Stand-Alone Renewable Energy Systems with a Squirrel Cage Induction
Generator
PED9 Report
by
David Tremblay
Under the Supervision of
Assoc. Prof. Florin Iov, Aalborg University
Uffe Jakobsen, Aalborg University
As Part of the Master of Engineering at
Université du Québec à Rimouski
Directed by
Prof. Jean-François Méthot, UQAR
4th December 2007
Aalborg, Denmark
Contents
1 Introduction
1.1
Stand-Alone Renewable Energy Systems
1.2
Chapter Organisation
2 Problem Identification
2.1
Problem Statement
I Squirrel-Cage Induction Generator
3 Basic Knowledge
3.1
Synchronous Speed
3.2
Slip
3.3
Rotor Frequency
3.4
Electrical Radians
3.5
Steady-State Model
3.6
Power Assessment
3.7
Torque Characteristics
4 Self-Excitation of an Induction Generator
5 Dynamic Modelling
5.1
ABC/abc Model Excluding the Saturation
5.2
dq Model Including the Saturation
6 Simulink Model
6.1
Model without Saturation Considerations
II Laboratory Tests
7 Theory
7.1
DC Test
7.2
No-Load Test
7.3
Locked Rotor Test
7.4
Determination of the Magnetizing Inductance
8 Evaluation of the Simulink Model
8.1
Model without Saturation Considerations
III Project Closure
9 Discussion
10 Conclusion
Bibliography
A dq0 Transformation
B Simulink Model without Saturation Considerations
C DC Test
D No-Load Test
E Locked Rotor Test
F DC Method to Determine the Magnetizing Inductance
G Results of the Simulation of the Model
2
3
3
6
7
7
9
10
10
11
11
12
12
13
14
16
18
18
23
28
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29
30
30
31
32
34
36
36
40
41
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52
67
70
79
84
87
Chapter 1
Introduction
1.1 Stand-Alone Renewable Energy Systems
Because the world population increases faster and faster, a large number of people moved
from massive agglomeration to more isolated areas and the progresses in transportation
and communication fields at the end of the 1980s are one of the reasons that helped this
movement. This enlargement of the population distribution led to a boost of the energy
demand, to supply the remote areas. By concerns of the quality of the environment and
limitation of the resources, new types of power plants, which are more efficient and less
polluting, became more and more popular. So, this phenomenon increases the researches
about these new generation systems, which are often micro hydro and wind energy
systems. A micro energy systems means that the system produces less than 100 kW of
power [1]. The micro energy systems are very popular because they are not legally
regulated as much as the bigger energy systems. Moreover, they are most of the time
stand-alone systems and are often used to reduce the energy cost of a particular user of
the public network.
The induction generators are often found to be more suitable for micro energy systems.
Indeed, they do not need a separate DC source [2] or special concerns related to
synchronisation [1]. They can work with variable speeds and have a self-protection
against overloads and short circuits [1]−[3]. The voltage and frequency of induction
generators can also be automatically regulated if added to a large distribution network
[1]. However, even if they have many advantages, they also have some drawbacks. For
example, they spend reactive power and have a poor voltage regulation under variable
speed operations [2], but the improvements of the last years concerning power electronics
and control systems make their use much simpler.
Because they are more practical for variable speed than squirrel-cage machines [4], the
wound rotor machines could be a good choice for small wind energy conversion systems.
3
Nevertheless, the squirrel-cage induction generators bring several advantages. First, the
rotor of squirrel-cage machines is more robust and solid [1],[2],[5]−[8]. The squirrel-cage
machines are also simpler [1],[7], less expensive [2]−[8] and are relatively smaller
[1],[2],[6]. Because they do not have any brushes, they require less maintenance
[2]−[5],[7],[8]. Finally, the squirrel-cage machines are more available [5].
As the induction generators need reactive power to maintain the terminal voltage
constant, they can be self-excited by switching capacitors (figure 1.1(a)) or pulse-width
modulation (PWM) converters (figure 1.1(b)). The switching capacitors are cheaper and
are suitable to regulate de voltage, but it appears that they do not have any considerable
effect on the frequency regulation [9]. As for the PWM converters, they can regulate both
the frequency and the voltage, but they are more expensive and are less rugged for standalone systems [9]. A combination of capacitor bank and PWM converter, like in figure
1.3, can also be used for the self-excitation of the induction generator.
Figure 1.1:
General bloc diagram of energy conversion systems self-excited by a
capacitor bank (a) or by power electronics (b).
In any of these generating systems using induction generators, a minimum air gap flux
linkage and load impedance are necessary to preserve the self-excitation [10]. When the
machine is operating under saturation conditions, the magnetizing inductance depends
upon the magnetizing flux [7], so the minimum impedance needed to maintain the
4
saturation effect is not constant. Thus, the effect of the saturation plays an important role
and must be considered for self-excited induction generator applications.
Figure 1.2:
Example of a stand-alone pico-hydro power generation system [11].
Figure 1.3:
Example of a stand-alone wind energy conversion system [6].
Like explained earlier, a large number of hydro and wind energy conversion systems
exist in the literature for micro energy systems. Figure 1.2 and 1.3 are 2 examples of
them. In figure 1.2, the rotor of the induction generator is pulled by a constant water flow.
Then, the capacitor bank supplies a part of the reactive power needed by the generator
and the electronic load controller supplies the rest, which depends upon the non constant
consumer load. Also, the electronic load controller and the control system provide a
constant voltage for the consumer load. In figure 1.3, the rotor of the induction generator
5
is that time pulled by a non constant wind, through a gear box that increase the rotational
speed. Then, a capacitor bank is also used to supply a part of the reactive power needed
by the generator. The rest of the reactive power is provided by the voltage source inverter
(VSI) that is used to regulate the terminal voltage of the load. Finally, the dump load is
used to ensure a constant frequency for the terminal voltage.
1.2 Chapter Organisation
This report is mainly separated in 4 parts. The first one concerns the problem
identification. This part contains the problem statement and the principal objectives of the
project. The second part is all about the squirrel-cage induction generators. In this part,
the basic knowledge concerning the squirrel-cage induction generators is presented.
Indeed, the synchronous speed, the slip, the rotor frequency, the electrical radians, the
steady-state model, the power assessment and the torque characteristics are discussed.
Then, the main theory of the self-excitation is briefly introduced. Afterward, 2 different
dynamic models are explained. The first one is an ABC/abc model without saturation
considerations and the second one is a dq model including saturation. At the end of that
part, the Simulink model of the ABC/abc model is showed. Then, the third part concerns
the laboratory tests. In this part, the theory of 4 laboratory tests is explained. The 4 tests
are the DC test, the no-load test, the locked rotor test and the magnetizing inductance
curve test. Thus, the Simulink model is evaluated using the measurements from the lab.
Finally, the fourth and last part of that report is about the project closure. This final part
contains a discussion on all results of the project and a conclusion that explains the future
work related to that project.
6
Chapter 2
Problem Identification
As it has been seen in the previous chapter, the squirrel-cage induction generators are
suitable for micro energy generating systems, but special cares most be taken to regulate
the voltage and the frequency. Indeed, when the generator is separated from the grid, the
magnitude and the frequency of the terminal voltage vary with the rotor speed [3]−[6],
the amount of excitation [6] and with the magnitude of the load and its power factor
[3]−[6]. Therefore, it is necessary to control these parameters to obtain quality energy,
and the implementation of a control system requires computer simulations, which
necessitate a quality induction generator model. For that reason, this project is all about
the modelling of a squirrel-cage induction machine.
2.1 Problem Statement
Accordingly, the aim of this project is to simulate a squirrel-cage induction machine with
Simulink. The resulting model must be useful for further studies about self-excitation and
voltage and frequency regulation, so the effect of the saturation is an important factor that
must be investigated, as stated in the previous chapter. To achieve this main purpose, the
following objectives are defined.
2.1.1 Dynamic Model Without Saturation Considerations
The first objective of that project is to develop the theory of the squirrel-cage induction
machine. So, the basic knowledge of these machines has to be studied to obtain a
dynamic model, which can be used to build a Simulink model. This first model will not
consider the saturation of the machine. After, typical tests will be done in lab to
determine the different parameters of the model of the machine. At the end of that stage,
the model will be compared with the lab measurements. The model will be accepted
when the stator currents of the model will be within 0.5% of the measured stator currents
7
in lab for no-load and locked rotor tests. This error corresponds to the unbalance allowed
for a power supply according to the IEEE Standards [12].
2.1.2 Dynamic Model Including Saturation Considerations
When the first model will be acceptable, as the saturation effect have a considerable
influence on the reactive power needed by the induction generator, an extension will be
added to the theory of the model to consider the effect of the saturation. Then, additional
tests will be done in lab to measure the saturation curve of the machine. A new model
will be built in Simulink and this model will be compared again with the lab
measurements. The model will be accepted when the stator currents of the model will be
within 0.5% of the measured stator currents in lab for no-load and locked rotor tests.
2.1.3 Study of the Self-Excitation
If the previous objectives are achieved and there is enough time to go through another
stage, the self-excitation of an induction machine will be studied. The variation of the
reactive power needed by the machine will be investigated and this variation will be
observed in lab. The purpose of this part of the project is to be able to compute the
reactive power required by the induction machine for stand-alone applications in any
running conditions.
8
Part I
Squirrel-Cage Induction Generator
9
Chapter 3
Basic Knowledge
The main objective of this chapter is to present the main characteristics of the squirrel
cage induction machine. Firstly, the synchronous speed, the slip, the rotor frequency and
the electrical radians are discussed. Afterward, the steady-state model is introduced.
Subsequently, the power flow through the machine is presented. Finally, the torque
characteristics of the induction machine are examined.
3.1 Synchronous Speed
The induction machines are found to be suitable for low power generation, notably
because of their robustness and their ability to operate in high speed applications [1]. The
rotor of these machines can be squirrel-cage or wound rotor, but the construction of the
first one is quite simpler. Independently of the rotor type, each induction machine has a
wounded stator in which a magnetic field turns around an axis perpendicular to the plane
of the machine. Because the speed of this rotational movement, called the synchronous
angular speed (ωsync), directly depends upon the electrical frequency fe of the stator
voltage, it can be expressed as:
ω sync =
4πf e
p
(3.1)
Where p is the number of poles per phase in the stator windings. It is also important to
state that the synchronous angular speed ωsync is expressed in rad/s. In case of a squirrelcage rotor, the magnetic field created by the stator winding induces a voltage in the shortcircuited rotor bars and the induced voltage depends upon the rotor speed. For example, if
the rotor is rotating at the synchronous speed, there is no induced voltage in the bars,
because the speeds of the magnetic field and the rotor are the same and there is no
induced torque on the rotor.
10
3.2 Slip
To simplify the relation between the movements of the magnetic field and the rotor, it is
helpful to introduce the slip. The slip angular speed ωslip is simply the difference between
the synchronous angular speed ωsync and the rotor mechanical angular speed ωm:
ω slip = ω sync − ω m
(3.2)
And then, the slip s, which is the ratio between the slip speed and the synchronous speed,
can be expressed by:
s=
ω slip
ω −ω
(× 100 % ) = sync m (× 100 % )
ω sync
ω sync
(3.3)
Conversely, the mechanical speed of the rotor can be found by:
ω m = (1 − s )ω sync
(3.4)
Therefore, if the rotor is rotating at synchronous speed, the slip is 0 and if the rotor is at
rest, the slip is 1. When the slip is between 0 and 1, the machine transforms the electrical
power of the stator into mechanical power on the rotor shaft. As explained earlier, when
the rotor is rotating at synchronous speed, the induced torque is 0. Over this limit, the
torque is inverted and the machine begins to transform the mechanical power of the rotor
into electrical power, thus the machine starts to work as a generator.
3.3 Rotor Frequency
An asynchronous machine can be seen as a rotating transformer, but unlike a transformer,
the electrical frequency at the secondary terminals (rotor) is not necessarily the same as at
the primary terminals (stator). If the slip is 0, the electrical frequency of the rotor is the
same as the synchronous frequency and if the slip is 1, the frequency is 0. Between these
limits, the electrical frequency fr of the rotor can be expressed as:
f r = sf e
11
(3.5)
3.4 Electrical Radians
The models of induction generators are often built for machines with 2 poles per phase in
the stator windings, but it is also possible the find machines with 4 poles per phase or
more. So, to generalize the construction of the machine models, it is possible to define the
angular positions θ and speeds ω in terms of electrical radians, which are independent of
the number of poles per phase. Thus, the following equations can be used to convert the
different angular variables to electrical radians:
θ s=
p
p
p
θ sync , ω s = ω sync , θ r= θ m
2
2
2
and
ωr=
p
ωm
2
(3.6)
Where p is the number of poles per phase and θs, ωs, θr and ωr are the angular positions
and speeds with electrical radians for the stator (subscripts s) and the rotor (subscripts r).
3.5 Steady-State Model
Because of the similarity with the transformer, the per phase equivalent steady-state
model of the induction machine can be drawn as a transformer model, as shown in the
following figure.
Figure 3.1:
Equivalent steady-state circuit of the asynchronous machine [13].
12
The voltage Vφ is the per phase voltage at the stator terminals and ω is the stator magnetic
field angular speed (in rad/s). The resistances Rs, Rc and Rr represent respectively the
copper losses in the stator windings, the losses in the magnetic core and the losses in the
conducting bars of the squirrel-cage rotor. The inductances Lℓs, Lℓr and Lm are
correspondingly the leakage inductances of the stator and the rotor and the magnetizing
inductance. The variables Emφ, Isφ, Irφ and Imφ are in that order the air gap voltage, the per
phase stator and rotor currents and the magnetizing current. Finally, ωm and ωslip are the
rotor mechanical and the slip angular speeds (also in rad/s).
3.6 Power Assessment
After obtaining the per phase steady-state equivalent model of the induction machine, it is
possible to process the power assessment for both the motor and the generator. First, the
electrical power Pe at the stator terminals can be found as:
Pe = 3Vφ I sφ cos (θ ) = 3Vl I sφ cos (θ ),
where Vl = 3Vφ
(3.7)
In this equation, cos(θ) is the power factor and Vℓ is the line-to-line voltage of the stator.
