ICEM 2010, XIX International Conference on Electrical Machines, Roma, Italy, 6-8 September 2010.
Complex Dynamic Model of a Multi-phase Asynchronous Motor
Roberto Zanasi, Giovanni Azzone
Abstract— The paper presents a new complex and reduced
dynamical model of a multi-phase asynchronous motor using
the Power-Oriented Graphs modeling technique. The new reduced model is obtained in the state space using a complex and
rectangular “congruent” transformation. The model describes
the electric motor using a reduced number of complex state
space variables. The characteristic polynomial of the new model
has complex coefficients: its roots and the complex conjugate
of these roots are exactly the eigenvalues of the considered
asynchronous motor. The complex steady-state equations of the
new reduced model are given and discussed. Some simulation
results end the paper.
I. INTRODUCTION
The main topic of this paper is the dynamic modeling of a
multi-phase asynchronous motor with an arbitrary number of
phases using complex state space variables. The steady-state
model of a three-phase asynchronous motor is well known
in literature, see for example [1], and its extension to the
multi-phase case is an interesting research topic, see [2] and
[3]. The dynamic equations of a multi-phase asynchronous
motor using the Power-Oriented Graphs (POG) modeling
technique has been given in [4] and [5] using “real” state
space variables. In this paper a new “complex and reduced”
dynamical model of a multi-phase asynchronous motor is
obtained using a “complex and congruent” state space transformation. The electrical part of the motor is modeled using
a “complex” second-order linear system: the eigenvalues of
the motor coincide with the roots of the reduced system
and their complex conjugate values. The paper is organized
as follows: Section II describes the basic properties of the
POG technique in the complex case. Section III shows the
details of the POG modeling and the expression of the system
eigenvalues. Finally, in Section IV some simulations are
reported.
II. P OWER -O RIENTED G RAPHS BASIC PRINCIPLES
The Power-Oriented Graphs technique, see [6] and [7], is
suitable for modeling physical systems. The POG is based
on the same “energetic ideas” of the Bond Graphs technique,
see [8], but it uses a different and specific graphical representation. The POG are normal block diagrams combined with
a particular modular structure essentially based on the use of
the two blocks shown in Fig. 1.a and Fig. 1.b: the elaboration
block (e.b.) stores and/or dissipates energy (i.e. springs,
masses, dampers, capacities, inductances, resistances, etc.);
the connection block (c.b.) redistributes the power within the
R. Zanasi and G. Azzone are with Faculty of Engineering, DII Information Engineering Department, University of Modena e Reggio
Emilia, Via Vignolese 905, 41100 Modena, Italy {roberto.zanasi,
giovanni.azzone}@unimore.it.
a) elaboration block
b) connection block
- x1 - K - x2
x1
x2
?
G(s)
y Fig. 1.
?-y
y1 K ∗ y2
POG basic blocks: a) elaboration block; b) connection block.
u- B-- ?
L-1
?
6
Fig. 2.
L ẋ = −Ax + Bu
y = B∗ x
6
-
(1)
⇓ (x = Tz)
A
1
s
y B∗ ?x
(
Lż = −Az+Bu
∗
y=B z
(2)
POG block scheme of a generic dynamic system.
system without storing or dissipating energy (i.e. any type
of gear reduction, transformers, etc.). The c.b. transforms
the power variables imposing the constraint x∗1 y1 = x∗2 y2 .
