CHAPTER 1.
INDUCTION MOTOR MODEL. GENERALITIES.
1.1 - Equations of the induction motor model.
1.1.1 – Introduction.
A dynamic model of the machine subjected to control must be known in order to understand
and design vector controlled drives. Due to the fact that every good control has to face any
possible change of the plant, it could be said that the dynamic model of the machine could be
just a good approximation of the real plant. Nevertheless, the model should incorporate all the
important dynamic effects occurring during both steady-state and transient operations.
Furthermore, it should be valid for any changes in the inverter’s supply such as voltages or
currents [ROM 1].
Such a model can be obtained by means of either the space vector phasor theory or two-axis
theory of electrical machines. Despite the compactness and the simplicity of the space phasor
theory, both methods are actually close and both methods will be explained.
1.1
Induction motor model. Generalities.
For simplicity, the induction motor considered will have the following assumptions:
§
Symmetrical two-pole, three phases windings.
§
The slotting effects are neglected.
§
The permeability of the iron parts is infinite.
§
The flux density is radial in the air gap.
§
Iron losses are neglected.
§
The stator and the rotor windings are simplified as a single, multi-turn full pitch coil
situated on the two sides of the air gap.
sA'
sB
sC
ra'
rb
rc
rc'
rb'
ra
sC'
sB'
sA
Figure 1.1. Cross-section of an elementary symmetrical three-phase machine.
1.1.2 – Voltage equations.
The stator voltages will be formulated in this section from the motor natural frame, which is
the stationary reference frame fixed to the stator. In a similar way, the rotor voltages will be
formulated to the rotating frame fixed to the rotor.
In the stationary reference frame, the equations can be expressed as follows:
u sA (t) = R s i sA (t) +
dψ sA (t)
dt
(1.1)
u sB (t) = R s i sB (t) +
d ψ sB (t)
dt
(1.2)
u sC (t) = R s i sC (t) +
dψ sC (t)
dt
(1.3)
1.2
Induction motor model. Generalities.
Similar expressions can be obtained for the rotor:
u ra (t) = R r i ra (t) +
dψ ra (t)
dt
(1.4)
u rb (t) = R r i rb (t) +
d ψ rb (t)
dt
(1.5)
u rc (t) = R r i rc (t) +
dψ rc (t)
dt
(1.6)
The instantaneous stator flux linkage values per phase can be expressed as:
ψ sA = Ls i sA + M s isB + M s isC + M srcosθm i ra + M srcos(θm + 2 π 3 )irb + M srcos(θm + 4 π 3 )i rc
(1.7)
ψ sB = M s i sA + L s i sB + M s i sC + M sr cos(θ m + 4 π 3 )i ra + M sr cosθ m i rb + M sr cos(θ m + 2 π 3 )i rc
(1.8)
ψ sC = M s i sA + M s i sB + L s i sC + M sr cos(θ m + 2 π 3 )i ra + M sr cos(θ m + 4 π 3 )i rb + M sr cos θ m i rc
(1.9)
In a similar way, the rotor flux linkages can be expressed as follows:
ψ ra = M srcos(- θ m )i sA + M srcos(- θ m + 2 π 3 )isB + M srcos(- θ m + 4 π 3 )isC + L ri ra + M r irb + M r irc
(1.10)
ψ rb = M sr cos(- θ m + 4 π 3 )i sA + M sr cos(- θ m )i sB + M sr cos(- θ m + 2 π 3 )i sC + M r i ra + L r i rb + M r i rc
(1.11)
ψ rc = M sr cos(- θ m + 2 π 3 )i sA + M sr cos(− θ m + 4 π 3 )i sB + M sr cos(- θ m )i sC + M r i ra + L r i rb + M r i rc
(1.12)
Taking into account all the previous equations, and using the matrix notation in order to
compact all the expressions, the following expression is obtained:
u sA   R s + pL s
u  
pM s
 sB  

 u sC 
pM s

=
 u ra   p M sr cosθ m
 u rb   p M sr cosθ
m2

 
 u rc   p M sr cosθ m1
pM s
pM s
p M sr cosθ m
pM sr cosθ m1
R s + p Ls
pM s
pM s
R s + pL s
p M sr cosθ m2
p M sr cosθ m1
p M sr cosθ m
p M sr cosθ m2
pM sr cosθ m1
pM sr cosθ m
p M sr cosθ m2
pM sr cosθ m2
p M sr cosθ m1
p M sr cosθ m
R r + pL r
pM r
pM r
pM r
R r + pL r
pM r
pM sr cosθ m2  i sA 
  
p M sr cosθ m1  i sB 
pM sr cosθ m  i sC 
⋅  
pM r
  i ra 
 i 
pM r
  rb 
R r + p L r   i rc 
(1.13)
1.3
Induction motor model. Generalities.
1.1.3 – Applying Park’s transform.
In order to reduce the expressions of the induction motor equation voltages given in equation
1.1 to equation 1.6 and obtain constant coefficients in the differential equations, the Park’s
transform will be applied. Physically, it can be understood as transforming the three windings
of the induction motor to just two windings, as it is shown in figure 1.2 [VAS 1].
sQ
sB
rβ
ra
Wm
rα
Wm
rb
sA
sD
rc
sC
Figure 1.2 Schema of the equivalence physics transformation.
In the symmetrical three-phase machine, the direct- and the quadrature-axis stator magnitudes
are fictitious. The equivalencies for these direct (D) and quadrature (Q) magnitudes with the
magnitudes per phase are as follows:
1
1
u s 0 
 1 2
 u sA 
2
2
 

