 Open Access
 Total Downloads : 246
 Authors : Omonowo (David) Momoh
 Paper ID : IJERTV2IS110665
 Volume & Issue : Volume 02, Issue 11 (November 2013)
 Published (First Online): 21112013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Dynamic Simulation of Squirrel Cage Induction MachineA Simplified and Modular Approach
Omonowo (David) Momoh
Indiana UniversityPurdue University Fort Wayne, Indiana, USA
Abstract
This paper presents an easy to understand dynamic simulation model of Induction Machine Model on Simulink/MATLAB software platform. A somewhat different induction machine model equations in which flux linkage equations are expressed in terms of reactances rather than inductances is used. Also, the model equations for the three phase carrierbased pulsewidth voltage source inverter for driving the induction machine are presented in a comprehensible and detailed manner. The Simulink blocks developed are divided into fifteen subsystems with each sub – system solving one of the model equations in a detailed, easy to understand form. The results obtained correlates with the results obtained using inductance based flux linkage approach and those obtained in the original paper.

Introduction
In many applications, the dynamic behavior of the induction machines has an important effect upon the overall performance of the system of which it is a part [1]. Dynamic simulation of induction machines using Simulink allows the transient behavior to be studied. The use of Simulink for dynamic simulation of electrical system places it at an advantage over other circuit simulators that shows only the steady state behavior of the circuit. Simulink is an extension of MATLAB that allows engineers to rapidly and accurately build computer models of dynamical systems using block diagram notation [2]. The block diagram notation for modeling complex nonlinear systems makes Simulink an effective analytical tool for engineering technology students.
A model of the induction motor and other electric machine models are available in the Simulink Power System Blockset, these models provide access to a
limited number of machine parameters [3]. Also, many Simulink induction machine models in existence do not give detailed explanation concerning how the model equations are derived and simulated. In most cases, the authors skip many steps thereby leaving the readers to figure out the missing links. While this approach may be considered justifiable for experts in the field due to the nature of their training, it is imperative and necessary that detailed, more comprehensive and clearer model equations be presented in works that focus on engineering students that are just learning to develop research tools in electrical machine analysis.
This work is a reproduction of part of earlier work done in [4]; it is also an extension of the earlier work done by this author in [5]. The motivation for embarking on reproducing the original paper stems from the fact that similar results obtained in [4] can be achieved using a more simplified and understandable but yet shorter method. The author in [3] has also made a commendable attempt in simplifying the work done in [4], but he failed to include the model equations for the inverter drive employed for his machine. In this work, the model equations for the three phase carrier based pulsewidth voltage source inverter are developed and presented in a clear and easily implementable manner. The approach adopted in [6] is employed to develop the model equations for the three phase voltage source inverter. Also, the induction machine model equations are restructured in such a way that makes it easier for engineering and technology students to follow.
In this paper, the Simulink model is configured into subsystems, with each subsystem solving one model equations in a comprehensive and easy to understand form. A double click on each of the subsystem gives a detailed model equation and the machine parameters used. The method used to transform the voltages and currents from their natural (ABC) form to the synchronously rotating reference frame (dq) form in the original paper was somehow long and strenuous. The authors first transformed the three phase voltages to a
two phase stationary reference frame ( V s andV s ),
From (1),
qs ds
and thereafter multiply by unit vector block to achieve
the desired synchronously rotating reference frame transformation. In this paper, a more straight forward
Sin 1 Sip
By substituting (4) into (3), we have the following:
(4)
method was used by choosing the reference frame transformation speed (angle) that is equal to the
vio
1 v
2 dc
2S
1
(5)
ip
ip
fundamental frequency (speed) of the variables (voltages and currents) associated with the rotating circuits [8, 9]. Also, the flux linkages are expressed in
For the three phase voltage source inverter, incorporating (2) with (5) translates to
terms of reactances rather than inductances. This is achieved by simply multiplying the inductances by the base speed of the inductance machine.
van vno
1 v 2S 1
dc ap
dc ap
2

