Advanced Electrical Drives:
supplementary
Prof. Dr. ir. Rik de Doncker
Prof. Dr. ir. Duco Pulle
Dr. ir. André Veltman
Springer
Abstract
As with any book, including ‘Advanced Electrical Drives’ (AED), limits
are placed on the amount of material that can be included. This implies
that some material could not be included in the book. This supplementary
chapter of our book extends the IRTF based asynchronous machine models
to allow phenomenon such as skin effect and main inductance saturation
to be modeled. Both dynamic and steady state operating conditions are
discussed and results given are based on an actual machine. Furthermore an
IRTF based model is introduced which may be used to examine asymmetric
supply conditions.
The motivation for presenting this material is based on questions from
industrial users regarding the suitability of IRTF based models for handling
skin effect and saturation. It is hoped that the material presented will
alleviate these concerns. A set of simulation tutorials has been added to
demonstrate the theory discussed in this supplementary chapter.
Contents
1 Extended modeling of voltage source connected asynchronous
machines
1
1.1 A universal IRTF based model adaptation for skin effect . . .
1
1.1.1 Second order skin effect adaptation for universal IRTF
based model . . . . . . . . . . . . . . . . . . . . . . . .
4
1.1.2 Steady-state analysis of model with skin-effect . . . .
8
1.1.3 Third order skin effect adaptation for universal IRTF
based model . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2 Modelling the influence of saturation . . . . . . . . . . . . . . 16
1.2.1 Steady-state analysis . . . . . . . . . . . . . . . . . . . 20
1.3 Modeling machines for operation under asymmetrical conditions 24
1.4 Tutorials for this Supplementary chapter . . . . . . . . . . . . 29
1.4.1 Tutorial 1: Line start of induction machine with skin
effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.4.2 Tutorial 2: Steady state characteristics, grid connected
induction machine with skin effect . . . . . . . . . . . 31
1.4.3 Tutorial 3: Grid connected induction machine with
main inductance saturation . . . . . . . . . . . . . . . 36
1.4.4 Tutorial 4: Steady state characteristics, grid connected
induction machine with main inductance saturation . 42
1.4.5 Tutorial 5: Grid connected induction machine with
one phase open-circuited . . . . . . . . . . . . . . . . . 48
List of Figures
52
i
Chapter 1
Extended modeling of
voltage source connected
asynchronous machines
In this chapter a series of dynamic and steady state models are discussed
which will allow the user to represent a wide range of voltage source connected
asynchronous machines with specific characteristics. The work presented
is based on the IRTF models introduced in our book ‘Advanced Electrical
Drives’ [1]. In this chapter these models are extended in order to represent
specific aspects of the machine such as skin-effect and main inductance
saturation. Attention is also given to the issue of representing machine
operation under asymmetric operation which occurs in certain types of softstarters. Machine operation under dynamic and steady state conditions is
discussed. A set of ‘build and play’ tutorials is provided at the end of this
chapter which contain Caspoc based dynamic models as well as MATLAB
based steady-state models of the various machines discussed in this chapter.
1.1
A universal IRTF based model adaptation for
skin effect
Skin-effect in the context of this book refers to the rotor side of the machine
only and is popularly described as the tendency of alternating current to
concentrate in areas of lowest impedance. For rotor bars that form part of
a ’squirrel cage’ rotor the outermost part of the bar acts as an area of low
impedance. This means that for low slip frequencies the entire bar cross-
1
sectional area contributes to the effective rotor resistance RR and leakage
inductance LσR (see figure 8.12). At higher slip frequencies the current
distribution is no longer uniform across the rotor bar cross-sectional area.
Under these conditions the current vacates the inner parts of the bars and
concentrates more on the regions closer to the outer diameter of the rotor
bars. Consequently, the effective rotor resistance will increase and the rotor
leakage inductance will decrease. The higher rotor resistance is beneficial for
asynchronous machines designed for direct on line starting as the starting
torque can be increased (in case the designer has chosen to use skin-effect for
this purpose). For converter connected asynchronous machines there is no
need to operate in the high slip region of the motor, consequently, skin-effect
is not exhibited for the fundamental operating frequency ωs . However, higher
converter harmonics occur in this case which are susceptible to skin-effect.
Consequently, the higher converter current harmonics cause higher rotor
losses (due to the increased rotor resistance) and most importantly a higher
torque ripple, given that these currents experience a lower effective leakage
inductance. Given the above it is prudent to determine the ability of the universal IRTF based model (as discussed in section 8.3.2) in terms of handling
rotor skin-effect and to consider topology adaptations.
According to skin-effect theory [2] the complex admittance Gskin (ω) of
a sheet of isotropic material with resistance RR and inductance LσR may be
expressed in the following normalized form
q
tanh
g skin =
j 3ω
ωc
q
j 3ω
ωc
(1.1)
G
R
where ωc = LRσR
and g skin = G skin(0) . Furthermore the term Gskin (0) = R1R
skin
represents the admittance at zero frequency. It may be argued that the
outer rotor shell (including all the rotor bars) consists of copper or aluminum
segments which are interleaved with sections of laminated iron. As such the
admittance of this rotor shell will exhibit a frequency dependency according
to equation (1.1).
The rotor circuit of the universal IRTF based model as represented in
figure 8.12 of our book, by circuit elements RR , LσR , has a normalized (with
respect to R1R ) admittance which may be written as
g1 =
1
1 + j ωωc
2
(1.2)
A Nyquist plot which shows the normalized admittance functions according
to equations (1.1) and (1.2) is given in figure 1.1 for ωωc = 0 → ωωc = 104 . Also
shown in this figure are three discrete normalized ( ωωc ) frequency points with
values 0.1, 1 and 10 respectively. An observation of figure 1.1 learns that a
first order approximation of a rotor circuit which consists of the rotor leakage
inductance LσR and rotor resistance RR is not capable of accurately modeling
the skin-effect phenomenon for rotor frequencies in excess of ± 0.1 ωc . In
practical terms the two normalized rotor frequency points 0.1, 1 correspond
(under motoring conditions) to a shaft speed of 1478 rpm and 1281 rpm
respectively for the four pole machine associated with the torque/speed
curve given in figure 8.48 (page 302). The remaining normalized frequency
point equal to 10 is reached with the machine in question at −670 rpm, i.e
‘plugging’ mode of operation. These three values may be put in perspective
by considering the shaft speed that corresponds to peak motoring torque of
this machine, which is at approximately 1400 rpm.
0.1
0
ω/ωc=0.1
imag(gskin,g1)
−0.1
g
(ω/ω )
c
−skin
−0.2
ω/ωc=10
−0.3
−0.4
ω/ωc=1
−0.5
g1 (ω/ωc)
−
−0.6
0.1
0.2
0.3
0.4
0.5
0.6
real(gskin,g1)
0.7
0.8
0.9
Figure 1.1: Nyquist diagram of the complex admittance: g skin , g 1 as function
of ωωc
3
1.1.1
Second order skin effect adaptation for universal IRTF
based model
From the previous discussion it is apparent that a higher order approximation of the rotor circuit is warranted in order to obtain a better correlation
with equation (1.1). The approach taken here is to replace the rotor based
leakage inductance LσR with a series network in the form of a inductance
L0σR positioned on the stator side of the IRTF module and a ‘skin-effect’
dependent impedance in series with the rotor resistance RR .
In this section a ‘second order’ model adaptation is considered where the
‘skin-effect’ impedance is represented in terms of a parallel network which
consists of an inductance L1σR and resistance Rp1 as shown in figure 1.2.
The sum of the two inductances L0σR , L1σR must be equal to the leakage
Figure 1.2: Universal, IRTF based induction machine model, with 2nd order
skin effect adaptation
inductance LσR of the ‘original’ model (see figure 8.12, of our book) given
that this topology must re-appear for the case Rp1 → ∞ . Introduction of a
parallel resistance/inductance network on the rotor side of the IRTF leads to
a second order rotor circuit which consists of an inductance Lrσ0 , resistance
RR and parallel network with elements L1σR , Rp1 . The equation set which
corresponds with the revised symbolic model according to figure 1.2 may be
4
written as
~s
dψ
~us = ~is Rs +
dt
~
~
~
ψs = ψM + is LσS
~
~M − ~iR L0
ψR1 = ψ
σR
~M
ψ
LM
(1.3a)
(1.3b)
(1.3c)
= ~is − ~iR
(1.3d)
~ xy
~ xy
dψ
dψ
R1
R
=
−
dt
dt
= ~ixy
p1 Rp1
∆~uxy
R1
∆~uxy
R1
∆~uxy
R1
=
L1σR
~xy
d ~ixy
R − ip1
(1.3e)
(1.3f)
dt
~ xy
dψ
R
R
+
0 = −~ixy
R R
dt
n
o
~ ∗ ~iR
Te = = ψ
R1
(1.3g)
(1.3h)
(1.3i)
~xy
where the space vectors ∆~uxy
R1 and ip1 are introduced, which respectively represent the voltage across the parallel network Rp1 , L1σR and current through
the resistance Rp1 .
The normalized (with respect to R1R )) complex admittance of the rotor
circuit in its present form (which includes rotor leakage inductance L0σR
located on the stator side of the IRTF module) may be written as
g2 =
1
z2
z2 = j
(1.4a)
1
rp1 j ωωc lσR
ω 0
lσR +
1 +1
ωc
rp1 + j ωωc lσR
(1.4b)
with
0
lσR
=
L0σR 1
L1
Rp1
; lσR = σR , rp1 =
LσR
LσR
RR
(1.5)
In equation (1.5) a normalization of the rotor network elements is introduced
with respect to the rotor leakage inductance and rotor resistance of the
‘original’ model (see figure 8.12). The problem of determining the normalized
0 , l1
0
1
inductances lσR
σR and resistance rp1 with constraint lσR + lσR = 1 is
5
solved by the use of a MATLAB minimization algorithm which
h minimizes thei
0
1
lσR
rp1
function J (X ) as given by expression (1.6) in which X = lσR
represents the set of normalized variables.