 

2π
2π
u sD  = c ⋅  cos θ cos (θ − 3 ) cos(θ + 3 )  ⋅  u sB 
u sQ 
− sin θ − sin (θ − 2 π 3 ) − sin (θ + 2 π 3 )  u sC 
 
(1.14)
1
u sA 
u  = c ⋅  1

 sB 
1
u sC 

(1.15)
2
2
2
cos θ
− sin θ   u s 0 
 

cos (θ − 2 π 3 ) − sin (θ − 2 π 3 ) ⋅ u sD 
cos (θ + 2 π 3 ) − sin (θ + 2 π 3 ) u sQ 
Where "c" is a constant that can take either the values 2/3 or 1 for the so-called non-power
invariant form or the value
2
3
for the power-invariant form as it is explained in section
1.3.3. These previous equations can be applied as well for any other magnitudes such as
currents and fluxes.
Notice how the expression 1.13 can be simplified into a much smaller expression in 1.16 by
means of applying the mentioned Park's transform.
1.4
Induction motor model. Generalities.
u sD  
R s + pL s
- L s pθ s
pL m
- L m ( P ⋅ w m + p θ r )  isD 
  
u  
L s pθ s
R s + pL s
L m ( P ⋅ w m + pθ r )
pL m
 ⋅  isQ 
 sQ  = 
 u rα  
  i rα 
pL m
- L m ( pθ s - P ⋅ w m )
R r + pL r
- L r pθ r
  

 
u
pL m
L r pθ r
R r + pL r
 rβ   L m ( p θ s - P ⋅ w m )
  i rβ 
Where Ls = L s − M s , Lr = Lr − M r and Lm =
3
2
(1.16)
M sr .
1.1.4 – Voltage matrix equations.
If the matrix expression 1.16 is simplified, new matrixes are obtained as shown in equations
1.17, 1.18 and 1.19 [VAS 1].
1.1.4.1 – Fixed to the stator.
It means that ws = 0 and consequently wr = -wm.
u sD   R s + pL s
u  
0
 sQ  = 
 u rd   pL m

 
 u rq  - P ⋅ w m L m
0
pLm
R s + pLs
0
P ⋅ w m Lm
R r + pL r
pL m
- P ⋅ w m Lr
 i sD 
 
pL m  i sQ 
⋅
P ⋅ w m L r   i rd 
  
R r + pL r   i rq 
0
(1.17)
1.1.4.2 – Fixed to the rotor.
It means that wr = 0 and consequently ws = wm.
u sD   R s + pLs
u  
 sQ  =  L s P w m
 u rd   pL m

 
0
 u rq  
- Ls P wm
pL m
R s + pL s
L m P wm
0
R r + pL r
pL m
0
− L m P w m   isD 
 i 
pLm
 ⋅  sQ 
  i rd 
0
  
R r + pL r   i rq 
(1.18)
1.1.4.3 – Fixed to the synchronism.
It means that wr = sws.
u sD   R s + pLs
u  
 sQ  =  L s w s
 u rd   pL m

 
 u rq   L m sw s
- Ls w s
pL m
R s + pLs
Lm w s
− L m sw s
R r + pL r
pL m
L r sw s
− L m w s  i sD 
 
