RESEARCH PAPER
International Journal of Recent Trends in Engineering, Vol 1, No. 3, May 2009
Steady State and Transient Performance
Analysis of Three Phase Induction Machine
using MATLAB Simulations
Prof. Himanshu K. Patel
Assistant Professor, Instrumentation & Control Engineering Department,
Institute of Technology, Nirma University of Science & Technology. India.
E-mail : hkpatel@nirmauni.ac.in, hkpatel@hkpatel.com,himu21@yahoo.com
Abstract: DC drives have hitherto been widely used in
various Industrial applications but recent device technology
and high speed microprocessors have tilted the balance and
the induction machine is becoming the workhorse of the
electrical power industry. Computer based modeling and
simulation of induction machine has certainly opened new
horizons for the performance analysis. A good mathematical
model can help in predicting the behavior of an induction
machine under different operating conditions and in
selecting the appropriate machine for a specific application.
This paper presents a quality mathematical model
of induction machine based on the steady state and dynamic
equations and D-Q transformation technique. This model
can be used for steady state as well as transient analysis of
squirrel cage or wound rotor structure. The model is used to
investigate the effects of variations in the machine size and
parameter values on the dynamic performance of induction
machine. Different operating conditions are simulated using
MatLab code. Also, the behavior of the induction machine is
observed with and without supply harmonics.
used to excite the induction motor are rich in harmonics.
These time harmonics produce respective rotor current
harmonics, which in turn interact with the fundamental
air gap flux, generating harmonic torque pulsations. The
torque pulsations are undesirable: they generate audible
noise, speed pulsations, and losses thus decreasing the
thermal capabilities of the motor and eventually derating
the motor. In this paper the induction motor performance
is analyzed for the effects of the 3rd, 5th & 7th harmonics
at the supply side. MATLAB is used to solve the
differential equations. MATLAB is a prominent software
package for computer simulation [3].
II. MODELING OF THE 3-PHASE INDUCTION
MACHINE:
Fig.-1 shows the detailed single-line diagram of a threephase induction machine with all the components referred
to the stator side [2], [4], [6].
Index Terms -- 3-φ Induction Motor, Steady state &
Transient Response, Supply Harmonics, Synchronous
speed.
I. INTRODUCTION
The dynamic model of induction machine and its
simulation plays a vital role in the validation of design
process of the motor-drive systems, eliminating
inadvertent design mistakes and the resulting errors in the
prototype constructions and testing. The dynamic model
of the Induction machine in d-q-o axes is derived from
fundamentals.
This paper presents a computer program, which is
developed to analyze the performance of induction motor
[1]. The power of the proposed tool lies in the ability to
study the dynamic behavior of the induction machine in
the absence of complicated mathematics. The program
was designed to illustrate clearly the effects of the Park’s
transformation. This concept is normally very difficult to
learn, but by representing the model of the induction
motor in a generic reference frame rotating with an
angular speed ω, and simulating some transients, with
different operating conditions, one can learn quite well
this concept. Normally the inverter voltage waveforms
Fig. 1. Single line diagram of a 3 – phase induction machine.
The induction machine equations are derived from basic
principles [6] and applied to induction motors [1] and the
relationship between the induction machine stator and
rotor quantities can be presented as:
Prof. Himanshu K. Patel, Assistant Professor, Institute of Technology,
Nirma University of Science & Technology, Ahmedabad, Gujarat,
INDIA. (e-mail : hkpatel@nirmauni.ac.in, himu21@yahoo.com).
