A New Electromagnetic Model for PM Synchronous Machines
DAJAKU Gurakuq
A New Electromagnetic Model for PM Synchronous Machines
Gurakuq Dajaku, Dieter Gerling
Institute for Electrical Drives, University of Federal Defense Munich
Werner-Heisenberg-Weg 39, D-85577 Neubiberg, Germany
tel: +49 89 6004 3708, fax: +49 89 6004 3718,
e-mail: gurakuq.dajaku@unibw.de, dieter.gerling@unibw.de
Homepage: http://www.unibw.de/EAA
Keywords
«Permanent Magnet Motor», «Finite Element Calculation», «New Electromagnetic Model»,
«Space-Vector Theory»
Abstract
In this paper a new and effective electromagnetic model for PM machines is presented and analysed.
The new mathematical model is based on the modified equivalent circuit per phase of the PM machine
and is developed based on the space-vector theory. Using the new model the performances of the
electrical machines for the linear and saturated case can be described with high accuracy. The
accuracy of this model is verified and validated by comparing with other methods (measurements and
FEM calculations).
Introduction
Permanent magnet synchronous machines (PMSM) gain more and more importance for special drive
applications. Up to recent years, PMSM were known for small drives, e.g. for servo applications. In
the last years, PMSM are increasingly applied in several areas such as traction, automobiles, etc. An
accurate electromagnetic analysis of electric machines is of greatest importance for a successful design
and effective utilization of the system at normal operation. The analysis and control of electrical
machines requires an accurate mathematical model for performance assessment and system
simulation. During mathematical modeling of the electrical machines, we try to establish functional
relationships between entities which are important. As is known, various mathematical models for the
salient-pole PM machines exist today. Three-phase models in stator reference frame, and two-phase
models in rotor reference frame for the linear and non-linear operation conditions of the PM machines
are widely analysed by many authors [3-5]. Based on the complexity of these models, especially when
the saturation effects have to be taken into account, in [1] a simple mathematical model for PM
machines is developed. In analogy with the three-phase model, the new mathematical model also
describes the PM machine in the stator reference frame. During developing of this model the idea was
to have a simple model that consists of few parameters but, at the same time to regard the magnetic
properties of the machine in a satisfactory manner. To validate the accuracy of the new model in the
fourth section of this paper, the electromagnetic torque obtained with the new model is compared with
finite element method (Maxwell’s stress tensor method) and measurements.
EPE 2007 - Aalborg
ISBN : 9789075815108
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A New Electromagnetic Model for PM Synchronous Machines
DAJAKU Gurakuq
Electromagnetic modeling of PM machines in dq- and uvw coordinate
system
Usually, during analysis of the electrical machines different reference frames can be used. Depending
on the considered reference frame, different mathematical models for the electrical machines can be
obtained. Three-phase model in stator reference frame, and two-phase model in rotor reference frame
are two well know models, which are widely analysed and studied in many literatures. In the following
the performances of these models (existing models) for the linear and non-linear operation condition
of the PM machines are described briefly.
Three-phase model in stator reference frame
The three-phase model based on the stator reference frame (uvw model) is often used during the
analysis of electrical machines. Since each set of phase coils has self-inductance effect and
mutual-inductance effect, the voltage and torque equations obtained from this model are unwieldy and
complicated. As the magnetic properties of the iron core material are linear, and the air-gap has a
constant length (non-salient pole machines), the self- and mutual inductances are constant parameters.
In the case of salient pole machines, where the effective air-gap length (reluctance of the magnetizing
path) isn’t constant along the circumferential direction, the inductances of such machines change with
the rotor position, that leads to complex equations with time-varying coefficients [3]. Taking into
consideration the saturation effect, the parameters of the three-phase model depend also on the stator
current and the load angle δ . Therefore, the equations of the three-phase model are too complex and
not suitable for analysis and control of electrical machines.
Two-phase model in the dq-rotor reference frame
In an electrical machine where inductances vary as a function of rotor angle, the two-phase (d-q)
equivalent circuit model is commonly used for simplicity. The dq transformation provides the machine
inductances independent of rotor position by performing the equations in a frame of reference that
rotates in synchronism with the rotor. Under steady-state AC conditions the currents and voltages in
this frame of reference are constant. As long as the saturation effect does not occur this model is very
simple and attractive. When the motor operates with high torque, hence high current, saturation effects
are not negligible. Cross-magnetization (coupling) effects between d- and q-axis appear inside the
machine, so the dq-parameters become involved function of both d- and q-axis current [4, 5]. In order
to take into consideration the saturation effects in the PM machine, the dq-model should be further
extended and improved. It is known that such a model that consists of many non-linear parameters
isn’t appropriate for the modeling and control of electrical machines. Two dq-models for the saturated
PM machines are analysed and compared in [1] (improved dq-model and the simplified dq-model for
the saturated PM machine).
A new-mathematical model for PM machine
Based on the complexity of these models, especially when the saturation effects have to be taken into
account, and thanks the new derived expression for the phase inductance of the salient pole machine
[1, 2], in this section, a simple mathematical model for PM machines is presented and analysed. In
analogy with the three-phase model, the new mathematical model also describes the PM machine in
the stator reference frame. During development of this model the idea was to have a simple model that
consists of few parameters but, at the same time to regard the magnetic properties of the machine in a
satisfactory manner. The new mathematical model is based on the modified equivalent circuit per
phase of the PM machine, see figure 1, and is developed based on the space-vector theory.
In the new model, each stator phase is represented with its phase resistance R, phase inductance L(δ ) ,
and induced back emf. Also an additional component denoted with e − jγ has to be taken into account
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ISBN : 9789075815108
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A New Electromagnetic Model for PM Synchronous Machines
DAJAKU Gurakuq
during the analysis of the PM machines with this model. This component represents an additional
phase shift between the voltage drop through the phase inductance u L and the phase current i [1].
Figure 1: Modified equivalent circuit per phase of the PM machine
Characteristics of the L(δ ) and γ parameters
The L(δ ) parameter in the above equivalent circuit represents the effective inductance seen by one
phase under the balanced 3-phase conditions of normal machine operation. This parameter is analysed
accurately both by analytical and finite element methods in [2]. Deriving a correct formula for the total
phase inductance of the salient pole machines, it is shown that the phase inductance depends only on
the load angle δ . Compared with the self- and mutual inductances, the total phase inductance is
constant independent from the rotor position θ . The correct expression for the phase inductance is:
L (δ ) =
2
2
3
 LA + LB ⋅ cos ( 2δ )  +  LB ⋅ sin ( 2δ )
2
(1)
where LA is the average value of the self-inductance, and LB is the amplitude of the sinusoidal
harmonic part in the self-inductance curve.
According to the analysis made in [1], it is shown that under balanced three-phase excitation
conditions the current flux-linkage isn’t always in phase with current but is shifted away for the angle
γ depending on the load angle δ . The angle between flux-linkage and phase current can be derived
as