pL m  i sQ 
⋅
− L r sw s   i rd 
  
R r + pL r   i rq 
(1.19)
1.5
Induction motor model. Generalities.
1.2 – Space phasor notation.
1.2.1 – Introduction.
Space phasor notation allows the transformation of the natural instantaneous values of a threephase system onto a complex plane located in the cross section of the motor. In this plane, the
space phasor rotate with an angular speed equal to the angular frequency of the three phase
supply system. A space phasor rotating with the same angular speed, for example, can
describe the rotating magnetic field. Moreover, in the special case of the steady state, where
the supply voltage is sinusoidal and symmetric, the space phasor become equal to three-phase
voltage phasors, allowing the analysis in terms of complex algebra. It is shown in figure 1.3
the equivalent schematic for this new model.
Ws
s
sB
ra
Wm
r
Wr
rb
sA
rc
sC
Figure 1.3. On the right the equivalent two rotating windings induction motor.
In order to transform the induction motor model, in natural co-ordinates, into its equivalent
space phasor form, the 120º operator is introduced:
a = ej
2π
3
, a2 = ej
4π
(1.20)
3
Thus, the current stator space phasor can be expressed as follows:
[
]
i s = c ⋅ 1 ⋅ i sA (t ) + a ⋅ i sB (t ) + a 2 ⋅ i sC (t )
The factor "c", takes usually one of two different values either
(1.21)
2
3
or
2
3
. The factor
2
3
makes the amplitude of any space phasor, which represents a three phase balanced system,
equal to the amplitudes of one phase of the three-phase system. The factor
2
3
may also be
used to define the power invariance of a three-phase system with its equivalent two-phase
system (see section 1.3.3).
1.6
Induction motor model. Generalities.
1.2.2 – Current space phasors.
During this section the induction machine assumptions introduced in the section 1.1.1 will be
further considered.
It is represented in figure 1.4 the model of the induction machine with two different frames,
the D-Q axis which represent the stationary frame fixed to the stator, and the α-β axis which
represent rotating frame fixed to the rotor.
sQ
rβ
sA'
α
sB
sC
ra'
θ
wm
rα
rb
θm
rc
rc'
rb'
ra
sD
sC'
sB'
sA
Figure 1.4. Cross-section of an elementary symmetrical three-phase machine, with
two different frames, the D-Q axis which represent the stationary frame fixed to the
stator, and α-β axis which represent rotating frame fixed to the rotor.
The stator current space phasor can be expressed as follows:
is =
2
3
[i
sA
(t ) + aisB (t ) + a 2i sC (t )] = i s e jθ
(1.22)
Expressed in the reference frame fixed to the stator, the real-axis of this reference frame is
denoted by sD and its imaginary-axis by sQ.
The equivalence between the stator phasor and the D-Q two-axis components is as follows:
i s = isD ( t ) + j ⋅ i sQ ( t )
(1.23)
or:
[ (i
Im (i ) = Im[ (i
Re( is ) = Re
s
]
)] = i
2
3
sA
+ aisB + a 2 isC ) = i sD
2
3
sA
+ aisB + a 2 isC
(1.24)
sQ
1.7
Induction motor model. Generalities.
The relationship between the space phasor current and the real stator phase currents can be
expressed as follows:
[ (i + ai + a i )] = i
Re( a i ) = Re[ (a i + i + ai )] = i
Re( ai ) = Re[ ( ai + a i + i )] = i
Re( is ) = Re
2
2
3
sA
2
2
3
s
sC
sA
2
sA
sB
sC
sB
2
2
3
s
sB
sA
sB
sC
(1.25)
sC
In a similar way, the space phasor of the rotor current can be written as follows:
ir =
2
3
[i
ra
(t) + airb (t ) + a 2 irc (t )] = ir e jα
(1.26)
Expressed in the reference frame fixed to the rotor, the real-axis of this reference frame is
denoted by rα and its imaginary-axis by rβ.
The space phasor of the rotor current expressed in the stationary reference frame fixed to the
stator can be expressed as follows:
'
j α +θ
i r = ir e jθ = ir e ( m )
(1.27)
The equivalence between the current rotor space phasor and the α-β two-axis is as follows:
ir = ir α ( t ) + j ⋅ irβ ( t )
(1.28)
or:
[ (i
Im (i ) = Im[ (i
Re( ir ) = Re
r
]
)] = i
2
3
ra
+ airb + a 2 i rc ) = irα
2
3
ra
+ airb + a 2 i rc
(1.29)
rβ
The relationship between the space phasor current and the real stator currents can be
expressed as follows:
[ (i + ai + a i )] = i
Re( a i ) = Re[ ( a i + i + ai )] = i
Re( ai ) = Re[ (ai + a i + i )] = i
Re( ir ) = Re
2
2
3
ra
2
2
3
r
r
2
3
rb
rc
ra
2
ra
rb
rc
2
ra
rb
rc
rb
(1.30)
rc
The magnetising current space-phasor expressed in the stationary reference frame fixed to the
stator can be obtained as follows:
i m = i s + ( N re N se )ir '
1.8
(1.31)
Induction motor model. Generalities.
1.2.3 – Flux linkage space phasor.
In this section the flux linkages will be formulated in the stator phasor notation according to
different reference frames.
1.2.3.1- Stator flux-linkage space phasor in the stationary reference frame fixed to the stator.
Similarly to the definitions of the stator current and rotor current space phasors, it is possible
to define a space phasor for the flux linkage as follows:
ψs =
2
3
(ψ
sA
+ aψ sB + a 2 ψ sC )
(1.32)
If the flux linkage equations 1.7, 1.8, 1.9 are substituted in equation 1.32, the space phasor for
the stator flux linkage can be expressed as follows:
(
ψs =
2
3
)
(
)
(
)
i L + a M s + a 2 M s + i M s + a L + a 2 M s + i M s + a M s + a 2 L + 
sB
s
sC
s
 sA s

+ i M cos θ + a M cos θ + 4 π + a 2 M cos θ + 2 π +

sr
sr
( m 3)
sr
( m 3)
m
 ra



2
2
π
4
π
+ irb M sr cos(θ m + 3 ) + a M sr cos θ m + a M sr cos( θ m + 3 ) +



2
+ irc M sr cos(θ m + 4 π 3 ) + a M sr cos( θ m + 2 π 3 ) + a M sr cos θ m

(
(
(
)
)
)
(1.33)
Developing the previous expression 1.33, it is obtained the following expression:
(
)
(
)
(
)
i L + a M s + a 2 M s + a ⋅ i a 2 M s + L + a M s + a 2 i a M s + a 2 M s + L +
sB
s
sC
s
 sA s

 + i M cos θ + a M cos θ + 4 π + a 2 M cos θ + 2 π +

sr
sr
( m 3)
sr
( m 3)
ra
m

2
ψs = 3 

2
2
π
4
π
 + a ⋅ irb a M sr cos (θ m + 3 ) + M sr cos θ m + a M sr cos (θ m + 3 ) +



2
2
 + a ⋅ irc a M sr cos( θ m + 4 π 3 ) + a M sr cos (θ m + 2 π 3 ) + M sr cos θ m