Model Equations

Modelling of CarrierBased Voltage Source Inverter
vbn
vcn

vno

vno
1 v
2 dc
1 v
2 dc
2S
bp
bp
cp
cp
2S
1
1
(6)
ip
ip
The turnon and turnoff of the switching semiconductor devices on the three phase voltage
The Fourier series approximation of the existence function are approximated as follows [6, 7]
source inverter are represented by an existence
Sip
0.51 m
(7)
function. The existence function has a value of unity
and zero when the device is turned and turned off
For the three phase inverter, this translates to
respectively [6]. The existence function can be designated as Sij where i = a, b, c represents each of
the three phases and j = p, n, represents the top and
sap
s
0.51 m
ap
ap
0.51 m
(8)
bottom switches on any of the three inverter legs
bp bp
respectively.
scp
0.5 1 mcp
In other to avoid short circuit on any of the inverter legs, Kirchhoffs Voltage Law (KVL) must be obeyed in the operation of the switching devices. Consequently the existence function must be constrained as follows: Sip Sin 1 (1)
The voltage between the ith inverter phase and the center point of the dclink capacitor otherwise known as the switched inverter phase voltage is related to the inverter output phase voltage as follows:
map, mbp, and mcp are the respective carrierbased modulation waveforms for each of the three phases. The magnitude ranges between 1 and 1 and are made up of fundamental frequency components [6]. These modulation signals are compared with the carrier signal such that the switches turn on when the magnitude of the corresponding modulation signal is greater than the triangle carrier signal and vice versa.
For a balanced system, the equation for the modulation
signal for the top switching devices can be obtained by combining (5) and (7) as follows:
vio vin vno
(2)
where vio is the switched inverter phase voltage, vin is
m vip
dc
dc
(9)
the inverter output voltage, and vno is the neutral
ip 0.5v
voltage. Also, there is a relationship between the inverter switched voltage, the existence functions, and the dc bus voltage as follows:
where mip is the magnitude of the phase modulation signal, vip s the magnitude of the fundamental inverter
output phase voltage and vdc is the dc bus voltage.
vio
1 v
S
S
2 dc ip

Sin
(3)
where vdc is the dc bus voltage.



Induction MachineSynchronous Reference
V
Vas
Frame Transformation
qs K (
)V
(16)
V
e bs
ds
V
The output phase voltages from the voltage source inverter are used to drive the induction motor. The
cs
V '
three phase voltages feeding the stator of the induction
V '
ar
qr K ( ) V '
(17)
motor are:
V ' e
br
Vas
Vms
Cos(et)
(10)
dr
V '
cr
cr
Vbs Vms Cos(et )
(11)
Vcs
Vms Cos(et )
(12)
The rotor voltage equations referred to the stator side
cos(e )
cos(e )
cos(e )
are:
V '
Ns V
e
e
Cos( t)
(13)
K ( ) 2 sin( )
sin(
)
sin(
)
(18)
N
N
ar mr e
r
3
e
1
e
1
e
1
2
2
2
3
e
1
e
1
e
1
2
2
2
' Ns
e e dt (0)
(19)
N
N
Vbr
r
Vmr Cos(et )
(14)
V ' Ns V Cos( t )
(15)
The quadrature and direct axes equivalent circuit for
N
N
cr mr e
r
the induction motor are shown in Fig. 1 and Fig. 2
The solution of the voltage equation is very
respectively.
iqs
L L L
Llr Lr Lm
iqr
involving and complex due to the fact that some of the
Rs ls
s m Rr
machine inductances (self and mutual) are functions of
+ – – +
the rotor speed which automatically make them time
(e r )dr
varying. This undesirable feature can be eliminated by
e ds
Fqs Lm
Fqr
Vqr
a proper change of variables which in effect transforms the voltages and currents of both the stator and rotor to a common frame of reference [1]. Also, reference frame transformation of induction machines voltage, current and flux linkage equations, allows for the controllability induction machine like their direct current (dc) machine counterpart. Power electronics
qs qr
b b
Fig.1. q axis equivalent circuit of an induction machine
converters have made induction machine control easy
ids
R
R
s
Lls Ls Lm
Llr Lr Lm
idr
R
R
r
and it has relatively replaced dc machine. In vector
+ – – +
control, the induction motor is run like a separately
excited dc motor [10]. Although, the transformed equations of induction machines are now developed in
e qs
Lm
(e r )qr
the one size fits them all reference name otherwise known as arbitrary reference, type of reference frame transformation would need to be specified during simulation .
ds
Fds
b
dr
Fdr
b
The synchronous reference frame was used in this work. It has the unique property that during balanced steady steady state operation, the system variables (voltages, currents, and flux linkages) are constants [11]. The stator and rotor referred voltages in synchronous reference frame is as expressed in equation (16) and (17) respectively.
Fig.2. d axis equivalent circuit of an induction machine