LB ⋅ sin ( 2δ )


 LA + LB ⋅ cos ( 2δ ) 
γ = atan 
(2)
Mathematical equations based on the new model
The space vector form of the stator voltage equation on the stationary reference frame is:
us = R⋅ is +
dψ s
(3)
dt
where, with u s , i s , and ψ s are denoted the corresponding space vectors for the voltage, current and
total flux-linkage.
The flux linking the stator winding consists of the contribution of the flux created by the stator
currents, and the flux created by the permanent magnets. Therefore the stator flux-linkage may be
expressed as follows
ψ s = ψ si + ψ sm
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A New Electromagnetic Model for PM Synchronous Machines
DAJAKU Gurakuq
where, ψ si and ψ sm are the corresponding space vectors for the flux-linkages due to the currents and
magnets, respectively.
Using the correct expression for the total phase inductance L (δ ) and angle γ , the flux-linkage
equations due to current in complex form can be written as [1]:
ψ si = L (δ ) ⋅ i s ⋅ e − jγ
(5)
Also, the space vector for the PM flux-linkage is:
ψ = ψˆ PM ⋅ e
m
s
 π
j θ − 
 2
(6)
With ψˆ PM is denoted the flux-linkage (peak value) due to magnets. Using equ. (5) and (6) into (4)
leads to:
ψ s = L (δ ) ⋅ i s ⋅ e − jγ + ψˆ PM ⋅ e
 π
j θ − 
2