J (X ) =
k=200
X
k=1
s
|g 2
ωk
ωc
− g skin
ωk
ωc
|
(1.6)
The function in question is minimized for a specific set of equally spaced
(on a logarithmic scale) normalized frequencies ωωkc ; k = [1..200] with ωω1c =
10−2 , ωω200
= 104 . Furthermore, a weighting function is introduced in the
c
form of a square root function which serves to weight the smaller errors
in comparison with the larger errors. The set of normalized rotor circuit
parameters as determined via the minimization procedure was found to be
0
1
lσR
= 0.2896, lσR
= 0.7104, rp1 = 2.532
(1.7)
Use of equation (1.7) with equation (1.5) allows the calculation of the rotor
elements L0σR , L1σR and Rp1 of the revised IRTF model shown in figure 1.2.
The extend of the correlation between equation (1.1) and equation (1.4), with
varying normalized frequency ωωc = 0 → ωωc = 104 and parameters according
to equation (1.7) may be observed with the aid of figure 1.3. A comparison
between the Nyquist diagram shown in figure 1.3 and figure 1.1 learns that
the 2nd order rotor model admittance g 2 complies with equation (1.1) for a
normalized frequency range 0 ≤ ω/ωc < 10. This normalized frequency range
is generally sufficient to accommodate skin-effect phenomena over the slip
range −1 ≤ s ≤ 1, hence the adapted IRTF model is (for example) suitable
for line start transient analysis purposes.
For the development of a dynamic model, a generic diagram is required
which complies with the symbolic model according to figure 1.2 and equation (1.3). An implementation example of such a diagram, as given in
figure 1.4, resembles the universal based model shown in figure 8.13 [1].
In both cases the inverse matrix L−1 needs to be calculated using equation (8.17) [1]. However in the skin-effect based model the rotor inductance
must be set to LR = LM + L0σR , given that only a part L0σR (which is not
affected by skin-effect) of the total leakage inductance LσR is used for the
computation of the inverse matrix elements. Furthermore the new model has
an additional gain module and integrator module with gain values of Rp1
and 1/L1σR respectively which are linked to the parallel resistance/ leakage
inductance network present on the rotor side of the IRTF (see figure 1.2).
It is instructive to compare the transient results linked to a line start of a
6
0.1
gskin (ω/ωc)
−
0
ω/ωc=0.1
ω/ωc=10
imag(gskin,g2)
−0.1
−0.2
g2 (ω/ωc)
−
ω/ωc=1
−0.3
−0.4
−0.5
−0.6
0.1
0.2
0.3
0.4
0.5
0.6
real(gskin,g2)
0.7
0.8
0.9
Figure 1.3: Nyquist diagram of the complex admittance: g skin , g 2 as function
of ωωc
7
Figure 1.4: Generic model representation of a universal IRTF based induction
machine with 2nd order skin-effect
22kW machine, with out skin effect (as discussed in section 8.6.7 [1]) with
those obtained with the skin-effect adapted model. A Caspoc based tutorial
as given in section 1.4.1 is based on the generic model shown in figure 1.4.
The excitation and machine parameters used to obtain the transient results
of the four parameter model (without skin effect, see section 8.6.6 [1]) are
applied to the new model using the approach set out in this section. The
results given in figure 1.5 show the torque, shaft speed and line current of
the 22 kW induction machine for a IRTF model without and with skin effect,
under identical conditions. observation of the transient waveforms generated
with the aid of both IRTF based models show that the high slip behavior of
the skin-effect based machine is markedly different. For example the time
required for the machine to reach its synchronous speed after the voltage
source is connected is approximately halved when skin-effect is taken into
account. The steady-state behavior will, as may be expected, also be affected
by the changes to the model as will become apparent in the next section.
1.1.2
Steady-state analysis of model with skin-effect
The steady-state analysis may be undertaken using the approach given in
section 8.3.6, page 273 [1], where the transformation from symbolic to steady
state model was affected by introducing a slip dependent rotor resistance.
8
T
mec
(Nm)
200
100
0
−100
−200
0
0.2
0.4
0.6
0.8
1
(a) time (s)
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1
(b) time (s)
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1
(c) time (s)
1.2
1.4
1.6
1.8
2
nmec (rpm)
2000
1500
1000
500
0
line current (A)
400
200
0
−200
(a) without skin effect
T
mec
(Nm)
400
200
0
−200
0
0.2
0.4
0.6
0.8
1
(a) time (s)
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1
(b) time (s)
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1
(c) time (s)
1.2
1.4
1.6
1.8
2
nmec (rpm)
2000
1500
1000
500
0
line current (A)
400
200
0
−200
(b) with 2nd order skin effect
Figure 1.5: Line start example of 22 kW delta connected machine
9
For the transformation process considered here two slip dependent resistive
elements must be introduced which leads to a phasor based model given
in figure 1.6. An indication of its validity may be obtained by setting the
resistance Rp1 to infinity, in which case the model according to figure 8.31,
page 274 [1] appears, provided that the transformation variable is set to
a = LLmr .
Figure 1.6: Equivalent circuit of a skin-effect adapted asynchronous machine
and voltage source connected (steady-state version of dynamic model in
figure 1.2)
The process of determining the stator current phasor trajectory is as function
of the slip s is initiated by considering the input impedance z in of the model
according to figure 1.6 which is of the form
z in = Rs + jωs LσS +
jωs LM z R
jωs LM + z R
(1.8)
where z R represents the rotor impedance of the steady-state model which
may be written as
z R = jωs L0σR +
RR
jωs L1σR Rp1/s
+
1
s
jωs LσR + Rp1/s
(1.9)
Use of equations (1.8), (1.9) leads to the current phasor is = zûs , which makes
in
use of the supply voltage phasor us = ûs as introduced in section 8.3.6, page
273 [1]. It is instructive to reconsider the Heyland diagram example given in
figure 8.32, page 274 [1] and examine the impact of using a skin-effect adapted
steady-state model for the grid connected 22kW machine. The Heyland
diagram of the steady model, with skin effect, is shown in figure 1.7(a) for a
slip range of −1 ≤ s ≤ 1 . As with the previous example a normalization
is introduced in figure 1.7(a) which is of the form ins = is / (ûs/ωs LσS ), where
LσS represent the leakage inductance of the four parameter, rotor flux based,
10
IRTF model. Also shown in figure 1.7(a) is the Heyland diagram which
corresponds to a machine without skin-effect, i.e realized by setting the
parameter Rp1 → ∞ in equation (1.9). A comparison of the two current
loci learns that they are substantially different in the high slip region of
operation.
(a) Normalized stator current phasor
(b) Torque versus speed
Figure 1.7: Steady-state characteristics of voltage source connected asynchronous machine, with skin-effect, model according to figure 1.6
Computation of the torque speed curve is readily achieved by making use
of expression 8.33, page 275 [1], which in turn requires access to the stator
current phasor is as calculated above. Figure 1.7(b) shows the torque versus
speed characteristics of the 22kW machine used for this example , with a
skin-effect adapted IRTF model. Also shown in this diagram, for comparison purposes is the torque/speed characteristic of the same machine model,
without skin-effect. These results confirm that the use of a skin-effect model
can have a significant impact on the transient and steady state behavior of
the machine when operating under high slip conditions.
In the sequel to this subsection the issue of choosing the transformation coefficient Γa is addressed. For the universal IRTF based model representation
as discussed in section 8.3.2, page 252 [1] the transformation coefficient can
be freely chosen within the range −100% ≤ Γa ≤ 100% without affecting
the outward electrical of mechanical characteristics. With the skin-effect
adaptation part of the rotor inductance circuit has been purposely modified
11
to model skin-effect behavior. Varying the value of Γa provides a means of
tuning the model given that this variable controls (among others) the value
of the rotor leakage inductance LσR which in turn defines the parameters
L0σR , L1σR as discussed in this section. In the example shown in figure 1.7 the
value of the transformation variable Γa was to Γa = 0% which is deemed to
be its ‘default’ value, as it assumes equal rotor and stator leakage inductance.
However experimental data, in the form of for example the steady-state
torque/speed curve or transient shaft speed during a line start may be used
to tune the simulation model.
The effect of altering the transformation coefficient Γa is illustrated with
the aid of figure 1.8, which shows how the normalized torque/slip torque
of the 22kW used in the example discussed above is affected. In figure 1.8
Figure 1.8: Steady-state torque/speed curve with adjustable transformation
coefficient Γa
three torque/speed curves are shown where the transformation variable is
set 20% above and below the default setting Γa = 0%. An observation of
these results shows that the use of the transformation variable as a tuning
parameter is effective. However its value cannot be set to Γa = −100 in this
case, given that this value corresponds to a machine with zero rotor leakage.
12
1.1.3
Third order skin effect adaptation for universal IRTF
based model
In converter connected motors considerably higher rotor frequencies may
appear as mentioned earlier, which may exceed the value of 10 ωc shown in the
normalized admittance diagram according to figure 1.3. An observation of this
diagram learns that there is a significant error between the admittance g skin
(see equation (1.1)) and the 2nd order normalized rotor circuit admittance g 2
(see equation (1.4)) for rotor frequencies in access of 10 ωc . Consequently it is
prudent to consider a second skin-effect adaptation which is in the form of a
3rd order rotor model with a normalized rotor admittance that complies with
the admittance g skin at normalized frequencies beyond 10 ωc . In this section
such a model adaptation is examined by way of a second parallel network
(2)
with elements LσR , Rp2 as shown in figure 1.9. In this figure only the IRTF
module and rotor part of the model are shown given that the remaining
stator part of the machine remains unchanged when compared to the previous
case (see figure 1.2). The new rotor network is in this case formed by the
Figure 1.9: Rotor section of universal, IRTF based induction machine model,
with 3nd order skin effect adaptation
stator based inductance L0σR and rotor based network which consists of the
(2)
inductances L1σR , LσR and resistances RR , Rp1 , Rp2 as shown in figure 1.9.
As with the 2nd order model adaptation, the sum of the inductances must
equal the leakage inductance LσR of the ‘original’ (see figure 8.12 [1] ) model.