(
(
(
)
)
)
(1.34)
And finally, expression 1.34 can be represented as follows:
(
= (L
) (
)
ψ s = L s + a M s + a 2 M s i s + M sr cos θ m + a M sr cos(θ m + 4 π 3 ) + a 2 M sr cos(θ m + 2 π 3 ) i r =
s
)
(
)
(
)
− M s i s + 1.5 cos θ m M sr i r = L s − M s i s + 15
. M sr i r e j θm = L s − M s i s + 15
. M sr i
= Ls i s + L m i
'
r
'
r
=
(1.35)
Where Ls is the total three-phase stator inductance and Lm is the so-called three-phase
magnetising inductance. Finally, the space phasor of the flux linkage in the stator depends on
two components, being the stator currents and the rotor currents.
Once more, the flux linkage magnitude can be expressed in two-axis as follows:
ψ s = ψ sD + jψ sQ
(1.36)
1.9
Induction motor model. Generalities.
Where its direct component is equal to:
ψ sD = Ls isD + Lm ird
(1.37)
And its quadrature component is expressed as:
ψ sQ = Ls i sQ + Lm i rq
(1.38)
The relationship between the components ird and irα and irq and irβ may be introduced as
follows:
(1.39)
'
i r = ird + jirq = i r e j θm
The compactness of the notation in the space phasor nomenclature compared to the two-axis
notation in 1.1. is noticeable.
1.2.3.2- Rotor flux-linkage space phasor in the rotating reference frame fixed to the rotor.
The rotor flux linkage space phasor, fixed to the rotor natural frame can be defined as follows:
ψr =
2
3
(ψ
ra
+ aψ rb + a 2 ψ rc )
(1.40)
If the flux linkage equations 1.10, 1.11, 1.12 are substituted in equation 1.40, the space phasor
for the rotor flux linkage can be expressed as follows:
(
)
(
)
(
)
i L + a M r + a 2 M r + i M r + a L + a 2 M r + i M r + a M r + a 2 L + 
sB
r
sc
r
 ra r

+ i M cos θ + a M cos θ + 2 π + a 2 M cos θ + 4 π +

sr
sr
( m 3)
sr
( m 3)
sA
m

2
ψr = 3 

2
4
π
2
π
+ isB M sr cos( θ m + 3 ) + a M sr cos θ m + a M sr cos(θ m + 3 ) +



2
+ isC M sr cos( θ m + 2 π 3 ) + a M sr cos(θ m + 4 π 3 ) + a M sr cos θ m

(
(
(
)
)
)
(1.41)
By re-arranging the previous expression 1.41, it can be expressed as:
(
)
(
)
(
)
i L + a M r + a 2 M r + a ⋅ i a 2 M r + L + a M r + a 2 i a M r + a 2 M r + L + 
rb
r
rc
r
 ra r

2
+ i M cos θ + a M cos θ + 2 π + a M cos θ + 4 π +

sr
sr
( m 3)
sr
( m 3)
m
 sa

ψ r = 23 

2
4
π
2
π
+ a ⋅ isb a M sr cos( θ m + 3 ) + M sr cos θ m + a M sr cos(θ m + 3 ) +

 2

2
+ a ⋅ i sc a M sr cos( θ m + 2 π 3 ) + a M sr cos( θ m + 4 π 3 ) + M sr cos θ m

