Simulation and speed control of induction motor drives

ABSTRACT
Induction motors are the most widely used electrical motors due to their reliability, low cost
and robustness. However, induction motors do not inherently have the capability of variable
speed operation. Due to this reason, earlier dc motors were applied in most of the electrical
drives. But the recent developments in speed control methods of the induction motor have led
to their large scale use in almost all electrical drives.
Out of the several methods of speed control of an induction such as pole changing, frequency
variation, variable rotor resistance, variable stator voltage, constant V/f control, slip recovery
method etc., the closed loop constant V/f speed control method is most widely used. In this
method, the V/f ratio is kept constant which in turn maintains the magnetizing flux constant
so that the maximum torque remains unchanged. Thus, the motor is completely utilized in
this method.
During starting of an induction motor, the stator resistance and the motor inductance (both
rotor and stator) must be kept low to reduce the steady state time and also to reduce the jerks
during starting. On the other hand, higher value of rotor resistance leads to lesser jerks while
having no effect on the steady state time.
The vector control analysis of an induction motor allows the decoupled analysis where the
torque and the flux components can be independently controlled (just as in dc motor). This
makes the analysis easier than the per phase equivalent circuit.
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C O N T E N T S
CERTIFICATE.................................................................................................................................. I
ACKNOWNLEDGEMENT................................................................................................................ II
ABSTRACT................................................................................................................................... III
LIST OFTABLES ...........................................................................................................................VI
LISTOFFIGURES..........................................................................................................................VI
LIST OFSYMBOLS........................................................................................................................IX
CHAPTERS
1.INTRODUCTION........................................................................................................................1
2.LITERATURE REVIEW ................................................................................................................2
2.1 Three phaseinduction motor and their Torque-Speedanalysis ..................................................2
3.TRANSIENTS DURING STARTING OFA 3- INDUCTION MOTOR ..................................................5
3.1 Lowstator inductance (~0.05mH)................................................................................................ 6
3.2 Medium stator inductance (~0.7mH).........................................................................................10
3.3 High stator inductance (~2mH) ..................................................................................................14
3.4 LowRotor Resistance (~0.1) ...................................................................................................18
3.5 High Rotor Resistance (~0.5)................................................................................................... 22
3.6 LowStator Resistance(~0.16 )................................................................................................. 26
3.7 High Stator Resistance (~0.8).................................................................................................. 30
4.ANALYSIS OFVARIOUS METHODS FOR SPEED CONTROL OFIM................................................35
4.1 Variable Rotor Resistance........................................................................................................... 35
4.2 Variable Stator Voltage ............................................................................................................... 36
4.3 Constant V/f Control ...................................................................................................................37
4.3.1 Closed LoopV/f speed control method ...............................................................................38
4.3.2 OpenLoop V/f speed control method using PIcontroller...................................................42
4.3.3 Closed Loop V/f speed control method using PIcontroller.................................................44
4.4 Vector Control Method............................................................................................................... 47
4.4.1 d-q Equivalent Circuit........................................................................................................... 47
4.4.2 Axes Transformation............................................................................................................48
5.CONCLUSIONS........................................................................................................................54