Voltages and Flux Linkages Equations
F F '
F x'
ds dr
'
'
(33)
Based on the equivalent circuits of Fig.1 and Fig.2, the following equations for stator and rotor referred
md ml
x
x
x
x
ls
x
x
x
x
(x
(x

x

x
x
lr
x' x
voltages and flux linkages for dynamic simulation are
x' m lr ls
obtained.
1
ml
ls
1
' '
lr m lr
x
x
x
x
1 1

xm
1
1
x
x
xls )
(34)
Vqs rs Iqs
Fds
b b
pFqs
(20)
ls
'
lr m
V r I e F 1 pF
(21)


Electromagnetic Torque Equations
ds s ds qs ds b b
' ' '
(e r ) ' 1 '
The torque equations are given in equations (35) and
Vqr rr Iqr
b
Fdr
b
pFqr
(22)
(36).
V ' r ' I '
(e r ) F '

1 pF '
(23)
T 3 P 1 (F I

F I )
(35)
dr r qr
qr dr
b b
e 4
ds qs
qs ds
e rs
b
b
T T 2J p
(36)
x
x
pFqs b Vqs Fds (Fmq Fqs )
b ls
(24)
e L P r
pF e F


rs (F
F )
(25)
x
x
ds b Vds qs
b
md
ls
ds


Simulation Results
pF '
' (e r ) F '

rr (F
F ' ) (26)
The Simulink block developed for the simulation of
qr b Vqr
x
x
dr mq
b lr
qr
the induction motors is shown in Fig. 3.
pF '
' (e r ) F '

rr (F
x
x
F ' ) (27)
dr b Vdr
qr md
b lr
dr
For a squirrel cage induction machine (which is the
qr
qr
case in this paper), V '
and V '
in equations (22), (23),
dr
dr
(26) and (27) are set to zero since the rotor bars are shortcircuited at the end ring.
The equations for self and mutual flux linkages are given as follows:
Fqs
Fds
xls
xls
Iqs
Ids

xm

xm
(Iqs
(Ids
qr ) xls

I

I

I

I
'
'
'
'
dr ) xls
Iqs
Ids


Fmq

Fmd
(28)
(29)
F ' x' I '

x (I I '

) x' I ' F
(30)
qr lr qr
m qs qr
lr qr mq
F ' x' I ' x (I I '
) x' I ' F
(31)
Fig.3. Simulink block for three phase voltage
dr lr dr
F
m qs dr
F '
lr dr md
source inverter driven squirrel cage induction motor simulation
ml
ml
mq
mq
F x'
<> qs
qr
(32)
x
x
'
'
xls lr
Equations (1) through (9) are used for the Simulink simulation of the three phase carrierbased PWM voltage source inverter. The magnitude and frequency of the modulation signals are 0.8 and 60 Hz
respectively. The dc bus voltage utilized to realize the 180 V input voltage to the motor is 450 Vdc. This can be determined using (9). The inverter output phase voltages for the three phases are shown in Fig. 4. It can be seen from figure 4 that each output phase voltage exhibits threesteps in their respective quartercycle. This is a characteristic of a three phase, twolevel, voltagesource inverter and it confirms the switching operations in each of the threelegs of the inverter.
Fig.5. Plots of combined stator current versus time
Figure 6 shows the plot of the electromagnetic torque of the induction motor. It can be seen from the figure that the machine has reached steady state at about 0.14 second. A 50 Nm load torque was applied to the machine at 0.16 second. The figure shows how the electromagnetic torque rises from zero to 50 Nm to counter the 50 Nm load torque applied and a new steady state established.
500
van (V)
van (V)
0
500
500
vbn (V)
vbn (V)
0
0 0.005 
0.01 
0.015 
0.02 
0.025 
0.03 
0.035 
0.04 
0.045 
0.05 

0 0.005 
0.01 
0.015 
0.02 
0.025 
0.03 
0.035 
0.04 
0.045 
0.05 
0 0.005 
0.01 
0.015 
0.02 
0.025 
0.03 
0.035 
0.04 
0.045 
0.05 

