Research Journal of Applied Sciences, Engineering and Technology 4(17): 2846-2853, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: November 10, 2011
Accepted: December 09, 2011
Published: September 01, 2012
Modelling of Permanent Magnet Synchronous Motor Incorporating Core-loss
K. Suthamno and S. Sujitjorn
Control and Automation Research Unit, Power Electronics, Machines and Control
Research Group, School of Electrical Engineering, Suranaree University of Technology,
Nakhon Ratchasima, 30000, Thailand
Abstract: This study proposes a dq-axis modelling of a Permanent Magnet Synchronous Motor (PMSM) with
copper-loss and core-loss taken into account. The proposed models can be applied to PMSM control and drive
with loss minimization in simultaneous consideration. The study presents simulation results of direct drive of
a PMSM under no-load and loaded conditions using the proposed models with MATLAB codes. Comparisons
of the results are made among those obtained from using PSIM and SIMULINK software packages. The
comparison results indicate very good agreement.
Key words: Dq-axis, losses, modelling, permanent magnet synchronous motor
INTRODUCTION
Present-day industry has widely utilized Permanent
Magnet Synchronous Motors (PMSM) because of their
several advantages. These include high efficiency, low
maintenance, high power-factor and ruggedness. Some
applications, such as hybrid vehicles, aircrafts, automated
machines etc., usually require accurate control of speed
and position. Control difficulty arises since a PMSM has
complex dynamic that results in complicated modelling.
In classical machine textbooks, readers often find
equivalent circuit, phasor and differential equation
models, for instance books authored by Fransua and
Magureanu (1984) and Ong, (1998). DQ-axis models, or
dq-models, of PMSMs are also available (Pillay and
Krishnan, 1989; Monajemy and Krishnan, 2001;
Krishnan, 2010). The dq-models are useful for machine
control in the sense that relevant ac-signals are presented
as dc-signals. To achieve minimum loss drive of a
PMSM, its models with core-loss taken into account is
desired. Equivalent circuit models for the purpose are
available (Morimoto et al., 1994; Fernandez-bernal et al.,
2011; Cavallaro et al., 2005; Lin et al., 2009). The
equivalent circuit models provide an insight for loss
minimization, however cannot provide accurate dynamic
performances of the machines. In contrast, the dq-models
provide accurate information concerning machine
dynamic and control. However, the available dq-models
do not incorporate loss terms. Use of the models is thus
limited to machine control and drive without an extension
to cover loss minimization.
This study proposes the dq-models of PMSMs that
incorporate core-loss as well as the equivalent circuit
models. The proposed models are applied to simulations
using MATLABTM. The results are compared with those
of commercial software packages including PSIMTM and
SIMULINKTM with PowerSim Blockset.
MATERIALS AND METHODS
Modelling of a PMSM herein considers both copperand core-losses designated as RS and RC, respectively.
Figure 1 illustrates the diagram representing a PMSM, the
symbols in which are listed by the end of the study. vSabc
= [vsa vsb vsc]T and iSabc = [isa isb isc]T are the voltage and the
current vectors, respectively, at the motor terminals. The
voltage and the current vectors at the motor cores are
represented by vSabc =[v ca v cb v cc]T and iCabc = [ica icb icc]T,
respectively. ioabc = [i0a i0b i0c]T is the rotor current vector.
The following matrices designate the inductances and
flux:
1





 L0  L ms cos 2 (re )  2 Lo  Lms cos 2  re  3 



 1



  2 Lo  Lms cos 2  re  3 



 1

2 




  L0 Lms cos 2 re   L0  Lms cos 2 re 

3
2
3

Lmsabc  

 1

  2 L  Lms cos 2re   


 1


 1
cos
cos
2
2

L

L


L

L








ms
ms
re

 2 0
 re 3  2 0



2





 L0  Lms cos 2 re  3 


(1)
Corresponding Author: S. Sujitjorn, Control and Automation Research Unit, Power Electronics, Machines and Control Research
Group, School of Electrical Engineering, Suranaree University of Technology, Nakhon Ratchasima,
30000, Thailand
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Res. J. App. Sci. Eng. Technol., 4(17): 2846-2853, 2012
Fig. 1: Diagram representing a PMSM
 Lls
Lls   0
 0
0
Lls
0