The normalized (with respect to the ‘original’ leakage inductance LσR and
rotor resistance RR ) values, admittance and impedance for this circuit may
be found with the aid of figure 1.9 and by taking into account that the stator
13
based leakage inductance L0σR is also part of this rotor circuit which leads to
g3 =
1
z3
(1.10a)
(2)
z3
1
rp2 j ωωc lσR
rp1 j ωωc lσR
ω 0
+1
= j lσR +
1 +
(2)
ωc
rp1 + j ωωc lσR
rp2 + j ωωc lσR
(1.10b)
where
(2)
0
lσR
=
L
L0σR 1
L1
Rp1
Rp2
(2)
; lσR = σR , lσR = σR , rp1 =
, rp2 =
LσR
LσR
LσR
RR
RR
(1.11a)
The process of determining the values of the parameter set (equation(1.11)
is similar to that discussed for the previous case. However, in this case
the minimization function (1.6) is used with the function g 3 (see equation
(1.10a) instead of g 2 . Furthermore, the minimization vector is expanded
to include
the additional two rotor
elements and is therefore of the form
h
i
(2)
0
1
X = lσR
.
The
constraint imposed is that the sum
lσR lσR rp1 rp2
of the inductances must equal the ‘original’ (without skin-effect) leakage
inductance, which for the normalized inductance parameters translates to
0 + l1 + l(2) = 1.
the following condition: lσR
σR
σR
The corresponding set of normalized rotor circuit parameters as determined
via the minimization procedure was found to be
(2)
0
1
lσR
= 0.0993, lσR
= 0.2622, lσR = 0.6384, rp1 = 6.992, rp2 = 2.145 (1.12)
A comparison between the admittance functions g skin , g 3 for the normalized
frequency range used for the previous case is given in the Nyquist diagram
(figure (1.10). A comparison of figure (1.10) with figure (1.3) learns that the
third order model complies with the function g skin for rotor frequencies well
in excess of 10 ωc . Consequently the revised skin-effect model is capable of
handling skin-effect based phenomena at higher (in comparison to the 2nd
order model) rotor frequencies. The development of a generic model which
is based on the revised symbolic machine concept discussed in this section
proceeds along the lines discussed in section 1.1.1. This leads to a similar
model as shown in figure 1.4. where the L1σR integrator and Rp1 gain modules
where introduced to model skin-effect. The adaptation of the generic model
for 3rd order skin-effect phenomena requires the use of a second integrator
and gain module on the rotor side of the IRTF with gain settings of 1/L(2)
σR
and Rp2 respectively.
14
0.1
gskin (ω/ωc)
−
0
ω/ωc=0.1
ω/ωc=10
imag(gskin,g3)
−0.1
−0.2
g3 (ω/ωc)
−
−0.3
ω/ωc=1
−0.4
−0.5
−0.6
0.1
0.2
0.3
0.4
0.5
0.6
real(gskin,g3)
0.7
0.8
0.9
Figure 1.10: Nyquist diagram of the complex admittance: g skin , g 3 as
function of ω/ωc
15
1.2
Modelling the influence of saturation
The symbolic model according to figure 8.9, page 252 [1] features a set of
three linear inductances, which represent the stator leakage inductance Lσs ,
rotor leakage inductance Lσr and magnetizing inductance Lm respectively.
In this section we will consider the implications for the case where the
relationship between the magnetizing flux and magnetizing current is not
linear. This implies that the magnetizing inductance Lm cannot be considered
to be constant as its value will depend on the saturation encountered in the
machine. In this context we will assume that saturation effects in the machine
will not affect the leakage inductances Lσs , Lσr which is a realistic assumption.
Note that the conversion to a ‘universal’ type inductance model as discussed
in section 8.3.2 [1] is not advisable (for conversion factors other then Γ = 0)
given the fact that (among others) the leakage inductances LσS , LσR of this
model will also become non-linear given that these parameters are a function
of the magnetizing inductance Lm . The revised inductive components of
the IRTF based model as given in figure 1.11 show the presence of the two
leakage inductances and a non-linear inductive element which represents
the non-linear relationship between magnetizing flux and current. Also
Figure 1.11: Inductance circuit of IRTF based asynchronous machine model
shown in figure 1.11 are the voltage/current notations as used in the IRTF
model given in figure 8.9, page 252 [1]. In the interest of readability it is
convenient to reconsider the space vector based flux/current relationships
for the inductance circuit in its present form (see figure 1.11)
~s − ψ
~m = Lσs~is
ψ
~m − ψ
~r = Lσr~ir
ψ
~is − ~ir = ~im
(1.13a)
(1.13b)
(1.13c)
Equation (1.13) may also be rewritten in terms of the space vector flux vari~m , ψ
~s , ψ
~r and magnetizing current ~im which after some mathematical
ables ψ
16
manipulation leads to
ψ~m =
Lσr
~s +
ψ
Lσs + Lσs
|
{z
Lσs
Lσs Lσr ~
~r −
ψ
im
Lσs + Lσs
Lσs + Lσs
}
ψ~
mo
|
{z
Leq
(1.14)
}
Examination of equation (1.14) learns that the linear part of the inductance
circuit shown in figure 1.11 can be represented by a Thevenin equivalent
~
circuit with voltage source dψdtmo and leakage inductance Leq . This equivalent
circuit is in turn connected to the non-linear magnetizing inductance as
shown in figure 1.12.
Figure 1.12: Equivalent inductance circuit
The process of determining the magnetizing flux and current is initiated
by considering a ‘dq’ coordinate transformation where the real ‘d’ axis is
~m = ψm ejρm , hence ψ
~ dq = ψm in
aligned with the magnetizing flux vector ψ
m
which case the latter represents the flux-linkage magnitude. Furthermore
the variable ρm is introduced which represents the instantaneous angle of
~m with respect to a stationary coordinate system. Coordinate
the vector ψ
conversion of equation (1.14) to its new coordinate system, with variables
ψmo , Leq leads to a linear expression in terms of the flux ψm and current im
as given by equation (1.15).
ψm = ψmo − Leq im
(1.15)
The intersection of this linear flux-linkage/current function with the nonlinear magnetization curve ψm (im ) shown in figure 1.13 determines the value
of the flux ψm and current im for a given value of ψmo . Note that for the
magnetically linear case the magnetization curve ψm (im ) shown in figure 1.13
is reduced to ψm = Lm im (‘blue’ curve in figure 1.13) in which case the
17
Figure 1.13: Numerical determination of the magnetization flux and current
for a given value of ψmo
relationship between ψm and ψmo is reduced to
ψm =
Lm
Lm + Leq
!
ψmo
(1.16)
For the non-linear case the relationship ψm (ψmo ) may be found with the
aid of the generic diagram shown in figure 1.14. This approach makes use of
a non-linear module in the form of a look-up table im (ψmo ), the contents
of which may be found by calculating the current at the intersection of the
non-linear function ψm (im ) and the function according to equation (1.15)
(see figure 1.13) for a specified range of values for the variable ψmo . The
output of the module im (ψmo ) is used together with equation (1.15) to find
the corresponding value of ψm .
Figure 1.14: Non-linear module ψm (ψmo )
A suitable starting point for the development of a generic dynamic model
of the saturated machine is the universal IRTF model shown in figure 8.13,
page 258 [1], for the case a = 1. The universal model in question has, under
18
~s , ψ
~r as
these conditions, an inverse matrix module L−1 , with flux vectors ψ
~
~
input variables and current vectors is . ir as output variables which must be
replaced by an alternative set of modules as given in figure 1.15. Note that
in this figure the generic modules related to the rotor circuit and mechanical
part of the machine are not shown given that they are unaffected by the
~s , ψ
~r shown in figure 1.15 are
changes proposed here. The flux vectors ψ
Figure 1.15: Generic model of asynchronous machine with main inductance
saturation
~mo with the aid of equation (1.14).
now used to calculate the flux vector ψ
A coordinate transformation of this vector is achieved by making use of
a cartesian to polar conversion module which generates the variable ψmo
~mo and the required
and instantaneous angle ρmo . Note that the vector ψ
~
vector ψm are aligned, i.e have the same instantaneous angle ρmo , given that
the non-linear and linear elements are both inductances (see figure 1.12).
The non-linear module given in figure 1.14 generates the flux ψm which
corresponds to the input value ψmo . The ‘dq’ to stationary coordinate transformation is realized with a polar to cartesian module, which generates the
~m . Computation of the current vectors ~is , ~ir is carried out using
vector ψ
~s , ψ
~r and ψ
~m together with equations (1.13a), (1.13b) as may
the vectors ψ
be observed from figure 1.15. Note that the new model will yield identical
results with the model according to figure 8.13, page 258 [1], in case the flux
ψm is calculated using equation (1.16) and the universal machine model is
used with a conversion factor of Γ = 0% (a = 1).
19
The tutorial given in section 1.4.3 is concerned with the introduction of
main inductance saturation for the 22 kW used in this chapter. Figure 1.16
shows the torque, shaft speed and line current of the 22 kW machine, with
and without main inductance saturation during a line start.
A comparison between the results given in figure 1.16(b) and those
obtained with the machine without saturation (see figure 1.16(a)) learns that
the line current magnitude has increased noticeably in the low slip region
(t > 0.4 s) of the simulation. The torque and shaft speed curves remain
virtually unaffected by the introduction of saturation in the dynamic model.
1.2.1
Steady-state analysis
The steady-state analysis of the saturated machine is undertaken with the aid
of figure 1.17 which shows the presence of a slip dependant rotor resistance
Rr
s and a non-linear magnetizing reactance which may also be written as
jωs Lm = jωs
ψm (im )
im
(1.17)
where ψm (im ) represents the non-linear magnetization curve as shown in
figure 1.13. The approach taken for the dynamic analysis was based on
introducing a Thevenin equivalent circuit, in the form of a linear and nonlinear component. A similar approach may also be taken in this case by
grouping the linear components which consist of a stator based impedance Z s
and rotor based impedance Z r which are of the form shown in equation (1.18).