The e.b. and the c.b. are suitable for representing both scalar
and vectorial systems. In the vectorial case, G(s) and K
are matrices: G(s) is always a square matrix of positive
real transfer functions; matrix K can also be rectangular,
time varying and function of other state variables. The circle
present in the e.b. is a summation element and the black spot
represents a minus sign that multiplies the entering variable.
The main feature of the Power-Oriented Graphs is to keep
a direct correspondence between the dashed sections of the
graphs and real power sections of the modeled systems: the
real part of the scalar product x∗ y of the two power vectors x
and y involved in each dashed line of a power-oriented graph,
see Fig. 1, has the physical meaning of the power flowing
through that particular section. Another important aspect
of the POG technique is the direct correspondence between
the POG representations and the corresponding state space
descriptions. For example, the POG scheme shown in Fig. 2
can be represented by the state space equations given in (1)
where the energy matrix L is symmetric and positive definite:
L = L∗ > 0. When an eigenvalue of matrix L tends to
zero (or to infinity), system (1) degenerates towards a lower
dimension dynamic system. In this case, the dynamic model
V1
V2
V3
V ms
1
Vrr
Rs
9
Ls
Ir1 ?
Ir2 ?
Ir3
?
Ms12 Ms23 Ms3ms
Mr12 Mr23 Mr3mr
Vs0
Vr0
τm
5
4
3
4
6
Fig. 4.
3
2
7
Lr
Msr
ωm
8
Irmr
?
5
Multi-phase asynchronous motor with ms = 9, mr = 5 and p = 1.
bm
Jm
Fig. 3.
1
Rr
Is1 ?Is2 ?Is3 ?Isms ?
2
τe
Basic physical structure of a multi-phase asynchronous motor.
(2) of the “reduced” system can be directly obtained from
(1) by using a simple “congruent” transformation x = Tz
(matrix T can also be complex and/or rectangular) where
L = T∗ LT, A = T∗ AT and B = T∗ B. If matrix
T is time-varying, an additional term T∗ LṪz appears in
the transformed system. When matrix T is rectangular, the
system is transformed and reduced at the same time. The
POG schemes maintain theirs physical meaning even when
the dynamic system is described using complex variables.
III. C OMPLEX DYNAMIC M ODEL OF THE M OTOR
The basic structure of a multi-phase star-connected asynchronous motor is shown in Fig. 3. The system is characterized by the following parameters:
ms : number of stator phases;
mr : number of rotor phases;
p : number of rotor and stator polar expansions;
2π
γs : stator angular phase displacement (γs = m
);
s
2π
γr : rotor angular phase displacement (γr = mr );
θm : rotor angular position;
ωm : rotor angular velocity;
θs : stator voltage angular position;
ωs : stator voltage frequency;
θ : electric angle (θ = p θm );
Rs : stator phases resistance;
Ls : stator phases self inductance;
Ms0 : maximum mutual inductance of the stator phases;
Rr : rotor phases resistance;
Lr : rotor phases self inductance;
Mr0 : maximum mutual inductance of the rotor phases;
Msr0 : maximum value of the mutual inductance between
stator and rotor phases;
Jm : rotor inertia momentum;
bm : rotor linear friction coefficient;
τm : electromotive torque acting on the rotor;
τe : external load torque acting on the rotor.
All the electrical parameters of the motor have been obtained
connecting in series the p polar couples of the motor. An
example of a multi-phase asynchronous motor with ms = 9,
mr = 5 and p = 1 is shown in Fig. 4. In this paper the
following matrix notation will be used:


R11 R12 · · · R1m
 R21 R22 · · · R2m 
i
j


|[ Ri,j ]| =  ..
..
..  .
.
.
.
.
.
. 
1:n
1:m 
Rn1 Rn2 · · · Rnm
and symbol Im will denote the identity matrix of order m.
Let us denote t Vs , t Is , t Vr and t Ir as stator and rotor
voltage/current vectors in the external reference frame Σt :








Ir1
Vr1
Is1
Vs1
 Ir2 
 Vr2  t
 Is2  t
 Vs2  t
t







Vs =
 ... , Is = ... , Vr = ... , Ir = ... .
Vsms
Isms
Vrmr
Irmr
where Vsi = Vi − Vs0 for i ∈ {1, 2, . . . , ms } and Vri =
Vrr − Vr0 for i ∈ {1, 2, . . . , mr }. Using the following
generalized state vector t q̇ and extended input vector t V:
t

t 
t t
Vs
Is
Ie
Ve
t
t
,
V =  t Vr  =
q̇ =  t Ir  =
ωm
−τe
ωm
−τe
and applying the “Lagrangian” approach discussed in [4],
one obtains the following dynamic equations of the multiphase asynchronous motors:
t
t d t Le 0 t Ie
Re + t Fe t Ke t Ie
Ve
=−
+
t T
0
J
ω
−
K
b
ω
−τ
dt
m
m
m
m
e
e
{z
} | {z } | {z }
| {z } | {z }
|
t
t
t
t
L(t q) t q̇
R + tW
q̇
V
(3)
The structures of energy matrix t L(t q), dissipating matrix
t
R and power redistribution matrix t W are the following:


t
t
t
Ls
MTsr (θm ) 0
Le 0
t
t
t
t


0
Msr (θm )
Lr
L( q) =
,
=
0 Jm
0
0
Jm

 t
t
0
Rs 0
Re 0
t
,
R =  0 t Rr 0 =
0 bm
0
0 bm


t
T
1 ∂ Msr t
0
− 21 t ṀTsr
I
r
2 ∂θm


t
1 t
1 ∂ t Msr t
W=
Ṁ
0
Is 
−
sr


2
2 ∂θm
t
∂ t MT
− 21 t ITr ∂ ∂θMmsr − 21 t ITs ∂θmsr
0
tV
Ē
ωm
0
ω
e @
- - @
K̄e Re(·) 6
@
6
6
6
1
?
Jm
ω V̄
e
ω
- Re(·) -- t T∗ω - - e ?
1
s
?
ω R̄
ω Ω̄
e
e
ω L̄-1
e
tI
e
ω Ī
t
Complex/Real
conversion
Electrical part
(complex variables)
Σt ↔ Σ̄ω
2
1̄
i
where:

j
Ls = Ls0 Ims + Ms0 |[ cos((i − j)γs ) ]| ,
1:ms
1:ms
i
j
ω
1:mr
j
ω
Msr (θ) = Msr0 |[ cos(θ + iγr − jγs ) ]| ,
0:mr −1
0:ms −1
t
Rs = Rs Ims ,
T̃ω (m, θ) =
r
2
m
Rr = Rr Imr
h
k
ej(θ−khγm ) 0:m−1
1:2:m−2
t
where γm = 2π
m and let Tω denote the following matrix:

t
T̃ω (ms , θs )
t
Tω = 
0
0
0
t
T̃ω (mr , θp )
0

0
0
1
where θp = θs − θ. It can be easily shown that all the
columns of matrix t Tω ∈ R(ms +mr +1)×(ms +mr )/2 are
orthogonal complex vectors. This complex matrix can be
used to perform a state space transformation from the original
frame Σt to a new complex rotating frame Σ̄ω .
The dynamic equations in the new complex transformed
frame Σ̄ω are:
ω
ω R̄e + ω F̄e + ω Ω̄e ω K̄e ω Īe
V̄e
L̄e 0 ω Ī˙ e
+
=−
0 Jm ω̇m
ωm
−τe
− ω K̄∗e
bm
{z
} | {z } | {z }
| {z } | {z }
|
ω
ω
L
ω
ω
R+ ω W
ω
q̇
ω
V
(4)
∗
∗
where: ω L = t Tω t L t Tω , ω R = t Tω t R t Tω = t R and
∗
∗
ω
W = t Tω t W t Tω . The complex vectors ω V = t Tω t V
∗
and ω q̇ = t Tω t q̇ have the following structure:
ω

ω 
ω
ω V̄s
Īs
V̄e
Īe
ω
ω
V =  ω V̄r  =
,
q̇ =  ω Īr  =
−τe
ωm
−τe
ωm
q̈


V̄s=
ω
and Ls0 = Ls − Ms0 , Lr0 = Lr − Mr0 . Let t T̃ω (m, θ)
denote the following rectangular “complex” matrix:
t

