International Journal of Electrical, Electronics and Computer Systems (IJEECS)
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Modelling &Simulation of a Three-Phase Electric Traction Induction
Motor Using Matlab Simulink
1
Naveed Rahaman, 2H.V. Govindraju
M-tech Student1, Associate Professor2, Dept. of EEE,
Dr. AIT, Bangalore, Karnataka, India
Email: 1naveedrahaman86@yahoo.in; 2govind.raju37@yahoo.com;
Abstract– The theory of reference frames has been
effectively used as an efficient approach to analyze the
performance of the induction electrical machines. This
paper presents a step by step Simulink implementation of
an induction machine using dq0 axis transformations of the
stator and rotor variables in the arbitrary reference frame.
For this purpose, the relevant equations are stated at the
beginning, and then a generalized model of a three phase
induction motor is developed and implemented in an easy
to follow way. The obtained simulated results provide clear
evidence that the reference frame theory is indeed an
attractive algorithm to demonstrate the steady-state
behavior of the induction machines. The designed induction
motor model is then given the input supply from an IGBT 2
level inverter controlled by space vector PWM. The speed
control of the designed traction induction motor is achieved
by PI controller based on closed loop voltage control.
Keywords: Induction motor, D-Q transformation, Electric
traction, Flux linkage, Speed, Torque, SVPWM, Inverter.
A. INDUCTION MOTOR MODELLING
I. INTRODUCTION
The voltage and torque equations that describe the
dynamic behaviour of an induction motor are timevarying. It is successfully used to solve such differential
equations and it may involve some complexity. A
change of variables can be used to reduce the
complexity of these equations
by eliminating all
time-varying inductances, due to electric circuits in
relative motion, from the voltage equations of the
machine[1,2,3,4]. By this approach, a poly phase
winding can be reduced to a set of two phase windings
(q-d)with their magnetic axes formed in quad rature. In
other words, the stator and rotor variables (voltages,
currents and flux linkages) of an induction machine are
transferred to a reference frame, which may rotate at any
angular velocity or remain stationary. Such a frame of
reference is commonly known in the generalized
machines analysis as arbitrary reference frame[5,6, 7].
Figure1The dq0 equivalent circuit of an induction motor
The dynamic analysis of the symmetrical induction
machines in the arbitrary reference frame has been
intensively used as a standard simulation approach from
which any particular mode of operation may then be
developed.
Matlab/ Simulink has an advantage over other machine
simulators in modelling the induction machine using
dq0 axis transformation [8,9]. It can be a powerful
techniqueinimplementingthemachineequationsastheyaret
ransferredto a particular reference frame. Thus, every
single equation among the model equations can be easily
implemented in one block so that all the machine
variables can be made available for control and
verification purposes. In this paper, Matlab/Simulink is
used to simulate the dynamic performance of an
induction motor model whose stator and rotor variables
are referred to an arbitrary reference frame [1, 6, 8, 10].
The provided machine model is simulated in a way that
makes it easy for the reader to follow and understand the
implementation process since it gives full details about
Simulink structure of each of the model equations. The
equivalent circuit of the induction machine in the
arbitrary reference frame is shown in figure 1 below [11,
12].
II. INDUCTION MOTOR MODEL
Driving the model equations can be generated from the
dq0 equivalent circuit of the induction machine shown
in figure 1. The flux linkages equations associated with
this circuit can be found as follows:
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International Journal of Electrical, Electronics and Computer Systems (IJEECS)
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dλqs
dt
dλds
dt
dλqr
dt
dλdr
dt
= ωb [Vqs −
= ωb [Vds −
ωe
λds +
ωb
ωe
ωb
λqs +
= ωb [Vqr −
(ωe−ωr)
= ωb [Vdr −
(ωe−ωr)
ωb
ωb
Rs
Xls
Rs
Xls
2π
(λmq − λqs)]. . . . . . .(1)
Vb = √2 Vrms sin(ωt − ). . . . .. . . . . . . . . . . . . .(15)
(λmd − λds)]. . . . . . .(2)
Vc = √2 Vrms sin(ωt + ). . . . ... . . . . . . . . . . . . (16)
λdr +
λqr +
Rr
Xlr
Rr
Xlr
(λmq − λqr)]. . .(3)
(λmq − λdr)]. . .(4)
Where
3
2π
3
These three-phase voltages are transferred to a
synchronously rotating reference frame in only two
phases (d-q axis transformation). This can be done using
the following two equations.