(
(
)
(
)
)
(1.42)
And finally:
(
= (L
) (
)
ψ r = Lr + a M r + a 2 M r i r + M sr cos θ m + a M sr cos(θ m + 2 π 3 ) + a 2 M sr cos(θ m + 4 π 3 ) i s =
r
)
= Lr i r + Lm i
1.10
(
)
(
)
− M r i r + 15
. cos( − θ m ) M sr i s = Lr − M r ir + 1.5 M sr i s e− jθ m = Lr − M r i r + 1.5 M sr i s =
'
s
'
(1.43)
Induction motor model. Generalities.
Where Lr is the total three-phase rotor inductance and Lm is the so-called three-phase
'
magnetising inductance. i s is the stator current space phasor expressed in the frame fixed to
the rotor.
Once more the flux linkage magnitude can be expressed in the two-axis form as follows:
ψ r = ψ rα + jψ rβ
(1.44)
Where its direct component is equal to:
ψ rα = Lr ir α + L m i sα
(1.45)
And its quadrature component is expressed as:
ψ rβ = Lr ir β + Lm i sβ
(1.46)
1.2.3.3- Rotor flux-linkage space phasor in the stationary reference frame fixed to the stator.
The rotor flux linkage can also be expressed in the stationary reference frame using the
previously introduced transformation ejθm, and can be written as:
ψ
'
r
(
)
= ψ rd + jψ rq = ψ r e j θm = ψ r α + jψ rβ e jθ m
(1.47)
The space phasor of the rotor flux linkage can be expressed according to the fixed coordinates as follows:
'
'
'
'
ψ r = Lr i r + Lm i s e j θm = Lr i r + Lm i s
(1.48)
The relationship between the stator current referred to the stationary frame fixed to the stator
and the rotational frame fixed to the rotor is as follows:
'
i s = i s e j θm
'
i s e − jθ m = i s
(1.49)
Where
i s = isD + ji sQ
i s = is α + jis β
'
(1.50)
From figure 1.5, the following equivalencies can be deduced:
i s = is e jθ
i s = is e jα = is e j(θ − θ m ) = i s e− j θm
'
'
'
(1.51)
1.11
Induction motor model. Generalities.
sQ
rβ
is,is'
wm
rα
α
θ
θm
sD
Figure 1.5. Stator-current space phasor expressed in accordance with the rotational
frame fixed to the rotor and the stationary frame fixed to the stator.
1.2.3.4- Stator flux-linkage space phasor in the rotating reference frame fixed to the rotor.
Similarly than 1.2.3.3 section, it can be deduced the following expression:
'
− jθ m
ψ s = ψ se
'
'
− jθ
=  L s i s + L m i r  e m = Ls i s + L m i r
(1.52)
1.2.4. – The space phasors of stator and rotor voltages.
The space phasors for the stator and rotor voltages can be defined in a similar way like the
one used for other magnitudes.
us =
2
3
ur =
2
3
[u
[u
sA
ra
( t ) + ausB ( t ) + a 2usC ( t ) ] = usD + jusQ =
2
3
(u
sA
− 12 usB − 12 usC ) + j
( t ) + aurb( t ) + a 2 urc ( t ) ] = urα + jurβ = 23 (ura − 12 urb − 12 urc ) + j
1
3
1
3
(u
(u
sB
rb
− usC )
− urc )
(1.53)
Where the stator voltage space phasor is referred to the stator stationary frame and the rotor
voltage space phasor is referred to the rotating frame fixed to the rotor.
Provided the zero component is zero [VAS 1], it can also be said that:
usA = Re( u s )
usB = Re( a 2 u s )
usC = Re( aus )
Equivalent expressions can also be obtained for the rotor.
1.12
(1.54)
Induction motor model. Generalities.
1.2.5 - Space-phasor form of the motor equations.
The space phasor forms of the voltage equations of the three-phase and quadrature-phase
smooth air-gap machines will be presented. Firstly, the equations will be expressed in a
general rotating reference frame, which rotates at a general speed wg, and then to the
references frames fixed to the stator, rotor and synchronous speed.
1.2.5.1 - Space-phasor voltage equations in the general reference frame.
If the vector in the figure 1.6 is the stator current, then its formulation in the space phasor
form is as follows:
i sg = i s e
− j θg
= i sx + ji sy
(1.55)
sQ
rβ
y
wg
x
θg
αs
αr
wm
θm
rα
sD
Figure 1.6. It is shown a magnitude represented by means of the vector, and its angle
referred to the three different axis. The three different axis are: sD-sQ fixed to the
stator, rα-rβ fixed to the rotor whose speed is wm, and finally the general frame
represented by means of the axis x-y whose speed is equal to wg .
In a similar way and for other magnitudes, it can be written the following equations:
u sg = u s e
− jθ g
ψ sg = ψ s e
= usx + jusy
− jθ g
= ψ sx + jψ sy
(1.56)
Where the magnitudes are the voltage space phasor and the stator flux linkage respectively.
However, if the magnitude in the figure 1.6 is for instance the rotor current, its space phasor
notation will be:
i rg = i r e
(
− j θ g −θ m
) = i + ji
rx
ry
(1.57)
1.13
Induction motor model. Generalities.
and for other magnitudes:
u rg = u r e
(
− j θ g −θ m
ψ rg = ψ r e
)
(
− j θ g −θ m
= u rx + ju ry
)
(1.58)
= ψ rx + jψ ry
Manipulating the previous equations yields the following stator and rotor space phasor
voltage equations in the general reference frame.
u sg e
j θg
= Rs i sg e
(
u rg e
j θ g −θ m
jθ g
+
(
d ψ sg e
jθ g
s sg e
dt
) = R i rg e (
r
j θ g −θ m
)=R i
)+
d  ψ r e

j θg
(
j θ g −θ m
+e
d ψ sg
jθ g
dt
) 

dt
= Rr i rg e
+ je
(
j θ g − θm
jθ g
w g ψ sg
) + e j( θ
g
−θ m
) d ψ rg + je j ( θ
dt
g
−θ m
) w − P⋅ w ψ
( g
m ) rg
(1.59)
Simplifying equation 1.59, it is obtained equation 1.60.
u sg = R s i sg +
u rg = Rr i rg +
d ψ sg
dt
d ψ rg
dt
+ jw g ψ sg
(
(1.60)
)
+ j w g − P ⋅ wm ψ rg
Where, the flux linkage space phasors are:
ψ sg = Ls i sg + Lm i rg
(1.61)
ψ rg = Lr i rg + Lm i sg
Using the two-axis notation and the matrix form, the voltage equations can be represented by:
R s + pLs
u sx  

u 
w g Ls
 sy  = 
u rx  
pL m
  
u ry   w g - P ⋅ w m L m
(
)
- w g Ls
pL m
R s + pLs
w g Lm
(P ⋅ w
m
)
- w g Lm
pLm
(w
R r + pL r
g
  isx 
  
  isy 
⋅
P ⋅ w m - w g L r   i rx 
  
R r + pL r
  i ry 

- w g Lm
)
- P ⋅ w m Lr
(
pLm
)
(1.62)
1.2.5.2 - Space-phasor voltage equations in the stationary reference frame fixed to the stator.
If wg = 0, the matrix expression obtained is 1.63, being equal to the expression 1.17.
u sD   R s + pL s
u  
0
 sQ  = 
 u rd   pL m