Īs=
ω
Lr = Lr0 Imr + Mr0 |[ cos((i − j)γr ) ]| ,
i
t
3
Energy
Conversion
3̄
τe
τm
τr
Mechanical part
Complex/Real
conversion
(real variables)
4
5
POG graphical representation of a multi-phase asynchronous motor in the complex transformed rotating frame Σ̄ω .
1:mr
t
1
s
e
where:
t
bm
6
6
6
Re(·) t Tω ?- - - - - - -- ω K̄∗e -- Re(·) - - ?
1
Fig. 5.
6
@ω
@
F̄e
@
6
ω
ω
I¯s1
I¯s3
.
.
.
I¯s(ms−1)
ω
ω
V̄s1
V̄s3
.
.
.

"

=
V̄s(ms−1)

#
ω¯
Is1
ω
Īsx
"

=
ω
ω
,
#
V̄s1
,
ω
V̄sx
ω


Īr=
ω


V̄r=
I¯r1
I¯r3
.
.
.

V̄r1
V̄r3
.
.
.

ω
ω
ω
I¯r(mr−1)
ω
ω
V̄r(mr−1)
"
#
ω¯
Ir1

,
= ω
Īrx
"
#
ω
V̄r1

= 0.
= ω
V̄rx
A POG graphical representation of system (4) is shown in
Fig. 5: the connection blocks present between sections 1 and
2 represent the state space transformation Σt ↔ Σ̄ω . The
c.b. defined by function “Re(·)” represents the “complex to
real conversion” of the input vectors. The elaboration blocks
between sections 2 and 3 represent the Electrical part of
the system. This part is composed only by complex matrices
and complex variables (see the lightly shaded section of
Fig. 5). The Mechanical part of the motor is described by
the blocks present between sections 4 and 5 . The c.b.
between sections 3 and 4 represents the energy and power
conversion (without accumulation nor dissipation) between
the electrical and mechanical domains. The expanded form
of system (4) is shown in Fig. 6 where:
ms
Ms0 ,
2
√
Msr0 ms mr
Msre =
,
2
Lse = Ls0 +
Lre = Lr0 +
mr
Mr0 ,
2
ωp = ωs − ω,
m̄s3 = (ms − 3)/2 and m̄r3 = (mr − 3)/2. It can be easily
proved that in (4) the two terms ω Ke ωm and ω F̄e ω Īe
simplify each other. These two terms have been left in the
POG scheme of Fig. 5 and have been eliminated in eq. (5).
Putting ω q̈ = 0 in system (5), one obtains the following
complex steady-state equations of the asynchronous motor:
 ω
V̄s1 = (Rs + jωs Lse ) ω I¯s1 + jωs Msre ω I¯r1




ω

V̄sx = Rs ω Īsx



0 = jωp Msre ω I¯s1 + (Rr + jωp Lre ) ω I¯r1
(6)