266
© 2009 ACADEMY PUBLISHER
V abcs = rs i abcs + p λ abcs
= r r i abcr + p λ abcr
⎡ Vas ⎤
⎡ i as ⎤
Vabcs = ⎢⎢ Vbs ⎥⎥ , i abcs = ⎢ i bs ⎥ ,
⎢
⎥
⎢⎣ Vcs ⎥⎦
⎢⎣ i cs ⎥⎦
V abcr
v abcr
⎡ v ar
= ⎢⎢ v br
⎢⎣ v cr
⎡ λ abcs
⎢λ
⎣ abcr
⎤
⎥ , i
abcr
⎥
⎥⎦
⎤
⎡ Ls
⎥ = ⎢L
⎦
⎣ sr
=
(1)
⎡
⎢
⎢
⎢⎣
i ar ⎤
i br ⎥⎥
i cr ⎥⎦
L sr ⎤ ⎡ i abcs
⎢
L r ⎥⎦ ⎣ i abcr
⎤
⎥ (3)
⎦
(2)
RESEARCH PAPER
International Journal of Recent Trends in Engineering, Vol 1, No. 3, May 2009
⎡ L ls + L ms
L s = ⎢⎢ − . 5 L ms
⎢⎣ − . 5 L ms
⎡ L lr + L mr
L r = ⎢⎢ − .5 L mr
⎣⎢ − .5 L mr
− . 5 L ms
L ls + L ms
− . 5 Lms
− .5 L mr
L lr + L mr
− .5 Lmr
V ' dr = r ' r i ' dr − ( ω − ω r ) λ ' qr + ρλ ' dr
− .5 L m ⎤
− . 5 L ms ⎥⎥
L ls + L ms ⎥⎦
− .5 L m ⎤
− .5 L mr ⎥⎥
L lr + L mr ⎦⎥
(7)
V ' or = r ' r i ' or + p λ ' or
Where,
λ qs = L ls i qs + M ( i qs + i 'qr )
(4)
λ ds = L ls i ds + M (i ds + i ' dr ) ,
λ os = L ls i os
and,
⎡
cos θ r
⎢
L sr = l sr ⎢⎢ cos( θ − 2 π )
r
3
⎢
⎢ cos( θ + 2 π )
r
⎢⎣
3
cos( θ r +
2π
)
3
cos θ r
cos( θ r −
In (1)-(5) “p” denotes the “
2π
)
3
2π ⎤
)
3 ⎥
2π ⎥
cos( θ r +
)⎥
3 ⎥
cos( θ r ) ⎥⎥
⎦
cos( θ r −
(5)
λ ' qr = L ' lr i ' qr + M ( i qs + i ' qr )
λ ' dr = L' lc i' dr + M (i ds + i' dr )
d
” operator, and λ, L1, Lm,
dt
and Lsr, denote the flux, leakage inductance, mutual
inductance and rotor-stator inductance respectively.
After applying d-q transformation, a set of non-linear
first-order differential equations will result. There is one
more equation that relates the input and output torques to
the speed of the rotor:
⎛ 2 ⎞
(6)
T e = J ⎜
⎟ ρω r + T L
⎝ P ⎠
λ ' or = L ' lr i ' or
(8)
Also, the developed electrical torque can be written as:
⎛ 3 ⎞⎛ P ⎞
) (9)
= ⎜
T
i'
− i
i'
⎟⎜
⎟ M (i
e
⎝ 2 ⎠⎝ 2 ⎠
qs
dr
ds
dr
Solving (6), (7) and (9), simultaneously, gives the stator
and rotor currents, as well as the rotor speed.
Fig. 2 shows the flow chart of the Matlab program
developed for the analysis of induction machine
performance using equations (1) to (9).
III. EXPERIMENTAL DATA:
The proposed model of the machine is used to study the
performance of two induction machines of different sizes.
One of the machines is much bigger than the other one.
Simulation results are obtained for both machines and
compared to illustrate the effects of the machine
parameters on the outputs, as well as the steady state and
transient behaviors. Furthermore, the effects of changing
the load on the machine’s performance are shown.
