II
with coordinates
x' = x cos θ + y sin θ ;
mal derivatives of Az is automatically performed by
finite element method.
y' = – x sin θ + y cos θ, (11)
where θ is the rotation angle of the rotor about the sta
tor.
The points (x', y') at the boundary of domain II are
not necessarily the discretization points. The value of
Az in the discretization points of domain II is stated by
means of polynomial interpolation. Joining of the nor
If magnets are insulated from the magnetic circuit,
this partial differential equation should be added by
Nm algebraic equations due to the presence of the
0
0
additional variables E 1 , …, E Nm . These equations
express the equality of current flowing through the
magnet cross section to zero:
∫∫J
= 0,
z
(13)
Ω iM
where ΩiM are the domains corresponding to magnets.
CALCULATION OF THE THRUST MOMENT
The voltage integration is a method for calculating
the torque. To increase the accuracy of finite element
methods, the surface gap instead of curvilinear inte
gral over the line is calculated throughout the whole
domain [7]:
2
M = L
∫∫
Ω air
2
2
2
xy ( B y – B x ) + ( x – y )B x B y
dS,
2
2
δ air μ 0 x + y
(14)
where Ωair is the domain of the air gap, which is limited
by circles of the rotor and stator surfaces; δair is the
thickness of the air gap; and L is the length of the
machine.
THE EQUILIBRIUM EQUATION
OF THE ELECTRIC CIRCUIT
The equilibrium equation of the electric circuit at a
series connection of coils in the phase has the form
U i + EMF i = rI i ,
RUSSIAN ELECTRICAL ENGINEERING
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(15)
No. 3
2012
A FINITE ELEMENT MODEL OF ELECTRIC MACHINE
where Ui is the voltage (lateral emf) of the ith phase;
EMFi is the electromotive force; r is the phase resis
tance;
−
EMF i = NL
+
S k(i)
∂A
dS,
∑ ∫ ∫ ∂t
(16)
The initial state of the magnet field is described by
the equation obtained from (11) by substituting E0 = 0
∂A
and z = 0:
∂t
e
∂ 1 ∂A
∂ 1 ∂A
J z = – ⎛ z⎞ – ⎛ z⎞
∂x ⎝ μμ 0 ∂x ⎠ ∂y ⎝ μμ 0 ∂y ⎠
Ωk
where L is the length of the stator pack (z coordinate);
and N is the number of vortices in the slot side of coil.
Summing in expression (16) is carried out over the
domains corresponding to the conducting parts of the
slot Ωk relative to the phase i and the negative and pos
itive signs are selected for arbitrarily positive and neg
ative domains.
Upon dynamic modeling of the FSM as a part of
electrical drive, the equations of the FSM model are
complemented by the motion equation of the mobile
part, which determines the rotation angle θ and the
·
angular velocity Ω = θ and the control system equa
tions that define the controls Ui or Ii.
FORMULATION AND SOLUTION
OF THE CAUCHY PROBLEM
To find the unique solution, the model equation
needs to be supplemented by initial and boundary
conditions.
The absence of the field at the boundary of the cal
culation domain Az = 0 is conveniently considered as
the boundary condition. The calculation domain nec
essarily contains the air adjacent to motor, in which
some of the magnet fluxes are closed.
The initial conditions are the value of field Az and
the additional degrees of freedom V1, …, V Nm in the
initial time, when the magnets are insulated from the
magnetic circuit. For correct formulation of the
Cauchy problem, the state in which the degrees of
freedom do not depend on the history of system and
can be directly calculated on the basis of solving the
stationary problem should be taken as the initial time.
During the rotor rotation, dissipative processes in
steel and magnets containing derivatives with respect
to time occur even under open winding. These pro
cesses exist when there is alternating current flowing
through the stator winding as well. Therefore, the state
with a stationary rotor and steady currents in windings
can only be selected for the initial time.
Due to the stationary of state of the initial time,
emf is absent in magnets, wherein current does not
flow, and, then, the potential component of the elec
tric field
0
Ei = 0
(17)
does not exist.
RUSSIAN ELECTRICAL ENGINEERING
Vol. 83
129
r
(18)
r
Bx
∂ B
∂ – y + .
∂x μμ 0 ∂y μμ 0
e
Here, J z is specified according to (15), and Ui = rIi due
to the absence of emf.
Based on the described mathematical model, the
program complex for modeling the transient processes
in the stator–PM EM using the finite element method
is developed.
AN EXAMPLE OF CALCULATION RESULTS.
THE LOADING OPERATION
In order to demonstrate the possibilities of the
model, the test calculation of the generator mode of
FSM in which the power supply terminals were con
nected with resistors by the circuit “star coupler with
common wire” was carried out. The motion of the
rotor was taken to be uniformly accelerated with the
angular acceleration ε = 377 rad/s2 corresponding to
60 rpm. Then,
U i = – r load I i ;
ω = εt;
2
.