ω

0 = Rr Īrx




∗ ω¯
Ir1 ) = τe + bm ωm
τm = Re(jpMsre ω I¯s1

Msre
0
Lse
0
0
0
 0 Ls0 Im̄s3

0
Lre
0
 Msre
 0
0
0 Lr0 Im̄r3
0
0
0
0
{z
|
ω
L
Fig. 6.


 ω I¯˙s1

jωs Msre
0
Rs +jωs Lse
0
0
ω ˙ 
0
Rs Im̄s3
0
0
0  Īsx 




0
Rr +jωp Lre
0
0  ω I¯˙r1 = − jωp Msre


0
Rr Im̄r3
0
0
0 
 ω Ī˙ rx 
1
∗
∗
Jm
j p Msre ω I¯r1
0
0
− 21 j p Msre ω I¯s1
2
} ω̇m
{z
|
| {z }
ω
ω
R+ W
ω
q̈
Im
ωp =0
1
π+βr
v
− 21
(
1− 21a
ωp =0+
βr
2βr
r
φs1
βr
)
v
φs2
Re
2βr
βs
φs
v
(1− a1 )
φs
ωp →∞
tan βs
Fig. 7.
Phase displacement vector
rφ
s
as function of ωp (ωs = cost).
From the third equation of (6) one obtains the following
complex relation between the currents ω I¯r1 and ω I¯s1 :
ω¯
Ir1
=
Msre r ω ¯
φs Is1
Lre
(7)
where the phase displacement vector r φs is defined as:
r
φs = − cos βr ejβr ,
βs = arctan
Rs
.
ωs Lse
V̄s1 = ωs Lse v φs ω I¯s1
Fig. 8.
(8)
(9)
ωs cos βr j(π+βr )
e−jβs
+j
e
φs = j
cos β
a {z
}
| {z s} |
vφ
s1
(10)
vφ
s2
Lse Lre
.
a=
2
Msre
p Rr ω ¯ 2
| Ir1 | .
ωp
rx
r
rx
which represent two linear systems of m̄s3 and m̄r3 dimension, whose eigenvalues are λs = − LRs0s and λr = − LRr0r ,
respectively. From the second equation of (12), it follows
that ω Īrx = 0 if the initial conditions are zero. Without
equations (12), system (5) reduces to:
"
# "ω ¯˙ #
Lse Msre 0
Is1
˙
ω I¯
Msre Lre 0
(13)
r1 =
0
0 Jm
ω̇m#"
"
#
"
#
ω I¯
ω V̄
Rs +jωs Lse
jωs Msre
0
s1
s1
ω I¯
jωp Msre
Rr +jωp Lre
0
0
=−
+
.
r1
1
∗ − 1 jpM
ω ¯∗
−τe
jpMsre ω I¯r1
ωm
sre Is1 bm
2
2
Supposing the mechanical dynamics slower than the electrical one, i.e. assuming a constant velocity ωm , from (13) one
obtains the following complex second-order reduced system:
#
"
ω
ω
Lse Msre ω I¯˙s1
I¯s1
Rs +jωs Lse jωs Msre
V̄s1
+
.
=
−
Msre Lre ω I¯˙r1
jωp Msre Rr +jωp Lre ω I¯r1
0
{z
} | {z } |
{z
} | {z } | {z }
|
ω L̄
(11)
Vector v φs links the stator current vector ω I¯s1 to the stator
voltage vector ω V̄s1 . A graphical representation of vector
v
φs as function of ωp is shown in Fig. 8. From (6), (7) and
(8) one obtains the following expression for the mechanical
torque τm :
∗ ω¯
τm = Re(j p Msre ω I¯s1
Ir1 ) =
as function of ωp (ωs = cost).
and parameters βr and a are defined as follows:
Rr
βr = arctan
,
ωp Lre
vφ
s
Re
The same result involving only real variables is given in [4].
From second and fourth equations of system (5) one obtains
the following uncoupled dynamics:
Ls0 ω Ī˙ sx = −Rs ω Īsx + ω V̄sx ,
(12)
L ω Ī˙ = −R ω Ī
where the phase displacement vector v φs is:
v
Phase displacement vector
r0
A graphical representation of vector r φs as function of ωp
is shown in Fig. 7. Substituting (7) in the first equation of
(6) one obtains the following relation between the complex
vectors ω V̄s1 and ω I¯s1 :
ω
(5)
The complex dynamic equations of a multi-phase asynchronous motor in the transformed rotating frame Σ̄ω
Im
−1
ωp →∞