If the machine is operating in the motoring mode, the electrical power Pe is the input
power. Thus, it is possible to obtain the air gap power PAG by subtracting the stator
copper losses PSCL and the magnetic core losses Pcore of the electrical input power.
PSCL = 3I s2φ Rs
Pcore = 3
E m2 φ
(3.8)
(3.9)
Rc
PAG = Pe − PSCL − Pcore
(3.10)
Afterward, the equations for the rotor copper losses PRCL and the converted power Pconv
can be described. The last one can also be expressed with the air gap power.
PRCL = 3 I r2φ Rr
⎛ω
Pconv = 3 I r2φ Rr ⎜ m
⎜ω
⎝ slip
13
⎞ ωm
⎟=
P
⎟ ω AG
⎠
(3.11)
(3.12)
After all, the mechanical output power Pm on the rotor shaft depends upon the mechanical
friction and windage losses PF&W and also on the miscellaneous losses Pmisc.
Pm = Pconv − PF &W − Pmisc
(3.13)
Finally, the induced torque, or electromagnetic torque, can be expressed in terms of the
converted power or in terms of the air gap power.
Te =
Pconv
ωm
=
PAG
ω sync
(3.14)
If the machine is operating in the generating mode, the mechanical power Pm becomes the
mechanical input power. After having removed PF&W and Pmisc of that input power, the
converted power becomes the input power of the equivalent circuit. Then, it is possible to
obtain the electrical output power Pe by subtracting the rotor and the stator copper losses
and the core losses of the converted power.
3.7 Torque Characteristics
The figure 3.2 is an example of profile of the torque/speed curve. It can be seen that the
induced torque Te is zero at synchronous speed. Also, between no-load and full load, the
curve can be estimated as a linear curve. There is also a maximum torque Tmax, the
pullout torque, that can be induced in the machine. But the must interesting characteristic
of this curve is that the torque is negative above the synchronous speed ωsync. This change
of sign means that the torque is inverted and then the induction machine becomes to work
as a generator. Finally, there is also a maximum torque that can be induced in the
generating mode.
14
Figure 3.2:
Induced torque in function of the rotor mechanical speed [13].
15
Chapter 4
Self-Excitation of an Induction Generator
During the excitation process of an induction generator, the residual magnetism in the
iron core generates a voltage that produces a capacitive current. Subsequently, the
capacitive current increases the voltage which also increases the capacitive current and so
on. That process continues until the complete build up of the voltage, which occurs at the
saturation of the magnetic core. But if there is no residual magnetism in the core, the
induction generator cannot generate any voltage.
If the residual magnetism is completely lost, there are 2 kinds of technique to recover it,
which are run the machine as a motor at no-load to rebuild the magnetism or cause a
current surge in the stator windings [1]. In the second case, the current surge can be
caused by a battery, by capacitors or by a rectifier fed from the network.
Figure 4.1:
Magnetizing curve of the induction machine [1]
16
When a capacitor bank is used for the self-excitation of an induction generator, the
magnetizing curve can be used to understand the excitation process (see figure 4.1). In
the previous figure, Vmφ is the air gap phase voltage and Im is the magnetizing current.
The curve 1 is the saturation curve of the generator, which depends upon the construction
of the machine. The curve 2 is a linear curve defined by the reactance XC of the excitation
capacitance, which is:
XC =
1
2πfC
(4.1)
The intersection point between curves 1 and 2 is the point at which the capacitor current
IC supplies the entire reactive power needed by the generator. Curves 3 and 4 are
limitation points, so the curve 2 must be kept inside these limits. Indeed, at curve 3, an
infinite number of common points between curves 1 and 2 leads to an unstable voltage.
Finally, curve 4 is determined by the maximum current ICmax that the induction machine
can support.
17
Chapter 5
Dynamic Modelling
5.1 ABC/abc Model Excluding the Saturation
To obtain the dynamic model of the induction machine, it is necessary to take into
account some assumptions. For example, the air gap diameter of the machine is assumed
to be constant, so the reluctance of the air gap is constant at any point. Furthermore, the
effect of the stator and the rotor slots is neglected. As well, the effect of the temperature
is neglected.
If it is also assumed that the magnetomotive forces are sinusoidal, the effect of the
saturation is negligible and there are no losses in the magnetic core, the fundamental
equations of the machine become [14]:
[V ] = [R ]× [I ] + d [λ ]
(5.1)
dt
[λ ] = [L]× [I ]
(5.2)
Where V, I and λ are respectively the voltage, the current and the linked flux matrices, R
is the resistance matrix and L is the inductance matrix. The 3 first ones are defined as:
[V ] = [v A
[I ] = [i A
[λ ] = [λ A
vB
iB
λB
vC
iC
va
ia
λC
vb
ib
λa
vc ]
T
ic ]
T
λb
λc ]T
In the equations above, the subscripts A, B and C refer to each phase of the stator and
subscripts a, b and c are the same for the rotor. Also, the resistance matrix R is defined
as:
18
⎡ Rs
⎢0
⎢
⎢0
[ R] = ⎢
⎢0
⎢0
⎢
⎣⎢ 0
0
0
0
0
Rs
0
0
0
0
Rs
0
0
0
0
Rr
0
0
0
0
Rr
0
0
0
0
0⎤
0 ⎥⎥
0⎥
⎥
0⎥
0⎥
⎥
Rr ⎦⎥
Hence, if equations 5.1 and 5.2 are combined, it is possible to obtain the equation below:
d [I ]
d [L ]⎫
⎧
−1
−1
= [L ] × ⎨− [R ] −
⎬ × [I ] + [L ] × [V ]
dt
dt ⎭
⎩
(5.3)
Then, it is not convenient to derivate the inductance matrix in function of the time,
because most of the terms in that matrix are in function of the rotor position θr expressed
in electrical radians. This is why it is suitable to use the chain rule as follow:
d [L] d [L] dθ r d [L]
=
×
=
× ωr
dt
dθ r
dt
dθ r
(5.4)
In the previous equation, ωr is the angular speed of the rotor in electrical rad/s.
Afterward, equations 5.3 and 5.4 can be combined to obtain the usual form of control
systems:
d [I ]
= [A]× [I ] + [B ]× [V ]
dt
(5.5)
Where,
[A] = [L]−1 × ⎨− [R] − ω r d [L]⎬
[B] = [L]−1
⎧
⎫
⎩
dθ r ⎭
To use that form, the inductance matrix L most be defined. Therefore, if the stator
windings and the rotor equivalent windings are assumed to be electrically and
magnetically symmetrical, the inductance matrix can be written as the matrix below [15].
It is also important to mention that the windings of the rotor are equivalent ones, because
it is already known that the rotor is of the squirrel-cage type of construction, so this one is
not really made of windings. This fact will be taken as another assumption.
19
⎡
⎢ Lls + M s
⎢ 1
⎢ − Ms
⎢ 2
⎢−1M
s
[L ] = ⎢⎢ 2
⎢ M sr f1
⎢
⎢ M f
sr 2
⎢
⎢
⎢ M sr f 3
⎣
−
1
Ms
2
1
Ms
2
1
− Ms
2
−
Lls + M s
−
1
Ms
2
Lls + M s
M sr f1
M sr f 2
M sr f 3
M sr f1
M sr f 2
M sr f 3
1
− Mr
2
M sr f 3
M sr f 2
Llr + M r
M sr f1
M sr f 3
−
M sr f 2
M sr f1
1
Mr
2
1
− Mr
2
Llr + M r
−
1
Mr
2
⎤
M sr f 3 ⎥
⎥
M sr f 2 ⎥
⎥
M sr f1 ⎥
⎥
⎥
1
− Mr ⎥
2
⎥
1
− Mr ⎥
⎥
2
⎥
Llr + M r ⎥
⎦
(5.6)
Where,
f1 = cos(θ r ),
2π ⎞
⎛
f 2 = cos⎜θ r +
⎟ and
3 ⎠
⎝
2π ⎞
⎛
f 3 = cos⎜θ r −
⎟
3 ⎠
⎝
And Lℓs and Lℓr are in that order the stator and the rotor leakage inductances and are the
same as in section 3.5. The variables Ms, Mr and Msr are the stator, the rotor and the
stator-rotor mutual inductances. Then, if the stator windings and the rotor equivalent
windings are considered to have the same number of turns in the model, the 3 mutual
inductances Ms, Mr and Msr are equal. Moreover, if the currents of the 3 phases are
balanced as:
i A + i B + iC = 0 and ia + ib + ic = 0
The inductance matrix in equation 5.6 can then be simplified as:
⎡ Ls
⎢ 0
⎢
⎢ 0
[L ] = ⎢
⎢ M sr f1
⎢ M sr f 2
⎢
⎣⎢ M sr f 3
0
0
M sr f1
M sr f 2
Ls
0
M sr f 3
M sr f1
0
Ls
M sr f 2
M sr f 3
M sr f 3
M sr f1
M sr f 2
M sr f 3
Lr
0
0
Lr
M sr f 2
M sr f1
0
0
Where,
20
M sr f 3 ⎤
M sr f 2 ⎥⎥
M sr f1 ⎥
⎥
0 ⎥
0 ⎥
⎥
Lr ⎦⎥
(5.7)
Ls = Lls +
3
Ms
2
Lr = Llr +
and
3
Mr
2
Thus, the derivate of the inductance matrix in function of the position of the rotor can be
performed.
⎡0
⎢0
⎢
⎢0
d [L ]
= − M sr ⎢
dθ r
⎢ g1
⎢g2
⎢
⎢⎣ g 3
0
0
g1
g2
0
0
g3
g1
0
0
g2
g3
g3
g1
g2
g3
0
0
0
0
g2
g1
0
0
g3 ⎤
g 2 ⎥⎥
g1 ⎥
⎥
0⎥
0⎥
⎥
0 ⎥⎦
(5.8)
Where,
2π ⎞
⎛
g1 = sin (θ r ), g 2 = sin ⎜θ r +
⎟ and
3 ⎠
⎝
2π ⎞
⎛
g 3 = sin ⎜θ r −
⎟
3 ⎠
⎝
Finally, the last problem to solve about the equation 5.5 is to determine the inverse of the
inductance matrix. This problem has been solved in [14] and consequently the resulting
inverse of the inductance matrix is:
[L ]−1
⎡ b11
⎢b
⎢ 21
⎢b
= [B ] = ⎢ 31
⎢b41
⎢b51
⎢
⎣⎢b61
b12
b13
b14
b15
b22
b23
b24
b25
b32
b33
b34
b35
b42
b52
b43
b53
b44
b54
b45
b55
b62
b63
b64
b65
Where,
b11 = b22 = b33 =
K2
Ls
b12 = b13 = b21 = b23 = b31 = b32 =
b44 = b55 = b66 =
21
K2
Lr
K3
Ls
b16 ⎤
b26 ⎥⎥
b36 ⎥
⎥
b46 ⎥
b56 ⎥
⎥
b66 ⎦⎥
(5.9)
b45 = b46 = b54 = b56 = b64 = b65 =
K3
Lr
b14 = b25 = b36 = b41 = b52 = b63 = K 4 f1
b15 = b26 = b34 = b43 = b51 = b62 = K 4 f 2
b16 = b24 = b35 = b42 = b53 = b61 = K 4 f 3
And,
3
3
−
L s Lr
4, K =
4
, K2 =
K1 =
3
9
9
M sr2
K1 −
K1 −
4
4
K1 −
1
M sr
K4 =
9
K1 −
4
−
and
Because the rotor angular speed ωr is part of equation 5.5 and also because its integral,
the rotor angular position θr, is part of equations 5.8 and 5.9, it is necessary to define
these parameters for the model. Thus, the derivative of the rotor angular speed is:
d ω m Te − TL
=
dt
J
(5.10)
Because the angular speed of the rotor ωm is not expressed in electrical rad/s in this
equation, it is necessary to used equation 3.6 to obtain ωr. Moreover, Te and TL are
respectively the electromagnetic and the load torques and J is the moment of inertia of
the rotor. Knowing the stator and rotor currents in matrix I and the rotor angular position
θr, it is possible to find the electromagnetic torque as:
⎧
⎫
⎪(i i + i i + i i )sin (θ )
⎪
r
⎪ Aa Bb Cc
⎪
⎪
2π ⎞ ⎪
⎛
Te = − pM sr ⎨+ (i Aib + iB ic + iC ia )sin ⎜ θ r +
⎟⎬
3 ⎠⎪
⎝
⎪
⎪
2π ⎞ ⎪
⎛
⎟⎪
⎪+ (i Aic + iB ia + iC ib )sin ⎜ θ r −
3 ⎠⎭
⎝
⎩
(5.11)
Where p is the number of poles per phase. With these last equations, the ABC/abc model
is completed and ready to be implemented in a Simulink model, by using equations 5.5
and 5.8 through 5.11. Like explained earlier, the leakage inductances Lℓs and Lℓr are the
22
same as in the equivalent circuit of section 3.5 and the mutual inductances can be
calculated as below:
M s = M r = M sr =
2
Lm
3
(5.12)
Because the rotor of the machine is of the squirrel-cage type, a no-load test and locked
rotor test must be performed to estimate the value of each inductance. These tests will be
introduced in the next chapter.
5.2 dq Model Including the Saturation
In the modeling of section 5.1, one of the main assumptions is that the saturation effect is
negligible. However, it is well known that the magnetic saturation is considerable during
the start-up and operations near rated conditions [16]. So, if the saturation is considered,
the main magnetizing flux, its harmonics and the leakage fluxes can all be considered for
saturation [17]. But, to keep the problem simple, only the main magnetizing flux is
considered in this model. So, like the dynamic ABC/abc model, the model of this section
is obtained by considering some assumptions, that time from [18]. First, the model is
again defined as in equation 5.1, with 3 stator windings and 3 equivalent symmetrical
rotor windings. The stator and the rotor windings are still considered to be sinusoidally
distributed. One more time, the air gap is considered constant. For the magnetomotive
forces, only the fundamental parts are considered. Finally, the skin effect is neglected.
