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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 35, NO.4, NOVEMBER 1988 537 'r,:,
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Modeling of Permanent Magnet Motor Drives '..'
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PRAGASAN PILLAY, MEMBER,
IEEE, ANDR. KRISHNAN, MEMBER,
IEEE
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Abstract-Recent research has Indicated (hat (he permanent magnet I. INTRODUCTION
r,
motor drives, which include the permanent magnet synchronous motor
(PMSM) and the brushless dc motor (BDCM) could become serious DECENT research [1]-[3] has indicated that the permanent f;'
competitors to the induction motor for servo applications. The PMSM .Rmagnet motor drives, which include the permanent .,.
has a sinusoidal back emf and requires sinusoidal stator currents to magnet synchronous motor (PMSM) and the brushless dc ;".:
produce constant torque while the BDCM has a trapezoidal back emf and --
motor (BDCM) could become serious competitors to the :,1
requires rectangular stator currents to produce constant torque. The
PMSM is very similar to the wound rotor synchronous machine except induction motor for servo applications. The PMSM has a i,-
that the PMSM that is used for servo applications tends not to have any sinusoidal back emf and requires sinusoidal stator currents to ,I,'
damper windings and excitation is provided by a permanent magnet produce constant torque while the BDCM has a trapezoidal r~'
instead of a fi~ld winding. Hence the d, q model of the PMSM can be back emf and requires rectangular stator currents to produce
derived from the well-known model of the synchronous machine with the j:
constant torque. Some confusion exists, both in the industry '!'
equations of the damper windings and field current dynamics removed. ~'
Because of tbe nonsinusoidal variation of the mutual inductances between and in the university research environment, as to the correct
the stator and rotor in the BDCM, it is also shown In this paper that no models that should be used in each case. The PMSM is very i
~.
particular advantage exists in transforming the abc equations of the similar to the standard wound rotor synchronous machine ",..
BCDM to the d, q frame. Hence the solution of the original abc equations except that the PMSM has no damper windings and excitation ,<
is proposed for the BDCM.
is provided by a permanent magnet instead of a field winding.
Hence the d, q model of the PMSM can be derived from the
NOMENCLATURE well-known [4] model of the synchronous machine with the
equations of the damper windings and field current dynamics ,.,
B damping constant, Nlradls (in Newtons per removed.
,,'
radian per second) ,, ;
ea, eb, ec a, b, and c phase back emfs (in volts) As is well known, the transformation of the synchronous
Ep peak value of back emf (in volts) machine equations from the abc phase variables to the d, q
ia, ib, ic a, b, and c phase currents, (in amperes) variables forces all si~usoidally varying inductances in the abc
d, q axis stator currents (in amperes) frame to become constant-in the d, q frame. In the BDCM
id, iq'
J motor, since the back emf is nonsinusoidal, the inductances do i':
moment of inertia, kg - m2 'I;
La, Lb, Lc self inductance of a, b, and c phases (in not vary sinusoidally in the abc frame and it does not seem
henrys) advantageous to transform the equations to the d, q frame
since the inductances will not be constant after transformation. -
Lab mutual inductance between phases a and b (in
henrys) Hence it is proposed to use the abc phase variables model for
Ld, Lq d, q axis inductances (in henrys) the BDCM. In addition, this approach in the modeling of the
p derivative operator -
BDCM allows a detailed examination of the machine's torque
P number of pole pairs behavior that would not be possible if any simplifying ~-
R stator resistance (in ohms) assumptions were made. p
~,.
The d, q model of the PMSM has been used to examine the
Te electric torque (in newton meters)
T, transient behavior of a high-performance vector controlled
load torque, (in newton meters)
Va, Vb, Vc a, b, and c phase voltages (in volts)
PMSM servo drive [5]. In addition, the abc phase variable
model has been used to examine the behavior of a BDCM
Vd, Vq d,. q axis voltages (in volts)
Vdc dc bus voltage (in volts) speed servo drive [6]. Application characteristics of both
Aaf mutual flux due to magnet (in webers) machines have been presented in [7]. The purpose of this
d, q axis flux linkages (in webers) paper is to present these two models together and to show that
Ad, Aq
Wr rotor speed (in radians per second) the d, q model is sufficient to study the PMSM in detail while
Ws synchronous speed, (in radians per second) the abc model should be used in order to study the BDCM. It
Or angle between stator phase A and the rotor (in is therefore tutorial in nature and summarizes previously
radians) published work on the PMSM and BDCM.