 ω ¯   ω
0
Is1
V̄s1
0  ω Īsx   ω V̄sx 


 
0 
 ω I¯r1 + 0 
0  ω Īrx   0 
bm
−τe
ωm
}| {z } | {z }
ω
ω
V
q̇
c
˙
ω Ī
c
ω Ā
c
ω Ī
c
ω V̄
c
The eigenvalues of the reduced system ω L̄c ω Ī˙ c =
ω
Āc ω Īc + ω V̄c are the eigenvalues of matrix A =
ω −1 ω
L̄c
Āc . For the linear POG systems described in the
form Lẋ = Ax + Bu, with L = L∗ > 0, the following
property holds.
Property 1: if matrix L is non singular, the eigenvalues
of matrix L−1 A, i.e. the roots of polynomial ∆L-1 A (s) =
det(sI − L-1 A), coincide with the roots of polynomial
∆L,A (s) = det(sL − A).
Phase vector r φs
Eigenvalues λi
60
0.2
0.1
40
0
20
wp =0
wp →∞
Im
−0.1
βr
Im
−0.2
ωs = 0
0
−0.3
ωs = 10
−0.4
−20
ωs = 20
−0.5
ωs = 30
−40
−0.6
ωs = 40
ωs = 50
−60
−60
−1
−0.8
−0.6
−0.4
−0.2
0
Re
−50
−40
−30
−20
−10
0
Fig. 11.
Re
rφ
Phase displacement vector
Fig. 9. Eigenvalues s1,2 and s3,4 = s∗1,2 when rs = rr = r = 10,
a = 1.25, p = 1, ωs ∈ [0 : 10 : 50] rad/s and ω ∈ [0, 50] rad/s.
s
as function of ωp (ωs =cost).
Phase vector v φs
10
wp =0
Eigenvalues λi
9
60
βr
8
40
7
6
Im
Im
20
0
5
4
3
−20
2
wp →∞
βs
−60
−50
−40
−30
−20
rs = 0
rs = 2
rs = 4
rs = 6
−60
1
rs = 8
rs = 10
−40
−10
0
0
0
Re
Proof. The following mathematical relations hold:
∆L,A (s) = det(sL − A) = det[L(sI − L-1 A)]
= det(L) det(sI − L-1 A) = det(L) ∆L-1 A (s)
If matrix L is non singular, the polynomials ∆L,A (s) and
∆L-1 A (s) have the same roots. The property is proved.
From Prop. 1 it follows that the eigenvalues of matrix A =
ω −1 ω
L̄c
Āc coincide with the roots of polynomial ∆(s):
∆(s) = ∆ω L̄c ,ω Āc (s) = det(s ω L̄c − ω Āc )
sLse + Rs + jωs Lse s Msre + jωs Msre
= det
sMsre + jωp Msre sLre + Rr + jωp Lre
= (sLse + Rs + jωs Lse )(sLre + Rr + jωp Lre )
−(sMsre + jωp Msre )(sMsre + jωs Msre )
Using the following parameters a, rs and rr :
Lse Lre
> 1,
2
Msre
rs =
2
3
4
5
6
7
Re
Fig. 10. Eigenvalues s1,2 and s3,4 = s∗1,2 when a = 1.25, p = 1,
rs ∈ [0 : 2 : 10], rr = rs + 1, ω ∈ [0, 50] rad/s and ωs = ω + r rad/s.
a=
1
Rs
,
Lse
rr =
Rr
,
Lre
Fig. 12.
Phase displacement vector
vφ
s
as function of ωp (ωs =cost).
the polynomial ∆(s) can be expressed as follows:
h
r +rs )
2
(a−1) s2 + a(r
∆(s) = Msre
+
j(ω
+
ω
)
s+
p
s
(a−1)
i
a(ω rr +ωp rs )
ars rr
+ (a−1)
+ j s(a−1)
− ωs ωp
Note that the coefficients of polynomial ∆(s) are complex
and therefore not necessarily its roots are complex coniugate.
The roots s1,2 of the complex polynomial ∆(s), that is the
eigenvalues of matrix ω L̄-1c ω Āc , are:
s1,2 = −
ω
a(rs + rr )
− j(ωs − )
2(a − 1)
2
s
2
ars rr
a(rs − rr )
ω
±
+
+j
.
2(a − 1)
2
(a − 1)2
(14)
The complex conjugate values s3,4 = s∗1,2 of roots s1,2
are the other two eigenvalues of the original fourth-order
electrical system. When rs = rr = r, from (14) one obtains
the following eigenvalues:
s
ω2
ω
a r2
ar
±
−
−j(ωs − ).
s1,2 = −
2
(a − 1)
(a − 1)
4
2
Current Re(ω I¯s1 )
Currents t Is
40
10
0
−10
−20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Currents t Ir
Re(ω I¯s1 ) [A]
t