So, by applying the dq0 transformation of appendix A, the voltage equations are
transferred from a 3-axis to a 2-axis frame of reference and 2 zero-sequence equations
appear in the model. But, because there are no zero-sequence voltages in delta connection
configurations and no zero-sequence currents in wye connection configurations, there is
no power related to the zero-sequence equations, so both are neglected. Moreover, the
effect of the zero-sequence currents on the saturation is neglected, like the effect of the
saturation on the zero-sequence currents. After all, the following equation shows the
resulting voltage equations in matrix form, which are non linear because of the
electromotive force, the torque and the saturation.
23
[V ] = [ R ] × [ I ] +
d [λ ]
− [W ] × [λ ]
dt
(5.13)
Where,
⎡v sd ⎤
⎡i sd ⎤
⎡λ sd ⎤
⎡ Rs
⎢v ⎥
⎢i ⎥
⎢λ ⎥
⎢0
sq ⎥
sq ⎥
sq ⎥
⎢
⎢
⎢
[V ] =
, [I ] =
, [λ ] =
, [ R] = ⎢
⎢v rd ⎥
⎢ird ⎥
⎢λ rd ⎥
⎢0
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢
⎢⎣ v rq ⎥⎦
⎢⎣irq ⎥⎦
⎢⎣ λ rq ⎥⎦
⎣0
⎡ 0
⎢− ω
and [W ] = ⎢ s
⎢ 0
⎢
⎣ 0
ωs
0
0
0
0
0
− (ω s − ω r )
0
0
0
Rs
0
0
Rr
0
0
0⎤
0 ⎥⎥
0⎥
⎥
Rr ⎦
⎤
⎥
⎥
(ω s − ω r )⎥
⎥
0
⎦
0
0
The variables v, i, λ and R are again the voltages, the currents, the linked fluxes and the
resistances, with subscripts s and r for the stator and the rotor and d and q for the direct
and the quadrature axis. Also, ωs and ωr are the synchronous angular speed and the rotor
angular speed in electrical rad/s, since the dq frame of reference is rotating at
synchronous speed. As the flux linkage matrix depends upon the leakage flux and the
magnetizing flux, the linkage matrix is redefined as:
⎡λlsd + λmd ⎤
⎢λ + λ ⎥
lsq
mq ⎥
(5.14)
[λ ] = ⎢
⎢λlrd + λmd ⎥
⎢
⎥
⎢⎣ λlrq + λmq ⎥⎦
Where the subscripts ℓ and m refer to the leakage and the magnetizing flux components.
Then, if the fluxes are expressed in terms of currents i and inductances L and if the
magnetizing flux saturation is considered, but not the leakage fluxes saturation, equation
5.14 becomes:
⎡ Lls isd
⎢L i
ls sq
[λ ] = ⎢
⎢ Llr ird
⎢
⎣⎢ Llr irq
+ Lm (im )imd ⎤
+ Lm (im )imq ⎥⎥
+ Lm (im )imd ⎥
⎥
+ Lm (im )imq ⎦⎥
24
(5.15)
The stator and rotor leakage inductances Lℓs and Lℓr are the same as in section 3.5, but the
magnetizing inductance Lm is now in function of the magnetizing current im and this
relation is not linear (see curve 1 of figure 5.1). Since the air gap is constant, the
magnetizing current can be defined according to the direct and the quadrature axis.
imd = isd + ird
imq = isq + irq
(5.16)
And,
2
2
im = imd
+ imq
Figure 5.1:
(5.17)
Magnetizing curve of the induction machine [18]
Thus, with the new definition of the fluxes matrix, equation 5.13 becomes:
[V ] = [ R ] × [ I ] +
d [λ ]
− ω s × [G1 ] × [ I ] + ω r × [G 2 ] × [ I ]
dt
Where,
25
(5.18)
Ls
Lm (im )⎤
0
0
0
⎡ 0
⎡ 0
⎢
⎥
⎢ −L
− Lm (im )
0
0
0
0
0 ⎥
s
and [G2 ] = ⎢
[G1 ] = ⎢
⎢ 0
⎢ 0
Lm (im ) 0
Lm (im )
Lr ⎥
0
⎢
⎥
⎢
− Lr
− Lr
0
0
0 ⎦
⎣− Lm (im )
⎣− Lm (im )
0⎤
0 ⎥⎥
Lr ⎥
⎥
0⎦
And,
Ls = Lls + Lm (im ) and
Lr = Llr + Lm (im )
This expression must be reformulated again, to eliminate the derivative of the fluxes
matrix. For this, the space-vector formulations of the magnetizing current and flux (im and
λm) are needed. So, the space-vector form of these variables is:
r
im = imd + jimq = im e jµ
r
r
λm = Lm (im )im = λmd + jλ mq
Where µ is the angle between im and the d-axis and the direct and quadrature axis
components of the magnetizing flux λmd and λmq are defined as:
λmd = Lm (im )im cos(µ )
λmq = Lm (im )im sin (µ )
Afterward, with some manipulations, the derivative expressions of λmd and λmq can be
obtained as in [18].
dimq
dλmd
di
= (L0 + L2 c ) md + L2 s
dt
dt
dt
dimq
dλmq
di
= L2 s md + (L0 − L2 c )
dt
dt
dt
(5.19)
Where,
L0 =
Lmt + Lm
L − Lm
, L2c = mt
cos(2 µ ) and
2
2
26
L2 s =
Lmt − Lm
sin (2 µ )
2
The variables Lm and Lmt are respectively the chord-slope (curve 2 of figure 5.1) and the
tangent-slope (curve 3 of figure 5.1) inductances and are defined as:
Lm =
λm
and
im
Lmt =
dλ m ∆λ m
=
dim
∆im
After all, with the last definitions, the equation 5.18 can be redefined as:
[V ] = [ R ] × [ I ] + [ L ] ×
d[I ]
− ω s × [G1 ] × [ I ] + ω r × [G2 ] × [ I ]
dt
(5.20)
Where,
⎡ Lsd
⎢L
2s
[ L] = ⎢
⎢ Ld
⎢
⎢⎣ L2 s
L2 s
Lsq
Ld
L2 s
L2 s
Lrd
Lq
L2 s
L2 s ⎤
Lq ⎥⎥
L2 s ⎥
⎥
Lrq ⎥⎦
And,
Lsd = Lls + L0 + L2c , Lrd = Llr + L0 + L2c , Ld = L0 + L2c ,
Lsq = Lls + L0 − L2 c , Lrq = Llr + L0 − L2 c
and
Lq = L0 − L2c
Finally, to simulate the model, the rotor angular speed and the electromagnetic torque are
again necessary. So, equation 5.10 can be employed for the rotor angular speed and the
following equation can be used for the electromagnetic torque Te [18]:
Te =
3
p × [ I ]T × [G 2 ] × [ I ]
4
Where p is again the number of poles per phase.
27
(5.21)
Chapter 6
Simulink Model
6.1 Model without Saturation Considerations
Based on the dynamic model of section 5.1, a Simulink model has been built in Matlab.
This model can be seen in figure 6.1 and all the details about the construction of the
model are available in appendix B.
Figure 6.1:
Simulink dynamic model of a squirrel-cage induction machine.
28
Part II
Laboratory Tests
29
Chapter 7
Theory
The objective of this chapter is to present the procedure to obtain the value of the
parameters discussed in the previous chapters. Firstly, the DC test, the no-load test and
the locked rotor test are introduced based on IEEE standards [12]. After all, a method is
showed to determine the magnetizing curve of an induction machine.
7.1 DC Test
This test is employed to find the value of the resistance Rs of the stator windings and the
appendix C can be used to perform it. Because it has already been assumed in the
previous chapter that the stator resistance is the same for each phase, 3 different measures
can be taken to obtain an average value of this resistance.
To perform this test, a small dc voltage Vdc has to be applied between 2 stator phases and
it is only necessary to measure the dc current Idc that flows through both windings. It is
important that the applied voltage has to be enough small to avoid currents upper than the
current limitations for the machine. Because the current flows between 2 phases of the
stator, the resistance for a single phase is half of the total resistance calculated from that
test, so the resistance Rs can be calculated as:
Rs =
1 Vdc
2 I dc
(7.1)
This procedure can be repeated for the 3 stator combinations (AB, AC and BC) to obtain
an average value of the stator resistance. This measure is considered at ambient
temperature, thus the value of the resistance can be corrected later on with the following
equation from [12]:
Rb =
Ra (tb + k1 )
t a + k1
(7.2)
Where Ra and Rb are respectively the winding resistances at temperature ta and at
corrected temperature tb (in Ω). The variables ta and tb are in that order the temperatures
30
of the windings when the resistance Ra was measured and to which the resistance is to be
corrected (in °C). Finally, k1 is a constant that equals to 234.5 for 100% IACS
conductivity copper.
7.2 No-Load Test
As the DC test is used to determine the resistance of the stator windings, the 2 next tests
are used to calculate the rest of the parameters of the equivalent circuit. Indeed, it is
necessary to combine the results of the no-load test (appendix D) and the locked rotor test
(appendix E) to find the equivalent rotor resistance Rr, the stator and rotor leakage
inductances Lℓs and Lℓr and the core losses Pcore. It is also possible to determine the
magnetizing inductance Lm with the same tests, but this one will be evaluated more
accurately with the method discussed in section 7.4.
Thus, to perform the no-load test, it is necessary to run the machine in motoring mode at
rated voltage and frequency. Naturally, the motor is free of connected load. During this
test, the measurements of the no-load stator rms line-to-line voltages and currents have to
be taken, like the total input power of the machine P0. Afterward, the average value of the
voltages and currents is calculated to obtain the no-load stator line-to-line voltage and
current Vs0ℓ and Is0ℓ. Like it has been explained previously, these values are used for
calculations with other values obtained from the locked rotor test. These values are the
stator line-to-line voltage and current VsLℓ and IsLℓ and the total input power PL for the
locked rotor test.
So, as the calculation of the parameters is explained in section 7.3, it is still necessary to
determine the core losses from the no-load test. For that reason, it is necessary to measure
at least 3 more points of the relation between the electrical input power Pe (P1, P2, P3, …)
and the line-to-line stator voltage Vsℓ (V1, V2, V3, …). These measures have to be
performed with lower voltages. It is also necessary to note the stator phase currents Isφ (I1,
I2, I3, …) for each point, in order to calculate the stator copper losses PSCL. These losses
can be calculated with equation 3.8. Then, the stator copper losses can be removed from
the electrical input power for each point, and a plot of the line-to-line stator voltage
squared Vsℓ2 in function of the resulting power P can be traced. Consequently, if the curve
31
is extended to intercept the zero voltage axis, this point of interception is thus the friction
and windage losses PF&W. After all, the core losses Pcore can be obtained by subtracting
the stator copper losses and the friction and windage losses of the electrical input power
for a given voltage. As a result, the core losses calculated for the rated voltage Vrated will
be used in the next section to determine the value of the resistance Rc.
7.3 Locked Rotor Test
The aim of this test is to prevent the rotor rotation, to obtain a slip equal to 1 and then
there is no magnetizing current. A small voltage has to be applied to obtain the full-load
current, but it is important to proceed carefully to avoid overheating the windings. Also,
the frequency of the input voltage must be less than 25% of the rated frequency. During
this test, the measurements of the stator rms line-to-line voltages and currents have to be
taken, like the total input power of the machine PL. The average value of the voltages and
currents is calculated to obtain the stator line-to-line voltage and current VsLℓ and IsLℓ for
the locked rotor test. These values are used for calculations with other values obtained
from the no-load test. These values are the stator line-to-line no-load voltage and current
Vs0ℓ and Is0ℓ, the no-load total input power P0 and the core losses Pcore.
After all, the following calculations can be achieved to obtain the rotor resistance Rr and
the leakage reactances Xℓs and Xℓr.
First, it is necessary to assume a relationship between the leakage inductances.
Consequently, the ratio Xℓs/Xℓr is 1 for design A and D and for wound rotor motors, 0.67
for design B motors and 0.43 for design C motors. Then, the line-to-line voltages Vs0ℓ and
VsLℓ have to be converted into phase voltages by simply dividing all line-to-line voltages
by √3. The phase currents and the line-to-line currents Is0ℓ and IsLℓ are exactly the same,
so the phase values Vs0, Is0, VsL and IsL are used in the following calculations. Thus, the
total reactive powers Q0 and QL are computed for both no-load and locked rotor tests.
Q0 =
(mV s 0 I s 0 )2 − P02
(7.3)
QL =
(mVsL I sL )2 − PL2
(7.4)
32
Where m is the number of phases. Subsequently, the magnetizing reactance Xm can be
calculated as:
Xm =
mV s20
1
×
2
Q0 − mI s 0 X ls ⎛
X
⎜⎜ 1 + ls
Xm
⎝
(
)
⎞
⎟⎟
⎠
2
(7.5)
To obtain a first value of the magnetizing reactance, it is necessary to assume an initial
value for Xℓs/Xm and Xℓs. After, the stator leakage reactance XℓsL at locked rotor test
frequency can be computed as:
X lsL =
⎡X
X ⎤
QL
× ⎢ ls + ls ⎥
X
Xm ⎦
⎡
X
X ⎤
mI sL2 × ⎢1 + ls + ls ⎥ ⎣ lr
X lr X m ⎦
⎣
(7.6)
Afterward, the stator leakage reactance can be converted for rated frequency.
X ls =
f rated
× X lsL
fL
(7.7)
Where frated and fL are respectively the rated and locked rotor test frequencies. After all,
the equation 7.5 can be use again to calculate Xm, but that time using Xℓs from equation
7.7 and Xℓs/Xm from equations 7.5 and 7.7. Finally, this iteration method has to be used
until obtaining errors on Xℓs and Xm values within 0.1%.
Then, the rotor leakage reactance XℓrL at locked rotor test frequency can be calculated.
X lrL =
X lsL
⎛ X ls ⎞
⎜⎜
⎟⎟
⎝ X lr ⎠
(7.8)
And the rotor leakage reactance can be converted for rated frequency.