* The paper is arranged as follows: Section II presents the
superscript indicating reference value
mathematical model of the PMSM. Section III presents the
/.
mathematical model of the BDCM. Section IV uses these
Manuscript received September 4, 1987; revised March 21, 1988.
The authors are with the Electrical Engineering Department, Virginia
models to present some key results to illustrate the use of these
Polytechnic Institute and State University, BIacksburg, V A 24061. models to study both transient and steady state behavior of
IEEE Log Number 8823354. these drive systems. Section IV has the conclusions.
0278-0046/88/1100-0537$01.00 @ 1988 IEEE -'.

2. 538 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS. VOL. 35. NO.4. NOVEMBER t988
II. MATHEMATICAL MODEL OF THE PMSM
The stator of the PMSM and the wound rotor synchronous
motor (SM) are similar.' In addition there is no difference '" ~
e q
between the back emf produced by a permanent magnet and
that produced by an excited coil. Hence the mathematical
i vd
model of a PMSM is similar to that of the wound rotor SM.
The following assumptions are made in the derivation:
1) Saturation is neglected although it can be taken into
account by parameter changes; R L
q
2) The back emf is sinusoidal;
3) Eddy currents and hysteresis losses are negligible. we~d
With these assumptions the stator d, q equations in the rotor ivq
reference frame of the PMSM are [6], [7]
Fig. 1. PMSM equivalent circuit from dynamic equations.
Vd= Rid+ PAd- (J),Aq (1)
where
Vq = Riq + PAq - ":rAd (2)
~
Aq=Lqiq (3) i~".'q
and
Ad=Ldid+Aaj (4)
Aajis the magnet mutual flux linkage. .
r:=9
iq +
v '"~
The electric torque T~=3P[~iq+(Ld-Lq)idiq]12 (5) q e d
For constant flux operation when id equals zero, the electric
Fig. 2. PMSM equivalent circuit from steady state equations.
torque T, = 3Aafiq12 = K,iq where K, is the motor torque
constant. Note that this torque equation for the PMSM
resembles that of the regular dc machine and hence provides inverse of the Park transform defined below
ease of control. .
Hence in state space form Va cos (0) sin (0)
Vb = cos(0-21r/3) sin (0-27r/3)
pid= (vd-Rid+ (J),Lqiq)ILd (6)
[ Vc] . ['cos (0 + 21r/3) sin (0+ 27r/3) m::]
piq = (Vq- Riq+ (J),Ldid~(J),Aaf)1
Lq (7) (11)
p(J),=(T~-B(J)r- ~)IJ (8) where 0 is the rotor position.
pO,=(J),. (9) The total input power to the machine in terms of abc
variables is
Vdand Vqare obtained from Va, Vb, and Vc-hrough the Park
t Power = vaia + vbib + vcic- (12)
transform defined below
while in terms of d, q variables
m =213
Power = 3 (vdid+ vqiq)l2.
The factor 3/2 exists because the Park transform defined above
(13)
COS (0) cos (0-21r/3) cos (0+ 27r/3)
is not power invariant.
sin (0) sin (0-27r/3) sin (0 + 21r/3)
From the dynamic equations of the PMSM, the equivalent
[ 1/2 . 1/2 1/2 ]
circuit in Fig. 1 can be drawn. During steady state operation,
the d, q axis currents are constant quantities. Hence .the
(10) dynamic equivalent circuit can be reduced to the steady state
[::] circuit shown in Fig. 2. The advantage of modeling the
machine using the d, q axis equations then is the subsequent
abc variables are obtained from d, q variables through the ease with which an equivalent circuit can be developed.
