Optimum Control for Interior Permanent Magnet Synchronous Motors (IPMSM) in Constant
Torque and Flux Weakening Range
Michael Meyer, Joachim Böcker
Paderborn University, Institute of Power Electronics and Electrical Drives, Paderborn, Germany
Abstract —Interior Permanent Magnet Synchronous Motors (IPMSM) gain importance due to their high torque per volume ratio particularly for hybrid electrical vehicles. However, unlike to standard control theory, the torque control strategy for these motors is not apparently due to their reluctance torque, which is typical with interior magnet design. In this contribution, a control strategy is presented, which enables optimal torque control both in the lower speed range as well in the full flux weakening range. Operation during flux weakening, however, causes stress to the magnets of the motor with the risk of permanent demagnetization. The relations between the crucial design parameters are shown.
I.II.I NTRODUCTION
The development of hybrid electric vehicles (HEV) is proceeding rapidly. Requirements for electrical motors used in the drive-train of an HEV are a high starting torque and a wide constant-power speed ar
ea. A
widespread electrical machine used for this purpose is the
Interior Permanent Magnet Synchronous Motor (IPMSM).The popularity of this motor increased in recent years due advances of rare earth permanent magnet materials as NdFeB  or SmCo . IPMSM offer a good torque
per volume ratio as well as a high efficiency. An inherent property of IPMSM is a significant degree of saliency. To tap the full potential of IPMSM and ensure an operation with a maximum torque to current ratio, the reluctance torque cannot be neglected. Operating with an optimum torque to current ratio, even in the constant torque area, the d -axis current will not be chosen equal to zero. The reluctance torque is proportional to the product of the direct (d ) and quadrature (q ) axis currents. Thus, given a d -axis current unequal to zero, the system to be controlled is nonlinear and the strict separation between flux generating and torque generating currents is not possible. The problem is to determine the loss minimal d - and q -axis current reference values for a given rotor speed, DC
link voltage and reference torque. Because of the required
constant power range, current and voltage limits have to
be taken into consideration. Below the rated speed the
motor torque is only limited by the maximum current. For speeds above rated speed, the voltage limit has also to be taken into consideration.
In [1] the conditions for an operation of an IPMSM with a maximum torque to current ratio are derived for
constant torque and constant power speed range. The
authors proposed to store the points of optimum operation (i d * and i q *) dependant on reference torque and rotor
speed in a Look Up Table (LUT). In [2] the same authors implemented a loss minimal control without LUT.
In [3] a control structure for loss minimal operation of an IPMSM without the use of LUT is proposed. A feedback method is used, which calculates a new reference current setpoint at time k  based on th
e setpoint at time k -1. The calculation of the new setpoint is based on a function i d,opt *=f (i q ) where i d,opt  is the optimum d -axis current for all i q values on the maximum-torque-per-current-curve. In the field weakening area, a voltage controller is keeping the required voltage close below the voltage limit. Eventually, an optimum setpoint will be reached in the constant torque region as well as in the partial field weakening region. However, in the full field weakening region, a loss-minimal setpoint will not be reached. A major drawback of this method is the poor
dynamic behavior, which is inherent to the feedback method, as the reference current values are not directly calculated from the rotor speed and reference torque. In [4], the calculation of the optimal setpoint is subject to a numerial optimization procedure, which includes also iron losses, but does not yield much insight of the technical interrelationsships.
