Comparison of Different FE Calculation Methods for
the Electromagnetic Torque of PM Machines
Dieter Gerling
Institute for Electrical Drives, University of Federal Defense Munich
Neubiberg, Germany
Summary:
Generally, the torque of electrical machines can be calculated applying different methods. In this
paper, the Maxwell’s stress tensor method, the magnetic co-energy method, and the lumpedparameter method are investigated. As an example, a PM machine with surface mounted magnets in
the rotor is analyzed by means of the FE-software package ANSYS. For the lumped-parameter
method, the dq-parameters of the electric machine are derived with the Fixed Permeability Method
(FPM). With this method the parameters of the PM machine can be calculated with high precision. The
obtained results for the electromagnetic torque applying the different calculation methods are
compared concerning accuracy, ease of use and computing time.
Keywords:
Torque Calculation, Finite Element Method, ANSYS, PM Machine, Fixed Permeability Method (FPM)
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1.
Introduction
The basic task of any electric machine is to generate torque to accelerate and drive a load over a
specific range of speeds. Thus, torque is a very important consideration for both analysis and design
of electrical machines. The finite element method provides an accurate approach to torque evaluation
from the derivation of electromagnetic field distribution. This calculation method allows precise
determination of machine parameters through the magnetic field solutions as it takes into account the
actual distribution of windings, details of geometry, and non-linearity of magnetic materials of an
electrical machine.
In this paper, a PM machine with surface mounted magnets in the rotor is analysed with ANSYS.
Figure 1 shows the geometry of the studied PM machine. The main geometrical data are presented in
the Table 1.
TABLE 1- GEOMETRY DATA
Rotor radius
[mm]
Air gap length
[mm]
Thickness of magnet
[mm]
Magnet pole arc
[deg]
Number of poles
[ -- ]
Stator outer radius
[mm]
Stack length
[mm]
21.6
0.9
4
127.8
4
40
35
Fig. 1: Geometry of the studied PM machine
Different methods based on finite element solutions have been used for the calculation of the
electromagnetic torque of this machine. The methods analysed here are Maxwell’s stress tensor-, dqmodel of PM machine-, and magnetic co-energy method. The use of Maxwell’s stress tensor is
probably the simplest method, since it requires only the local flux density distribution along a specific
contour around the air-gap of the machine. The accuracy of torque calculation with this method relies
on model discretization and on contour selection.
According to the virtual work principle, the electromagnetic torque equals the derivative of the
magnetic co-energy with respect to angular position at constant current. With this method at least two
finite element solutions are required to obtain the co-energy change due to an incremental
displacement, and this inevitably increases the computing time.
Calculation of the electromagnetic torque with the third method is based on the dq-mathematical
model of the PM machine. The dq-parameters of the machine are derived accurately using fixed
permeability method (FPM) [2-4]. This method ensures that the saturation effect which occurs in the
load model is not ignored during calculation of the
•
•
dq-axes inductances (single current excitation of the stator windings),
back emf (single magnet excitation).
The calculation algorithm for the dq-parameters of the PM machine with the fixed permeability method
is presented in the fourth section of this paper.
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2.
Electromagnetic Torque
2.1
Maxwell’s Stress Tensor Method
The Maxwell’s stress tensor method is probably the most commonly used since it is uncomplicated to
apply and needs a relatively small calculation time. But, on the other hand, the accuracy of this
method is obviously dependent on the type of the problem to be solved, on the model discretization
and on the selection of the integration line or contour. If a Maxwell’s stress tensor is used, it is
important to force at least three layers of elements in the small air-gap and to calculate the torque in
the middle element layer.
The electromagnetic torque Te in the motor air-gap, on a closed surface of radius r, can be calculated
by integrating the Maxwell’s stress tensor. For two-dimensional electromagnetic field models:
Te =
2π
L
µ0
∫r
2
Br Bθ dθ
(1)
0
where Br and Bθ are radial and tangential components of the flux density, and L is the active length of
the machine. Taking advantage of the specific periodicity of electrical machines, the integral can be
performed from 0 to 2π electrical radians and the result multiplied by the number of pole pairs p.
2.2
Co-Energy Method
The second most frequently used method is based on the stored magnetic co-energy change or the
virtual work with a small displacement. According to the virtual work principle, the electromagnetic
torque Te equals the derivative of the magnetic co-energy Wco-eng with respect to angular position at
constant current,
Te =
∆Wco-eng
∆θ
(2)
i = const
In co-energy method the calculation time is doubled, because with this method the field is solved twice
to get an energy difference.
