"Sharpening Skills.....
Serving Nation"
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459 (Online), Volume 4, Special Issue 1, February 2014)
International Conference on Advanced Developments in Engineering and Technology (ICADET-14), INDIA.
Methodology to Estimate The D-Q Axis Model Parameter For
Three Phase Induction Motor Using MATLAB
Mugdha Mishra1, Nitin Saxena2, Amit Singhal3, Sandeep Nain4
Moradabad Institute of Technology, Moradabad (U.P., INDIA)
mugdha1990@gmail.com1,nitinsaxena.iitd@gmail.com2,aks0207@rediffmail.com3,
sandeep.om.nain@gmail.com4
Abstract— This paper addresses a d-q model of three
phase induction motor. It is the objective of this paper to
derive and explain induction motor model in relatively
simple terms by using the concept of d-q variables. This
method reduces the three-phase system to a two-phase
system. Using MATLAB programming we have modeled
three-phase induction motor and obtained its direct and
quadrature axis parameters. The basic purpose of using d-q
model approach is to control the motor parameters
independently i.e. torque of induction motor.
Index Terms— d-axis, d-q transformation. induction
motor, q-axis.
I. INTRODUCTION
By the help of Clarke and park transforms, three-phase
induction motor can be modeled in an arbitrary two axis (d
& q-axis) rotating reference frame. It will be shown that
when we choose a synchronous reference frame in which
rotor flux lies on the d-axis, this model can estimate the
stator current along the direction of two-reference axis as
shown in fig.1 along D & Q axis. In the rotating frame of
reference the frame of reference in regard to the phase A is
named the d-axis (for direct axis) and the other axis is
named the q-axis (for quadrature axis).
fig.1: Two reference model for rotating machine
In many cases, analysis of induction motors with space
vector model is complicated due to the fact that we have to
deal with variables of complex numbers. When induction
motors are controlled by a vector drive, control
computation is often done in the synchronous frame. Since
actual stator variables either to be generated or to be
measured are all in stationary a-b-c frame, frame
transform should be executed in the control. The most
popular transform is between stationary a-b-c frame
quantities to synchronously rotating d-q quantities.
What remains is to define a method for performing the
phase transformations to the rotating frame of reference.
The transformation is done by defining a
transformation-matrix for the systems as,
f dq  Tabcdq f abc
Where, f denote currents, voltages, flux-linkage, etc.
The dynamic model of the induction motor is necessary
for understanding and analyzing the three-phase inductor
motor for the purpose of speed control. In this paper
induction motor has been modeled by MATLAB program.
II. MATHEMATICAL MODELING OF THREE-PHASE
INDUCTION MOTOR
There are a number of AC induction motor models. The
three-phase induction is modeled into a two-phase model.
We can look at the motor as a 2-phase machine. Utilizing
of the 2-phase motor model reduces the number of
equations and simplifies the control design. The model
used for vector control design can be obtained by utilizing
space-vector theory. Such a model is valid for any
instantaneous variation of voltage and current, and
adequately describes the performance of the machine
under both steady-state and transient operations. Complex
space vectors can be described using only two orthogonal
axes.
For the demonstration of the analysis of an induction
motor in the phase frame let us consider the power rating,
slip, stator and rotor impedances and voltage rating.
Firstly we have to develop the positive and negative
sequence network for the machine in which the value of
load impedances is only the difference.
Forward rotating magnetic field treated as positive
sequence and has slip s1 given by
(1)
And backward rotating magnetic field is treated as
negative sequence and has slip s2 given by,
(2)
The positive sequence load impedance is,
Lord Krishna College of Engineering (An ISO 9001:2008 Certified Institute) Ghaziabad, Uttar Pradesh, INDIA.
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"Sharpening Skills.....
Serving Nation"
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459 (Online), Volume 4, Special Issue 1, February 2014)
International Conference on Advanced Developments in Engineering and Technology (ICADET-14), INDIA.
Three-phase line-to-neutral voltages,
(3)
The negative sequence load impedance is,
[
]
[
]
(4)
Input complex power to the motor,
The net input positive sequence impedances of the
network is,
(
)
(
(5)
)
The net input negative sequence impedances of the
network is,
(
)
(
(6)
)
∑
(14)
Input power to motor =| |
power factor of the motor,
= cos(
(
))
(15)
For rotor currents and voltage evaluation we have to
obtain the ABCD parameters of the machine,
From Eq.5 & 6 we can evaluate the input sequence
admittances as,
(16)
(17)
(7)
(18)
and
and
(8)
(19)
Now the sequence admittance matrix will be formed as,
[
]
(9)
This sequence matrix will be transformed into the phase
admittance matrix,
[
]
[ ][
][ ]
1 1 1 

2
  α=
A= 1 

1   2 
,
[
]
(11)
][
]
(12)
The stator phase voltages can be evaluated from
,[
]
[ ][
[
[
]
[
]
[
]
[
]
]
[
[
The stator phase current is calculated as,
]
]
]
(20)
(21)
(22)
and
Let Vab, Vbc and Vca are the three phase voltages,
line-to-line voltages will be,
[
[
(10)
Where,
[
We can form sequence ABCD parameters as,
]
(13)
]
(23)
We have to transform these sequence ABCD
parameters into a-b-c phase parameters,
[
]
[ ][
][ ]
(24)
[
]
[ ][
][ ]
(25)
[
]
[ ][
][ ]
(26)
]
[ ][
][ ]
(27)
and
[
Rotor phase voltage and current can be evaluated as,
Lord Krishna College of Engineering (An ISO 9001:2008 Certified Institute) Ghaziabad, Uttar Pradesh, INDIA.