 
 u rq  - P ⋅ w m L m
0
pLm
R s + pLs
0
P ⋅ w m Lm
R r + pL r
pL m
- P ⋅ w m Lr
 i sD 
 
pL m  i sQ 
⋅
P ⋅ w m L r   i rd 
  
R r + pL r   i rq 
0
(1.63)
The stator voltage space phasor can be expressed as follows:
u s = Rs i s +
1.14
dψ s
dt
(1.64)
Induction motor model. Generalities.
The rotor voltage space phasor can be written as:
'
u r e − jθ m
'
=
'
Rr i r e − jθ m
'
u r = Rr i r +
'
dψ r
dt
'
d  ψ r e − jθ m 


+
dt
(1.65)
'
− j ⋅ P ⋅ wm ψ r
And the flux linkage space phasors can be expressed as follows
'
ψ s = Ls i s + L m i r
'
'
(1.66)
ψ r = Lr i r + L m i s
1.2.5.3 - Space-phasor voltage equations in the rotating reference frame fixed to the rotor.
If wg = wm, the matrix expression obtained is 1.67, being equal to the expression 1.18.
u sD   R s + pLs
u  
 sQ  =  L s P w m
 u rd   pL m

 
0
 u rq  
- Ls P wm
pL m
R s + pL s
L m P wm
0
R r + pL r
pL m
0
− L m P w m   isD 
 i 
pLm
 ⋅  sQ 
  i rd 
0
  
R r + pL r   i rq 
(1.67)
The stator voltage space phasor can be expressed as follows:
(1.68)
'
'
us
=
'
Rs i s
+
dψs
dt
'
+ jψ s ⋅ P ⋅ wm
The rotor voltage space phasor can be written as:
u r = Rr i r +
dψr
(1.69)
dt
And the flux linkage space phasors can be expressed as follows
'
(1.70)
'
ψ s = Ls i s + L m ir
'
ψ r = Lr ir + L m i s
1.2.5.4 - Space-phasor voltage equations in the rotating reference frame at synchronous speed.
If wg = ws, the matrix expression obtained is 1.71, being equal to expression 1.19.
u sD   R s + pLs
u  
 sQ  =  L s w s
 u rd   pL m

 
 u rq   L m sw s
- Ls w s
pL m
R s + pLs
Lm w s
− L m sw s
R r + pL r
pL m
L r sw s
− L m w s  i sD 
 
pL m  i sQ 
⋅
− L r sw s   i rd 
  
R r + pL r   i rq 
(1.71)
1.15
Induction motor model. Generalities.
The stator voltage space phasor can be expressed as follows:
u sg = Rs i sg +
d ψ sg
dt
+ jψ sg w s
(1.72)
The rotor voltage space phasor can be written as:
u rg = R r i rg +
d ψ rg
dt
+ j ψ rg ( w s − P ⋅ w m )
(1.73)
And the flux linkage space phasors can be expressed as follows
ψ sg = Ls i sg + L m i rg
ψ rg = L r i rg + L m i sg
1.16
(1.74)
Induction motor model. Generalities.
1.3 – Torque expressions.
1.3.1 - Introduction.
The general expression for the torque is as follows:
(1.75)
'
t e = cψ s × i r
Where the “c” is a constant, ψ s and i r are the space phasors of the stator flux and rotor
'
current respectively, both referred to the stationary reference frame fixed to the stator.
The expression given above can also be expressed as follows:
t e = c ψ s ⋅ i r sin γ
(1.76)
Where γ is the angle existing between the stator flux linkage and the rotor current. It follows
that when γ=90o the torque obtained is the maximum and its expression is exactly equal to the
one for the DC machines. Nevertheless, in DC machines the space distribution of both
magnitudes is fixed in space, thus producing the maximum torque for all different magnitude
values. Furthermore, both magnitudes can be controlled independently or separately. In an
AC machine, however, it is much more difficult to realise this principle because both
quantities are coupled and their position in space depends on both the stator and rotor
positions. It is a further complication that in squirrel-cage machines, it is not possible to
monitor the rotor current, unless the motor is specially prepared for this purpose in a special
laboratory. It is impossible to find them in a real application. The search for a simple control
scheme similar to the one for DC machines has led to the development of the so-called vectorcontrol schemes, where the point of obtaining two different currents, one for controlling the
flux and the other one for the rotor current, is achieved [VAS 1].
1.3.2 - Deduction of the torque expression by means of energy considerations.
Torque equation is being deduced by means of energy considerations. Therefore, the starting
equation is as follows:
Pmechanic = Pelectric − Ploss − Pfield
(1.77)
1.17
Induction motor model. Generalities.
Substituting the previous powers for its values, the equation can be expressed as follows:

 d ψ s *
2
*
t e ⋅ wr =  Re u s ⋅ i s − Rs is − Re
is
dt  



(
3
2
)

 d ψ'r '*   
'
'*
' 2

+  Re ur ⋅ i r − Rr ir − Re 
i r   
dt

  