Z s = Rs + jωs Lσs
Rr
Zr =
+ jωs Lσr
s
(1.18a)
(1.18b)
The equivalent model as given in figure 1.18 consist of the non-linear magnetizing reactance ωs Lm and an equivalent linear impedance Z eq which is of
the form
Z eq =
Z sZ r
Zs + Zr
(1.19)
where Z s , Z r are defined by equation (1.18). Furthermore a voltage source
uom is introduced in figure 1.18, which may be written as
uom =
Zr
u
Zs + Zr s
20
(1.20)
T
mec
(Nm)
200
100
0
−100
−200
0
0.2
0.4
0.6
0.8
1
(a) time (s)
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1
(b) time (s)
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1
(c) time (s)
1.2
1.4
1.6
1.8
2
nmec (rpm)
2000
1500
1000
500
0
line current (A)
400
200
0
−200
(a) without saturaturation
T
mec
(Nm)
200
100
0
−100
−200
0
0.2
0.4
0.6
0.8
1
(a) time (s)
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1
(b) time (s)
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1
(c) time (s)
1.2
1.4
1.6
1.8
2
nmec (rpm)
2000
1500
1000
500
0
line current (A)
400
200
0
−200
−400
(b) with saturation
Figure 1.16: Line start example of 22 kW delta connected machine, with
21
and without main inductance saturation
Figure 1.17: Steady-state model of an asynchronous machine with main
inductance saturation
Figure 1.18: Steady-state equivalent Thevenin represenation
where us represents the applied voltage phasor, of which the amplitude is
taken to be |us | = ûs . It is helpful at this point of proceedings to consider
the real and imaginary power balance of the equivalent circuit according to
figure 1.18 which may be represented as
n
o
(1.21a)
n
o
(1.21b)
< {i∗m uom } = i2m < Z eq
= {i∗m uom } = i2m = Z eq + im ωs ψm (im )
In order to simplify the ensuing analysis it is prudent to assume the magnetizing current phasor to be real, in which case the phasor i∗m shown in
equation (1.21) is reduced to i∗m = im . The immediate consequence of this
assumption is that the phasors uom , us will be dependant on the current im .
Use of of phasor representation i∗m = im with equation (1.21a) allows the real
part of the phasor uom to be written in the form given by equation (1.22a).
The imaginary component of the voltage phasor uom may be found with the
aid of equation (1.20) and taking into account that the supply magnitude
ûs is defined as an input variable. Subsequent mathematical manipulation
of equation (1.21b) by considering the amplitude relationship between the
22
phasors uom , us leads to equation (1.22b).
n
< {uom } = im < Z eq
= {uom }
s
=
o
(1.22a)
Zr
ûs |
|
Zs + Zr
2
− (< {uom })2
(1.22b)
Use of equation (1.22b) with equation (1.21b) and the previously made
assumption i∗m = im allows the latter expression to be rewritten as
= {uom }
ωs
n
− im
= Z eq
ωs
o
= ψm (im )
(1.23)
Expression (1.23) must be solved numerically in order to find the value
of the magnetizing current im . Once this value has been obtained the
real and imaginary components of the phasor uom may be calculated using
equation (1.22). The corresponding value of the phasor um may for example
be found with the aid of figure 1.22a which gives the expression
um = uom − im Z eq
(1.24)
which takes advantage of the fact the current phasor im has been purposely
chosen as real. The computation of the stator current phasor is is carried
out with the aid of figure 1.17, which shows that the stator current may be
found using
is =
us − um
Zs
(1.25)
where the phasors um , us are found using equation (1.24) and equation (1.20)
respectively. The Heyland diagram for the saturated machine requires access
to the normalized current ins as introduced in the previous section which is
of the form ins = is / (ûs/ωs Lσ ), where ûs represent the supply phasor which
was taken to be real in previous proceedings. In this section the supply
phasor is not assumed real, in order to simplify the mathematical handling
hence the normalization must be undertaken with respect to actual supply
phasor us . The Heyland diagram given in figure 1.19(a) shows the normalized
stator current over the slip range −1 → 1 for the 22kW machine used in
this chapter. For each slip value of this locus the current and torque are
calculated using the approach outlined above. Also included in the Heyland
diagram is the normalized current locus for the unsaturated machine. This
locus may be found using the mathematical approach given in section 8.3.6,
23
page 273 [1]. Alternatively this locus may be found by replacing the nonlinear ψm (im ) function of the machine (as given in equation (1.23)) with
the relationship ψm = Lm im , where Lm represents the (linear) magnetizing
inductance of the 22kW machine. Also shown in figure 1.19 is the normalized
(a) Normalized stator current phasor
(b) Torque versus speed
Figure 1.19: Steady-state characteristics of voltage source connected asynchronous machine, with main inductance saturation
torque speed characteristic of the saturated machine as derived with the
aid of equation 8.33, page 275 [1] ) and the phasors us , is as calculated in
this section. The torque normalization used is conform the approach given
in the previous section. A comparison between the torque/speed curve of
the non-saturated machine, which has also been included in figure 1.19(b),
learns that the torque is only marginally affected by saturation of the main
inductance. This in contrast to the stator current which has (in this particular
machine) been significantly affected by saturation, as may be observed from
the Heyland diagram. The tutorial given in section 1.4.4 provides further
insight with regard to the computation steps and M-file linked to the steadystate analysis of the saturated 22kW machine.
1.3
Modeling machines for operation under asymmetrical conditions
In some cases there is a need to consider the machine under asymmetric
conditions, i.e where, for example, the sum of the stator currents is not
24
zero, as is the case in certain types of soft-starters[3]. Alternatively the user
may for example need to consider the implication on the current and torque
in case one of the stator machine phases is disconnected. In each of these
cases the sum of the stator currents is no longer zero, which implies that
a homopolar current will occur which needs to be accommodated in the
machine model. Furthermore the homopolar inductance of the machine must
be incorporated in this extended dynamic model of the inductance machine
which is to be developed in this section.
A convenient platform for the development of a dynamic model which can
accommodate asymmetrical conditions is the generic machine model according to figure 1.4, with LσR = L0σR + L1σR (located on the stator side) and
Rp1 → ∞, i.e skin effect ignored in this case. For squirrel cage machines
as considered here, no homopolar conditions will occur on the rotor side,
which implies that the space vector rotor related modules shown in figure 1.4
remains unchanged. The stator circuit must however be redefined by introducing stator phases variables, rather then space vectors. This approach
may be readily initiated by considering expression (8.18) [1] which may also
be written as
h
h
~is
i
~iR
i
=
i
1 h
1 − a1 LLmr
σu Ls
=
1 h
σu Ls
1
a
|
Lm
Lr
{z
L−1
B
− a12
"
Ls
Lr
~s
ψ
~R
ψ
i
#
(1.26a)
"
~s
ψ
~
ψR
#
(1.26b)
}
The stator based equation (1.26a) can be directly converted to a three-phase
equivalent format namely
1
σu Ls isi = ψsi −
a
Lm
ψRi ; i = 1, .., 3
Lr
(1.27)
Observation of figure 1.4 learns that the IRTF module generates a stator
based rotor flux-linkage space vector which may be converted to three phase
o , ψ o , ψ o using a three to two conversion module. Note that
variables ψR1
R2
R3
the rotor flux phase variables in question are provided with a superscript
‘o’ to indicate the fact that their algebraic sum will be zero, i.e there is no
zero-sequence component present. The ‘universal’ rotor flux-linkage variable
ψRi , for i = 1, ..3 can with the aid of equation (8.8b) [1] be written as
ψRi = aψri , where a represent the ‘universal’ transformation variable. The
introduction of a zero-sequence rotor flux component may be undertaken by
25
o , as shown in
adding a term Lhom ihom to each rotor flux-linkage variable ψri
equation (1.28)
o
ψRi = a ψri
+a Lhom ihom ; i = 1, .., 3
(1.28)
| {z }
o
ψRi
where Lhom represent the homopolar inductance of the machine and ihom
the homopolar stator current which may be written as
ihom =
1
(is1 + is2 + is3 )
3
(1.29)
Subsequent use of equations (1.28), (1.29) with equation (1.27) and grouping
the stator current phase terms allows the latter expression to be written as



σu Ls + LH
LH
lH

LH
σu Ls + LH
LH


ψs1 −
LH
is1


 
LH
  is2  =  ψs2 −

σu Ls + LH
is3
ψs3 −
1
a
1
a
1
a
Lm
ψo
Lr R1
Lm
ψo
Lr R2
Lm
o
Lr ψR3




(1.30)

where the variable LH = 13 LLmr Lhom is introduced to facilitate the readability of the expression in question. Equation (1.30) also shows the transformation variable a which may be conveniently set to a = LLmr (which implies
transformation to a ‘rotor flux’ based model), as discussed in section 8.3.2.1
[1], which further simplifies the expression in question. Under these circumstances the term σu Ls reduces to LσS , which represents the stator based
combined leakage inductance of the machine. The process of determining
an explicit expression for the stator phase currents may be undertaken by
inverting the inductance matrix which leads to equation (1.31).



L
+2L
σS
H
is1
H
1  LσS +3L


LH
 is2  =
 − LσS +3LH
LσS
H
is3
− LσSL+3L
H
|
H
− LσSL+3L
H
LσS +2LH
LσS +3LH
H
− LσSL+3L
H
{z
H
o
− LσSL+3L
ψs1 − ψR1
H


LH
o (1.31)
− LσS +3LH   ψs2 − ψR2

o
LσS +2LH
ψ
−
ψ
s3
R3
L +3L

σS

H
}
L−1
A
Shown in equation (1.31) is the matrix gain module L−1
A which must be
generated for the dynamic model of the machine. Inputs to this module are
o , i = 1, ..3 where the rotor flux-linkage phase
the phase variables, ψsi , ψRi
variables are provided by a two to three phase conversion module as shown
in the dynamic model of the machine given in figure 1.20. The corresponding
26
matrix L−1
B as defined by equation (1.26b) reduces to expression (1.32) in
case the transformation variable is set to a = Lm/Lr .
h
~iR
i
i
1 h
1 − LLMs
=
LσS
|
{z
L−1
B
"
~s
ψ
~
ψR
#
(1.32)
}
Note that the gain module L−1
A is reduced to a diagonal matrix with coefficients L1σ in case the homopolar inductance is set to zero. In the event
that the operating conditions are such that no-zero sequence current can
occur then the dynamic behavior of the model according to figure 1.20 will
be identical to that obtained with the model given in figure 8.15, page 259
for p = 1 (2 pole machine) [1]. The ‘red’ wide lines shown in figure 1.20
Figure 1.20: Generic model representation of a IRTF based induction machine
with homopolar inductance
represent colom matrices which in turn represent the stator voltage/current
and stator/rotor flux-linkage phase variables. These lines should not be
confused with the ‘green’ and ‘blue’ space vector lines, as used for example
to depict the rotor current space vector ~iR . The latter is calculated with the
aid of the gain matrix L−1
B as defined by equation (1.32).