[
λmq = Xml [
λqs
Xls
+
λqr
Xlr
−1⁄
2
√3⁄
2
1
Vα
2
]= [
3
Vβ
0
]. . . . .. . . . .. . . . .. . . . . .(5)
−1⁄
Va
2
] [Vb]... . . . . . .. . . . . (17)
−√3⁄
2 Vc
Then, the direct and quadrature axes voltages are
λds
λmd = Xml [
Xls
+
λdr
Xlr
]. . . . . . . . . . . . . . . . . . . . .(6)
[
Xml =
1
. .. . . . .. . . . .. . . . . . . . . . . . (7)
1
1
1
+ + ]
Xm Xls Xlr
[
Then substituting the values of the flux linkages to find
the currents;
Iqs =
Ids =
Iqr =
Idr =
1
Xls
[λqs − λmq]. . . . . . . . . . . . . . . . . . . . . . . . .(8)
1
[λds − λmd]. .. . . . .. . . . .. . . . . . . . . . . . .(9)
Xls
1
Xlr
1
Xlr
[λqr − λmq]. . . . . . . . . . . . . . . . . . . . . . . . (10)
[λdr − λmd]. .. . . . .. . . . .. . . . . . . . . . . . (11)
Based on the above equations, the torque and rotor
speed can be determined as follows:
3 P
Te = ( )
1
2 2 ωb
[λdsIqs + λqsIds]. . .. . . . .. . . . . . . .(12)
P
ωr = ∫ (Te − Tl). . .. . . . .. . . . . . . . . . . . . . . . . . (13)
2J
Vd
cosθ
]=[
Vq
−sinθ
sinθ Vα
] [ ]... . . . . . . . . . .. . . . . . . (18)
cosθ Vβ
The instant aneous values of the stator and rotor currents
in three-phase system are ultimately calculated using the
following transformation:
Iα
cosθ
[ ]=[
Iβ
sinθ
1
Ia
2 −1⁄
[Ib] =
2
3
Ic
−1
[ ⁄2
−sinθ Id
] [ ]... . . . . . . . . . . . . . . . . . . (19)
cosθ Iq
0
√3⁄
2 ... . . . . . . . . .. . . . . . . . . . (20)
−√3⁄
2]
III MATLAB/SIMULINK
IMPLEMENTATION
In this section, the three phase induction machine model
is simulated by using the Matlab/Simulink. The Model is
implemented using the same set of equations provided
above in sections II. Figure 2 depicts the complete
Simulink scheme of the described induction machine
model.
Where P: number of poles; J: moment of inertia (Kg/m2)
For squirrel cage induction motor, the rotor voltages Vqr
and Vdr in the flux equations are set to zero since the
rotor cage bars are shorted. After driving the torque and
speed equations in term of d-q flux linkages and current
soft he stator, the d-q axis transformation should now be
applied to the machine input(stator)voltages[1,13,14].
The three-phase stator voltages of an induction machine
under balanced conditions can be expressed as:
Va = √2 Vrms sin(ωt). . . . .. . . . . . . . . . . . . . . . . . (14)
Figure 2 the 3-phase induction motor Matlab/Simulink
model
In this model the simulation starts with generating a
three-phase stator voltages according to the equations
(14, 15, 16), and then transforming these balanced
voltages to two phase voltages referred to the
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International Journal of Electrical, Electronics and Computer Systems (IJEECS)
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synchronously rotating frame using Clarke and Park
transformation as in equations (17, 18). After that the dq flux linkage and current equations were implemented
as to be demonstrated below. Figure 3 illustrates the
internal structure of the induction machine d-q model by
which the flux linkages, currents, torque and the rotor
angular speed are calculated.
Figure. 6 the implementation of the torque equation Te
(12) and the angular speed equation ωr (13)
Figure.7 shows the internal structure of the blocks (1- 4)
in figure 4 in which the equations (1)-(4) are
implemented in Matlab/Simulink format.
Figure 3 the internal structure of the 3-phase induction
motor d-q model
The Matlab/Simulink model to find the Flux linkage λ qs,
λqr, λds, λdr, as stated in equations (1)-(4) is shown in
figure 4.
Figure 4 the internal structure of the block to calculate
the flux linkages
Figures.5 show the Simulink blocks used to calculate the
currents Iqs, Ids, Iqr,Idr, according to the equations (8) –
(11), also λmq, λmd in equations (5), (6). Figures 6 show
the implementation of torque Te and angular speed ωr as
expressed in equations (12), (13) respectively.
Figure 7 the implementation of the equation (1)-(4)
IV. MATLAB/SIMULINK RESULTS
The simulation results are given for the induction motor
with the following specifications:
Figure 5 the internal structure of the block to calculate
the currents Iqs, Ids, Iqr,Idr, and the fluxes λmq, λmd.
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KW = 285, VL = 1140, P = 4, f = 50, Rs = 0.131, Xls =
0.589,Rr = 0.0905, Xlr = 0.589, J = 4, Xm = 19.11, rpm
= 3000
Figure8 Torque speed characteristics for given induction
motor
Figure10Torque & speed characteristics of induction
motor
Finally, the machine parameters should be defined to the
simulated machine system in order to complete the
simulation process. There are many ways to input the
required data. The method used here is the graphical
user interface (GUI). The machine parameters are
entered through the convenient graphical user interface
(GUI) available in Matlab/Simulink, where you can
Figure9Machine variables during free acceleration of
induction motor
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right click with your mouse and then choose parameters
to be added (edit mask). Figure 11 shows the GUI of the
induction machine d-q model shown earlier in figure 2.