In this paper an open loop control structure based on LUT is proposed. The LUT are obtained from measurement data. Hence – at least in steady state operation – the proposed open loop control structure inherently accounts for saturation effects. Given the reference torque and flux limit, the desired currents are directly obtained from the LUT. Thus, a high dynamic performance of the open loop control structure can be ensured. The overcompensation of the permanent magnet flux in parts of the field weakening area that may affect the magnets, a disadvantage of the loss minimal control, i
s addressed afterwards. The paper is organized as follows: Section II: Modeling of the IPMSM
Section III Loss minimal operation of an IPMSM considering current and flux limit Section IV Control Structure  Section V Magnetic Reluctance Model Section VI Conclusion M ODELING OF THE IPMSM In the rotor-fixed d/q  reference frame, the voltage
equations of the IPMSM are given by :
p RS d d RS q q q q q RS d d d i L i
L Ri u i L i
L Ri u \Z Z Z          (1)
The motor torque is described by:
q d q d p i i L L p T )(2
3
\(2)
Where q
d u u ,d - and q -axis voltag
e components q
d i i ,d - and q -axis current components R Stator resistance
q
d L L ,d - and q -axis inductances RS Z Electrical angular velocity p
\Permanent magnet flux
max I Maximum length of the stator current space vector dc U DC link voltage p
Number of pole pairs
The voltage and current limits of an IPMSM are restricted by the following constraints:
2
max
2
2
I i i q d d  (3)
2
max
2
2
U u u q d d  (4)
With
3
max dc U U
(5)
Considering an IPMSM operating in steady state and neglecting the ohmic voltage drop, equation (6) follows directly from equations (1) and (4)
RS
q q d d P U i L i L Z \max
22)()(d
(6)
Equation (6) expresses the voltage limit in terms of stator currents. The left hand side of equation (6) is the length of the effective flux space vector. The right hand side describes the maximum flux space vector amplitude of an IPMSM revolving with electrical angular frequency Z RS , without exceeding the voltage limit.
RS
RS U Z Z \max
max )(
(7)
In Fig. 1 two constant-current circles and two constant-flux ellipses, which correspond to the maximum allowable flux at the rotor speeds Z 1and Z 2are depicted. The dashed lines are lines of constant torque.
The constant-flux ellipses have the following geometric parameters in the i d /i q  plane:
¸
¸¹·¨¨©§ 0,d P L \Center point
d RS L U Z max
,q
RS L U Z max
Semi-major axis, semi-minor axis
The d -axis current component in the center point of the constant flux ellipse is equal to the negative short circuit current I 0.
d
p
L I \
0(8)
Fig.1 Characteristic curves of an IPMSM in terms of stator currents
III.A.B.L OSS -M INIMAL O PERATION OF AN IPMSM C ONSIDERING C URRENT AND F LUX L IMITS
For a given reference torque T *, the motor is operating in a loss-minimal way,if the amplitude of the current space vector used to apply this torque is minimal. Each constant torque curve can be reache
d with a unique minimum-amplitude current space vector. The geometric location of all minimum-amplitude current space vectors forms the minimum-torque-per-current curve (curve F 1-G 1-A 1-B 1-C 1in Fig. 1). For current combinations on this curve the torque per current ratio is at maximum.
Operation in constant torque range
In the constant torque range, the feasible operation area of the motor is not restricted by the voltage or flux limits,respectively. For a given reference torque, the associated operating point on the maximum-torque-per-current curve (F 1-G 1-A 1-B 1-C 1) is chosen. The maximum applicable torque T 1 is only restricted by the maximum-current circle (points C  and F  in Fig.1).This holds for rotor speeds below rated speed Z <Z 1.
Operation in the partial field weakening range
For speeds above rated speed,Z >Z 1, the allowable area of operation is restricted by the voltage limit. Assume that the motor depicted in Fig. 1 is revolving at the rotor speed Z 2. Given a constant DC link voltage, the effective
flux in the machine must not exceed \max (Z 2). For low reference torques T * < T 3, operation points
on the maximum torque-per-current curve are still located within the allowable operation area.For a reference torque T 3, the flux limit is reached. Higher torques can be applied, if the operation point is moved along the constant-flux ellipse towards values of higher demagnetizing current.The maximum torque is reached at the intersection point between constant-flux-ellipse and constant-current-circle.The trajectory of optimum currents for the flux limit \max (Z 2) is given by the curve (E 1-G 1-A 1-B 1-D 1).