X lr =
f rated
× X lrL
fL
(7.9)
Moreover, the resistance Rc can be determined knowing the total core losses Pcore from
the no-load test:
Rc =
1
Pcore ⎛
X ⎞
× ⎜⎜1 + ls ⎟⎟
2
mV s 0 ⎝
Xm ⎠
33
2
(7.10)
Finally, the resistance RrL for the locked rotor test can be computed as:
R rL
⎛ P
⎞ ⎛
X
= ⎜⎜ L2 − R sL ⎟⎟ × ⎜⎜ 1 + lr
Xm
⎝ mI sL
⎠ ⎝
2
⎞ ⎛ X lr
⎟⎟ − ⎜⎜
⎠ ⎝ X ls
2
⎞ ⎛ X l2sL
⎟⎟ × ⎜⎜
⎠ ⎝ Rc
⎞
⎟⎟
⎠
(7.11)
It is important to remind that the resistance Rs has to be converted to the temperature of
the test to obtain RsL, using equation 7.2. The same equation can be used to obtain Rr
from RrL.
7.4 Determination of the Magnetizing Inductance
Two different kinds of method can be used to determine the magnetizing inductance of an
induction machine. Indeed, it can be obtained from a DC method or from an AC method
[19]. For a DC method, the inductance profile can be found more exactly by measuring
the rise or the fall of the current that flows between 2 phases of the stator when a DC
voltage is applied between these phases. Because of the DC voltage input, there is no
induction in the rotor and no core losses, so from the equivalent circuit of figure 3.1 it is
possible to obtain the following circuit:
Figure 7.1:
Reduced equivalent circuit used for the determination of the
magnetizing inductance.
In the previous figure, Vin(t) is the instantaneous input voltage between 2 phases of the
stator, i(t) is the instantaneous stator phase current and Req and Leq are the equivalent
resistance and inductance of the series RL circuit. Therefore, if the fall of the current i(t)
is used to determine the inductance profile, it is assumed that a DC voltage was applied
enough long to reach a steady-state value of the current, which is considered as the initial
34
current I(0). Thus, at time t = 0, i(t) = I(0). Then, for the falling of the current that occurs
after t = 0, the equivalent inductance Leq can be computed as in [20].
Req t
Leq =
(7.12)
⎛ I (0 ) ⎞
ln⎜⎜
⎟⎟
⎝ i (t ) ⎠
And because the time constant τ of that relation is Leq/Req, Leq can also be found by
measuring the time τ when i(t) is 37% of I(0), and then:
Leq = τReq
(7.13)
Finally, the magnetizing inductance Lm is half of Leq and a variable resistance Rv can be
used to obtain Lm for different stator currents. All the details about how to obtain figure
7.1 and equations 7.12 and 7.13 are presented in appendix F.
35
Chapter 8
Evaluation of the Simulink Model
8.1 Model without Saturation Considerations
With the lab tests of appendixes C to E, it is possible to extract the parameters from the
equivalent circuit of figure 3.1 for the induction machine used in this project. The
following table regroups all the parameters.
Table 8.1:
Parameter
Value
Unit
Resistance of the stator windings (Rs)
2.8899
Ω
Rotor equivalent resistance (Rr)
1.9247
Ω
Equivalent resistance of the magnetic core (Rc)
157.2490
Ω
Stator leakage inductance (Lℓs)
14.3252
mH
Rotor leakage inductance (Lℓr)
14.3249
mH
Magnetizing inductance (LM)
365.3526
mH
Friction and windage losses (PF&W)
14.9024
W
Core losses (Pcore)
664.2626
W
Parameters of the induction machine used in this project.
The inductances are obtained from the reactances of the appendix E by using the
following equation at rated frequency:
L=
X
2πf
(7.1)
Where L is the inductance, X is the reactance and f is the frequency. From appendix D, it
is also possible to use the graphs of figures 8.1 and 8.2 to compare with the simulation of
the Simulink model of appendix B.
36
Rms phase current in function of the rms phase voltage
Average phase curren
2.5
2
1.5
1
0.5
0.0098x
y = 0.3101e
0
0
25
50
75
100
125
150
175
200
Average phase voltage (in V)
Figure 8.1:
Rms phase current in function of the rms phase voltage for the noload test.
Total input power in function of the rms phase voltage
Total input power (i
800
600
400
200
0.019x
y = 18.789e
0
0
25
50
75
100
125
150
175
200
Average phase voltage (in V)
Figure 8.2:
Total input power in function of the rms phase voltage for the no-load
test.
37
So, with the same conditions of in lab, the model as been simulated and the results can be
seen in appendix G. For every test at no-load, it is possible to see that after the transient,
the rotor currents and the electromagnetic torque reach 0 at steady-state, which is correct
because there is no induction at synchronous speed. Moreover, the rotor angular speed
reaches 314.16 in electrical rad/s. That is also correct for a 50 Hz input voltage. Finally,
the steady-state rms phase currents have been observed from the graphs of the simulation
and are compared in table 8.2 with the rms phase currents measured in lab. From the
calculated relative errors, it can be seen that the Simulink model is not really accurate.
The same comparison has been done with the measured values of the locked rotor test
(table 8.3). That time, the relative error is lower, but still over the specified target, which
is 0.5%. The graphs of appendix G show that the rotor currents and the electromagnetic
torque do not reach 0 anymore at steady-state, which is correct for a locked rotor test.
Point
Rms stator phase
voltage (Vs)(in V)
Rms stator phase current
(Is)(in A)
Measured
Simulated
Relative
error (in %)
1
19.5164
0.3948
0.1636
-58.56
2
29.5238
0.3870
0.2474
-36.07
3
39.5620
0.4492
0.3317
-26.16
4
49.4635
0.5209
0.4146
-20.41
5
193.9127
2.0717
1.6193
-21.84
Table 8.2:
Comparison between the measured and the simulated rms stator
phase currents for the no-load test.
Rms stator phase
voltage (Vs)(in V)
38.6170
Table 8.3:
Rms stator phase current
(Is)(in A)
Measured
Simulated
6.832
7.0145
Relative
error (in %)
2.67
Comparison between the measured and the simulated rms stator
phase currents for the locked rotor test.
38
Because it is know from appendixes C to E that the temperature measurements were a
major problem during the parameter tests, a sensibility test can be processed for all
parameters, to know the influence of the temperature on each parameter. So, with the
same calculation technique of chapter 7, the parameters have been computed again with
all temperatures raised by 100%. Only the ambient temperature took during the DC test
remains the same. The new values of the parameters can be seen in table 8.4. It can be
seen that only the rotor equivalent resistance can be really affected by an error on the
measurements of the temperature. A different rotor resistance changes the transient
results of the Simulink model, but it does not change the steady-state value of the stator
phase currents enough to obtain better results. Then, the temperature measurements are
not the source of the errors in tables 8.2 and 8.3. When a no-load test is performed, it is
important that the machine runs for at least 30 minutes before taking the measurements. It
is possible that this time has not been respected, because the running time has not been
calculated before the measurements. So, another parameter tests can be necessary to
obtain better results. Also, it is also important to mention that the moment of inertia J has
been estimated for the simulation. Thus, the measurement of this parameter can also
improve the results of the simulation.
Parameter
Value
Value
Unit
Resistance of the stator windings (Rs)
2.8899
2.8899
Ω
Deviation
(in %)
0
Rotor equivalent resistance (Rr)
Equivalent resistance of the magnetic
core (Rc)
Stator leakage inductance (Lℓs)
1.9247
1.3525
Ω
-29.73
157.2490
158.3802
Ω
0.72
14.3252
14.3252
mH
0
Rotor leakage inductance (Lℓr)
14.3249
14.3249
mH
0
Magnetizing inductance (LM)
365.3526
365.3526
mH
0
Friction and windage losses (PF&W)
14.9024
14.7876
W
-0.77
Core losses (Pcore)
664.2626
659.5184
W
-0.72
Table 8.4:
Sensibility test on the parameters of the induction machine.
39
Part III
Project Closure
40
Chapter 9
Discussion
At this point of the project, it is necessary to evaluate the results obtained and to look at
the advantages and the drawbacks of the built model, in order to determine the future
work that will be necessary to continue the project.
In chapter 2, it has been explained that the 3 main objectives of that project are to build a
model of a squirrel-cage induction generator without considering the saturation, another
one that considers the saturation and finally to investigate and test the self-excitation of
an induction machine. Actually, the theory of all these parts have been developed, but
only the induction machine model without saturation considerations have been built in
Simulink and compared with the lab measurements. Because the differences between the
results obtained from the simulation of that model and those obtained from the lab are
huge, it has been complicated to move to the next part of the project. Indeed, the chapter
8 shows relative errors up to 58.56% on the stator rms phase currents for the no-load test.
Then, it is important to evaluate this Simulink model well, in order to determine the
upcoming work to do. Still in chapter 8, it has been explained that the nature of the
transient of the simulated results of appendix G seems to be correct, because the rotor
currents and the electromagnetic torque reached 0 at steady-state for the no-load test
conditions. Indeed, at no-load, there is no induction in the rotor, so these values must be
0. Also, the steady-state value of the rotor angular speed was 314.16 electrical rad/s,
which is also correct for a 50 Hz input voltage. For the locked rotor test, the
electromagnetic torque reached a value different to 0 at steady-state, and that result is
also correct. Therefore, only the values of the steady-state stator rms phase currents are
questionable after the simulation. It has been explained that the temperature
measurements have not been accurate during the lab tests, but it has also been explained
in chapter 8 that the value of the measured temperatures does not affect the steady-state
stator currents enough to explain the differences between the measurements and the
simulation. It is possible that the machine did not run enough long before the
measurements during the no-load test and at this moment the equivalent inductances are
not accurate. An error on the value of the inductances of the model is well enough to
change the steady-state of the stator currents and that can be the reason of the large
relative errors between the measurements and the simulation. Also, the locked rotor test
must be performed before the no-load test to prevent the overheating of the windings. A
lab measurement of the moment of inertia can also help to get better results from the
simulation. So, another parameter tests will be necessary to improve the simulation
results.
To compare the quality of the model itself, the only real way to do it is to compare
measured transients with simulated transients, which is not possible because no measured
transients have been taken in lab. But, it can be interesting to discuss about the
assumptions, the advantages and the disadvantages of the 2 models, including the model
that consider the saturation. Thus, in both models, the air gap diameter is assumed to be
constant, the effect of the stator and rotor slots is neglected, like the effect of the
temperature and the core losses, the electromotive force is considered to be sinusoidal,
the windings or equivalent windings are assumed to be magnetically and electrically
symmetrical, the rotor is considered as equivalent windings and the currents of the 3
phases are considered to be balanced. So, the big difference between the 2 models is that
one of them considers the saturation of the magnetizing flux. Indeed, the ABC/abc model
does not consider the saturation and the dq model considers only the saturation of the
main magnetizing flux, by neglecting the harmonics of the magnetizing flux and the
saturation of the leakage fluxes. Also, since the dq model is a reduced order one, it
considers that the zero-sequence currents and the saturation do not affect each other.
Thus, the drawbacks of the 2 models are that the temperature is not considered, like the
core losses and that the three phases are considered to be perfectly balanced. Indeed, it is
well known that the temperature affects the value of the different parameters of a model.
Also, the 3 phases are rarely perfectly balanced. But, all these assumptions make the
models simple and usable, which is an important advantage of them. The major drawback
of the ABC/abc model is that the saturation is not considered, but it has the advantage that
the 3 phase currents are directly observable. For the dq model, the disadvantages are the
42
fact that the saturation of the harmonics of the magnetizing flux are not considered, like
the saturation of the leakage fluxes and the dq variables have to be transformed to
ABC/abc variables to be observable. Nevertheless, the saturation of the main magnetizing
flux is considered, which is a major advantage.
So, as future works, it is first important to obtain better lab measurements, to be able to
have a better evaluation of the ABC/abc model. So, the DC test, the no-load test and the
locked rotor test must be performed again. Then, a test must be performed in lab to obtain
the magnetizing inductance curve and another test can be performed to obtain a measured
value of the moment of inertia of the rotor. Moreover, the Simulink model of the dq
model considering saturation must be built and simulated, so it will be possible to
evaluate the quality of it. Then, if the 2 models are not enough accurate, the addition of
the effect of the temperature on the parameters or the core losses will be investigated.
Finally, after having obtained an acceptable model of a squirrel-cage induction generator,
it will be possible to extend the investigation on the self-excitation and to work on the
regulation of the voltage and the frequency of a stand-alone energy conversion system.
43
Chapter 10
Conclusion
This project is about stand-alone renewable energy systems with squirrel-cage induction
generators, but more particularly about the construction of an accurate dynamic model
that can be used for voltage and frequency regulation. So, 3 principal objectives have
been stated, which are the construction of a Simulink model without saturation
considerations, the construction of another Simulink model with saturation considerations
and the investigation of the self-excitation of an induction machine.
Thus, the basic knowledge concerning the squirrel-cage induction generators has been
presented. Indeed, it has been question of the synchronous speed, the slip, the rotor
frequency, the electrical radians, the steady-state model, the power assessment and the
torque characteristics. After, the self-excitation of an induction generator has been
explained and the theory of 2 dynamic models has been presented, one without saturation
and the other with saturation. Then, the Simulink model of the first model have been built
and presented in this report. The theory of the laboratory tests that are necessary to obtain
the different parameters of the models have been presented, so the DC test, the no-load
test, the locked rotor test and the magnetizing inductance curve test have been presented.
The first 3 tests have also been performed in lab. Afterward, the Simulink model has been
evaluated and it has been shown that additional lab work is necessary to increase the
quality of the model. Indeed, it is necessary to be sure that the machine runs at least 30
minutes before the no-load measurements. Finally, all the work done have been discussed
and the future work that will be necessary has been explained.
So, all the parameter tests will be performed again and that time the magnetizing
inductance curve test will be performed. Also, the inertia of the rotor will be measured in
lab. Then, the Simulink model with saturation considerations will be built and evaluated.