The tutorial given in subsection 1.4.5 is linked to the material presented in
this section. In this tutorial the line start of a delta connected 22kW prototype machine (as used for previous examples in this chapter), is considered,
27
whereby a single stator phase is purposely open-circuited. For simulation
purposes this type of machine fault is simulated by setting one of the stator
phase resistance values to a value which is notably higher then the actual
value Rs of the machine, as discussed in the tutorial. An example of the
results achieved with this Caspoc based simulation model, as given in figure 1.21, show the instantaneous torque, shaft speed and line current as
function of time. Furthermore the homopolar stator current is also added to
this figure (see subplot (c), ‘red’ trace) in order to demonstrate the ability of
the model to handle asymmetric conditions.
Tmec (Nm)
400
200
0
−200
0
0.2
0.4
0.6
0.8
1
(a) time (s)
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1
(b) time (s)
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1
(c) time (s)
1.2
1.4
1.6
1.8
2
(rpm)
2000
mec
1000
n
0
−1000
iline, ihom(A)
400
200
0
−200
−400
Figure 1.21: Line start example of a delta connected machine, with one phase
disconnected
28
1.4
Tutorials for this Supplementary chapter
1.4.1
Tutorial 1: Line start of induction machine with skin
effect
This tutorial is concerned with extending the dynamic model discussed in
tutorial 1 in order to accommodate rotor skin effect. For this purpose it
is prudent to make use of the machine parameters presented in table 8.2
(tutorial 8.6.6 of [1]) given the need to develop a universal based dynamic
model as discussed in section 1.1.1 where the transformation coefficient Γa
is now used a ‘tuning’ parameters for this purpose of this tutorial choose
a value of Γa = 0%. The machine remains delta connected and the three
phase grid parameters remain unaltered for this tutorial in order to facilitate
a comparison with the machine without skin-effect. Consequently the same
load conditions will be assumed and the same waveform plots are to be
derived from this Caspoc based simulation.
A possible solution to this tutorial may be realized by implementing the
generic diagram according to figure 1.4 which requires access to the inverse
inductance matrix L−1 parameters as well as the parameters Rp1 and L1σR .
With the present choice of transformation coefficient, namely a = 1 the
parameter set given in table 1.1 may be found with the aid of table 8.2 [1]
and the computational approach outlined in section 1.1.1. The Caspoc based
Parameters
Magnetizing inductance
Leakage inductance
Leakage inductance
Leakage inductance
Rotor resistance
Rotor resistance
Stator resistance
LM
LσS
L0σR
L1σR
RR
Rp1
Rs
Value
260.7 mH
11.7 mH
3.4 mH
8.3 mH
0.5377 Ω
1.361 Ω
0.5250 Ω
Table 1.1: Machine parameters for a universal model, with a = 1
model as given in figure 1.22 clearly shows the inverse inductance matrix on
the stator side of the IRTF module. The required parameters for this module
are: LM , LR = LM + L0σR and Ls = LM + LσS , which may be derived using
table 1.1. Furthermore the rotor circuit is provided with a set of additional
modules as required to model ‘second order’ skin effect. A set of SCOPE
29
Excitation
u
Frequency
50
@
s
314.159
supply
415
1
RMS
i
I
x
supply
54.965
2
U
supply
239.600
x
supply
415
u
supply
31.734
RMS
Real
Power(Invariant)
@
1
P
supply
19.379k
i
Supply
415
u
P
Asynchronous machine, with skin effect
u
supply
415
@
s
2.243
u
s
718.801
1
_
IRTF-Current
@
R1
2.053
@
1
xy
@
R1
2.053
R
p1
_
1
L
-1
1
r
@
R
s
L
i
i
s
31.734
R
29.437
i
i
i
supply
54.965
2
3
s1
-23.672
2
P
3
T
e
T
_
em
120.006
Power Invariant
1
0
SCOPE1
i
s1
-23.672
R
R
1
xy
i
R
29.437
SCOPE2
@
me
153.175
1
@
me
153.175
T
L
120
SCOPE3
T
em
120.006
J
SCOPE4
@
s
2.243
Figure 1.22: Simulation of connected asynchronous machine, with 2nd order
skin effect
30
modules are shown in this figure, which are used to display the results of
the simulation. These results are, for the sake of readability, represented
by a set of MATLAB subplots, as may be observed from figure 1.23. The
effect of varying the transformation coefficient on the simulation results may
be examined by recomputing the parameters in table 1.1 (where the stator
resistance is unaffected) for a given value of a. The revised parameter set
must then be used with the Caspoc simulation.
1.4.2
Tutorial 2: Steady state characteristics, grid connected
induction machine with skin effect
The tutorial given in section 8.6.8 [1] was concerned with the steady-state
analysis of the asynchronous machine without skin effect. A similar analysis
is to be undertaken in this tutorial, with the aid of a model that is able to
accommodate ‘second order’ skin effect as discussed in section 1.1.1. For the
purpose of the analysis use the same machine parameters and excitation as
given in section 8.6.6 (table 8.2),[1]. Provide your results in the form of a
set of subplots as presented for the previous (without skin effect) case (see
figure 8.48 [1]). In addition provide an m-file for this problem which outlines
the phasor based analysis of a machine with skin-effect.
A suitable starting point for this analysis is the equivalent circuit model of
the machine as shown in figure 1.2 which has been adapted to accommodate
so called ‘second order’ skin-effect. The parameters for this model are given
by table 1.1 and correspond to a transformation coefficient value of Γ = 0%.
Note that for the analysis undertaken here the transformation coefficient
is used as a tuning parameter as discussed with the aid of figure 1.8 hence
the MATLAB analysis should be able to recompute the parameters of the
model for different values of the transformation coefficient. Computation
of the stator current is proceeds along the lines discussed for the machine
without skin-effect. In this case the terminal impedance needs to be adapted
to handle the additional skin-effect based elements of the model. Computation of the electro-mechanical torque may undertaken with the aid of
equation 8.33, page 275 [1], which in turn leads to the mechanical torque.
The machine in question is connected in delta ( unchanged with respect to
the tutorial without
skin-effect) hence the input voltage phasor is of the
√
form us = 415 3. The M-file as shown at the end of this tutorial provides
shows in details the mathematical steps required to calculate the steady-state
performance characteristics of the machine as given in figure 1.24 for slip
range of s : 1 → 0.
31
SCOPE 1
400
is1
(A)
200
0
−200
0
0.2
0.4
0.6
0.8
1
SCOPE 2
1.2
1.4
1.6
1.8
2
200
(rad/s)
150
100
50
0
ωme
0
0.2
0.4
0.6
0.8
1
SCOPE 3
1.2
1.4
1.6
1.8
2
400
(Nm)
Te
200
0
−200
0
0.2
0.4
0.6
0.8
1
SCOPE 4
1.2
1.4
1.6
1.8
2
6
(Wb)
ψs
4
2
0
0
0.2
0.4
0.6
0.8
1
t(s)
1.2
1.4
1.6
1.8
2
Figure 1.23: Simulation results of grid connected asynchronous machine,
with 2nd order skin effect
32
140
2.5
2
100
flux ψs (Wb)
RMS line current (A)
120
80
60
1.5
1
40
0.5
20
0
0
500
1000
(a) shaft speed (rpm)
0
1500
0
500
1000
(b) shaft speed (rpm)
1500
500
1000
(d) shaft speed (rpm)
1500
4
3
output power pout (W)
mech. torque Tem (Nm)
200
150
100
50
0
0
500
1000
(c) shaft speed (rpm)
2.5
2
1.5
1
0.5
0
1500
x 10
0
Figure 1.24: Steady-state characteristics of 22 kW asynchronous machine,
with skin-effect
A comparison between the results shown above and those calculated for
the machine without skin-effect (see figure 8.48, page 302 [1]) clearly shows
that the low shaft speed (high slip) region of operation is severely affected by
skin-effect. The vertical ‘red’ line shown in these plots represent the rated
speed of the machine. As mentioned earlier the transformation parameter
may be varied to ‘tune’ the model to, for example, measured results.