C.IV.Operation in the full field weakening range
If the center point of the constant-flux ellipse is located within the circle of maximum current, i.e. if I 0 < I max  , the strategy described in paragraph B will not lead to the maximum torque for sufficient small \max  values.
Fig. 2 Characteristic curves and feasible area of operation of an
IPMSM in terms of stator currents
Looking at Fig. 2, the maximum torque T 5 with flux limit \max (Z 5)is not reached at the intersection point of constant-flux ellipse and constant-current circle any more.For all flux limits smaller than \max
(Z 4) the operation point of maximum torque will be reached with a current vector length smaller than I max . The performance of the motor in this speed range is only restricted by the flux and voltage limit.This speed range is called the full field weakening area. The curve of optimal stator currents at speed Z 5 is given by the curve C 2-A 2-B 2. The feasible operation area of the IPMSM is shaded gray in Fig. 2. C ONTROL S TRUCTURE
The classic flux oriented control structure, known from non-salient PMSM, is not applicable to implement the loss-minimal control strategy described in Chapter    3.Taking the reluctance torque into consideration, the separation between torque generating and flux generating current components is not possible. The problem is to
provide optimal d - and q -axis current setpoints for a given reference torque T * and given flux limit \max .
From measurement data the LUT of equation (9) can be created, i.e. a unique value of applied torque and effective flux space vector amplitude can be assigned to any rotor-oriented current vector (i d ,i q ).
)
,(~
),(1q d i i f T  \In the feasible area of operation, the constant-torque curves and the constant-flux ellipses have a unique point of intersection. Hence, an inversion of the LUT in the feasible area of operation is possible.
(9)
)
,(~),(lim lim 1
1\T f i i q d  (10)
The torque and flux values T lim  and \lim , which will be applied to the LUT of equation (10), have to be generated based on the reference Torque T * and the flux limit \max .For this purpose, two additional LUT are necessary.
In the constant torque range, the optimum amount of effective flux for a given reference torque T * in the machine is given by the correspondent flux values on the maximum torque-per-current curve. Th
ese flux values for a reference torque range between zero and rated torque are stored in the LUT of equation (11).
)
(~
2T f opt  \(11)
In the field weakening range, the flux limit value \max must not be exceeded. Thus, the flux value \lim  applied to the LUT of equation (10) is determined by equation (12).
)
,min(max lim \\\opt  (12)
The maximum torque T max , which can be applied by the IPMSM in the constant torque area, is equal to the rated torque. In the field weakening area, T max  depends on the flux limit \max . Considering these limitations, the LUT of eqn.(13) can be generated from the measurement data.
)
(~
lim 3max \f T  The magnitude of the reference torque is limited to the T (13)
max  value obtained from the LUT.
The open loop control structure for the determination of the reference current setpoint is depicted in Fig. 3. It is embedded in an overall control structure including subordinated current control, which is shown in Fig. 4. One advantage of the proposed control structure is, that the LUT based on measurement data do already account for saturation effects in steady state operation. However,dynamic cross saturation effects are not comprised in the LUT.
To ensure an exact operation at the voltage limit in the flux weakening area and guarantee the availability of sufficient actuating variable reserve for transient operations a modulation controller is installed which acts on the available flux \max .
V.M AGNETIC R ELUCTANCE M ODEL
The load of the permanent magnets can be derived from the d - and q -axis reluctance models. The magnetic equivalent circuits are shown in Fig. 5.