 PM abc   




PM
PM
PM
0
0 
Lls  (2)



2  

sin  re 


3  
2  

sin  re 


3 
Tqdo
Therefore, we can state that
(3)
-1
v sabc = T
d
iS  vCabc
dt abc
Let
vsqdo
abc

d
 abc or RC iC
abc
dt
, = [vsq vsd vs0] ,
qdo
(2re)
v sqdo
v sqdo
= Tqdo (2re)
(6)
v sabc and
, where (2re) stands for an electrical
reference angle of the rotor shaft. Since:
d
i  v Cqdo
dt Sqdo
 0 1 0


  re Lls   1 0 0 i Sqdo
 0 0 0
v S qdo  RS i
(4)
and Eq. (5) expresses the voltage across the motor cores:
vC
2 
2  


cos  re 
 cos   re 



3
3  
2 
2  


sin  re 
 sin   re 
 


3
3 

1
1

2
2

sin (re )
RS = diag [RS RS RS] and RC = diag [Rc Rc Rc] represent the
winding and the core resistances, respectively. Therefore,
Eq. (4) expresses the motor voltage:
vSabc  RS iSabc Lls 

cos ( re )

2
( re )  sin ( re )
3
 1

 2
sqdo
 Lls
(7)
the rate of change of the motor current wrt time in dq0frame can be expressed as:
(5).
d
1
i sq 
(v  ( Rs  Rc )i sq
dt
Lls sq
 RC ioq   re Lls i sd )
(8)
d
1
i sd 
(v  ( RS  RC ) i sd
dt
Lls sd
 RC iod   re Lls i sq )
(9)
isqdo , = [isq isd is0]
and  qdo = [8q 8d 80]T be the terminal voltages, currents
and magnetic flux in dq0-frame, respectively.
Transformation from the three-phase abc-frame to the
dq0-frame utilizes the transformation matrix:
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Res. J. App. Sci. Eng. Technol., 4(17): 2846-2853, 2012
Fig. 2: Equivalent circuits of a PMSM taking account of core-loss and separated inductances
(a)
(b)
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Res. J. App. Sci. Eng. Technol., 4(17): 2846-2853, 2012
(c)
Fig. 3: Simulation diagrams, (a) PSIM, (b) internal simulation diagram of the PMSM coupler module in (a), (c) SIMULINK with
PowerSim Blockset
Table 1: Parameters of a PMSM for simulations
Parameters
Proposed mode
SIMULINK
RS (e)
1.9
1.9
330
NA
RC (e)
0.77
NA
Lls (mH)
15.75
NA
Lmd (mH)
31.05
NA
Lmq (mH)
NA
16.52
Ld (mH)
NA
31.82
Lq (mH)
8 PM
0.31*
0.31*
0.0005
0.0005
J (kg-m2)
B(Nm.s / rad)
0.03
0.03
*:in V. s / rad; **:in V peak L-L/Krpm

d
1
i 
R i  RCiod   re Lmqioq
dt od Lmd C Sd
PSIM
1.9
NA
NA
NA
NA
16.52
31.82
112.45**
0.0005
0.03
(10)
Since
vC
qdo
d

 qdo   re M  qdo
dt
 dqo
0
Lmd
0

3
v i  v sd i sd
2 sq sq
(11)
 em 
0 i oq  0 
 
0 i od   PM 
0 i o 0  0 




 R i2  i2  R i2  i2
C cq
cd
 S sq sqd
3
   Lls i sq lsq  i sd l sd
2

  Lmq i sq lsq  L md lsd i sd





  Pout



3
Pout   re ( PM ioq  ( Lmd  Lmq ) ioq iod )
2
and
 Lmq
  0
 0
(15)
Based on the derived voltage and current equations, the
equivalent circuits as shown in Fig. 2 can be obtained. Eq.
(16) to (19) express the input power, output power,
electromagnetic torque and the rate of change of speed of
the motor, respectively.
Pin 
d
1
i 
(v  R S iso )
d t so L ls so