2.3
Electromagnetic Torque Based on the dq-Mathematical Model of PM Machine
The PM motor is usually modeled and analyzed by means of the space-vector theory, which is well
known and effective. The motor equations, valid also when the iron is saturated, are reported in the
following. The electromagnetic torque based on the dq -formulation is
Te =
3
p (ψ d ⋅ iq − ψ q ⋅ id )
2
(3)
while the voltage dynamic equations are
d
ψ d − ω ⋅ψ q
dt
d
uq = R ⋅ iq + ψ q + ω ⋅ψ d
dt
ud = R ⋅ id +
(4)
where, in general, the flux linkages depend on both current components, according to the following
magnetic model
ψ d = ψ d ( id , iq )
(5)
ψ q = ψ q ( id , iq )
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If the motor works in the magnetic linear region (low current), the d- and q-axis flux linkages vary
linearly with the corresponding d- and q-axis current components, i. e.
ψ d = Ld ⋅ id + ψ dm
ψ q = Lq ⋅ iq
(6)
where, ψ dm is the d-axis flux-linkage due to magnets. For the non-saturation case the total PM fluxlinkage flows through the d-axis.
When the motor operates with high torque, hence high current, saturation effects are not negligible
and the dq flux linkages become involved functions of both d- and q-axis current components. The
complete magnetic model for a saturated motor can be thus expressed as
ψ d ( id , iq ) = Ld ( id , iq ) ⋅ id + ψ dm ( id , iq )
(7)
ψ q ( id , iq ) = Lq ( id , iq ) ⋅ iq + ψ qm ( id , iq )
Compared with the equ. (6), an additional q-axis PM flux-linkage component is present in the
expression of the total flux-linkage. Like Ld and Lq , when the saturation occurs the ψ dm and ψ qm also
depend on the d- and q-axis current components. In an analysis made in [1] for a PM machine with
inset magnets in the rotor (IPM machine), it is shown that the ψ dm increases with id , while it decreases
with iq components. In the same way, when the saturation occurs, the magnitude of ψ qm (which is
negative for the motor operation condition of the PM machine) increases with iq , while it decreases
with id ; its variation reaches up to 24% of the ψ dm . This is the main effect of the saturation.
Using the equ. (3) and (7), the following expression for the electromagnetic torque is obtained
Te =
{
}
3
p  Ld (id , iq ) − Lq (id , iq )  ⋅ id ⋅ iq + ψ dm (id , iq ) ⋅ iq −ψ qm (id , iq ) ⋅ id 
2 
(8)
The above expression is used to calculate the electromagnetic torque with the third FE method. The
dq-parameters of the PM machine are calculated using the fixed permeability method (FPM) [2, 3, 4].
The main point of this method is that it transforms non-linear problems into linear ones by storing the
permeabilities from the non-linear analysis. For each operation point the magnetic permeability of
each element obtained from the previous FE simulation under the double excitation is fixed and stored
for the further analysis. With this fixed permeability, a second calculation is carried out either with
current or magnet single excitation. This ensures that the saturation effect which occurs in the load
model is not ignored during calculation of the dq-parameters of the PM machine.
3.
Finite Element Analysis Results
Based on the above calculation FE methods, in the following the electromagnetic torque is calculated
for I = 2.56 A (peak value) and δ = 0° operation condition. δ is the electrical angle between total flux
linkage due to the stator currents and the rotor q-axis (in the literature this angle is known as torque
angle or load angle). The flux line distribution and the flux density along the surface of a PM pole pair
for this operation condition are shown in the figure 2. It is shown that the rotor surface is subjected to a
relatively large flux density fluctuation due to the varying reluctance as teeth are passed.
All following finite element calculations have been performed using the software package ANSYS.
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B [T]
Fig. 2: Flux line distribution and the flux density along the surface of a PM pole pair.
Firstly, the electromagnetic torque is evaluated using the Maxwell’s stress tensor and the magnetic coenergy method for a fixed current excitation in the stator and for different incremental rotor positions.
Figure 3 compares the results obtained with these methods. It is shown that the results obtained from
the Maxwell’s stress tensor method are in good agreement with the results obtained with the magnetic
co-energy method. The torque curves vary sinusoidally with the load angle (for a fixed current in the
stator, the load angle is equivalent with the rotor position). As is known the PM machine with surface
mounted magnets in the rotor gives the maximum torque for the case when the load angle is zero
( δ = 0° , I = I q ). Due to the slotting effect some harmonics (torque pulsations) are added in the toque
curve of this type of the machine. This torque component is known as cogging torque. The cogging
torque is often the largest component of torque pulsation in permanent magnet motors. It is caused by
the interaction between the magnets and the stator teeth. This effect is normally not depending on the
stator current. Only if the teeth are saturated due to stator current, usually the cogging torque
increases because of the wider effective slot openings.
I=2.559 A; Torque-load angle
1,2
1
0,8
T_maxw
0,6
T_Coen
T [Nm]
0,4
0,2
0
-0,2
-0,4
-0,6
-0,8
-1
-1,2
0
20
40
60
80
100
120
140
160
180
delta [elec. degree]
Fig. 3: Electromagnetic torque versus load angle obtained with Maxwell’s stress tensor- and the
magnetic co-energy method.