-

3. PILLAY AND KRISHNAN: MODEUNG OF PERMANENT MAGNET MOTOR DRIVES 539
m. MATHEMATICALMODEL OF THE BDCM
The BDCM has three stator windings and a permanent
magnet on the rotor. Since both the magnet and the stainless
steel retaining sleeves have high resistivity, rotor induced
currents can be neglected and no damper windings are
modeled. Hence the circuit equations of the three windings in
phase variables are
Va R 0 0 ia
Vb = 0 R 0 ~b
[ Ve ] [0 0 R ] [ Ie ]
La Lba Lea ia ea
+p Lba Lb Leb ~b + eb (14)
[ Lea Leb Le ] [ Ie ] [ ee ]
where it has been assumed that the stator resistances of all the
windings are equal. The back emfs ea, eb' and ee have a
trapezoidal [6]. Assuming further that there is no change in the Fig. 3. BDCM equivalent circuit from dynamic equations.
rotor reluctance with angle, then
transformation can'be made from the phase variables to d, q
La=Lb=Le=L coordinates either in the stationary, rotor, or synchronously
Lab = Lea = Lbc = M rotating reference frames. Inductances that vary sinusoidally
in the abc frame become constants in the d, q frame. Since the
Va R 0 0 ia back emf is nonsinusoidal in the BDCM, it means that the
Vb = 0 R 0 ~b
mutual inductance between the stator and rotor is nonsinu-
[ Ve] [ 0 0 R ] [ Ie] soidal, hence transformation to a d, q reference frame cannot
be easily accomplished. A possibility is to find a Fourier series
L M M ia ea of the back emf in which case the back emf in the d, q
+ M L M P ~b + eb (15) reference frame would also consist of many terms. This is
[M M L ] [ Ie] [ ee] considered too cumbersome hence the abc phase variable
but model developed above will be used without further transfor-
ia+ib+ie=O. (16) mation.
The equation of motion is
Therefore
p(J),=(Te-Tf-B(J),)/J. (21)
Mib+Mie= -Mia (17)
From the dynamic equations of the BDCM the circuit in
Va R 0 O ia Fig. 3 can be drawn. ea, eb, and ee have the trapezoidal shapes
characteristic of the BDCM. Because of this nonsinusoidal
Vb = 0 R 0 ~b
[ Ve] [ 0 0 R ][ Ie] shape in the back emf, further simplifications in the model are
difficult.
L-M 0 0 ia ea IV. REsULTS
+ 0 L-M (18)
L ~M ] p
The models presented previously can be used to examine
[ 0 0 [ ~b ] + [ eb ]
Ie ee
both the transient and steady state behavior of PM' motor drive
Hence in state space form we have that systems. The models of the current controllers and inverter
switches have been presented previously [5J.
lI(L-M) 0
~a - Some typical results that can be generated using the above
p Ib - 0 lI(L-M) models are now given. Fig. 4 shows the results when a PMSM
[ ie] [ 0 ° lI(L~-M) ] is started up from zero to a speed of 1750 rpm. The response is
Va R ° 0 ia ea slightly underdamped in the design used here. The torque is
(19) held constant at the maximum capability of the machine while
the motor runs up to speed. At 0.025 s a load of 1 pu is added.
[[ ~: ] - [ .~ ~ ~ ] [ ~:] - [ :: ]] The electric torque of the machine increases in order to satisfy
and the load torque requirements. The sinusoidal currents required
Tt = (eaia + ebib 20
+ eeie)/ (J),. by the PMSM during the startup are shown including the
( ) voltage profile when a hysteresis band equal to 0.1 of the peak
The currents ia, ib, and ieneeded to produce a steady torque. rated current is used.
without torque pulsations are of 120' duration in each half Similar results are obtainable by using the dynamic model of
cycle [6]. With ac machines that have a sinusoidal back emf, a the BDCM as shown in Fig. 5. In the BDCM, the current

4. 540 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS. VOL. 35. NO.4. NOVEMBER 1988
"
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~ CURRENT ,.
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5P(ED
U
!.'
_~ I..'
~o X REF -8
:J.
..
~
:c8
<> I'>CTlA. 0...0
0
~ "
~. TORaUE
X REF
CI~
1:.