After achieving the Matlab/Simulink implementation of
the
described
machine
model
using
the
Matlab/Simulink, a Matlab code program was assigned
to the same model using the same set of equations. The
code provided similar results to those obtained by
Matlab/Simulink. However, it was found that
Matlab/Simulink is more convenient in terms of
simplicity in construction and control algorithms which
may be set forth for this model. The code compilation
steps can be stated by the flow chart shown below in
figure 11.
Figure 11 the graphical user interface (GUI) used to
define the input data to the simulated induction motor.
V. INDUCTION MOTOR WITH INVERTER
CLOSED LOOP VOLTAGE CONTROL
In this section, the three phase induction machine model
with inverter and closed loop voltage control system is
simulated by using the Matlab/Simulink. Here the
actual speed of the induction motor is compared with the
reference speed; the difference between them is error.
The error signal fed to the PI controller and then
controlled output is then given as reference to the
controlled dc voltage source for the inverter input.
Figure 12: Simulink model of PI controller based three
phase Induction motor voltage control
Figure 13: Internal structure of the 3-phase igbt based
inverter model
VI. MATLAB/SIMULINK RESULTS
The simulation results are given for the induction motor
with the closed loop control system at different
reference speed and load torque:
1. Reference speed NS = 1500 and Load Torque
TL=1000:
Figure 12 depicts the complete Simulink scheme of the
described model. Figures 13 show the 3 phase - 2 level
IGBT based inverter. It also provided with the 2 nd order
low pass filter to get pure sine wave output from the
inverter.
Figure 14: Torque speed characteristics for given
induction motor at NS=1500 & TL=1000
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The figures 18 and the figure 19 show the frequency
spectrum of the phase voltage and the line voltage after
2nd order low pass filter. The value of the THD for phase
voltage is 54.12% while for the line voltage it is 10.88%.
The value of the fundamental component of the phase
voltage is 548.2 V while for the line it is 948.4 V for the
dc input voltage of 1140 V.
Figure 15: Torque & speed characteristics of induction
motor at NS=1500 & TL=1000
Figure 18: THD in phase voltage Va at NS=1500 &
TL=1000
Figure 16: Machine current during closed loop at
NS=1500 & TL=1000
Figure 19: THD in LP filter output phase voltage Va at
NS=1500 & TL=1000
2. Reference speed NS = 1500 and Load Torque
TL=2035:
Figure 17: Inverter output voltages during closed loop at
NS=1500 & TL=1000
Figure 20: Torque speed characteristics for given
induction motor at NS=1500 & TL=2035
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International Journal of Electrical, Electronics and Computer Systems (IJEECS)
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Figure 24: THD in phase voltage Va at NS=1500 &
TL=2035
Figure 21: Torque & speed characteristics of induction
motor at NS=1500 & TL=2035
Figure 22: Machine current during closed loop at
NS=1500 & TL=2035
Figure 25: THD in LP filter output phase voltage Va at
NS=1500 & TL=2035
The figures 24 and the figure 25 show the frequency
spectrum of the phase voltage and the line voltage after
2nd order low pass filter. The value of the THD for phase
voltage is 53.03% while for the line voltage it is 6.80%.
The value of the fundamental component of the phase
voltage is 768.8 V while for the line it is 1330 V for the
dc input voltage of 1140 V.
VII. CONCLUSIONS
Figure 23: Inverter output voltages during closed loop at
NS=1500 & TL=2035
In this paper, an implementation and dynamic modeling
of a three-phase induction motor using Matlab/Simulink
are presented in a step-by-step manner and its speed
control is achieved using voltage control by closed loop
PI control system. The parameters are obtained from the
testing done on the electrical traction induction motor.
The model was tested at three different load torque of a
small and large induction motors. The simulated
machine has given a satisfactory response in terms of the
torque and speed characteristics. Also, the model was
controlled from an IGBT based 2 level inverter at
different load torque conditions. Both physical and
simulation methods have given almost same results for
the three phase induction motors used for traction
purpose. There is appreciable improvement in THD in
inverter line and phase voltage as the modulation index
is increased and close to unity. From the analysis we
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conclude that as the modulation index increased. This
concludes that the Matlab/Simulink is a reliable and
sophisticated way to analyse and predict the behaviour
of induction motors using the theory of reference
frames. Total harmonic distortion THD in line and phase
voltage decreases as the value of load torque increased.
The value of fundamental component in line and phase
voltage is increased with the increase in load torque.
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