θ = εt
2
(19)
The first relationship allows one to exclude Ui from
(15), while the field of the component of vector poten
tial Az and phase currents turned out to be independent
0
0
variables under insulation of magnets E 1 , …, E Nm .
Geometry of FSM
The external radius of stator is
0.13 m
The coefficient of filling the slot kz by winding is
0.45
The load resistance rload is
4.23 Ω
The residual magnetic induction Br is
1.33 T
The air gap is
0.75 mm
The specific resistance of the magnets is
1.4 × 106 Ω m
The length of the stator pack (z coordinate) is
0.2 m
The steel grade is
ST 2013
The load resistance is ten times higher than the
winding resistance. Consequently, the useful power is
only 91% of the power of the electric circuit of
machine, while the loss in winding is 9%.
Figure 2 shows the resultant relationships between
the time and instantaneous mechanical power (a),
torque (b), losses in magnets and magnet circuits
No. 3
2012
130
DMITRIEVSKII et al.
Power, W
14000
Torque, N m
(a)
(b)
250
12000
200
10000
150
8000
6000
100
4000
50
2000
0
0.04
0.08
Power, W
1400
0.12
0.16
0
Time, s
0.04
0.08
Power, W
0.10
(c)
0.12
0.16 Time, s
(d)
0.09
1200
0.08
1000
0.07
800
0.06
2
2
0.05
600
0.04
0.03
400
0.02
1
200
0
3
3
0.04
0.08
0.01
0.12
0.16
Time, s
Current, A
80
60
1
0
0.04
0.08
0.12
0.16 Time, s
0.12
0.16 Time, s
(e)
Phase C
40 Phase B
Phase A
20
0
–20
–40
–60
–80
0
0.04
0.08
Fig. 2.
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No. 3
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A FINITE ELEMENT MODEL OF ELECTRIC MACHINE
(c, curve 1), the portions of these losses of the instan
taneous power (d, curve 1), and the phase currents (e).
Curves 1 in Figs. 2a–2d are given for the case of insu
lated magnets. Curves 2 in Figs. 2c and 2d characterize
losses in magnets in the presence of an electric con
tact, while curves 3 in Figs. 2c and 2d define the losses
in magnetic circuits.
The proposed model allows one to calculate the
pulsation character of power, torque, and other values.
The power oscillations are evident in Figs. 2a and 2b.
From Figs. 2c and 2d, it follows that the losses in
steel exceed the losses in magnets at a low rotation fre
quency and vice versa at a high one. In the final period
(at rotation frequency 12 rps and current frequency
264 Hz), the steel losses under magnet insulation are
14% of mechanical power and the magnet losses are
1.7%. With the presence of a contact between the mag
nets and magnetic circuit, the losses in magnets are 9%
of the mechanical power, i.e., they increase by five
times. Therefore, magnet insulation at high rotation
frequencies results in a considerable increase in losses.
However, since there is a squared relationship between
the losses in magnets and the frequency of the low
speed EM, the losses in magnets turn out to be insig
nificant even under the presence of a contact with the
magnetic circuit.
RUSSIAN ELECTRICAL ENGINEERING
Vol. 83
131
The mathematical model presented and the devel
oped program complex for calculating the EM’s char
acteristics can be applied to the EM design.
REFERENCES
1. Ming Cheng, Wei Hua, Xiaoyong Zhu, et al., Stator–
Permanent Magnet Brushless Machines: Concepts,
Developments and Applications, Proc. 11th Int. Conf.
on Electrical Machines and Systems, ICEMS 2008,
Wuhan, 2008, pp. 2802–2807.
2. Sitin, D.A., Magnetic Systems for Synchronous Elec
tric Machines with Rare–Earth Permanent Magnets
and Increased Rotation Speed, Cand. Sci. (Eng.) Dis
sertation, Moscow, 2009.
3. European Patent EP 2 169 804 A2.
4. Kopylov, I.P., Proektirovanie elektricheskikh mashin
(Electric Machines Design), Moscow: Energiya, 1980.
5. Prakht, V.A., Sarapulov, F.N., and Dmitrievskii, V.A.,
Computer Simulation of AC Converter–Fed Induction
Motor with Permanent Magnets, Distantsionnoe Vir
tual’noe Obuchenie, 2010, no. 10, pp. 38–46.
6. Dmitrievskii, V.A., Research of the Induction
Machines with Open Magnetic Circuit on the Base of
Field and Circuit Theories, Cand. Sci. (Eng.) Disserta
tion, Yekaterinburg, 2007.
7. Rymsha, V.V., Radimov, I.N., and Poraiko, A.S., Elek
tromashinostroen. Elektrooborudovanie, 2003, issue 60,
pp. 35–38.
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2012