The parameters of the two 3-phase, 4-pole, 60Hz
machines, are as follows:
Machine 1: 3 hp, 220V , 1710 rpm, rs = 0.435 Ω,
Xls = 0.754 Ω XM = 26.13 Ω,
X’lr = 0.754 Ω r’r = 0.816 Ω
J = .089 kg.m2
Machine 2 : 2250 hp , 2300 V , 1786 rpm , rs = 0.029 Ω
Xls = 0.226 Ω , XM =13.04 Ω
X’lr = 0.226 Ω r’r= 0.022 Ω
J= 63.87 kg.m2.
Here, XM has been assumed to be constant due to the
fact that the machine is directly connected to the grid and
thus the output voltage of the machine is equal to the grid
voltage.
IV. SIMULATION RESULTS:
This section presents the simulation results for the
machines introduced in the previous section and
compared to investigate the effects of machine’s size &
different operating conditions on its performance.
Fig.2 : Flow Chart
Some of the curves are shown just to prove the accuracy
and dependability of the simulations. First, the larger
machine –2250 hp has been simulated.
The dynamic equations of the machine in d-q frame are:
V qs = rs i qs + ωλ ds + ρλ qs
V ds = rs i ds − ωλ qs + ρλ ds
V os = r s i os + p λ os
V ' qr = r ' r i ' qr + ( ω − ω r ) λ ' dr + ρλ ' qr
© 2009 ACADEMY PUBLISHER
267
RESEARCH PAPER
International Journal of Recent Trends in Engineering, Vol 1, No. 3, May 2009
150
1.4
Loadless
1.2
100
Torque(Nm)
Rotor speed(p.u)
1
0.8
0.6
Loadless
50
0.4
0
0.2
0
-50
0
2
4
Time(s)
6
8
0
0.5
Again comparison between Fig.3 and Fig.5 clearly proves
that the smaller machine has almost no overshoot while
the larger one has a considerable overshoot and a
transient period of almost 2-3 times longer than that of
the smaller machine.
2
Loadless
Torque(Nm)
1
0
-1
-2
-3
0
2
4
Time(s)
6
8
Fig. 4. Torque (Nm)
Fig.3 and Fig.4 correspond to the case where the machine
runs at no-load. Fig.4 clearly shows that the developed
torque goes to zero after the initial transition. Fig.3 shows
that the machine rotates at a speed very close to the
synchronous speed when operating at no-load.
Now, same curves for the 3 hp induction machine will be
presented. It can be seen that the oscillatory behavior of
the smaller machine is much more acceptable than that of
the larger machine. In fact, the best curves for
comparison are those for the no-load machine from
standstill to synchronous speed.
The transient behavior of the machine can be related to its
inertia. The larger the machine is, or the larger its inertia
(J) is, the larger is the torque required during the start-up
period to speed it up. After reaching the synchronous
speed (in the case of no-load machine) the larger
machine’s speed will overshoot, taking some time to
stabilize at around the synchronous speed. This implies
the possibility of instability in big machines if connected
directly to the grid. Furthermore, the very large inrush
currents at start-up of large machines can damage the
wiring of the machine.
1.4
Loadless
1.2
Loaded
1
0.8
0.6
0.4
0.2
1.5
Loadless
0
0
2
4
Time(s)
6
8
Fig. 7. Rotor speed (p.u.)
1
3
x 10
4
2
Loadless
Loaded
0.5
1
0
Torque(Nm)
Rotor speed(p.u)
2
4
Rotor speed(p.u)
x 10
1.5
Fig. 6. Torque (Nm)
Fig. 3. Rotor speed (p.u.)
3
1
Time(s)
0
0.5
1
Time(s)
1.5
0
-1
2
Fig. 5. Rotor speed (p.u.)
-2
-3
0
2
4
Time(s)
Fig. 8. Torque (Nm)
© 2009 ACADEMY PUBLISHER
268
6
8
RESEARCH PAPER
International Journal of Recent Trends in Engineering, Vol 1, No. 3, May 2009
3
1 .5
x 10
4
Loadless
Loaded
Loaded
2
1
1
Torque(Nm)
Rotor speed(p.u)
L o a d le ss
0
-1
0 .5
-2
-3
0
0
0 .5
1
Time (s)
1 .5
2
0
2
4
Time(s)
6
8
Fig.12: Torque (Nm)
Fig. 9. Rotor speed (p.u.)