If the models are not enough accurate, some additions will be investigated, like the
consideration of the effect of the temperature on the different parameters or the
consideration of the core losses. After all, it will be possible to work on the self44
excitation and on the regulation of the voltage and the frequency of a stand-alone
renewable energy conversion system.
45
Bibliography
[1] Félix A. FARRET, Marcelo Godoy SIMÕES, Renewable energy systems: Design and
Analysis with Induction Generators, Boca Raton, Florida, CRC Press, 2004, 408 p.
[2] R. C. BANSAL, T. S. BHATTI, D. P. KOTHARI, Bibliography on the Application of
Induction Generators in Nonconventional Energy Systems, IEEE Transactions on
Energy Conversion, September 2003, Volume 18, Issue 3.
[3] Jayanta K. CHATTERJEE, B. Venkatesa PERUMAL, Naveen Reddy GOPU,
Analysis of Operation of a Self-Excited Induction Generator with Generalized
Impedance Controller, Transactions on Energy Conversion, June 2007, Volume 22,
Issue 2.
[4] T. F. CHAN, K. A. NIGIM, L. L. LAI, Voltage and Frequency Control of SelfExcited Slip-Ring Induction Generators, IEEE Transactions on Energy Conversion,
March 2004, Volume 19, Issue 1.
[5] Bhim SINGH, S. S. MURTHY, Sushma GUPTA, Analysis and Design of
STATCOM-Based Voltage Regulator for Self-Excited Induction Generators, IEEE
Transactions on Energy Conversion, December 2004, Volume 19, Issue 4.
[6] Luiz A. C. LOPES, Rogério G. ALMEIDA, Wind-Driven Self-Excited Induction
Generator with Voltage and Frequency Regulated by a Reduced-Rating Voltage
Source Inverter, Transactions on Energy Conversion, June 2006, Volume 21, Issue 2.
[7] Jyoti SASTRY, Olorunfemi OJO, Zhiqiao WU, High-Performance Control of a Boost
AC-DC PWM Rectifier/Induction Generator System, IEEE Transactions on Industry
Applications, September/October 2006, Volume 42, Issue 5.
[8] Woei-Luen CHEN, Yuan-Yih HSU, Controller Design for an Induction Generator
Driven by a Variable-Speed Wind Turbine, Transactions on Energy Conversion,
September 2006, Volume 21, Issue 3.
[9] Olorunfemi OJO, Performance of Self-Excited Single-Phase Induction Generators
with Shunt, Short-Shunt and Long-Shunt Excitation Connections, IEEE Transactions
on Energy Conversion, September 1996, Volume 11, Issue 3.
[10] Olorunfemi OJO, Dynamics and System Bifurcation in Autonomous Induction
Generators, IEEE Transactions on Industry Applications, July/August 1995, Volume
31, Issue 4.
[11] Bhim SINGH, S. S. MURTHY, Sushma GUPTA, Analysis and Design of
Electronic Load Controller for Self-Excited Induction Generators, IEEE Transactions
on Energy Conversion, March 2006, Volume 21, Issue 1.
46
[12] IEEE Standard 112-2004, IEEE Standard Test Procedure for Polyphase Induction
Motors and Generators.
[13] Stephen J. CHAPMAN, Electric Machinery Fundamentals, McGraw Hill, New
York, 2005, 4th edition, 744 p.
[14] Clovis GOLDEMBERG, Aderbal DE ARRUDA PENTEADO Jr., Improvements
on the inductance matrix inversion simplifying the use of the ABC/abc induction
machine model, International Conference on Electric Machines and Drives, May
1999.
[15] P. PILLAY, V. LEVIN, Mathematical Models for Induction Machines, IEEE
Industry Applications Conference, October 1995, Volume 1.
[16] V. DONESCU, A. CHARETTE, Z. YAO, V. RAJAGOPALAN, Modeling and
Simulation of Saturated Induction Motors in Phase Quantities, IEEE Transactions on
Energy Conversion, September 1999, Volume 14, Issue 3.
[17] A. AH-JACO, H. YAHOUI, A. MAKKI, G. GRELLET, Transient Model of the
Three-Phase Cage Asynchronous Machine with Saturation and Space Harmonics
Effects, European Transactions on Electrical Power, March 2005, Volume 15, Issue
2.
[18] T. J. VYNCKE, F. M. L. L. DE BELIE, K. R. GELDHOF, L. VANDEVELDE,
R. K. BOEL, J. A. A. MELKEBEEK, A Simulink State-Space Model of Induction
Machines Including Magnetizing-Flux Saturation, International Symposium on Power
Electronics, Electrical Drives, Automation and Motion, May 2006.
[19] K. Y. LU, E. RITCHIE, P. O. RASMUSSEN, P. SANDHOLDT, A Simple
Method to Estimate Inductance Profile of a Surface Mounted Permanent Magnet
Transverse Flux Machine, IEEE International Conference on Power Electronics and
Drive Systems, November 2003, Volume 2.
[20] S. S. MURTHY, Bhim SINGH, Virendra KUMAR SHARMA, A Frequency
Response Method to Estimate Inductance Profile of Switched Reluctance Motor,
IEEE International Conference on Power Electronics and Drive Systems, May 1997,
Volume 1.
[21] Ned MOHAN, Advanced Electric Drives : Analysis, Control and Modeling using
Simulink®, Minneapolis, Minnesota, MNPERE, 2001.
[22] F. BARRET, Régime Transitoire des Machines Tournantes Électriques,
Collection des études de recherche Édition Eyrolles, Paris 1982.
[23] Marian P. KAZMIERKOWSKI, R. KRISHNAN, Frede BLAABJERG, Control in
Power Electronics: Selected Problems, Academic Press, 2002, ISBN 0-12-402772-5.
47
Appendix A
dq0 Transformation
To simply the dynamic model of the induction generator, it is possible to reduce the order
of the model by using the Park transformation. This transformation converts a 3-axis
ABC/abc model into a 2-axis dq0 model, where the subscripts d, q and 0 are related in
that order to the direct and the quadrature axis and to the zero-sequence component. The
comparison of the 3 frames of reference can be seen in the following figure.
Figure A.1:
Representation of the different frames of reference (abc, ABC and
dq0) used for the Park transformation [21].
48
The angles θdA and θda are respectively the angle between d and A (stator) axis and the
angle between d and a (rotor) axis. Then, the Park matrix can be used to obtain the
different variables from the ABC/abc frame of reference to the dq0 frame of reference
[21]:
[ P] =
⎡
2 ⎞
4 ⎞⎤
⎛
⎛
cos ⎜ θ − π ⎟ ⎥
⎢ cos (θ ) cos ⎜ θ − 3 π ⎟
3 ⎠⎥
⎝
⎝
⎠
⎢
2⎢
2 ⎞
4 ⎞
⎛
⎛
− sin (θ ) − sin ⎜ θ − π ⎟ − sin ⎜ θ − π ⎟ ⎥
3⎢
3 ⎠
3 ⎠⎥
⎝
⎝
⎢
⎥
2
2
2
⎢
⎥
⎢⎣ 2
⎥⎦
2
2
(A.1)
Where the angle θ corresponds to the angle θdA or θda, which depends if the
transformation is performed for the stator or the rotor. Also, the inverse of the Park
transformation can be used to bring back the variables to the original ABC/abc frame of
reference [21].
[ P ]−1 = [ P ]T =
⎡
− sin (θ )
⎢ cos (θ )
⎢
2⎢ ⎛
2 ⎞
2 ⎞
⎛
cos⎜θ − π ⎟ − sin ⎜ θ − π ⎟
⎢
3
3 ⎠
3 ⎠
⎝
⎝
⎢
⎢cos⎛⎜θ − 4 π ⎞⎟ − sin ⎛⎜ θ − 4 π ⎞⎟
⎢⎣ ⎝
3 ⎠
3 ⎠
⎝
2⎤
⎥
2 ⎥
2⎥
2 ⎥
⎥
2⎥
2 ⎥⎦
(A.2)
At this point, it is important to mention that the Park transformation can be performed for
the voltages, the currents and the fluxes [22],[23], so the following equations can be used
to perform the transformation:
[Vdq 0 ] = [ P ] × [Vabc ], [Vabc ] = [ P ]T × [Vdq 0 ],
[ I dq 0 ] = [ P ] × [ I abc ], [ I abc ] = [ P ]T × [ I dq 0 ],
(A.3)
[λdq 0 ] = [ P ] × [λabc ] and [λabc ] = [ P ]T × [λdq 0 ]
Where,
[
] , [V ] = [v v
[ I ] = [i i i ] , [ I ] = [i i
] = [λ λ λ ] and [λ ] = [λ
[Vdq 0 ] = v d
vq
v0
T
abc
a
b
ic ] ,
a
λb
T
dq 0
[λ dq 0
d
0
q
abc
a
T
d
q
0
abc
49
vc ] ,
T
b
T
λc ]T
Thus, from the general relation of equation 5.1, which is:
[Vabc ] = R × [ I abc ] +
d [λabc ]
dt
(A.4)
That time, only the rotor equations are considered, but the following is also true for the
stator equations. So, it is possible to obtain the following form using the relations of
equation A.3:
[Vabc ] = R × [ P] × [ I dq 0 ] +
T
(
)
d [ P ]T × [λdq 0 ]
dt
(A.5)
By expanding the derivatives, equation A.5 becomes:
[Vabc ] = R × [ P ]T × [ I dq 0 ] +
d [λdq 0 ]
d [ P ]T
× [λdq 0 ] + [ P ]T ×
dt
dt
(A.6)
Hence, using the following relation from equation A.3:
[Vabc ] = [ P]T × [Vdq 0 ]
The following equation can be found:
[Vdq 0 ] = R × [ I dq 0 ] + [ P ] ×
d [λdq 0 ]
d [ P ]T
× [λdq 0 ] +
dt
dt
(A.7)
And because,
⎡
⎢ 0
d [ P ]T ⎢ dθ
=⎢
[ P] ×
dt
⎢ dt
⎢ 0
⎢⎣
−
dθ
dt
⎤
0⎥
0 − ω 0⎤
⎥ ⎡
⎢
0⎥ = ⎢ω 0 0⎥⎥ = −[Ω]
⎥
0⎥ ⎢⎣ 0 0 0⎥⎦
⎥⎦
0
0
The final form of the voltage equations in dq0 coordinates is:
[Vdq 0 ] = R × [ I dq 0 ] +
d [ λ dq 0 ]
− [ Ω ] × [ λ dq 0 ]
(A.8)
dt
So, by applying the last result on both the stator and rotor voltage equations, the dq0 form
of these equations is:
50
[VDQ 0 ] = [ R ] × [ I DQ 0 ] +
d [λ DQ 0 ]
dt
− [WDQ 0 ] × [λ DQ 0 ]
(A.9)
Where,
[
] = [i
] = [λ
[VDQ 0 ] = v sd
v sq
[ I DQ 0
i sq
[λ DQ 0
sd
sd
λ sq
vs0
is0
λs0
v rd
ird
v rq
irq
λ rd
vr 0
ir 0
]
]
T
T
λ rq
λ r 0 ]T
And,
⎡ Rs
⎢0
⎢
⎢0
[ R] = ⎢
⎢0
⎢0
⎢
⎢⎣ 0
0
0
0
0
Rs
0
0
0
0
0
Rs
0
0
Rr
0
0
0
0
0
Rr
0
0
0
0
0⎤
⎡ 0
⎥
⎢− ω
0⎥
⎢ dA
⎢ 0
0⎥
⎥ and [WDQ 0 ] = ⎢
0⎥
⎢ 0
⎢ 0
0⎥
⎥
⎢
Rr ⎥⎦
⎢⎣ 0
ω dA 0
0
0
0
0
0
0
0
0
0
0
0
ω da
0
0 − ω da
0
0
0
0
0
0
0
0⎤
0⎥⎥
0⎥
⎥
0⎥
0⎥
⎥
0⎥⎦
The angular speeds ωdA and ωda are the derivatives of the angular positions θdA and θda
from figure A.1, so they are the angular speeds of the dq0 frame of reference compared to
the A and a axis. From the same figure, it is possible to define ωda in function of ωdA.
ω da = ω dA − ω m
(A.10)
In equation A.10, the angular speed ωm of the rotor can be recognized. To determine ωdA
and ωda, the speed of the dq0 frame of reference must be chosen. Thus, the rotation of the
dq plan can be related to 3 different speeds, which are the synchronous speed ωsync, the
rotor speed ωm or 0. Hence, the angular speeds ωda respectively equal to ωslip, 0, or –ωm.
51
Appendix B
Simulink Model without Saturation
Considerations
The Simulink model used to simulate the dynamic performances of a squirrel-cage
induction generator is presented in figure B.1. First, it is important to state that model is
not considering the saturation effect, but it has been built to be able to include easily a
non constant magnetizing inductance in the model.
Figure B.1:
Simulink dynamic model of a squirrel-cage induction machine.
The best way to understand the model is to look its different parts. Firstly, figure B.2
shows the initial code that must be executed before simulate the model. It contains the
parameters of the induction machine Rs, Rr, Lℓs, Lℓr and Lm (see table 8.1) and other input
parameters, which are the number of poles per phase p, the frequency f of the input
52
voltage, the load torque TL, the inertia J of the rotor and the magnitude VA, VB and VC of
the input voltages at the stator terminals.
Figure B.2:
Figure B.3:
Initial code of the Simulink model.
Input voltages (a), equivalent inductances (b) and resistance matrix
(c).
Secondly, figure B.3 shows some of the inputs of the model. In (a), 3 sine waves blocs,
vA, vB and vC, are used as the voltage inputs at the stator terminals. Their phase shifts are
respectively 0, 2π/3 and −2π/3 rad. These sine waves can also be seen using the scope
53
Vin. Finally, 3 constant blocs, va, vb and vc, are used as the rotor input voltages. They are
set to 0, because the rotor is of the squirrel-cage type. A multiplexer and a reshape bloc
are used to build the matrix V of equation 5.5 with the 6 inputs. In (b), the scalar Msr, Ls
and Lr are simply calculated with Lm, Lℓs and Lℓr, using equation 5.12 and the relations of
Ls and Lr in equation 5.7. In (c), a constant bloc is used to create the matrix R of equation
5.5.