(7)
By the substitution of the above equ. (7) into equ. (3) (space vector voltage equation), we obtain,

π
j θ − 
di
u s = R ⋅ i s + L (δ ) ⋅ s ⋅ e − jγ + jω ⋅ψˆ PM ⋅ e  2 
dt
(8)
Based on the space-vector theory, the electromagnetic torque for an AC machine can be defined as a
vector product of the stator flux-linkage and the stator current space vectors
T=
3
p ψs × i s
2
(
)
(9)
The cross product in the equation of the torque reveals that the equation is independent from the
coordinate system used; the cross product depends only on the angle between the vectors. Therefore
the torque may be calculated either from the quantities in the stator coordinates or in the rotor
coordinates.
According to [1], the above expression in the two-axis stationary reference frame attached to the stator
( αβ -frame) can be expressed as:
ψ sα = ψ siα + ψ smα = L (δ ) ⋅ iˆ ⋅ cos (ω t + δ − γ ) + ψˆ PM ⋅ sin (θ )
ψ sβ = ψ siβ + ψ smβ = L ( δ ) ⋅ iˆ ⋅ sin (ω t + δ − γ ) − ψˆ PM ⋅ cos (θ )
dψ sα
dψ siα dψ smα
= R ⋅ isα +
+
dt
dt
dt
i
dψ sβ
dψ sβ dψ smβ
= R ⋅ is β +
= R ⋅ is β +
+
dt
dt
dt
(10)
usα = R ⋅ isα +
us β
G 3
T = T ⋅ k = p (ψ sα ⋅ isβ − ψ sβ ⋅ isα )
2
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A New Electromagnetic Model for PM Synchronous Machines
DAJAKU Gurakuq
The new mathematical model is valid also for the saturated PM machine. When the saturation effect
occurs inside the machine the L (δ ) , γ , and ψˆ PM parameters depend on the stator current iˆ and load
angle δ . Also, a new parameter ∆ϑ , which represents the phase shifting of the PM flux from the real
rotor d-axis as result of the saturation effect, should be taken into account [1]. Therefore, the
mathematical equations of the new model for the saturated PM machine are:
ψ s (iˆ, δ ) = L(iˆ, δ ) ⋅ i s ⋅ e− jγ (i ,δ ) + ψˆ PM ⋅ e
ˆ
π

j  θ +∆ϑ ( iˆ ,δ ) − 
2

di
ˆ
u s = R ⋅ i s + L (iˆ, δ ) ⋅ s ⋅ e − jγ (i ,δ ) + jω ⋅ψˆ PM (iˆ, δ ) ⋅ e
dt
T=
3
p ψ sα (iˆ, δ ) ⋅ isβ − ψ sβ (iˆ, δ ) ⋅ isα
2
(
)
(10)
π