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"Sharpening Skills.....
Serving Nation"
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459 (Online), Volume 4, Special Issue 1, February 2014)
International Conference on Advanced Developments in Engineering and Technology (ICADET-14), INDIA.
[
]
[
][
[
]
[
][
]
[
][
]
(28)
[
][
]
(29)
250
200
Shaft power,
∑
150
three-phase supply voltage
]
(30)
100
50
0
-50
-100
-150
-200
-250
To convert three-phase voltages into two-phase
synchronously rotating frame, they are first converted into
two-phase stationary frame using the formula given
below,
0
0.2
0.4
0.6
0.8
1
time t
1.2
1.4
1.6
1.8
2
fig.2 three-phase supply voltage
150
100
[
]
[
][
√
]
(31)
stator voltages
50
0
-50
√
-100
and then from the stationary frame to the synchronously
rotating frame it is converted to give two-phase voltages,
-150
0
0.2
0.4
0.6
0.8
1
time t
1.2
1.4
1.6
1.8
2
fig.3 stator three-phase voltages
(32)
400
(33)
300
200
stator current
Where, s denotes stationary reference frame
and
100
0
-100
∫
-200
-300
0
0.2
stator angular electrical frequency
0.4
0.6
0.8
1
time t
1.2
1.4
1.6
1.8
2
fig.4 stator three-phase currents
We can obtain d-axis and q-axis currents from voltages,
150
(34)
100
rotor voltage
50
(35)
0
-50
√
-100
[ ]
[
[
√
]
-150
0
0.2
0.4
]
0.6
0.8
1
time t
1.2
1.4
1.6
1.8
2
fig.5 rotor voltages
III. RESULTS
150
100
50
rotor voltage
The basic purpose of this paper is to understand the
exact mathematical model used for induction motor for
finding the direct-axis and quadrature axis components of
voltage and current. Using MATLAB programming we
have modeled three-phase induction motor into two-phase
machine. The quadrature and direct axis voltage
components of modeled three-phase induction motor are
shown in fig.7.
0
-50
-100
-150
0
0.2
0.4
0.6
0.8
1
time t
1.2
1.4
1.6
1.8
2
fig.6 rotor currents
Lord Krishna College of Engineering (An ISO 9001:2008 Certified Institute) Ghaziabad, Uttar Pradesh, INDIA.
Page 87
"Sharpening Skills.....
Serving Nation"
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459 (Online), Volume 4, Special Issue 1, February 2014)
International Conference on Advanced Developments in Engineering and Technology (ICADET-14), INDIA.
Nomenclature
150
(d,q)
:
Rs, Xs
:
Rr, Xr
:
Vqr & Vdr :
Vqs & Vds :
100
Vqs & Vds
50
0
-50
-100
-150
0
0.2
0.4
0.6
0.8
1
time t
1.2
1.4
1.6
1.8
2
fig.7 voltage components of q-axis and d-axis
IV. CONCLUSION
In this paper a two-phase model of three-phase
induction motor has been modeled. By the help of
two-phase modeled the analysis of three-phase induction
motor becomes simple as well as this model is helpful for
the speed and torque control of induction motor. This
model is also essential when developing high performance
control techniques for the asynchronous motor drives such
as vector control or direct control (DTC) drives.
APPENDIX
Induction Motor Data
Specifications of Induction Motor
Power rating: 25 HP
Supply : 240V, 50Hz
Stator impedance: 0.0774+j0.1843 ohm
Reactance of core: j4.8384 ohm
Rotor impedance: 0.0908+j0.1843 ohm
Slip: 0.035
direct & quadrature axis
stator resistance and reactance
rotor resistance and reactance
d-axis & q-axis components of rotor voltage Vr
d-axis & q-axis components of stator voltage Vs
REFERENCES
[1 ] Wiliam H.Kersting, Distribution System Modeling and Analysis,
CRC press, Second Edition 2007.
[2 ] Burak Ozpineci, Leon M.Tolbert, “Simulink Implementation of
Induction Machine Model-A Modular Approach”
[3 ] M. G. Morshad, “Direct Torque Control in Induction Motor”,
IEEMA Journal of Electrical Engineering, Nov 2011.
[4 ] Sifat Shah, A. Rashid, and M.K.L Bhatti, “ Direct Quadrate (D-Q)
Modeling of 3-phase Induction Motor Using MATLAB Simulink”,
Canadian Journal on Electrical and Electronics Engineering , vol.
3, no. 5, May 2012.
[5 ] Power System Stability and Control, Pranha Kundur, TMH, Fifth
Edition, 2008.
[6 ] M. L. de Aguiar, M. M. Cad, “The concept of complex transfer
functions applied to the modeling of induction motors,” Power
[7 ] Engineering Society Winter Meeting, pp. 387–391, 2000.
[8 ] [7] P. C. Krause, Analysis of Electric Machinery, McGraw-Hill
Book Company, 1986.
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