Is [A]
20
Tω = 0.236s
30
ω
10
Ta = 0.727s
t
Ir [A]
20
0
10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time [s]
0
−10
Fig. 15.
−20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Stator current Re(ω I¯s1 ) in the transformed rotating frame Σ̄ω .
1
Time [s]
Angular velocity ωm
Currents ω Īs : Re(ω I¯s1 ), Im(ω I¯s1 ) and |ω Isx |.
20
Re(ω I¯s1 )
5
0
0
0
Im(ω I¯s1 )
0.1
0.2
0.3
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Currents ω Īr : Re(ω I¯r1 ), Im(ω I¯r1 ) and |ω Irx |.
0
0.9
1
Im(ω I¯r1 )
−20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.6
0.7
0.8
0.9
1
0.7
0.8
0.9
1
100
50
0
0.1
0.2
0.3
0.4
0.5
0.6
Time [s]
Re(ω I¯r1 )
0
0.5
0
|ω Irx |
−10
τm [Nm]
0
0.4
Mechanical torque τm
150
−20
Īr [A]
10
|ω Isx |
ω
Īs [A]
15
Stator and rotor currents in the original reference frame Σt .
ωm [rad/s]
Fig. 13.
ω
Iss
20
Fig. 16.
Angular velocity ωm and mechanical torque τm .
1
Time [s]
Fig. 14.
Stator and rotor currents in the transformed rotating frame Σ̄ω .
When r → 0, all the eigenvalues tend to the imaginary axis:
ω ω
ω ω
s1,2 = −j ωs − ∓
,
s3,4 = j ωs − ∓
.
2
2
2
2
Two examples of how the eigenvalues s1,2 and s3,4 = s∗1,2
move on the complex plane as function of parameters rs , rr ,
ωs and ω are shown in Fig. 9 and Fig. 10.
IV. S IMULATION RESULTS
The simulation results presented in this section have been
obtained in Matlab/Simulink using the following electrical
and mechanical parameters: ms = 7, mr = 7, p = 1, Ls =
0.12 H, Ms0 = 0.1 H, Rs = 3 Ω, Lr = 0.12 H, Mr0 =
0.1 H, Rs = 3 Ω, Msr0 = 0.09 H, Jm = 3 kg m2 , bm =
5 Nm s/rad, τe = 0 Nm, Vmax = 100 V (maximum value of
the stator voltage vector ω Vs ) and frequency ωs = 8π rad/s.
The phase displacement vectors r φs and v φs as function
of ωp are reported in Fig. 11 and Fig. 12: note how they
tend to the theoretical steady-state paths given in Fig. 7 and
Fig. 8. The time behaviors of stator and rotor currents t Is ,
t
Ir , ω Īs and ω Īr in frames Σt and Σ̄ω are shown in Fig. 13
and Fig. 14, respectively.
The time behavior of current Re(ω I¯s1 ) is shown in Fig. 15:
this transient is deeply related to the electrical dominant poles
of the system located, when ωm = 2 rad/s, in s2,4 = −4.4 ±
j 24.1. The corresponding settling time Ta and oscillation
period Tω are respectively:
Ta =
3
≈ 0.7 s,
|Re(s2,4 )|
Tω =
2π
≈ 0.25 s.
Im(s2,4 )
which are very close to the Ta and Tω periods obtained in
simulation and shown in Fig. 15.
V. C ONCLUSIONS
In this paper, the POG technique has been used for
modeling a multi-phase asynchronous motor. A complex and
reduced dynamical model of the system has been obtained in
the state space using a rectangular “complex” transformation.
The dynamic behavior of the reduced complex system has
been analyzed by exactly computing the eigenvalues of the
electrical part of the motor. The simulation results have
shown the effectiveness of the proposed dynamic model.
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