After, figure B.4 shows the 3 subsystems used for the matrices of equations 5.8 (a) and
5.9 (b) and for the calculation of the electromagnetic torque Te of equation 5.11 (c).
These sybsystems are explained at the end of this appendix.
Figure B.4:
Subsystems for equations 5.8 (a), 5.9 (b) and 5.11 (c).
Moreover, figure B.5 shows the Simulink diagram that illustrates equation 5.5 and figure
B.6 shows the process to obtain the angular speed of the rotor omega_r in electrical rad/s
and the angular position of the rotor theta_r in electrical rad, which are necessary in this
model. First, equation 5.11 is used to find the electromotive torque Te, which can be
observed with the scope Te. Then, equation 5.10 is used to get the mechanical speed ωm
of the rotor. After, equation 3.6 is used to obtain the angular speed ωr of the rotor in
electrical rad/s, which can be integrated to obtain θr. Finally, the angular speed ωr can be
observed using the scope omega_r.
After all, figure B.7 shows the scopes is and ir that can be used to observe the stator and
rotor currents.
54
Figure B.5:
Figure B.6:
Equation 5.5.
Calculation of theta_r and omega_r.
Figure B.7:
Scopes is and ir.
55
Finally, to complete the explanations about the Simulink model, the 3 submatrices of
figure B.4 must be defined. So, the subsystem d[L]/dtheta_r will be explained first. This
one can be seen in figure B.8 and the 2 inputs of that subsystem are the angular position
θr and the stator-rotor mutual inductance Msr.
Figure B.8:
Figure B.9:
Subsystem d[L]/dtheta_r.
Inputs and functions f1 through f3.
56
In this subsystem, the functions blocs f1, f2 and f3 are first calculated with the 2 inputs, as
shown in figure B.9. So, at this point, the signals f1 through f3 are defined as:
f1 = − M sr sin (θ r ),
2π ⎞
⎛
f 2 = − M sr sin ⎜θ r +
⎟ and
3 ⎠
⎝
2π ⎞
⎛
f 3 = − M sr sin ⎜θ r −
⎟
3 ⎠
⎝
Figure B.10: Matrices M1 through M3 (a) and output matrix d[L] (b).
Then, the signals f1 through f3 are each multiplied by a matrix, which are C1, C2 and C3
respectively (see figure B.10(a)). The results of these 3 multiplications give 3 new
matrices, M1, M2 and M3, that are defined as:
57
− M sr g1
0
0
0
0
0
⎡ 0
⎢ 0
⎢
⎢ 0
M1 = ⎢
⎢− M sr g1
⎢ 0
⎢
⎢⎣ 0
0
0
0
0
− M sr g1
0
⎡ 0
⎢ 0
⎢
⎢ 0
M2 = ⎢
⎢ 0
⎢− M sr g 2
⎢
⎢⎣ 0
⎡ 0
⎢ 0
⎢
⎢ 0
M3 = ⎢
⎢ 0
⎢ 0
⎢
⎣⎢− M sr g 3
0
0
0
0
0
− M sr g 2
0
0
0
− M sr g 2
0
0
0
0
− M sr g 2
0
0
0
− M sr g 2
0
0
0
0
0
0
0
0
− M sr g 3
0
0
0
0
− M sr g 3
0
− M sr g 3
0
0
− M sr g 3
0
0
0
0
0
0
0
− M sr g 1
0
0
0
0
0
0
− M sr g1
0
0
0
0
0
0
0
0 ⎤
0 ⎥⎥
− M sr g1 ⎥
⎥
0 ⎥
0 ⎥
⎥
0 ⎥⎦
0 ⎤
− M sr g 2 ⎥⎥
0 ⎥
⎥
0 ⎥
0 ⎥
⎥
0 ⎥⎦
− M sr g 3 ⎤
0 ⎥⎥
0 ⎥
⎥
0 ⎥
0 ⎥
⎥
0 ⎦⎥
Where,
2π ⎞
⎛
g1 = sin (θ r ), g 2 = sin ⎜θ r +
⎟ and
3 ⎠
⎝
2π ⎞
⎛
g 3 = sin ⎜θ r −
⎟
3 ⎠
⎝
After all, with the summation of the signals M1 through M3 (see figure B.10(b)), the
output matrix d[L] is obtained, which is the same as in equation 5.8.
The second subsystem of figure B.4, [L]-1, is shown in figure B.11. The 4 inputs of that
subsystem are the angular position θr, the stator-rotor mutual inductance Msr and the
stator and rotor equivalent inductances Ls and Lr (see figure B.12(a)). From Msr, Ls and
Lr, the signals K2, K3 and K4 are calculated (see figure B.12(b)) and are defined as:
58
L s Lr 3
3
−
−
2
M sr 4
4
, K3 =
K2 =
L s Lr 9
L s Lr 9
−
−
4
4
M sr2
M sr2
1
M sr
K4 =
L s Lr 9
−
4
M sr2
−
and
Figure B.11: Subsystem [L]-1.
Figure B.12: Inputs (a) and signals K2 through K4.
Thus, from the angular position θr, the signals f1 through f3 are computed (see figure
B.13(a)). These signals are defined as:
59
f1 = cos(θ r ),
2π ⎞
⎛
f 2 = cos⎜θ r +
⎟ and
3 ⎠
⎝
2π ⎞
⎛
f 3 = cos⎜θ r −
⎟
3 ⎠
⎝
Also, from Ls and Lr, the signals Ls-1 and Lr-1 becomes (see figure B.13(b)):
L−s1 =
1
Ls
and
L−r1 =
1
Lr
Figure B.13: Signals f1 through f3 (a) and signals Ls-1 and Lr-1 (b).
After that, 4 matrices are defined, which are matrices L1 through L4, and are shown in
figure B.14(a). These matrices are used with signals Ls-1 and Lr-1 to obtain matrices M1
through M4 (see figure B.14(b)). They are defined as:
⎡1
⎢L
⎢ s
⎢0
⎢
⎢
M1 = ⎢ 0
⎢
⎢0
⎢
⎢0
⎢⎣ 0
0
0
1
Ls
0
0
0
0
0
1
Ls
0
0
0
⎤
⎡0 0
0 0 0⎥
⎢0 0
⎥
⎢
⎢0 0
0 0 0⎥
⎥
⎢
⎥
⎢0 0
=
M
,
2
0 0 0⎥
⎢
⎥
⎢0 0
0 0 0⎥
⎢
⎥
⎢
0 0 0⎥
⎢0 0
⎢⎣
0 0 0⎥⎦
60
0
0
0
0
0
0
0
1
Lr
0
0
0
0
0
0
1
Lr
0
0
0
⎤
⎥
⎥
⎥
⎥
0⎥
,
⎥
0⎥
⎥
1⎥
⎥
Lr ⎥⎦
0
0
0
⎡
⎢0
⎢
⎢1
⎢ Ls
⎢
M3 = ⎢ 1
⎢ Ls
⎢0
⎢
⎢0
⎢⎣ 0
1
Ls
0
1
Ls
0
0
0
1
Ls
1
Ls
0
0
0
0
⎤
⎡0 0
0 0 0⎥
⎢0 0
⎥
⎢
⎢0 0
0 0 0⎥
⎥
⎢
⎥
⎢0 0
0 0 0⎥ and M 4 = ⎢
⎥
⎢0 0
0 0 0⎥
⎢
⎥
⎢
0 0 0⎥
⎢0 0
0 0 0⎥⎦
⎣⎢
0
0
0
0
0
0
0
0
0
0
1
Lr
1
Lr
0
0
0
1
Lr
0
1
Lr
0⎤
0 ⎥⎥
0⎥
1⎥
Lr ⎥⎥
1⎥
Lr ⎥
⎥
0⎥
⎦⎥
Figure B.14: Matrices L1 through L4 (a), M1 through M4 (b) and matrices M5 and
M6 (c).
61
After, matrices M1 through M4 are used to get matrices M5 and M6 as shown in figure
B.14(c). Thus, matrices M5 and M6 become:
⎡1
⎢L
⎢ s
⎢0
⎢
⎢
⎢0
M5 = ⎢
⎢
⎢0
⎢
⎢0
⎢
⎢
⎢0
⎢⎣
0
0
0
0
1
Ls
0
0
0
0
1
Ls
0
0
0
0
1
Lr
0
0
0
0
1
Lr
0
0
0
0
⎤
⎡
0⎥
⎢0
⎥
⎢
⎥
⎢1
0
⎥
⎢ Ls
⎥
⎢1
0⎥
⎢
⎥ and M = ⎢ Ls
6
⎥
⎢
0⎥
⎢0
⎥
⎢
⎥
⎢0
0
⎥
⎢
⎥
⎢
1
⎥
⎢0
Lr ⎥⎦
⎢⎣
1
Ls
1
Ls
1
Ls
0
0
0
0
0
0
0
0
0
0
1
Lr
0
0
0
0
0
1
Ls
1
Lr
1
Lr
0
1
Lr
⎤
0⎥
⎥
0⎥
⎥
⎥
0⎥
⎥
1⎥
Lr ⎥⎥
1⎥
Lr ⎥
⎥
0⎥
⎥⎦
Figure B.15: Matrices C1 through C3 (a), M7 through M11 (b) and output matrix
[L]-1 (c).
62
Hence, figure B.15(a) shows the 3 last matrices that are needed in that subsystem, which
are matrices C1 through C3. These matrices and the previous signals are used in 5
multiplication blocs, to obtain matrices M7 through M11. The multiplication blocs can be
seen in figure B.15(b) and the resulting matrices are defined as:
⎡ 0
⎢ 0
⎢
⎢ 0
M7 = ⎢
⎢ K 4 f1
⎢ 0
⎢
⎢⎣ 0
0
0
K 4 f1
0
0
0
0
0
0
0
K 4 f1
0
0
0
0
K 4 f1
0
0
0
0
K 4 f1
0
0
⎡ 0
⎢ 0
⎢
⎢ 0
M9 = ⎢
⎢ 0
⎢ 0
⎢
⎢⎣ K 4 f 3
0
0 ⎤
⎡ 0
⎢ 0
0 ⎥⎥
⎢
⎥
⎢ 0
K 4 f1
⎥, M 8 = ⎢
0 ⎥
⎢ 0
⎢K 4 f2
0 ⎥
⎥
⎢
0 ⎥⎦
⎢⎣ 0
0
0
0
0
0
0
0
0
K 4 f3
0
0
K 4 f3
K 4 f3
0
0
K 4 f3
0
0
0
0
0
0
0
0
and
⎡
⎢ 0
⎢
⎢ K3
⎢ Ls
⎢K
⎢ 3
L
M 11 = ⎢ s
⎢
⎢ 0
⎢
⎢ 0
⎢
⎢
⎢ 0
⎢⎣
K 4 f3 ⎤
0 ⎥⎥
0 ⎥
⎥ , M 10
0 ⎥
0 ⎥
⎥
0 ⎥⎦
K3
Ls
K3
Ls
K3
Ls
0
0
0
K 4 f2
0
0
0
0
0
K4 f2
0
0
0
K4 f2
0
0
0
0
0
0
K4 f2
0
0
0
⎡ K2
⎢L
⎢ s
⎢ 0
⎢
⎢
⎢ 0
=⎢
⎢
⎢ 0
⎢
⎢ 0
⎢
⎢
⎢ 0
⎢⎣
0
0
0
0
0
0
0
0
0
0
K3
Lr
0
0
0
0
0
K3
Ls
63
K3
Lr
K3
Lr
0
K3
Lr
0
0
0
0
K2
Ls
0
0
0
0
K2
Ls
0
0
0
0
K2
Lr
0
0
0
0
K2
Lr
0
0
0
0
⎤
0 ⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
K3 ⎥
⎥
Lr ⎥
K3 ⎥
Lr ⎥
⎥
0 ⎥
⎥⎦
0 ⎤
K 4 f 2 ⎥⎥
0 ⎥
⎥,
0 ⎥
0 ⎥
⎥
0 ⎥⎦
⎤
0 ⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
0 ⎥
⎥
K2 ⎥
⎥
Lr ⎥⎦
Figure B.16: Subsystem Eq 5.11.
Figure B.17: Input matrix I (a), multiplication blocs (b) and summation blocs (c).
64
After all, the summation of matrices M7 through M11 give the matrix of equation 5.9, as
shown in figure B.15(c).
Then, to finish the definition of the Simulink model, the third subsystem has to be
detailed. So, the subsystem of figure B.4(c) is presented in figure B.16. This subsystem
has 3 inputs, which are the currents matrix I (see figure B.17(a)), the angular position θr
and the stator-rotor mutual inductance Msr. From the currents matrix I, figures B.17(b)
and (c) show the process to obtain the 3 following functions:
F1 = i A ia + i B ib + iC ic , F2 = i A ib + i B ic + iC ia
and
F3 = i A ic + i B ia + iC ib
Figure B.18: Signals f1 through f3 (a) and multiplication blocs (b).
After, the signals f1 through f3 are computed with the input θr (see figure B.18(a)). So,
they are defined as:
65
f1 = sin (θ r ),
2π ⎞
⎛
f 2 = sin ⎜θ r +
⎟ and
3 ⎠
⎝
2π ⎞
⎛
f 2 = sin ⎜θ r −
⎟
3 ⎠
⎝
Thus, 3 more multiplication blocs (see figure B.18(b)) are used to modify function F1
through F3, which ones become:
F1 = (i A ia + i B ib + iC ic ) f 1 , F2 = (i A ib + i B ic + iC ia ) f 2
and
F3 = (i A ic + i B ia + iC ib ) f 3
After all, figure B.19 shows the rest of the subsystem that is used to finally obtain the
electromagnetic torque Te of equation 5.11. In the gain bloc, the variable p is again the
number of poles per phase.
Figure B.19: Output electromotive torque Te.