j  θ +∆ϑ ( iˆ ,δ ) − 
2

(11)
(12)
According to the modeling technique and the characteristics of the new mathematical model, the
following conclusions can be stated [1]:
•
•
•
•
•
•
The new mathematical model is developed based on the space-vector theory. The linear, and
non-linear operation cases of the PM machine are taken into consideration.
It is simple (consists of few magnetic parameters) and is very easy for manipulation. For the
saturation case the parameters of this model depend on the load condition of the machine.
The new mathematical model is verified and validated by comparing it with other models. The
torque results calculated with the new mathematical model agree with the results from the
improved dq-model.
Compared with the simplified dq-model it is valid for every operation condition of the
machine.
In analogy with the three-phase to two-phase transformation (uvw to dq), the equations of the
new mathematical model can be simply transformed in the dq-rotor reference frame.
Concerning the phase inductance the non-salient pole machine can be regarded as a special
case of the salient pole machine. Therefore, the new mathematical model for the salient-pole
PM machine presented in this paper directly can be used also for the analysis and control of
the other types of PM machines such as non-salient pole machines.
Electromagnetic torque of the PM machine
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 chapter, two calculation methods based on the results of
electromagnetic field FE analysis are used to derive the electromagnetic torque of different PM
machines; the Maxwell’s stress tensor and the new mathematical model for the PM machine. Using
these methods the electromagnetic torque versus load angle and rotor position for different operation
points are calculated. However, the accuracy of the new mathematical model is verified further by
comparison with measurements.
Torque calculation with different FE methods
In the following the electromagnetic torque versus rotor position is evaluated using Maxwell’s stress
tensor method and the new mathematical model of the PM machine. This calculation procedure gives
the electromagnetic torque under normal operation of the PM machine (the current magnetomotive
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ISBN : 9789075815108
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A New Electromagnetic Model for PM Synchronous Machines
DAJAKU Gurakuq
force (mmf) rotates in synchronism with the rotor). Figure 2 compares the results obtained with these
methods. The obtained results (average values) show a good agreement between these methods.
Figure 2: Electromagnetic torque versus rotor position of the PM-V1 machine
Comparison with measurements
From the previous section, it is shown that there exists a good agreement between the torque results
obtained from the new mathematical model and the Maxwell’s stress tensor. To prove further the
accuracy of the new mathematical model, in the following the electromagnetic torque obtained with
this model is compared with measurements. The FE simulations and the measurements are performed
for the IPM-2 machine. The FE simulations and measurements are performed for different operation
points and under high load condition ( iˆ in the region of 360-400 A). The measurements are done at
500 rpm rotor speed. In the figure 3 the obtained results are presented and compared. Also here, the
results obtained from the new mathematical model are in good agreement with the measurements.
Figure 5: Electromagnetic torque of the IPM-2 machine for different operation points
Conclusion
In this paper, different mathematical models for the salient-pole PM machines are presented and
analysed. Three-phase models in stator reference frame, and two-phase models in rotor reference
frame for the linear and non-linear operation conditions of the PM machines are described briefly in
the first part of this paper. Based on the complexity of these models, especially when the saturation
effects have to be taken into account, and thanks to the new derived expression for the phase
inductance, in the second part of this paper, a simple mathematical model for PM machines is
EPE 2007 - Aalborg
ISBN : 9789075815108
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A New Electromagnetic Model for PM Synchronous Machines
DAJAKU Gurakuq
presented and analysed. During developing of this model the idea was to have a simple model that
consists of few parameters but, at the same time to regard the magnetic properties of the machine in a
satisfactory manner. The new mathematical model is based on the modified equivalent circuit per
phase of the PM machine, and is developed based on the space-vector theory. In analogy with the
three-phase model, the new mathematical model also describes the PM machine in the stator reference
frame. The linear, and non-linear operation cases of the PM machine are taken into consideration. It is
simple (consists of few magnetic parameters) and very easy for manipulation. To prove the accuracy
of the new model it is validated by comparing it with other FE methods and measurements. The
electromagnetic torque derived with this model is compared with the FE Maxwell’s stress tensor
method and measurements. The comparisons are performed for different PM machines with inset
magnets in the rotor (salient pole machines). The obtained results show a good agreement between
these methods.
References
[1] Dajaku, G.: “Electromagnetic and thermal modeling of highly utilized PM machines” (ISBN: 978-3-83225415-5). Ph. D. Thesis 2006, University of Federal Defence Munich, Institute for Electrical Drives and
Actuators.
[2] Dajaku, G.; Gerling, D.: “The correct analytical expression for the phase inductance of the salient pole
machines”, 2007 IEEE International Electric Machines & Drives Conference (IEMDC 2007), 3-5 May Antalya,
Turkey.
[3] Mohamed, O. A.; Liu, S.; Liu, Z.: “Physical modelling of PM synchronous motors for integrated coupling
with machine drives”, IEEE Transactions on Magnetics, Vol. 41, May 2005.
[4] Bianchi, N.; Bolognani, S.: “Magnetic models of saturated interior permanent magnet motors based on finite
element analysis”, The IEEE IAS Annual Meeting, 1998.
[5] 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, Singapore Nov. 2003, (PEDS 2003).
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