I
BRCK EMF
.
f...
-
'"
0 I'>CTlA.
-51 + L o~ :'
~..;
Y
.
§8
1:. V
, 8 ::;'
0
~ '.
~
:::RENT r.
i'
-8 0 I'>CTlA.
~~ i
~o ~~'0'.25
CI 0.375 0.5
"'"
.;,..
1: TIME! S) .10" :.,
~8
:c.
. Fig. 6. PMSM steady state results.
,..
..
"
~ VDLT~ ".'
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~ CURRENT
:1lIJI
N
.
..'
..
§'0.00 0.025 0.05
-8
:J.
:c TIMEISI a.°
Fig. 4. PMSM transient results. 'z
0 ..
CIO ~ RCTURL i'.
I:U'!
o
N. .'
I
N BRCK EMF ..
SPEED
_:
".
'.
"
~o X REF -8
:J.
§ <> I'>CTlA. a.°
1:8 ,.
ci
~
CIO
,;
51 TORaUE
'U I: I'!
--, '..
"T x REF .,.
'0.14 0.32 0.50
-8 0 I'>CTlA. TIMEt S) -10"
~..; '
g
([51
+ L Fig. 7. BDCM steady stale results.
:c.
9 i
'.,
C.RRENT controllers are used to force the actual current to track the }'-'
~ i. .'
~
rectangular shaped current references. A difference between
-8 the two drives can be seen in the voltage profile. Each phase .,
~o x REF conducts for 120. in the BDCM and remains nonconducting
for 60. as shown in Fig. 5. In the PMSM on the other hand, ).
i~ <> I'>CTlA. each phase conducts continuously as shown in Fig. 4. .'
Steady state results of the current and back emf for example
0
VDLT~ can also be studied using these models as shown in Figs. 6 and
nm
7, respectively. Two possibilities exist. Either the transient
-: phase can be removed in order to facilitate study of the steady ".'
~o state or appropriate initial conditions found so as to run the
§o
:c-: model in steady state only.
'0.00 0.025 0.05
TIME.SI V. CONCLUSIONS
Fig. 5. BDCM transient results. This paper has presented the dynamic models and equivalent
circuits of two PM machines. It has shown that although the
----

5. PILLAY AND KRISHNAN: MODEUNG OF PERMANENT MAGNET MOTOR DRIVES 541
PMSM and the BDCM are similar in construction, their magnet synchronous machine." IEEE Trans. Industry Applications.
modeling takes different forms. The d, q model of the wound vol. 1A-21, no. 2, pp. 408-413, Mar.lApr. 1985.
[3] E. Richter. T.l. E. Miller, T. W. Neumann. and T. L. Hudson. "The
rotor SM is easily adapted to the PMSM while an abc phase ferrite PMAC motor-A technical and economic assessment," IEEE
variable model is necessary for the BDCM if a detailed study Trans. Industry Applications, vol. 1A-21, no. 4, pp. 644-650, Mayl
of its behavior is needed. lune 1985.
[4] P. Krause, Analysis of Electric Machinery. New York: McGraw-
Both the steady state and dynamic behavior can be studied Hill, 1986.
using these models. [5] P. Pillay and R. Krishnan, "Modeling analysis and simulation oCa high
perfonnance, vector controlled, pennanent magnet synchronous motor
REFERENCES drive," presented at the IEEE IAS Annu. Meeting, Atlanta, 1987.
[6] -. "Modeling simulationand analysisof a permanentmagnet
[I] R. Krishnan and A. 1. Beutler, "Perfonnance and design of an axial brushless dc motor drive," presented at the IEEE IAS Annual Meeting,
field pennanent magnet synchronous motor servo drive," in Proc. Atlanta, 1987.
IEEE IAS Annu. Meeting, pp. 634-640, 1985. [7] -, "Application characteristics of pennanent magnet synchronous
[2] M. Lajoie-Mazenc, C. Villanueva, and 1. Hector, "Study and and brushless dc motors for servo drives," presented at the IEEE IAS
implementation of a hysteresis controlled invener on a pennanent Annual Meeting, Atlanta, 1987.