(
)
(1.78)
Since in the stationary reference frame, the stator voltage space phasor u s can only be
balanced by the stator ohmic drop, plus the rate of change of the stator flux linkage, the
previous expression can be expressed as follows:
(
'
'*
)
(
'
'*
)
'
(1.79)
'
t e ⋅ wr = 32 Re − jwr ψ r i r = − 32 wr Re j ψ r i r = − 32 wr ψ r × i r
Expressing the equation in a general way for any number of pair of poles gives:
t e = − 32 Pψ r × i r
'
'
(1.80)
If equations 1.66 and 1.35 are substituted in equation 1.80, it is obtained the following
expression for the torque:
(1.81)
t e = 32 P ψ s × i s
If the product is developed, expression 1.81 is as follows:
(
t e = 32 P ψsD ⋅ i sQ − ψ sQ ⋅ i sD
)
(1.82)
Finally, different expressions for the torque can be obtained as follows:
'
'
'
'
t e = − 32 P L r i r + L m i s  × i r = − 32 PL m i s × i r = − 32 PL m i s × i r


L
'
'
3L
'
Lm
'
3
t e = − 32 P m  L m i r + L s i s  × i r = −P m ψ s × i r = − P
ψs × ψ r
2

Ls 
2 Ls
2 LsLr − Lm
(1.83)
1.3.3 – Torque constant.
The value of the torque constant can take two different values. These depend on the constant
used in the space phasor. Both possibilities are shown in table I.I.
Non power invariant
Torque constant
Space phasor
constant
3
3→2
2
3
Power invariant
1
2
2→3
3→2
2→3
1
2
2
3
3
Table I.I. Torque constant values.
"3→2" means the change from three axis to either two axis or space phasor notation,
and "2→3" either two axis or space phasor notation to three axis.
1.18
Induction motor model. Generalities.
1.4 – Simulink model.
1.4.1 - Equations used in the model.
The final expressions used in the implemented models are obtained from all the previously
introduced expressions.
All equations have been re-arranged in order to use the operator 1/s instead of the operator p
because the “Simulink” deals with the integrator better than with the derivation.
1.4.1.1 – Stator reference.
Stator and rotor fluxes can be expressed as follows:
ψ sD =
1
s
ψ sQ =
1
s
ψ 'rd = 1s
ψ 'rq = 1s
(usD − Rs isD )
(u
(u
(u
sQ
− Rs isQ
)
) (− R i
) = (− R i
'
rd
− Rr i' rd − P ⋅ w m ψ 'rq =
1
s
r
'
rq
− Rr i' rq + P ⋅ w m ψ 'rd
1
s
r
'
'
rd
− P ⋅ w m ψ 'rq
rq
+ P ⋅ w m ψ 'rd
)
)
(1.84)
Stator and rotor currents can be expressed as follows:
Lr
L
− ψ 'rd m
Lx
Lx
L
L
i sQ = ψ sQ r − ψ 'rq m
Lx
Lx
L
L
ird' = ψ 'rd s − ψ sD m
Lx
Lx
L
L
irq' = ψ 'rq s − ψ sQ m
Lx
Lx
i sD = ψ sD
where Lx = Ls Lr − L2m
(1.85)
1.4.1.2 - Rotor reference.
Stator and rotor fluxes can be expressed as follows:
ψ rd =
1
s
ψ rq =
1
s
ψ'sd =
1
s
ψ'sq = 1s
(urd − Rrird ) = 0
(u
(u
(u
)
rq
− Rr irq = 0
'
sd
− Rs i'sd + P ⋅ wm ψ 'sq
'
sq
− Rr i'sq − P ⋅ wm ψ 'sd
)
)
(1.86)
1.19
Induction motor model. Generalities.
Stator and rotor currents can be expressed as follows:
Lr
isD = ψ ' sD
Lx
Lm
Lx
Lr
L
− ψ 'rq m
Lx
Lx
isQ = ψ ' sQ
Ls
ird = ψ rd
− ψ 'rd
Lx
− ψ ' sD
Lm
Lx
Ls
L
− ψ ' sQ m
Lx
Lx
irq = ψ rq
where L x = Ls Lr −
(1.87)
L2m
1.4.1.3 - Synchronous reference.
Stator and rotor fluxes can be expressed as follows:
(
(u
(u
(u
ψ sx = 1s usx − Rs isx + ws ψ sy
ψ sy = 1s
ψ rx =
1
s
ψ ry =
1
s
)
)
sy
− Rr isy − ws ψ sx
rx
− Rr irx + ψ ry ( ws − P ⋅ wm ) = 1s − Rr irx + ψ ry ( ws − P ⋅ wm )
ry
− Rr iry − ψ rx
) (
(w − P ⋅ w )) = ( − R i
s
m
1
s
r ry
− ψ rx ( ws − P ⋅ wm
)
))
(1.88)
Stator and rotor currents can be expressed as follows:
isx = ψ
sx
Lr
L
− ψ rx m
Lx
Lx
isy = ψ
sy
Lr
L
− ψ ry m
Lx
Lx
irx = ψ rx
Ls
−ψ
Lx
sx
Lm
Lx
iry = ψ ry
Ls
−ψ
Lx
sy
Lm
Lx
where Lx = Ls Lr −
(1.89)
L2m
1.4.1.4 – Motion equation.
The motion equation is as follows:
te − tL = J
dw m
+ Dw m
dt
(1.90)
Where, te is the electromagnetic torque, tL is load torque, J is the inertia of the rotor, and
finally the D is the damping constant.
1.20
Induction motor model. Generalities.
Using the torque expressions 1.82, the previous motion equation can be expressed as follows:
P ⋅ c ⋅ (ψ sD ⋅ i sQ − ψ sQ ⋅ i sD ) = t L + w m (D + Js)
wr =
P ⋅ c ⋅ (ψ sD ⋅ i sQ − ψ sQ ⋅ i sD ) − t L
D + Js
(1.91)
Where P is the number of pair of poles and the torque constant take the values either 1 or 2/3
according to the table I.I shown in the previous section 1.3.3.
1.4.2 – Simulated results.
Figures 1.7 and 1.8 show the torque and speed responses obtained from equation 1.84 to
equation 1.91. It must be said that all three different references (stator, rotor and synchronous)
gave the same simulated results shown in figures 1.7 and 1.8. The validity of the motor model
is corroborated.
60
60
50
50
40
40
torque (Nm)
torque (Nm)
All simulations are done in Matlab/Simulink. Motor characteristics are listed in section 4.2.
30
20
30
20
10
10
0
0
-10
0
-10
0.2
0.4
0.6
0.8
time (s)
1
1.2
0
0.2
0.4
0.6
0.8
1
1.2
time (s)
Figure 1.7. Torque response without load. Left: Motor_1kW. Right: Motor_1.5kW.
Te= 0Nm, J=0.08Kgm2. Notice the transient at the beginning and the steady state
torque value, being 0Nm for this ideal case.
1.21
160
160
140
140
120
120
100
100
wm (rd/s)
wm (rd/s)
Induction motor model. Generalities.
80
60
80
60
40
40
20
20
0
0
0
0.2
0.4
0.6
0.8
time (s)
1
1.2
1.4
0
0.2
0.4
0.6
0.8
1
time (s)
Figure 1.8. Speed response without load. Left: Motor_1kW. Right: Motor_1.5kW.
Te=0Nm, J=0.08Kgm2. Notice the small ripple at the beginning due to the transient.
The final speed value is 157 rd/s, as expected from this ideal case, where Te=0Nm.
1.22
1.2
1.4
Induction motor model. Generalities.
1.5 – Steady state analysis.
1.5.1 - Steady state conditions.
This section deals with the conditions under which, induction motor operates in steady state.
When an induction motor operates in steady state and is supplied by symmetrical and
sinusoidal waveforms, the space vectors become identical to its phasors. Therefore, the
following assumptions expressed in 1.92, can be taken into account:
u s = Vs
'
u r = Vr' e j θ
d is
= jws I s
dt
'
d ir
= jwr I r' + j ⋅ P ⋅ wm I r'
dt
(1.92)
1.5.2 - Steady state equations.
From the expressions 1.64, 1.65 and 1.66, referred to the stationary reference frame fixed to
the stator, can be written the equations 1.93 for the stator and 1.94 for the rotor, valid in both
transient and steady state:
u s = R s i s + (L s1 + 3 2 Lsm )
'
dis 3
dir
+ 2 Lrm
dt
dt
(1.93)
'
'
dir 3
dis
u = R i + (Lr 1 + 2 L rm )
+ 2 L sm
− j ⋅ P ⋅ wm  (Lr 1 + 3 2 L rm )i r + 3 2 Lsm i s 