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For the same operation condition as before in the following the electromagnetic torque versus rotor
position is evaluated using Maxwell’s stress tensor method and the dq mathematical model of the PM
machine. The fixed permeability method is used to derive the motor parameters. Using this method
the saturation condition of the machine under the double excitation is fixed and stored for the further
analysis. The dq-inductances and flux linkages due to magnets are derived under fixed permeability
from the double excitation condition. The dq-parameters of the motor are: iq = 2.559 A, id = 0 A,
L q = 6.483 mH, L d = 6.483 mH, ψ dm = 1.531 ⋅10−1 Vs, ψ qm = −7.663 ⋅10−4 Vs .
Figure 4 shows the saturation condition of the iron parts of the PM machine for the given operation
condition. Three regions of the machine can be seen to have low relative permeability values: the
stator teeth (in the main flux path), the stator yoke (in the main flux path), and the rotor yoke (in the
main flux path).
Fig. 4: The saturation condition (relative permeability) of the PM machine at I = 2.56 A and δ = 0° .
The electromagnetic torque results versus rotor position obtained with the Maxwell’s stress tensor and
from the dq mathematical model of the PM machine are presented and compared in the figure 5. Also
here the obtained results show a good agreement between these methods.
delta=0°, I=2.559 A: Torque-rotor position
1,4
1,2
T [Nm]
1
0,8
T_maxw
T_dq
0,6
0,4
0,2
240
228
216
204
192
180
168
156
144
132
120
108
96
84
72
60
48
36
24
12
0
0
theta [elec. degree]
Fig. 5: Electromagnetic torque versus rotor position obtained with Maxwell’s stress tensor- method
and the dq-mathematical model of PM machine.
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As the calculation of the electromagnetic torque with the Maxwell’s stress tensor method gives the
total torque components in the air-gap, the dq model gives only the average components of the
electromagnetic torque. The dq-theory is established based on the assumption that both the winding
flux-linkage and the currents are sinusoidal. The air-gap harmonics are ignored using this model.
4.
Computing time
As is discussed in the previous sections, the Maxwell’s stress tensor method is probably the most
commonly used method for the torque calculation since it is uncomplicated to apply and needs a
relatively small calculation time. Only one simulation for each operation point is needed with this
method. In co-energy method the calculation time is doubled, because with this method the field is
solved twice to get an energy difference. Also, the torque calculation with dq-mathematical model of
the PM machine parameters of which are derived with fixed permeability method needs two
simulations for each operation point. The first simulation should be done under the double excitation,
from which the permeability of each element is fixed and stored for the further analysis. With this fixed
permeability, a second calculation is carried out either with current or magnet single excitation. The
calculation time for the second simulation with the “frozen” permeability condition is faster because the
electromagnetic problem is linearized. Figure 6 shows the calculation procedure of the dq-parameters
with fixed permeability method. If the saturation effect occurs it is required that the calculation
procedure to be repeated for different rotor positions, therefore to derive the average values for these
parameters. For this case this method is very time consuming.
start
Iˆ, δ
dq-transformation theory
θ = 0 : ∆θ : τ P
Lq , Ld , ∗ψ qm , ∗ψ dm
Double excitation condition
Torque, T
Total flux linkage: ψ
T
τ P − Pole pitch
θn < τ P
Fixed Permeability method
Current flux-linkage: ψ
Average values
i
Lq , Ld , ∗ψ qm , ∗ψ dm , T
Magnet flux-linkage:
end
ψ m = ψ T − ∗ψ i
∗
Fig. 6: Calculation flow-chart of the dq-parameters of the PM machine.
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5.
Conclusions
The torque is a very important parameter for both analysis and design of electrical machines. The
finite element method provides an accurate approach to torque evaluation from the derivation of
electromagnetic field distribution. In this paper, three calculation methods based on the results of
electromagnetic field FE analysis are used to derive the electromagnetic torque of a PM machine; the
Maxwell’s stress tensor, the co-energy method and the dq mathematical model for the PM machine.
Using these methods the electromagnetic torque versus load angle and rotor position for a given
operation point is calculated. The obtained results show a good agreement between these methods.
References
[1]
[2]
[3]
[4]
Gerling D., Dajaku G.: “Finite element analysis and torque calculation of the
IPM2910 Machine”, Institute for Electrical Drives, University of Federal Defense Munich, July
2005
Kwak S. J., Kim K. J., Jung H. K.: “The characteristics of the magnetic saturation in the
interior permanent magnet synchronous motor”, International Conference on Electrical
Machines, 2004, Cracow, Poland (ICEM 2004)
Bianchi N., Bolognani S.: “Magnetic models of saturated interior permanent magnet motors
based on finite element analysis”, IEEE IAS Annual Meeting, 1998, St. Louis, USA
Jianhui H., Jibin Z., Weiyan L.: “Finite element calculation of the saturation dq-axes
inductance for a direct-drive PM synchronous motor considering cross-magnetization”,
International Conference on Power Electronics and Drive Systems, 2003, Singapore (PEDS
2003)
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