(12)
 r 



3  P
   i  Lmd  Lmq ioq iod
2  2  PM oq
1
(  
 B r )
J em load

(16)
(17)
(18)
(19).
one can obtain that:
d
 qdo  R C (iSqdo  i0qdo )   re M  qdo
dt
(13)
Hence, the rate of change of the torque generating
currents in dq-frame wrt time can be expressed as:

d
1  RCisq  RCioq


ioq 
dt
Lmq   re Lmd iod  re PM 
(14)
The developed models described so far were coded in
MATLAB to simulate the machine dynamic based on a
set of parameters (Ong, 1998; Lin et al., 2009) tabulated
in Table 1. The results are compared to those obtained
from using PSIM and SIMULINK with PowerSim
Blockset, respectively. Figure 3 shows the simulation
diagrams. In an actual drive, the supply frequency is
varied in stepwise manner such that the machine gradually
gains its speed. This was conducted accordingly with a
frequency-step of 10 Hz in our simulations. For the
simulated PMSM, the supply is 110 Vrms, 60 Hz. The
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Res. J. App. Sci. Eng. Technol., 4(17): 2846-2853, 2012
25
Torque (N.m)
20
15
10
5
Simulink
Psim 9.0.3
Proposed model
0
-5
0
0.5
1
1.5
2
2.5
time (sec)
(a)
250
Velocity (rad/s)
200
150
100
50
Simulink
Psim 9.0.3
Proposed model
0
-50
0
0.5
1
1.5
2
2.5
time (sec)
(b)
60
Simulink
Psim 9.0.3
Proposed model
current magnitude(A)
50
40
30
20
10
0
0
0.5
1
1.5
2
2.5
time (sec)
(c)
Fig. 4: Motor responses, (a) torque, (b) speed, (c) magnitude of the resultant current
machine was driven at no-load from stand-still up to a
speed of 150 rad/s within 1.6-1.7 s. At the moment of 1 s,
10 Nm load was suddenly applied to the machine shaft. At
2 s, the supply frequency was reduced from 50 to 40 Hz.
RESULTS AND DISCUSSION
Figure 4 illustrates the simulation results including
torque and speed of the motor as well as the magnitude of
the resultant current of the dq-currents. The results
obtained from the three simulation approaches agree well
except that PSIM gives the mechanical torque at shaft
while the other two approaches give the electromagnetic
torque. Therefore, the torque curve from PSIM shown in
Fig. 4a is lower than the other two curves. The curves
indicate oscillatory transient responses at the instant the
frequency and the load-torque are changed. Figure 5
shows the motor phase-voltages and -currents. Notice that
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Res. J. App. Sci. Eng. Technol., 4(17): 2846-2853, 2012
200
Simulink
Psim 9.0.3
Proposed model
150
100
voltage (V )
50
0
-50
-100
-150
-200
0
0.5
1
1.5
2
2.5
time (sec)
200
Simulink
Psim 9.0.3
Proposed model
150
100
voltage (V)
50
0
-50
-100
-150
-200
0
0.5
1
1.5
2
2.5
time (sec)
200
Simulink
Psim 9.0.3
Proposed model
150
voltage (V)
100
50
0
-50
-100
-150
-200
0
0.5
1
1.5
2
2.5
time (sec)
(a)
60
Simulink
Psim 9.0.3
Proposed model
40
c urrent (A)
20
0
-20
-40
-60
0
0.5
1
1.5
time (sec)
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2.5
Res. J. App. Sci. Eng. Technol., 4(17): 2846-2853, 2012
60
Simulink
Psim 9.0.3
40
c urrent (A )
Proposed model
20
0
-20
-40
-60
0
0.5
1
1.5
2
2.5
time (sec)
60
Simulink
Psim 9.0.3
Proposed model
40
current (A)
20
0
-20
-40
-60
0
0.5
1
1.5
2
2.5
time (sec)
(b)
Fig. 5: (a) motor phase-voltages, (b) motor phase-currents (top, middle and bottom windows for phase-a, -b and -c, respectively)