Fig. 5 Magnetic equivalent circuits in d -direction (a) and q -direction
(b)
The quantities of the magnetic circuit are as follows:
c
4Coercive magnetomotive force of permanent magnets
q
d Ni Ni ,Magnetomotiv
e force o
f d - and q -axes N
Number of turns of stator winding mq md R R ,Reluctances in d - and q -axes q
d )),Fluxes along d - and q -axes q
d \\,Interlinked fluxes in d - and q -axes
From the magnetic equivalent circuit follows:
mq
q q md d
c d R Ni R Ni  ) 4
),(14)
The interlinked flux components in d -and q -direction
are given by:
q
q q q d d p d d i L N i L N  )  ) \\\(15)
The inductances and the permanent flux are given by:mqminimal
q md d R N L R N L 22
,
(16)md
c p R N 4
\(17)
md
R
mq
R (a)
(b)
The magnetization curves of the permanent magnet material are depicted in Fig. 6. If the flux weakening caused by the negative d -axis currents does not exceed a certain temperature dependant limit, the permanent magnets will not suffer an irreversible damage, i.e. that as long as the flux weakening is limited to the linear range of the characteristic curve, the permanent magnets will recover completely, see also [5].
Fig. 6 Magnetic points of operation dependant on the flux weakening
current in negative d -direction
Loss-minimal control of the IPMSM is based on the utilization of the reluctance torque, i.e. that the d -axis current component does not only have a flux weakening function, but is also used to apply an optimum level of reluctance torque. This is the reason, why the permanent magnet flux is even overcompensated in parts of the field weakening region resulting in negative flux density b ,which includes the risk of irreversible demagnetization,particularly with rising temperature -. The over-compensation of the permanent magnet flux is an inherent property of the optimum control for IPMSM.
Looking at Fig. 2, the part of the feasible area of operation on the left hand side of line D 2-G 2 is cha
racterized by an overcompensation of the permanent magnet flux. The points of operation on the curve D 2-G 2correspond with point B 6in Fig. 5. All operation points on the left hand side of curve D 2-G 2 in Fig. 2 could correspond,for example, with operation point C 6 in Fig. 5. The points of maximum flux overcompensation in Fig.2 are E 2 and F 2. They are characterized by the intersection of the maximum-torque-per-flux curve and the circle of maximum current.
The worst case current in negative d -direction is given by:
43220
max
1)48884(432
21g g g g g g g k g g g I I d
(18)
Where g is the degree of saliency
d
q L L g
(19)
and k  is the motor design parameter
max I I k
(20)
< the ratio of maximum current and short circuit current.
The standardized worst case demagnetizing current I d max /I 0 versus degree of motor saliency g  is shown in Fig.7.
Fig.7 Worst case demagnetizing current
That diagram gives insight into the design constraints. Since a saliency of g =2 is a typical value, I d max /I 0 and thus k  cannot be much larger 1, i.e. the short circuit current should not be much smaller than the maximum current,because that would usually result in irreversible demagnetization of the magnets.
VI.C ONCLUSIONS
A concept for optimal torque control of IPMSM has been presented, which is able to cope with the full flux weakening range, which is characterized by the fact that the maximum torque is no longer achieved with maximum current. The control structure is based on look-up tables, where saturations effects can be considered. A consideration of the stress of the magnets during flux weakening showed the restrictions between the system design parameters.
R EFERENCES
[1]
S. Morimoto, Y. Takeda, T.Hisara, and K. Taniguchi,“Expansion of Operating Limits for Permanent Ma
gnet Motor by Current Vector Control Considering Inverter Capacity”,IEEE Trans. Ind.Appl., vol. 26, no.5, Sept./Oct. 1990
[2]
S.Morimoto, Y. Tong, Y. Takeda, and T. Hirasa,“Loss Minimization Control of Permanent Magnet Synchronous Motor Drives”,IEEE Trans. Ind. Appl., vol. 41, no. 5, Oct. 1994
[3]
S. Bosga, and H. Zelaya de la Parra,“Field-weakening control of an interior permanent magnet motor for application in electric vehicles”,EPE, Lausanne, 1999
[4]
L. Chédot and G. Friedrich,“Optimal control of interior permanent magnet synchronous integrated starter-generator”,EPE, Toulouse, 2003.
[5]
P. Thelin, “Short circuit fault conditions of a buried PMSM investigated with FEM”, NORPIE2002, Stockholm, Sweden, 2002

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