0 0.005 
0.01 
0.015 
0.02 
0.025 
0.03 
0.035 
0.04 
0.045 
0.05 
500
500
vcn (V)
vcn (V)
0
500
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
200
150
Te [Nm]
Te [Nm]
100
50
0
Figure 4. Inverter output phase voltages
50
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Time[s]
The simulations results for direct – on line starting of the induction motor showing plots of stator current, electromagnetic torque, and rotor speed at no load and at load are shown below.
Figure 5 shows the plot of the combined three phase stator currents of the induction motor. The figure shows a slight change in the stator current when a 50 Nm load torque was applied 0.16 second.
300
200
100
iabcs [A]
iabcs [A]
0
100
200
Fig.6. Plot of electromagnetic torque versus time
Figure 7 shows the rotor speed versus time. The figure shows how the rotor speed dropped from the synchronous value of 625 rad/s at no load to about 525 rad/s when the 50 Nm load was applied at 0.16 second.
700
600
500
Wr [rad/s]
Wr [rad/s]
400
300
200
100
0
300
100
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Time[s]
400
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Time[s]
Fig.7. Plot of rotor speed versus time
Figure 8 shows the plot of the combined three phase rotor currents of the induction motor. At steady state following the free acceleration, rotor currents become zero. However, at 0.16 second when the 50 Nm load torque was applied, there is current flowing in the rotor due to the applied load. In practical situation, damper windings are incorporated in the rotor to suppress this current.
250
200
150
100
iabcr [A]
iabcr [A]
50
0
50
100
150
200
250
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Time[s]
Fig.8. Plots of combined rotor current versus time
6. Conclusions
simplified method of dynamic simulation of squirrel cage induction motor using Simulink has been implemented in this work. Modeling of the machine equations was done in a detailed and easy to understand fashion. A somewhat different induction machine model equations in which flux linkage equations are expressed in terms of reactances rather than inductances was used. The model equations for the three phase PWM voltagesource inverter was also developed. The model equations were grouped into modules. Each module was implemented using a subsystem in the Simulink block. The results obtained agree with those of earlier work done in references [3] and [4]. However, the approach presented in this work is more detailed, simpler and shorter.

J. B. Dabney, and T. L. Harman, Mastering Simulink, Upper Saddle River, NJ: Pearson / Prentice Hall, 2004, pp. 16

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8. Appendix
Vms = magnitude of stator voltage.
7. References
Vmr
= magnitude of rotor voltage.
[1] P. C. Krause, and C. H. Thomas, Simulation of symmetrical induction machinery, IEEE Trans. Power Apparatus and Systems, vol. PAS84, no. 11, pp. 10381053, 1965.= 120o (three phase)
Ns = Number of turns of stator windings
Nr = Number of turns of rotor windings
K (e ) = synchronous reference frame transformation
matrix.
e = synchronous speed (stator angular electrical frequency)
e = synchronous angle.
(0) = phase angle
Fmq = q axis magnetizing flux linkages Fmd = d axis magnetizing flux linkages xls = stator leakage reactance (e Lls )
xlr = rotor leakage reactance (e Llr )
Fqs = q axis stator selfmagnetizing flux linkages Fds = d axis stator selfmagnetizing flux linkages Fqr = q axis rotor selfmagnetizing flux linkages Fdr = d axis rotor selfmagnetizing flux linkages xm = magnetizing reactance
Rs = stator resistance
Rr = rotor resistance
Vqs = q axis stator voltage
Vds = d axis stator voltage
Lr = Llr+Lm; Tr = Lr/Rr; wb = 2*pi*fb Xls = wb*Lls Xlr = wb*Llr Xm = wb*Lm
Xmstar=1/ ((1/Xls) + (1/Xm) + (1/Xlr))
V
V
qr
qr
' = q axis rotor voltage
V
V
dr
dr
' = d axis rotor voltage
P = number of poles
p (small case) = differential operator ( d )
dt
J = moment of inertia (kgm2) Te = electromagnetic torque TL = load torque
b = motor base electrical speed
r = rotor electrical speed
MACHINE PARAMETER SPECIFICATION: Rr =0.39;
Rs = 0.19; Lls = 0.21e3H; Llr = 0.6e3H; Lm = 4e3H;
fb = 100Hz; p=4;
J = 0.0226 kgm2;