when the motor is driven at 50 Hz, it is almost unstable
during which the torque is about rated. The motor can
produce a stable rated torque with somewhat lower
driving frequency, i.e. 40 Hz, as indicated by the response
curves from 2 s onward.
CONCLUSION
This paper has presented the derivation of the dq-axis
models of a PMSM with copper- and core-losses under
consideration. The proposed models are useful for
optimized efficiency drive development of a PMSM.
Using MATLAB, PSIM and SIMULINK, simulations of
direct-driven machine have been conducted. Comparisons
of the results have indicated very good agreement among
the three approaches.
8 PM
Tr, Tre
Jem
vsd, vsq
vod, voq
isd,isq
icd, icq
iod,ioq
Pin, Pout
B
vsabc , isabc
vCabc , iCabc
ioabc
7 abc
7qdo
NOMENCLATURE
RS
Rc
Lls
L0, Lms
Lmd , Lmq
L d, L q
Stator winding resistance
Core loss resistance
Stator leakage inductance
Self and mutual inductances
d- and q-axis mutual inductances
d- and q-axis inductances
Permanent magnet flux
Mechanical and electrical rotor speeds
Electromagnetic torque
d- and q-axis stator voltages
d- and q-axis core-loss voltages
d- and q-axis stator currents
d- and q-axis core-loss currents
d- and q-axis torque generating currents
Input power and output power
Viscous friction coefficient
Three-phase stator voltage and current vectors
Three-phase core-loss voltage and current
vectors
Three-phase torque generating current vectors.
Three-phase stator flux leakage vector
dq-axis stator flux leakage vector
ACKNOWLEDGMENT
Financial supports from the following organizations
are greatly acknowledged: Ministry of Science and
Technology (Thailand), Office of Higher Education of
Thailand under NRU project and Suranaree University of
Technology.
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REFERENCES
Cavallaro, C., A.O. Ditommaso, R. Miceti, G.R. Galluzzo
and M. Trapanese, 2005. Efficiency enhancement of
permanent-magnet synchronous motor drives by
online loss minimization approaches. IEEE T. Indus.
Electr., 52: 1153-1160.
Fernandez-bernal, F., A. Garcia-cerrada and R. Faure,
2001. Determination of parameter in interior
permanent magnet synchronous motors with iron
losses without torque measurement. IEEE T. Indus.
Appl., 37: 1265-1272.
Fransua, A. and R. Magureanu, 1984. Electrical Machines
and Drive Systems. Technical Press, Romania.
Krishnan, R., 2010. Permanent Magnet Synchronous and
Brushless DC Motor Drives. CRC Press, US.
Lin, C.K., T.H. Liu and C.H. Lo, 2009. Sensorless
interior permanent magnet synchronous motor drive
system with a wide adjustable speed range. IET
Electric Power Applications, 3(2): 133-146.
Monajemy, R. and R. Krishnan, 2001. Control and
dynamics of constant-power-loss-based operation of
permanent magnet synchronous motor drive system.
IEEE T. Indus. Electr., 48(4): 839-844.
Morimoto, S., Y. Tong, Y. Takeda and T. Hirasa, 1994.
Loss minimization control of permanent magnet
synchronous motor drives. IEEE T. Indus. Electr.,
41(5): 511-517.
Ong, C.M., 1998. Dynamic Simulation of Electric
Machinery. Prentice Hall, US.
Pillay, P. and R. Krishnan, 1989. Modeling, simulation
and analysis of permanent-magnet motor drives, part
II: The brushless DC motor drives. IEEE T. Indus.
Appli., 25(2): 274-279.
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