1.4
150
Loadless
Rotor speed(p.u)
1.2
Torque(Nm)
100
L o a d le ss
50
Loaded
Loaded
1
0.8
0.6
0.4
0
0.2
0
-5 0
0
0 .5
1
Tim e (s )
1 .5
2
0
0.5
1
Time(s)
1.5
2
Fig. 13. Rotor speed (p.u.)
Fig. 10. Torque (Nm)
150
Fig. 7 and Fig. 9 illustrate the behaviors of the two
machines due to the same amount of change in the p.u.
applied torque. The change in the p.u. rotor speed for the
smaller machine is much larger than that of the larger
machine.
The behavior of the 2250-hp and 3-hp machines in
response to a step change in the applied torque can be
confirmed by examining the rotor speed to torque
sensitivities of the two machines.
The rotor speed to torque sensitivity, i.e., evaluated at the
operating point for the 2250-hp and 3-hp machines were
found to be 0.01 and 0.04, respectively. As seen, the
speed of the smaller machine is much more sensitive to a
change in torque.
1.4
Loadless
Loaded
1.2
Torque(Nm)
100
Rotor speed(p.u)
Loaded
50
0
-50
0
0.5
1
Time(s)
1.5
2
Fig. 14. Torque (Nm)
Fig.11 to Fig.14 illustrates the behavior of the two machines
due to the same amount of harmonics present in the stator
supply. The harmonics injected in the stator supply are 3rd, 5th &
7th. Here it is clearly observed that there will be pulsations in
the developed torque due to presence of supply harmonics.
CONCLUSIONS
1
This paper presents a comprehensive model for Induction
machine which can be used for studying the behavior under
different operating conditions. The model is based on the d-q
transformation and covers the steady-state and transient
behaviors of the machine. The model is quite versatile and
capable of simulating the machine during a sudden change in
load torque, with and without supply harmonics.
0.8
0.6
0.4
0.2
0
Loadless
0
2
4
Time(s)
6
Fig. 11. Rotor speed (p.u.)
© 2009 ACADEMY PUBLISHER
8
Two induction machines of different sizes were simulated to
illustrate the differences in machines' behaviors due to similar
changes in the applied torque and supply harmonics to the
induction motor.
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RESEARCH PAPER
International Journal of Recent Trends in Engineering, Vol 1, No. 3, May 2009
The general conclusion based on the comparison between the
2250-hp and 3-hp machines can be summarized as follows:
(a) The smaller machine has smaller inertia and shows shorter
transient period and less overshoot during start-up or
following any changes in the inputs. Generally speaking, it
has better transient behavior compared to the larger
machine.
(b)The 2250-hp induction machine is more stable after reaching
the steady-state, since sudden changes in input torque can
not accelerate or decelerate the machine as easily as in the
case of the 3-hp machine.
© 2009 ACADEMY PUBLISHER
REFERENCES
[1] Paul C. Krause, Analysis of electric machinery, McGrawHill, New York, 1986.
[2] Charles V. Jones, The Unified Theory of Electrical
Machines, Plenum Press 1967.
[3] Matlab Users Guide. Mathworks, 1993
[4] R. Krishnan, Electric Motor Drives: Modelling, Analysis
and Control, Prentice-Hall of India Private Ltd-2001.
[5] Li wang, Ching-Huei Lee, "A novel analysis on the
performance of an isolated self- excited induction
generator", IEEE Trans. On Energy Conversion, vol.12,
No.2, June 1993.
[6] Ion Boldea and Syed A. Nasar, The Induction machine
Handbook, CRC Press New-York 2002.
270