66
Appendix C
DC Test
C.1
Purpose
The DC test is simply used to determine the resistance Rs of the stator windings.
C.2
Setup and Equipment
See figure C.1 and table C.1 for test setup with a squirrel-cage induction machine
(SCIM).
Figure C.1:
Component
Apparatus
Producer
Type
Ameter
Multimeter
Fluke
179
7 25 03 C10
Vmeter
Multimeter
Fluke
179
7 25 03 C13
Vsource
DC Voltage Source
GW Instek
GPS-4303
7 27 01 B03
Temperature
Multimeter
Fluke
179
7 25 03 C14
Table C.1:
C.3
•
The DC test setup.
AAU-nr.
Used equipment.
Procedure
Make the setup of section C.2 with the stator windings Y-connected.
67
•
Turn on the voltage source, at first with 0 V. Then, raise slowly the voltage to obtain
a DC current IA as closest as possible to the rated current Irated of the machine.
•
Measure the DC source voltage VAB and the DC current IA.
•
Redo the same procedure with the DC voltage source between the B and C phases,
and also between the A and C phases, in order the measure VBC, IB, VCA and IC.
•
Note the ambient temperature of the test ta.
C.4
Results
The following table of calculations can be used to determine Rs.
Item Test Point
Source or Calculation
Value
Unit
1
DC source voltage between phases A and B (VAB)
Measurement
19.60
V
2
DC current in phase A (IA)
Measurement
3.376
A
3
DC source voltage between phases B and C (VBC)
Measurement
19.47
V
4
DC current in phase B (IB)
Measurement
3.373
A
5
DC source voltage between phases C and A (VCA)
Measurement
19.45
V
6
DC current in phase C (IC)
Measurement
3.376
A
7
Resistance of the stator windings of phase A (RA)
[(1)/(2)−(3)/(4)+(5)/(6)]/2
2.8973
Ω
8
Resistance of the stator windings of phase B (RB)
(3)/(4)−(5)/(6)+(11)
2.9084
Ω
9
Resistance of the stator windings of phase C (RC)
(5)/(6)−(11)
2.8639
Ω
10
Average DC source voltage (Vdc)
[(1)+(3)+(5)]/3
19.5067
V
11
Average DC current (Idc)
[(2)+(4)+(6)]/3
3.3750
A
12
Resistance of the stator windings (Rs)
0.5×(10)/(11)
2.8899
Ω
13
Ambient temperature of the test (ta)
Measurement
22.8
°C
Table C.2:
C.5
Table of calculations for the DC test.
Test Analysis
The main objective of this test was to obtain the equivalent resistance Rs of the stator
windings. So, according to the lab measurements and the calculations, which can be seen
68
in table C.2, the resistances of each phase of the stator RA, RB and RC are respectively
2.8973 Ω, 2.9084 Ω and 2.8639 Ω. By assuming that the resistances of the 3 phases are
equivalent, Rs is 2.8899 Ω. With that assumption, the maximum error on the resistance
value is not much that 0.9%, so the statement is correct. These measurements have been
taken at an ambient temperature ta of 22.8 °C. This information is important, because it is
known that the electrical resistance varies with the temperature and later on the value of
Rs will be corrected in accordance with the temperature.
Some observations have also been taken during the test, which can be different sources of
error on the calculated value. First of all, the DC voltage source used in this test could not
supply more than 3 A, so the measures was not taken with the rated current, which is 5 A.
However, 3 A is probably enough to obtain a correct value of Rs. Also, the technique used
to measure the temperature was not the most accurate one, because the multimeter takes a
long time to get to a steady-state value. Accordingly, the measure of the temperature can
be lower than the real temperature.
C.6
Conclusion
According to the DC test, the values of the resistance of each stator phase (RA, RB and
RC), the equivalent resistance Rs of the stator windings and the ambient temperature ta can
be seen in table C.3.
Test Point
Value Unit
Resistance of the stator windings of phase A (RA)
2.8973
Ω
Resistance of the stator windings of phase B (RB)
2.9084
Ω
Resistance of the stator windings of phase C (RC)
2.8639
Ω
Resistance of the stator windings (Rs)
2.8899
Ω
22.8
°C
Ambient temperature of the test (ta)
Table C.3:
Results of the DC test.
69
Appendix D
No-Load Test
D.1
Purpose
This test is used with the locked rotor test to obtain the parameters of the equivalent
circuit of an induction machine. Thus, the leakage inductances Lℓs and Lℓr, for the stator
and the rotor respectively, the magnetizing inductance Lm, the equivalent resistance of the
magnetic core Rc and the rotor equivalent resistance Rr can be obtained. In addition, the
friction and windage losses PF&W and the core losses Pcore can be determined.
D.2
Setup and Equipment
See figure D.1 and table D.1 for test setup with a squirrel-cage induction machine
(SCIM).
Figure D.1:
The no-load test setup.
70
Component
Apparatus
Producer
Type
Panalyser
Universal Power
Analyser
Voltech
PM 3000A
Vsource
5 kVA AC Power
Source
California Instruments
5001 ix
Temperature
Multimeter
Fluke
179
Table D.1:
D.3
AAU-nr.
7 27 03 D01
7 25 03 C14
Used equipment.
Procedure
•
Make the setup of section D.2 with the stator windings Y-connected.
•
Apply a line-to-line voltage lower than the rated voltage of the machine. The
frequency of input voltage must be the rated frequency. Let the machine rotate for a
period of time until the temperature of the stator winding become stable and notice
that temperature (ts).
•
Measure the currents in each phase of the stator (IA, IB and IC), the line-to-line
voltages between each phase pairs (VAB, VBC and VCA) and the powers between phase
pairs (PAB, PBC and PCA).
•
Perform the same measurements for different lower values of voltage and finish with
a voltage as closest as possible to the rated voltage. The number of point will depend
upon the needed precision.
D.4
Results
The following table of calculations can be used to determine the core losses Pcore. This
example contains 5 measuring points, but the method can be used with more measured
point to increase the precision of the technique.
71
Item Test Point
Source or Calculation
Value
Unit
1
Temperature of the stator windings (ts1) (point 1)
Measurement
30.6
°C
2
rms current in phase A (IA1) (point 1)
Measurement
350.0
mA
3
rms current in phase B (IB1) (point 1)
Measurement
396.8
mA
4
rms current in phase C (IC1) (point 1)
Measurement
437.6
mA
Measurement
33.35
V
Measurement
34.55
V
Measurement
33.51
V
5
6
7
rms line-to-line voltage between phases A and B (VAB1)
(point 1)
rms line-to-line voltage between phases B and C (VBC1)
(point 1)
rms line-to-line voltage between phases C and A (VCA1)
(point 1)
8
Power between phases A and B (PAB1) (point 1)
Measurement
11.532
W
9
Power between phases B and C (PBC1) (point 1)
Measurement
13.164
W
10
Power between phases C and A (PCA1) (point 1)
Measurement
14.526
W
11
Temperature of the stator windings (ts2) (point 2)
Measurement
31.1
°C
12
rms current in phase A (IA2) (point 2)
Measurement
330.7
mA
13
rms current in phase B (IB2) (point 2)
Measurement
405.9
mA
14
rms current in phase C (IC2) (point 2)
Measurement
424.3
mA
Measurement
50.69
V
Measurement
51.91
V
Measurement
50.81
V
15
16
17
rms line-to-line voltage between phases A and B (VAB2)
(point 2)
rms line-to-line voltage between phases B and C (VBC2)
(point 2)
rms line-to-line voltage between phases C and A (VCA2)
(point 2)
18
Power between phases A and B (PAB2) (point 2)
Measurement
15.492
W
19
Power between phases B and C (PBC2) (point 2)
Measurement
17.867
W
20
Power between phases C and A (PCA2) (point 2)
Measurement
20.21
W
21
Temperature of the stator windings (ts3) (point 3)
Measurement
31.9
°C
22
rms current in phase A (IA3) (point 3)
Measurement
402.5
mA
23
rms current in phase B (IB3) (point 3)
Measurement
490.4
mA
24
rms current in phase C (IC3) (point 3)
Measurement
454.7
mA
Measurement
68.43
V
Measurement
69.25
V
Measurement
67.89
V
25
26
27
rms line-to-line voltage between phases A and B (VAB3)
(point 3)
rms line-to-line voltage between phases B and C (VBC3)
(point 3)
rms line-to-line voltage between phases C and A (VCA3)
(point 3)
28
Power between phases A and B (PAB3) (point 3)
Measurement
21.50
W
29
Power between phases B and C (PBC3) (point 3)
Measurement
26.76
W
30
Power between phases C and A (PCA3) (point 3)
Measurement
27.23
W
72
31
Temperature of the stator windings (ts4) (point 4)
Measurement
32.7
°C
32
rms current in phase A (IA4) (point 4)
Measurement
473.1
mA
33
rms current in phase B (IB4) (point 4)
Measurement
579.1
mA
34
rms current in phase C (IC4) (point 4)
Measurement
510.6
mA
Measurement
85.58
V
Measurement
86.56
V
Measurement
84.88
V
35
36
37
rms line-to-line voltage between phases A and B (VAB4)
(point 4)
rms line-to-line voltage between phases B and C (VBC4)
(point 4)
rms line-to-line voltage between phases C and A (VCA4)
(point 4)
38
Power between phases A and B (PAB4) (point 4)
Measurement
28.57
W
39
Power between phases B and C (PBC4) (point 4)
Measurement
36.77
W
40
Power between phases C and A (PCA4) (point 4)
Measurement
36.10
W
41
Temperature of the stator windings (ts-rated) (point 5)
Measurement
33.6
°C
42
rms current in phase A (IA-rated) (point 5)
Measurement
1.960
A
43
rms current in phase B (IB-rated) (point 5)
Measurement
2.124
A
44
rms current in phase C (IC-rated) (point 5)
Measurement
2.131
A
Measurement
334.1
V
Measurement
337.8
V
Measurement
335.7
V
45
46
47
rms line-to-line voltage between phases A and B
(VAB-rated) (point 5)
rms line-to-line voltage between phases B and C
(VBC-rated) (point 5)
rms line-to-line voltage between phases C and A
(VCA-rated) (point 5)
48
Power between phases A and B (PAB-rated) (point 5)
Measurement
395.3
W
49
Power between phases B and C (PBC-rated) (point 5)
Measurement
400.0
W
50
Power between phases C and A (PCA-rated) (point 5)
Measurement
448.2
W
51
Average rms phase current (Is1) (point 1)
[(2)+(3)+(4)]/3
0.3948
A
52
Average rms phase current (Is2) (point 2)
[(12)+(13)+(14)]/3
0.3870
A
53
Average rms phase current (Is3) (point 3)
[(22)+(23)+(24)]/3
0.4492
A
54
Average rms phase current (Is4) (point 4)
[(32)+(33)+(34)]/3
0.5209
A
55
Average rms phase current (Is0) (point 5)
[(42)+(43)+(44)]/3
2.0717
A
56
Average rms phase voltage (Vs1) (point 1)
√3×[(5)+(6)+(7)]/9
19.5164
V
57
Average rms phase voltage (Vs2) (point 2)
√3×[(15)+(16)+(17)]/9
29.5238
V
58
Average rms phase voltage (Vs3) (point 3)
√3×[(25)+(26)+(27)]/9
39.5620
V
59
Average rms phase voltage (Vs4) (point 4)
√3×[(35)+(36)+(37)]/9
49.4635
V
60
Average rms phase voltage (Vs0) (point 5)
√3×[(45)+(46)+(47)]/9
193.9127
V
61
Total input power (P1) (point 1)
√3×[(8)+(9)+(10)]/3
22.6448
W
62
Total input power (P2) (point 2)
√3×[(18)+(19)+(20)]/3
30.9281
W
63
Total input power (P3) (point 3)
√3×[(28)+(29)+(30)]/3
43.5842
W
73
64
Total input power (P4) (point 4)
√3×[(38)+(39)+(40)]/3
58.5664
W
65
Total input power (P0) (point 5)
√3×[(48)+(49)+(50)]/3
717.9351
W
66
Resistance of the stator windings (Rs)
From DC test
2.8899
Ω
67
Ambient temperature (ta)
From DC test
22.8
°C
*(66)×[(1)+k1]/[(67)+k1]
2.9775
Ω
*(66)×[(11)+k1]/[(67)+k1]
2.9831
Ω
*(66)×[(21)+k1]/[(67)+k1]
2.9921
Ω
*(66)×[(31)+k1]/[(67)+k1]
3.0011
Ω
*(66)×[(41)+k1]/[(67)+k1]
3.0112
Ω
68
69
70
71
72
Stator resistance converted to test temperature (Rs1)
(point 1)
Stator resistance converted to test temperature (Rs2)
(point 2)
Stator resistance converted to test temperature (Rs3)
(point 3)
Stator resistance converted to test temperature (Rs4)
(point 4)
Stator resistance converted to test temperature (Rs0)
(point 5)
73
Stator copper losses (PSCL1) (point 1)
3×(51)2×(68)
1.3923
W
74
Stator copper losses (PSCL2) (point 2)
3×(52)2×(69)
1.3401
W
75
Stator copper losses (PSCL3) (point 3)
3×(53)2×(70)
1.8112
W
76
Stator copper losses (PSCL4) (point 4)
3×(54)2×(71)
2.4432
W
77
Stator copper losses (PSCL0) (point 5)
3×(55)2×(72)
38.7701
W
78
Resulting power (PR1) (point 1)
(61)−(73)
21.2526
W
79
Resulting power (PR2) (point 2)
(62)−(74)
29.5880
W
80
Resulting power (PR3) (point 3)
(63)−(75)
41.7729
W
81
Resulting power (PR4) (point 4)
(64)−(76)
56.1232
W
82
Stator line-to-line voltage squared (Vsℓ12) (point 1)
[√3×(56)]2
1142.6653
V2
83
Stator line-to-line voltage squared (Vsℓ22) (point 2)
[√3×(57)]2
2614.9587
V2
84
Stator line-to-line voltage squared (Vsℓ32) (point 3)
[√3×(58)]2
4695.4472
V2
85
Stator line-to-line voltage squared (Vsℓ42) (point 4)
[√3×(59)]2
7339.9200
V2
86
Friction and windage losses (PF&W)
See linear regression
analysis of section 6.2
14.9024
W
87
Core losses (Pcore)
(65)−(77)−(86)
664.2626
W
*For 100% IACS conductivity copper, k1 = 234.5
Table D.2:
Table of calculations for the no-load test.