M-file code:
%Tutorial 2, chapter 1, supplementary
%clear all
%%%%%2nd order skin effect parameters
% 22kW machine, delta connected
VsR=415;% RMS line voltage
33
IsR=33.4;%%rated RMS line current
p=2;% four pole machine
nR=1465;%rated shaft speed (RPM)
fs=50;%supply frequency (Hz)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
ws=2*pi*fs;%electrical freq rad/sec
wm=p*2*pi*nR/60;% rated electrical shaft freq
%%%%parameters
Ls=272.4e-3;%stator inductance
Lm=260.7e-3;%magnetizing inductance
Lsigs=11.7e-3;%stator leakage inductance
Lsigr=11.7e-3;%rotor leakage inductance
Rr=0.5377;%rotor resistance
Rs=0.525;%stator resistance
Lr=Ls;%assumption of equal leakage stator rotor
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%supply data
isR=sqrt(3)*IsR/sqrt(3);% power invariant vector is value
usR=sqrt(3)*VsR;%power invariant machine voltage vector
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%a=1;%transformation variable used to calculate universal parameters
Av=0;%percentage change from a=1 pos and negative
if Av>=0
a=1+Av/100*(Ls/Lm-1)
else
a=1+Av/100*(1-Lm/Lr)
end
Lssig=Lm*(Ls/Lm-a);%stator leakage inductance
Lrsig=a*Lr*(a-Lm/Lr);%rotor leakage inductance
LM=a*Lm;%magnetizing inductance
RR=a^2*Rr;%rotor resistance
%%%%%%%%%%%%
LssigREF=Lm*(Ls/Lm-Lm/Lr);% reference stator leakage inductance a=Lm/Lr
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
%%%%%%%%second order skin effect
lrsig0=0.2896;
lrsig1=0.7104;
rp1=2.532*1;
%%%%%%%%%new rotor parameters
Lrsig0=lrsig0*Lrsig;%leakage not affected by skin effect
34
Lrsig1=lrsig1*Lrsig;%leakage affected by skin effect
Rp1=rp1*RR;% resistance parallel to Lrsig1
%%%%%%%%%%%%%%%%%calculate steady state results
close all
%figure
slip=1:-0.001:0.001;% choose slip range
wsyn=ws/p; %synchronous speed machine
wm=wsyn*(1-slip);
nm=wm*60/(2*pi);%mechanical shaft speed range
RRv=(RR./slip)’;
Rpv=(Rp1./slip)’;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
Zrot=j*ws*Lrsig0+j*ws*Lrsig1*Rpv./(j*ws*Lrsig1+Rpv)+RRv;
Zin=(Rs+j*ws*Lssig)+j*ws*LM*Zrot./(j*ws*LM+Zrot);%input phase impedance
Is=usR./Zin;% RMS stator line current
Imax=max(abs(Is));
Tem=p*((real(conj(usR).*Is) -conj(Is).*Is*Rs))/ws;%mechanical torque
Temax=max(Tem);
po=Tem.*wm’;%mechanical output power
pomax=max(po);
psi_s=(usR-Is*Rs)/(j*ws);%stator flux
psimax=max(abs(psi_s));
figure
subplot(2,2,1)
plot(nm,abs(Is))
grid
hold on
plot([nR nR],[0 Imax],’r’)
xlabel(’(a) shaft speed (rpm)’)
ylabel(’RMS line current (A)’)
%%%%%%%%%%%%%%%%%%%%%%
subplot(2,2,2)
plot(nm,abs(psi_s))
grid
hold on
plot([nR nR],[0 psimax],’r’)
xlabel(’(b) shaft speed (rpm)’)
ylabel(’flux \psi_s (Wb)’)
%%%%%%%%%%%%%%%%%%%%%%%%%%%
subplot(2,2,3)
35
plot(nm,Tem)
grid
hold on
plot([nR nR],[0 Temax],’r’)
xlabel(’(c) shaft speed (rpm)’)
ylabel(’ mech. torque T_{em} (Nm)’)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
subplot(2,2,4)
plot(nm,po)
grid
hold on
plot([nR nR],[0 pomax],’r’)
xlabel(’(d) shaft speed (rpm)’)
ylabel(’output power p_{out} (W)’)
%%%%%%%%%%%%%%%%%%
1.4.3
Tutorial 3: Grid connected induction machine with
main inductance saturation
This tutorial is concerned with a dynamic model adaptation for the experimental 22 kW machine used for previous tutorials. In particular this section
looks to the steps which must be undertaken to accommodate saturation of
the magnetizing inductance as discussed in section 1.2. A delta connected
machine is again assumed, which is consistent with the dynamic modeling
simulation sequence introduced in previous tutorials in this chapter. A line
start is to be undertaken for the machine in question in which case the
torque and line current waveforms can be examined and compared to a
machine where saturation is not included. According to the theory presented
in section 1.2 it is important to consider a universal model representation
of the machine with a transformation coefficient value of Γ = 0%, i.e a = 1
the parameters of which are given in table 8.2, page 298[1]. The reader is
reminded that for the case a = 1, the following holds in terms of the machine
parameters: LσS = Lσs , LσR = Lσr , LM = Lm (when saturation is ignored)
and RR = Rr . In this case the relationship between magnetizing flux ψm
and im is taken to be non-linear as shown in figure 1.25 which also shows
the linear case with Lm = 260.7 mH. For the purpose of this tutorial a
saturation curve has been introduced with a degree of non-linearity which is
greater then found in the actual experimental machine. The reason for this
is to better visualize the effects of magnetic saturation. The magnetization
characteristic of the machine as measured or predicted using a finite- element
36
2
1.8
1.6
Magnetizing flux ψm (Wb)
1.4
1.2
1
0.8
0.6
linear representation
spline representation
experimental data
0.4
0.2
0
0
10
20
30
Magnetizing current im (A)
40
50
60
Figure 1.25: Linear and non-linear magnetization curves for 22 kW machine
j ; j = 1..N .
approach is usually in the form of a set of data points ijm , ψm
This data is normally presented in an ‘amplitude invariant format’, which
must be adapted given that space vector based models in this book are used
in the ‘power invariant’ format. For the purpose of mathematical handling
use is made of ‘cubic B-splines’ to represent and fit the data. The M-file
code: data-fit given at the end of this section shows how this process may
be realized. This file generates two ‘asci’ files which contain the spline array
variables ‘coefs’ and ‘knotso’ which represents the discrete input data. Note
that this M-file requires access to the MATLAB ‘Spline’ toolbox.
The process of determining the value of the magnetizing current im for
a given value of the flux ψmo as shown in figure 1.13 is undertaken with the
aid of a ‘function’ M-file:‘FIMzero’ given at the end of this section. The function file makes uses of the spline based function ψm (im ) and equation (1.15)
in order to evaluate expression (1.33).
F IM zero(im ) = ψmo − Leq im − ψm (im )
(1.33)
The MATLAB scalar nonlinear zero finding routine ‘fzero’ routine makes use
of the function file ‘FIMzero’ in order to find the value of the magnetizing
current which corresponds to the condition F IM zero(im ) = 0, for a given
37
value of ψmo . The M-file ‘IMPSIMO’, also given at the end of this section,
calculates the function im (ψmo ) as required for the lookup table shown in
figure 1.14 for the experimental machine in use. The relationship im (ψmo )
as shown in figure 1.26 is represented in a look-up table format as required
for the simulation model linked to this tutorial.
45
40
Magnetizing current im (A)
35
30
25
20
15
10
5
0
0
0.5
1
1.5
Flux−linkage ψmo (Wb)
2
2.5
Figure 1.26: Relationship im (ψmo ) for 22 kW machine
The look-up table is in turn used with a Caspoc sub-module ‘PSICIM’ which
implements the generic diagram responsible for generating the relationship
ψm (ψmo ), (see figure 1.14) for the machine in question. With the introduction of this module the remaining steps needed to arrive at a complete
dynamic model, with main inductance saturation are relatively transparent
as may be observed from the generic model according to figure 1.15. The
simulation model given in figure 1.27 clearly shows the non-linear fluxmodel
discussed earlier as well as the other modules depicted in figure 1.15. Note
that the results generated by this model, as shown in figure 1.31, will be
identical to those obtained with the model according to tutorial 8.6.7, page
301 [1], in the event that the relationship ψm (ψmo ) is defined by expression 1.16. Under these circumstances the models are only different in terms
of the transformation coefficient in use. For tutorial 8.6.7, page 300 [1], the
value is Γ = −100 % whereas in this case the value is Γ = 0 %.
38
Excitation
u
Frequency
50
@
s
314.159
supply
415
1
RMS
i
I
x
supply
79.707
2
U
supply
239.600
x
supply
415
u
supply
46.019
RMS
Real
Power(Invariant)
@
1
P
supply
19.962k
i
Supply
415
u
P
Asynchronous machine, with main inductance saturation
u
@
s
2.242
u
supply
415
s
718.801
IRTF-Current
@
R
1.789
1
1
_
2
@
xy
@
R
1.789
1
1
_
2
R
s
R
R
0<@<2@
r
@
@
r
R
33.543
_
r
Non linear
FLux function
m
24.904
1
L
@S
i
1
L
T
e
T
_
em
120.003
i
2
3
s1
-51.895
2
1
P
@R
s
46.019
supply
79.707
i
i
i
i
xy
R
33.543
i
r
1
3
0
J
@
me
152.038
T
L
120
Power Invariant
SCOPE1
i
s1
-51.895
SCOPE2
@
me
152.038
SCOPE3
T
SCOPE4
@
s
2.242
em
120.003
Figure 1.27: Simulation model of grid connected asynchronous machine, with
main inductance saturation
M-file code: data-fit, used to represent the input data in the form of
a spline:
%Spline for mag-curve IM machine
close all
clear all
Lm=260.7e-3;%magnetizing inductance
imexp=[0 5 10 15 20 25 30 35 40 45 50 55 60];
psimexp=[0 1.09 1.28 1.39 1.49 1.57 1.64 1.70 1.75 1.81
%%%%%%%%%%%%%%%%%%%%%%%%%%
psim=Lm*imexp;
plot(imexp,psim,’r’)
hold on
immax=max(imexp);
% Choose break points and determine knots for cubic spline fit
knots=[0 5 7 immax];
knotsx=augknt(knots,4);
39
1.86
1.9
SCOPE 1
400
is1
(A)
200
0
−200
0
0.2
0.4
0.6
0.8
1
SCOPE 2
1.2
1.4
1.6
1.8
2
200
(rad/s)
150
100
50
0
ωme
0
0.2
0.4
0.6
0.8
1
SCOPE 3
1.2
1.4
1.6
1.8
2
200
Te
(Nm)
100
0
−100
−200
0
0.2
0.4
0.6
0.8
1
SCOPE 4
1.2
1.4
1.6
1.8
2
6
(Wb)
ψs
4
2
0
0
0.2
0.4
0.6
0.8
1
t(s)
1.2
1.4
1.6
1.8
2
Figure 1.28: Simulation results of grid connected asynchronous machine,
with main inductance saturation
40
% Do a least square cubic spline fit to Iv, Phi curve
sp=spap2(knotsx,4,imexp’,psimexp’);
% interpolate a new spline to force the flux curves through zero at zero
% current
cur=[0 knots knots(4)];
flval=fnval(sp,knots);
fl=[0 flval(2)/knots(2) flval(2:4) (flval(4)-flval(3))/(knots(4)-knots(3))]
sp1=spapi(knotsx,cur,fl);
fnplt(sp1);
hold on
plot(imexp,psimexp,’x g’)
xlabel(’Magnetizing current i_m (A)’)
ylabel(’Magnetizing flux \psi_m (Wb)’)
grid
[knotso,coefs,n,k,d]=spbrk(sp1);%spline coefficients
axis([0 immax 0 2])
legend(’linear representation’, ’spline representation’, ’experimental data
save coeffM.dat coefs -ascii
save knotsM.dat knotso -ascii
%note:reassemble spine using
spn=spmak(knotso’,coefs);
vals=fnval(spn,10);%value at im=10
M-file code: for function: FIMzero
%Function to calculate value of im
function F=PsiZeroim(xx)
%%zxx is the im value to be found
%used for the dynamic model
%run file Satspline to obtain coeffients and knots for psi(im)
global psimo Lsigr Lsigs Lm
coefs=[0.0000 0.3658 1.1271 1.7072 1.7608
2.0369];
knotso=[0
0
0
0 5.0 7.0 63.6 63.6 63.6 63.6];
imm=xx;
%set up spline
spn=spmak(knotso,coefs);
psimSpline=fnval(spn,imm);%spline representation of sat curve
%psimSpline=Lm*imm;%linear psim to compare to non linear above
Leq=Lsigr*Lsigs/(Lsigr+Lsigs);%equivalent inductance
KIn=psimo-Leq*imm;%linear equation
F=KIn-psimSpline;
41
M-file code :IMPSIMO used to produce look-up table for CASPOC
submodel ‘non linear flux function’.