dt
dt
'
r
'
r r
3
(1.94)
Once the conditions described above in equations 1.92 are applied to the stator 1.93 and rotor
1.94 equations, stator 1.95 and rotor 1.96 steady state equations are obtained:
V s = Rs I s + jws (L s1 + 3 2 L sm )I s + jws
3
0 = R r s I r' + jws (Lr 1 +
3
3
2
L sm )I r' + jws
2
Lsm I r'
(1.95)
2
Lsm I s
(1.96)
1.23
Induction motor model. Generalities.
1.5.3 - Steady state equivalent circuit.
From the equations 1.95 and 1.96, the well-known equivalent circuit for an induction motor
can be drawn as shown in figure 1.9:
Rs
wsL s1
Is
wsL r1
3/2wsLsm
Rr/s
Ir'
Figure 1.9. Steady state equivalent circuit of Induction motor.
From the previous steady state induction motor model the following expressions for the
torque, stator current and stator flux can be obtained:
T = 3 P 2 Rr s
Is =
Rr
s
1
ws
I r'
2
+ jws (Lr 1 + 3 2 Lsm ) '
Ir
− jws 3 2 L sm
ψ s = (L s1 + 3 2 Lsm )I s + 3 2 Lsm I r'
1.24
(1.97)
(1.98)
(1.99)
Induction motor model. Generalities.
1.6 – Interim conclusions.
In the present chapter has been deduced the motor model. The model has been formulated by
means of the two-axis theory equations and the space phasor notation. Despite the fact that
both nomenclatures are valid, it has been proved that the space phasor notation is much more
compact and easier to work with. The model has been developed in both nomenclatures for
the stator, rotor and synchronous references. In further chapters, the motor model with stator
reference, introduced in section 1.4.1.1, will be the one most used.
Different torque expressions have been deduced.
The final concrete equations used in the Matlab/Simulink motor model have been presented
by the three different references. Some simulations are shown to prove the validity of the
model, being equal for the previously mentioned three references. Two different motors have
been used in the model.
Finally the steady state motor analysis has been introduced.
1.25