74
Phase current in function of the line-to-line voltage and the phase
2.5
Phase current (in
2
A
1.5
B
C
1
Exponentiel (B)
0.5
0.0055x
y = 0.3295e
0
0
50
100
150
200
250
300
350
Line-to-line voltage (in V)
Figure D.2:
Graph of the measured phase currents.
Line-to-line power in function of the line-to-line voltage and the
phase
Line-to-line power (i
500
400
A
300
B
C
200
Exponentiel (B)
100
0.0107x
y = 11.325e
0
0
50
100
150
200
250
300
350
Line-to-line voltage (in V)
Figure D.3:
Graph of the measured line-to-line powers.
75
Stator windings temperature in function of the average phase
voltage
Temperature (in
34
33
32
31
30
0
25
50
75
100
125
150
175
200
Average phase voltage (in V)
Figure D.4:
Graph of the measured stator windings temperatures.
Stator line-to-line voltage squared in function of the resulting
power
Stator line-to-line voltage squared (V
8000
7000
6000
5000
4000
3000
2000
1000
y = 177.2x - 2640.7
0
0
5
10
15
20
25
30
35
40
45
50
55
Resulting power (PR)(in W)
Figure D.5:
Graph of the linear regression used to obtain the friction and windage
losses.
76
60
D.5
Test Analysis
The main objective of this test was to find the core losses Pcore and the friction and
windage losses PF&W of the induction machine, but also to obtain a measured point at noload conditions, to be able to compute the rest of the parameters after the locked rotor
test. From table D.2 and the graphs of figures D.2 to D.5, Pcore are 664.2626 W and PF&W
are 14.9024 W. Also, for no-load conditions, the average rms stator phase current Is0 is
2.0717 A, the average rms stator phase voltage Vs0 is 193.9127 V and the total input
power P0 is 717.9351 W.
Some observations have also been taken during the test, which can be different sources of
error on the calculated value. First of all, the no-load conditions, which was supposed to
be measured for the rated line-to-line voltage of the machine (380 V), was finally
measured for line-to-line voltages about 336 V. During the test, few seconds after the
rated voltage was applied, the speed and the voltage decreased suddenly, so it was
impossible to run the motor at rated conditions. For that reason, the test was performed
for a voltage lower than the rated voltage. Also, the technique used to measure the
temperature was not the most accurate one, because the multimeter takes a long time to
get to a steady-state value. Accordingly, the temperature measurements are lower than the
real temperature. Indeed, the temperatures of figure D.4 should be higher than they are,
so a considerable error must be taken into account for this test.
D.6
Conclusion
According to the no-load test, the values of the core losses Pcore, the friction and windage
losses PF&W, the average rms stator phase current Is0, the average rms stator phase voltage
Vs0 and the total input power P0 can be seen in table D.3.
77
Test Point
Average rms phase current (Is0)
Value
2.0717
Unit
A
Average rms phase voltage (Vs0)
193.9127
V
Total input power (P0)
717.9351
W
Friction and windage losses (PF&W)
14.9024
W
Core losses (Pcore)
664.2626
W
Table D.3:
Results of the no-load test.
78
Appendix E
Locked Rotor Test
E.1
Purpose
This test is used with the no-load test to obtain the parameters of the equivalent circuit of
an induction machine. Thus, the leakage inductances Lℓs and Lℓr, for the stator and the
rotor respectively, the magnetizing inductance Lm, the equivalent resistance of the
magnetic core Rc and the rotor equivalent resistance Rr can be obtained.
E.2
Setup and Equipment
See figure E.1 and table E.1 for test setup with a squirrel-cage induction machine
(SCIM).
Figure E.1:
The locked rotor test setup.
79
Component
Apparatus
Producer
Type
Panalyser
Universal Power
Analyser
Voltech
PM 3000A
Vsource
5 kVA AC Power
Source
California Instruments
5001 ix
Temperature
Multimeter
Fluke
179
Table E.1:
E.3
AAU-nr.
7 27 03 D01
7 25 03 C14
Used equipment.
Procedure
•
Make the setup of section E.2 with the stator windings Y-connected.
•
Apply an AC voltage in order to obtain the full-load current as the phase currents.
The frequency must be less than 25% of the rated frequency. Take also in note all
frequencies of the stator line-to-line voltages (fAB, fBC and fCA).
•
Measure the currents in each phase of the stator (IA, IB and IC), the line-to-line
voltages between each phase pairs (VAB, VBC and VCA), the powers between phase
pairs (PAB, PBC and PCA) and the stator winding temperature ts. It is important that the
temperature of the machine does not exceed 40°C over the rated temperature, so the
measures must be taken as fast as possible, within 5 seconds after the voltage is
applied.
E.4
Results
The following table of calculations can be used to determine the parameters of the
equivalent circuit.
80
Item Test Point
rms line-to-line voltage between phases A
1
and B (VAB)
rms line-to-line voltage between phases B
2
and C (VBC)
rms line-to-line voltage between phases C
3
and A (VCA)
Source or Calculation
Value
Unit
Measurement
67.77
V
Measurement
74.32
V
Measurement
58.57
V
4
Frequency of VAB (fAB)
Measurement
15.94
Hz
5
Frequency of VBC (fBC)
Measurement
16.00
Hz
6
Frequency of VCA (fCA)
Measurement
16.07
Hz
7
rms current in phase A (IA)
Measurement
6.056
A
8
rms current in phase B (IB)
Measurement
7.119
A
9
rms current in phase C (IC)
Measurement
7.321
A
10
Power between phases A and B (PAB)
Measurement
367.3
W
11
Power between phases B and C (PBC)
Measurement
517.4
W
12
Power between phases C and A (PCA)
Measurement
302.5
W
13
Temperature of the stator windings (ts)
Measurement
34.4
°C
14
Average rms phase voltage (VsL)
√3×[(1)+(2)+(3)]/9
38.6170
V
15
Average test frequency (fL)
[(4)+(5)+(6)]/3
16.0033
Hz
16
Average rms phase current (IsL)
[(7)+(8)+(9)]/3
6.832
A
17
Total input power (PL)
√3×[(10)+(11)+(12)]/3
685.4302
W
18
Average rms no-load phase voltage (Vs0)
From no-load test
193.9127
V
19
Average rms no-load phase current (Is0)
From no-load test
2.0717
A
20
Total no-load input power (P0)
From no-load test
717.9351
W
21
Locked rotor reactive power (QL)
√{[3×(14)×(16)]2−(17)2}
395.7895
var
22
No-load reactive power (Q0)
√{[3×(18)×(19)]2−(20)2}
967.9865
var
23
Leakage reactances ratio (Xℓs/Xℓr)
Depends on design (see section 6.3)
1
24
Rated frequency (frated)
From the machine’s plate
50
Hz
25
Magnetizing reactance (XM)
See iterative method of section 6.3
114.7789
Ω
26
Locked rotor test stator leakage reactance
(XℓsL)
See iterative method of section 6.3
1.4404
Ω
27
Stator leakage reactance (Xℓs)
See iterative method of section 6.3
4.5004
Ω
28
Locked rotor test rotor leakage reactance
(XℓrL)
(26)/(23)
1.4404
Ω
29
Rotor leakage reactance (Xℓr)
(24)×(28)/(15)
4.5003
Ω
30
Core Losses (Pcore)
From no-load test
664.2626
W
31
Equivalent resistance of the magnetic core
(Rc)
1/{[(30)/[3×(18)2]]×[1+(27)/(25)]2}
157.2490
Ω
81
32
Resistance of the stator windings (Rs)
From DC test
2.8899
Ω
33
Ambient temperature (ta)
From DC test
22.8
°C
34
Stator resistance converted to test
temperature (RsL)
*(32)×[(13)+k1]/[(33)+k1]
3.0202
Ω
35
Locked rotor test rotor equivalent
resistance (RrL)
{(17)/[3×(16)2]−(34)}×[1+(29)/(25)]2
−[1/(23)]2×[(26)2/(31)]
2.0115
Ω
36
Rotor equivalent resistance (Rr)
*(35)×[(33)+k1]/[(13)+k1]
1.9247
Ω
*For 100% IACS conductivity copper, k1 = 234.5
Table E.2:
E.5
Table of calculations for the locked rotor test.
Test Analysis
The main objective of this test was to determine the stator and rotor leakage reactances
Xℓs and Xℓr and the magnetising reactance Xm at rated frequency. The core and rotor
equivalent resistances Rc and Rr are also determined in that test. From table E.2, Xℓs is
4.5004 Ω, Xℓr is 4.5003 Ω, Xm is 114.7789 Ω, Rc is 157.2490 Ω and Rr is 1.9247 Ω.
Some observations have also been taken during the test, which can be different sources of
error on the calculated value. First of all, to avoid the overheating of the stator windings,
the AC power source was not switched on more than 5 seconds. Also, because the current
is decreasing since the moment that the AC power source is switched on, every value has
to be taken as fast as possible. So, during the test, the AC power source has been
switched on and off several times and the different measurements have not been taken in
the exact same conditions. Also, the technique used to measure the temperature was not
the most accurate one, because the multimeter takes a long time to get to a steady-state
value. Accordingly, the temperature measurement is lower than the real temperature.
Indeed, the temperature should be higher than it is, so a considerable error must be taken
into account for this test.
E.6
Conclusion
According to the locked rotor test, the stator and rotor leakage reactances Xℓs and Xℓr, the
magnetizing reactance Xm, the core and rotor equivalent resistances Rc and Rr can be seen
in table E.3. These values are correct for the rated frequency frated (50 Hz).
82
Test Point
Rated frequency (frated)
Magnetizing reactance (XM)
Stator leakage reactance (Xℓs)
Value
50
Unit
Hz
114.7789
Ω
4.5004
Ω
4.5003
Rotor leakage reactance (Xℓr)
Equivalent resistance of the magnetic core
157.2490
(Rc)
1.9247
Rotor equivalent resistance (Rr)
Table E.3:
Results of the locked rotor test.
83
Ω
Ω
Ω
Appendix F
DC Method to Determine the Magnetizing
Inductance
The method explained in this appendix is about the de-energizing of the stator windings,
so the fall time of the stator current is used to determine the magnetizing inductance.
To find the value of the magnetizing inductance Lm for a given stator phase current, the
first step is to draw the equivalent circuit of the test. So, since the input voltage Vin(t) is
applied between 2 of the stator phases, the equivalent circuit is like the one in figure F.1.
Because the voltage applied is a DC voltage, there is no induction in the rotor and thus no
core losses.
Figure F.1:
Equivalent circuit used for the determination of the magnetizing
inductance.
In the previous figure, A, B and n refer to phases A and B of the stator and to neutral
connection, but the circuit is also true for the 2 other stator combinations (BC and CA).
Besides, Rv represents the variable resistance and Rs is the resistance of the stator
84
windings. Afterward, iA(t) is the stator phase current (in this example for phase A of the
stator). Subsequently, this circuit can be reduced to simplify the calculations:
Figure F.2:
Reduced equivalent circuit used for the determination of the
magnetizing inductance.
And the equivalent resistance and inductance are defined as:
Req = 2 Rs + Rv
(F.1)
Leq = 2 L m
(F.2)
From figure F.2, the relation between Vin(t) and i(t) can be established as:
Vin (t ) = Req i (t ) + Leq
di (t )
dt
(F.3)
Using the Laplace transform, equation F.3 becomes:
Vin (s ) = (Leq s + Req )I (s )
(F.4)
Then, equation F.4 can be manipulated to obtain the transfer function.
I (s )
1
=
×
Vin (s ) Req
Req
Leq
R
s + eq
Leq
(F.5)
After, if the DC voltage source is turned on at t = 0 and turned off at t = t1, the Laplace
transform of the input voltage is:
1 e − t1s
Vin (s ) = −
s
s
85
(F.6)
Replacing the input voltage of equation F.6 in equation F.5 and after some manipulations,
the following relation of the stator current is obtained in the Laplace domain.
Req
I (s ) =
Leq
1
×
1 − e −t1s
Req
⎛
R ⎞
s⎜ s + eq ⎟
⎜
Leq ⎟⎠
⎝
(
)
(F.7)
Bringing back to the time domain, equation F.7 becomes:
−
t⎞
1 ⎛⎜
L
i(t ) =
1 − e eq ⎟(u (t ) − u (t − t1 ))
⎟
Req ⎜
⎝
⎠
Req
(F.8)
At this point, if this is assumed that the switch is turned on enough long to obtain a
steady-state stator current, equation F.8 can be simplified.
−
t⎞
1 ⎛⎜
L
i(t ) = I (0) −
1 − e eq ⎟u (t )
⎟
Req ⎜
⎝
⎠
Req
(F.9)
In this new equation, I(0) is the steady-state value of the current before the DC voltage
source is turned off. Also, the previous equation is time shifted to obtain t1 = 0, which is
the moment when the DC voltage source is turned off. After all, the equivalent
inductance Leq can be solved from equation F.9.
Leq =
Req t
(F.10)
⎛ I (0 ) ⎞
ln⎜⎜
⎟⎟
⎝ i (t ) ⎠
The last equation can be used to determine Leq, but in this case it is more suitable to
measure the time constant τ, which is the moment when the instantaneous current is 37%
of the maximum current I(0). Then, Leq can be found as:
Leq = τReq
86
(F.11)
Appendix G
Results of the Simulation of the Model
G.1
No-Load Test (point 1)
87
88
G.2
No-Load Test (point 2)
89
90
G.3
No-Load Test (point 3)
91
92
G.4
No-Load Test (point 4)
93
94
G.5
No-Load Test (point 5)
95
96
G.6
Locked Rotor Test
97
98