%clear all
%close all
%%%%%saturation of machine,
%use of amplitude invariant flux/current curve with power invariant model
% 22kW machine, delta connected
% dynamic machine model
global psimo Lsigr Lsigs Lm
%%%%parameters
Ls=272.4e-3;%stator inductance
Lm=260.7e-3;%magnetizing inductance
Lsigs=11.7e-3;%stator leakage inductance
Lsigr=11.7e-3;%rotor leakage inductance
Rr=0.5377;%rotor resistance
Rs=0.525;%stator resistance
Lr=Ls;%assumption of equal leakage stator rotor
pmoV=[];
imV=[];
CCpsi=[];
for k=0:1:101
psimo=k/100*2;%range of no-load flux amplitude
im= fzero(’PsiZeroim’,0);
imV=[imV;im];
pmoV=[pmoV;psimo];
Cpsi=[psimo im];
CCpsi=[CCpsi;Cpsi];
end
plot(pmoV,imV,’r’)
xlabel(’Flux-linkage \psi_{mo}’)
ylabel(’Magnetizing current i_m (A)’)
%%%%make lookup table for Caspoc
save psi0im.dat CCpsi -ascii
1.4.4
Tutorial 4: Steady state characteristics, grid connected
induction machine with main inductance saturation
The aim of this tutorial is to outline the mathematical steps needs to arrive at
the steady-state performance characteristics of the mains connected 22 kW
42
machine used in previous tutorials in the event that the main inductance is
taken to the non-linear. A detailed steady-state qualitative analysis for a machine with main inductance saturation has been undertaken in section 1.2.1
henceforth the proposed modeling concepts will be based on this theory.
As with the dynamic analysis presented in the previous tutorial use is
again made of the (amplitude invariant) magnetization curve as shown in
figure 1.25. However in this case the non-linear main inductance is part of
a steady-state model as given in figure 1.17
√ which assumes a given supply
voltage space vector amplitude |us | of 415 3 V that is consistent with the
value used for the unsaturated case (see section 8.6.8, page 301 [1]). Use of
this supply vector with the same machine concept, with exception of the
magnetizing inductance, is helpful in terms of comparing the performance
characteristics. The steady-state model in question may be replaced by a
Thevenin type model according to figure 1.18 which is used to obtain the
magnetizing current im for a given value of the slip. For this purpose expression (1.23) must be solved numerically using the function zeroimrev(im ) as
defined by expression (1.34) which is used by a MATLAB scalar nonlinear
zero finding routine ‘fzero’, as introduced in the previous tutorial for the
dynamic analysis of the machine with main inductance saturation.
n
= Z eq
= {uom }
zeroimrev(im ) =
− im
ωs
ωs
o
− ψm (im )
(1.34)
Expression (1.34) requires knowledge of the phasor uom and impedance Z eq
which are found using equation (1.22) and equation (1.19) respectively. The
m-file linked to expression (1.34) is given at the end of this section. Once
the value of the magnetizing current has been found, using the approach
outlined above the phasors uom and um are calculated using equation (1.22)
and equation (1.24). Computation of the stator current is using expression (1.25) completes the steady-state analysis as outlined in the m-file given
in the sequel of this section. This m-file is also used to plot the performance
characteristics as presented in figure 1.29. Furthermore this m-file was used
to calculate the normalized Heyland and torque/slip curves as given by
figure 1.19. A comparison between the performance characteristics presented
in this section and those obtained with the machine without saturation (see
figure ??) underline the comments made in section 1.2.1 that saturation
effects marginally affect the torque. However in the low slip operating region
the influence of saturation becomes more apparent as may be observed from
the line current subplot given in figure 1.29.
43
2.5
80
2
flux ψs (Wb)
RMS line current (A)
100
60
40
20
0
1.5
1
0.5
0
500
1000
(a) shaft speed (rpm)
0
1500
0
500
1000
(b) shaft speed (rpm)
1500
500
1000
(d) shaft speed (rpm)
1500
4
3
output power pout (W)
mech. torque Tem (Nm)
200
150
100
50
0
0
500
1000
(c) shaft speed (rpm)
1500
x 10
2.5
2
1.5
1
0.5
0
0
Figure 1.29: Steady-state characteristics of 22 kW asynchronous machine,
with main inductance saturation
M-file: steady-state analysis of machine with saturation.
%Tutorial 4, Supplementary chapter
%revised to simplify maths by setting
%clear all
%%%%%saturation of machine,use of
% 22kW machine, delta connected
global Zeq ws usRampl km Lm
VsR=415;% RMS line voltage
IsR=33.4;%%rated RMS line current
p=2;% four pole machine
nR=1465;%rated shaft speed (RPM)
44
im real
fs=50;%supply frequency (Hz)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
ws=2*pi*fs;%electricalfreq rad/sec
wm=p*2*pi*nR/60;% rated electrical shaft freq
%%%%parameters
Ls=272.4e-3;%stator inductance
Lm=260.7e-3;%magnetizing inductance
Lsigs=11.7e-3;%stator leakage inductance
Lsigr=11.7e-3;%rotor leakage inductance
Rr=0.5377;%rotor resistance
Rs=0.525;%stator resistance
Lr=Ls;%assumption of equal leakage stator rotor
LssigREF=Lm*(Ls/Lm-Lm/Lr);% reference stator leakage inductance a=Lm/Lr
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%supply data
isR=IsR;% line current vector value
usRampl=sqrt(3)*VsR;% supply voltage vector amplitude
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%calculate steady state results
%close all
%figure
isV=[];nmV=[];Nm=[];psiM=[];usRV=[];
for slip=0:0.001:1;%choose slip range 0 to 1 or -1 to -1 for chapter plots
%slip=0.0001;
if slip == 0
continue
else
wsyn=ws/p; %synchronous speed machine
wm=wsyn*(1-slip);
nm=wm*60/(2*pi);%mechanical shaft speed range
Rrv=(Rr/slip)’;
%%%%%%%%%%%%%%%%%%calculate im nonlinear psim-im curve
Zs=Rs+j*ws*Lsigs;
Zr=Rrv+j*ws*Lsigr;
km=Zr/(Zr+Zs);
Zeq=Zs*Zr/(Zs+Zr);
im= fzero(’zeroimrev’,0);
%%%%%%compute the psim value
umor=im*real(Zeq);
umoi=sqrt((usRampl*abs(km))^2-umor^2);
45
umo=umor+j*umoi;%complex umo phasor
usR=umo/km;%complex supply phasor
um=umo-im*Zeq;% phasor voltage across Lm (must be imaginary)
psim=um/(j*ws);%flux vector
psiM=[psiM;psim];
Is=(usR-um)/Zs;% stator current complex form
isV=[isV;Is];nmV=[nmV;nm];usRV=[usRV;usR];
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
Tem=p*((real(conj(usRV).*isV)-conj(isV).*isV*Rs))/ws;%mechanical torque
TnREF=(usRampl)^2/(2*ws^2*LssigREF);
Ten=(Tem/p)/TnREF;
Temax=max(Tem);
psi_s=psiM+Lsigs*isV;% stator flux
psimax=max(abs(psi_s));
po=Tem.*nmV*2*pi/60;%mechanical output power
pomax=max(po);
%plot steady-state characteristics
Imax=max(abs(isV));
figure
subplot(2,2,1)
plot(nmV,abs(isV))
grid
hold on
plot([nR nR],[0 Imax],’r’)
xlabel(’(a) shaft speed (rpm)’)
ylabel(’RMS line current (A)’)
%%%%%%%%%%%%%%%%%%%%%%
subplot(2,2,2)
plot(nmV,abs(psi_s))
grid
hold on
plot([nR nR],[0 psimax],’r’)
axis([0 1500 0 2.5])
xlabel(’(b) shaft speed (rpm)’)
ylabel(’flux \psi_s (Wb)’)
%%%%%%%%%%%%%%%%%%%%%%%%%%%
subplot(2,2,3)
plot(nmV,Tem)
46
grid
hold on
plot([nR nR],[0 Temax],’r’)
xlabel(’(c) shaft speed (rpm)’)
ylabel(’ mech. torque T_{em} (Nm)’)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
subplot(2,2,4)
plot(nmV,po)
grid
hold on
plot([nR nR],[0 pomax],’r’)
axis([0 1500 0 3e4]);
xlabel(’(d) shaft speed (rpm)’)
ylabel(’output power p_{out} (W)’)
%%%%%%%%%%%%%%%%%%
%%%%plots for chapter figures
figure
Isn=isV./(usRV/(ws*LssigREF));% normalized stator rms line current
ix=real(Isn);%real part
iy=imag(Isn);%imag part
plot(ix,iy,’b’)
grid
axis equal
axis([-0.5 0.5 -1 0])
%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%plot normalised torque
figure
plot(nmV,Ten,’b’)
grid
axis([0 3000 -1 1])
M-file code: function ‘zeroimrev’:
%Function to calculate value of im
function F=zeroimrev(imm)
%%zxx is the im value to be found
%run fiule Satspline to get coeffients and knots for psi(im)
global Zeq ws usRampl km Lm
coefs=[-0.0000
0.3658
1.1271
1.7072
1.7608
2.0369];
knotso=[0
0
0
0
5.0
7.0
63.6
63.6
63.6
63.6];
imA=imm*sqrt(2/3);%amplitude invariant for use with magnetization curve
47
%set up spline
spn=spmak(knotso,coefs);
psimSpline=sqrt(3/2)*fnval(spn,imA);
%spline representation of sat curve factor sqrt(3/2) to
%return to power inv. form
%psimSpline=sqrt(3/2)*Lm*imA;%linear psim to compare to non linear above
umor=imm*real(Zeq);
umoi=sqrt((usRampl*abs(km))^2-umor^2);%positive value of sqrt,
%because inductive circuit
KIn=umoi-imm*imag(Zeq);
F=KIn/ws-psimSpline;
end
1.4.5
Tutorial 5: Grid connected induction machine with one
phase open-circuited
The purpose of this tutorial is to consider a Caspoc based model adaptation
for the 22 kW prototype machine which will allow the user to consider asymmetric stator based operating conditions according to the approach discussed
in section 1.3. For this type of simulation the homopolar inductance Lhom
is required for the machine in question. Furthermore the parameters for
a universal model representation of the 22 kW prototype machine, with
Γ = −100% are also required and these have been given earlier (see table 8.2,
page 298 [1], which includes a value of for the homopolar inductance). Central to modelling machines for asymmetric operating conditions is the need
to represent the stator based variables as individual scalar quantities as is
apparent from the generic diagram 1.20 where the latter are represented by
colom matrices such as for example [is ] which represent the stator currents
[is ] = [is1 is2 is3 ]T . Observation of figure 1.20 learns that the latter is generated with the aid of the matric gain module L−1
A as defined by equation (1.31).
In the simulation model as given in figure 1.30 the module is represented
by a submodule 1/LA module which has as inputs the phase flux variables
0 , i = 1, 2, 3. A set of Scope modules is used to present the results in
ψsi , ψRi
the simulation. For reader convenience a set of MATLAB based subplots, as
given in figure 1.31 is introduced. The same flux variables (used as input for
the 1/LA module), but in space vector format, are used to calculate the rotor
current ~iR , where use is made of the gain module L−1
B (see figure 1.20) as
defined by equation (1.26b). In the Caspoc based simulation this expression
is represented by 1/LB module and is used together with the space vectors
~s , ψ
~R in order to calculate the current vector ~iR . The rotor side of the
ψ
48
Excitation
Frequency
50
@
s
314.159
u
u
supply
415
1
s
718.801
2
@
1
Supply
415
Asynchronous machine, with scalar stator section
0
2
3
2
3
v
s1
v
-293.451
s2
v
-293.447
s3
586.899
_
Power Invariant
_
1
@
s1
-1.594
1
@
s2
1.536
1
_
_
3
_
@
s
2.300
2
3
2
Power Invariant
_
@
s3
-798.493m
@
R2
1.160
@
R1
-415.501m 2
@
3
R3
-744.749m
2
3
Power Invariant
IRTF-Current
@
R
1.440
@
i
R
s
R
s
i
s3
162.411m
s2
18.989
R
R
R
46.600
xy
R
46.600
1/L_A
i
i
i
s1
-49.165
T
_
em
115.663
su1
-68.154
1
i
zero
-10.005
0
SCOPE1
su1
-68.154
1
P
T
e
i
i
1
1/L_B
i
R
s
xy
@
R
1.440
0
SCOPE2
@
me
150.883
SCOPE3
T
em
115.663
J
@
me
150.883
T
L
120
SCOPE4
@
s
2.300
Figure 1.30: Simulation model of grid connected asynchronous machine,
suitable for handling asymmetric supply conditions
IRTF module is formed by a integrator and gain module with gain RR . Also
apparent for the simulation model is that the colom matrices have been
replaced by three individual stator phases in order to clearly identify the
individual stator phases and to introduce changes to accommodate specific
asymmetric conditions. For example in this tutorial the resistance of phase 1,
which is defined as Rs has been increased from its nominal value of 0.525 Ω
to 2000.0 Ω in order to simulate the effect of operating with two instead of
three phases during a line start. Note that simulations of this type are prone
to numerical instability in the event that the ‘open circuit’ resistance is not
chosen prudently. Note that the results from this simulation must match
those obtained with those given in figure 1.16(a) in case the ‘open circuit’
resistance is reset to its nominal value.
49
SCOPE 1
400
i
s1
iso
(A)
200
0
−200
0
0.2
0.4
0.6
0.8
1
SCOPE 2
1.2
1.4
1.6
1.8
2
(rad/s)
200
100
0
ωme
−100
0
0.2
0.4
0.6
0.8
1
SCOPE 3
1.2
1.4
1.6
1.8
2
400
(Nm)
Te
200
0
−200
0
0.2
0.4
0.6
0.8
1
SCOPE 4
1.2
1.4
1.6
1.8
2
6
(Wb)
ψs
4
2
0
0
0.2
0.4
0.6
0.8
1
t(s)
1.2
1.4
1.6
1.8
2
Figure 1.31: Simulation results of grid connected asynchronous machine,
with one phase open circuited
50
Bibliography
[1] R. D. Doncker, D. Pulle, and A. Veltman, Advanced Electrical Drives.
Springer, 2010.
[2] A. Veltman, The fish method: interaction between AC-machines and
switching power converters. Delft: Delft University Press, 1993.
[3] D. Pulle and A. Veltman, “Quantification of homopolar components in
machines connected to branch delta type soft-starters,” EPE2003, vol. -,
2003.
51
List of Figures
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12
1.13
1.14
1.15
1.16
Nyquist diagram of the complex admittance: gskin , g1 as
function of ωωc . . . . . . . . . . . . . . . . . . . . . . . . . . .
Universal, IRTF based induction machine model, with 2nd
order skin effect adaptation . . . . . . . . . . . . . . . . . . .
Nyquist diagram of the complex admittance: gskin , g2 as function of ωωc . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Generic model representation of a universal IRTF based induction machine with 2nd order skin-effect . . . . . . . . . .
Line start example of 22 kW delta connected machine . . . .
Equivalent circuit of a skin-effect adapted asynchronous machine and voltage source connected (steady-state version of
dynamic model in figure 1.2) . . . . . . . . . . . . . . . . . .
Steady-state characteristics of voltage source connected asynchronous machine, with skin-effect, model according to figure 1.6
Steady-state torque/speed curve with adjustable transformation coefficient Γa . . . . . . . . . . . . . . . . . . . . . . . . .
Rotor section of universal, IRTF based induction machine
model, with 3nd order skin effect adaptation . . . . . . . . .
Nyquist diagram of the complex admittance: gskin , g3 as
function of ω/ωc . . . . . . . . . . . . . . . . . . . . . . . . . .
Inductance circuit of IRTF based asynchronous machine model
Equivalent inductance circuit . . . . . . . . . . . . . . . . . .
Numerical determination of the magnetization flux and current
for a given value of ψmo . . . . . . . . . . . . . . . . . . . . .
Non-linear module ψm (ψmo ) . . . . . . . . . . . . . . . . . .
Generic model of asynchronous machine with main inductance
saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Line start example of 22 kW delta connected machine, with
and without main inductance saturation . . . . . . . . . . . .
52
3
4
7
8
9
10
11
12
13
15
16
17
18
18
19
21
1.17 Steady-state model of an asynchronous machine with main
inductance saturation . . . . . . . . . . . . . . . . . . . . . .
1.18 Steady-state equivalent Thevenin represenation . . . . . . . .
1.19 Steady-state characteristics of voltage source connected asynchronous machine, with main inductance saturation . . . . .
1.20 Generic model representation of a IRTF based induction machine with homopolar inductance . . . . . . . . . . . . . . . .
1.21 Line start example of a delta connected machine, with one
phase disconnected . . . . . . . . . . . . . . . . . . . . . . . .
1.22 Simulation of connected asynchronous machine, with 2nd order
skin effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.23 Simulation results of grid connected asynchronous machine,
with 2nd order skin effect . . . . . . . . . . . . . . . . . . . .
1.24 Steady-state characteristics of 22 kW asynchronous machine,
with skin-effect . . . . . . . . . . . . . . . . . . . . . . . . . .
1.25 Linear and non-linear magnetization curves for 22 kW machine
1.26 Relationship im (ψmo ) for 22 kW machine . . . . . . . . . . .
1.27 Simulation model of grid connected asynchronous machine,
with main inductance saturation . . . . . . . . . . . . . . . .
1.28 Simulation results of grid connected asynchronous machine,
with main inductance saturation . . . . . . . . . . . . . . . .
1.29 Steady-state characteristics of 22 kW asynchronous machine,
with main inductance saturation . . . . . . . . . . . . . . . .
1.30 Simulation model of grid connected asynchronous machine,
suitable for handling asymmetric supply conditions . . . . . .
1.31 Simulation results of grid connected asynchronous machine,
with one phase open circuited . . . . . . . . . . . . . . . . . .
53
22
22
24
27
28
30
32
33
37
38
39
40
44
49
50