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REFERENCES ..............................................................................................................................55
APPENDICES ..............................................................................................................................56
Appendix 1: MATLABCodefor SpeedControl of 3- Induction motor using Variable Rotor
Resistance .........................................................................................................................................56
Appendix 2: MATLABCodefor SpeedControl of 3- Induction motor using Variable StatorVoltage
.......................................................................................................................................................... 58
Appendix 3: MATLABCode for Speed Control of 3- Induction motor using Constant V/f control.60
Appendix 4: MATLABCodefor Closed LoopSpeedControl of 3- Induction motor using Constant
V/f .....................................................................................................................................................62
Appendix 5: MATLABCodeto observe the variations in q-axis and d-axis stator currents with
change in stator voltagefor a 3- induction motor..........................................................................65
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LIST OF TABLES
Table 1: Machine details used in MATLAB codes execution for variable rotor resistance,
variable stator voltage and constant V/f control
Table 2: Motor rating and parameters used in MATLAB code execution for Vector control
method
LIST OF FIGURES
Figure 1.1: Block diagram of an electrical drive
Figure 2.1: Per phase equivalent circuit of a 3- induction motor
Figure 2.2: Per phase approximate equivalent circuit of a 3- induction motor
Figure 3.1: SIMULINK model of a 3- Induction motor
Figure 3.2: Parameters of 3- induction motors (Low stator impedance)
Figure 3.3: Rotor Speed Vs Time graph for machine parameters as in Figure 3.2
Figure 3.4: Torque Vs Time graph for machine parameters as in Figure 3.2
Figure 3.5: Stator Current Vs Time graph for machine parameters as in Fig 3.2
Figure 3.6: Rotor Current Vs Time graph for machine parameters as in Fig 3.2
Figure 3.7: Torque-Speed Characteristics for machine parameters as in Fig 3.2
Figure 3.8: Parameters of 3- induction motors (Medium stator inductance)
Figure 3.14: Parameters of 3- induction motors (High stator inductance)
Figure 3.15: Rotor Speed Vs Time graph for machine parameters as in Fig 3.14
Figure 3.17:Stator Current Vs Time graph for machine parameters as in Fig 3.14
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Figure 3.18:Rotor Current Vs Time graph for machine parameters as in Fig 3.14
Figure 3.19:Torque-Speed Characteristics for machine parameters as in Fig 3.14
Figure 3.20: Parameters of 3- induction motors (Low Rotor Resistance)
Figure 3.26: Parameters of 3- induction motors (High Rotor Resistance)
Figure 3.32: Parameters of 3- induction motors (Low Stator Resistance)
Figure 3.38: Parameters of 3- induction motors (High Stator Resistance)
Figure 4.1: Torque-Speed characteristics of a 3- IM with variable rotor resistance
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Figure 4.2: Torque-Speed characteristics of a 3- IM with variable stator voltage
Figure 4.3: Torque-Speed characteristics of a 3- IM with constant V/f ratio
Figure 4.4: Block diagram for closed loop V/f control on a 3- IM
Figure 4.5: Input Data (Machine details) for Closed loop Constant V/f Speed Control Method
Figure 4.6 Torque-Speed Characteristics with Starting Load Torque 1.5 Nm and Reference
Speed 500 rpm
Figure 4.7 Torque-Speed Characteristics with Starting Load Torque 1 Nm and Reference
Speed 1200 rpm
Speed 1500 rpm
Figure 4.9: SIMULINK block of open loop constant V/f speed control using PI controller
Figure 4.10: Variation of Stator current of a 3- in case of open loop PI control for constant
V/f control method
Figure 4.11: Variation of DC bus voltage of a 3- in case of open loop PI control for constant
V/f control method
Figure 4.12: Variation of Torque of a 3- in case of open loop PI control for constant V/f
control method
Figure 4.13: Variation of Rotor Speed of a 3- in case of open loop PI control for constant
V/f control method
Figure 4.14: SIMULINK block of close loop constant V/f speed control using PI controller
Figure 4.15: Variation of Stator current of a 3- in case of closed loop PI control for constant
V/f control method
Figure 4.16: Variation of DC Bus Voltage of a 3- in case of closed loop PI control for
constant V/f control method
Figure 4.17: Variation of Torque of a 3- in case of closed loop PI control for constant V/f
control method
Figure 4.18: Variation of Rotor Speed of a 3- in case of closed loop PI control for constant
V/f control method
Figure 4.19: Angular relationships between reference axes
Figure 4.20: Variation of q-axis stator current with change in stator voltage
Figure 4.21: Variation of d-axis stator current with change in stator voltage
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LIST OF SYMBOLS
IM Induction Motor
Rs Stator Resistance
Rr Rotor Resistance
Rr’ Rotor Resistance Referred to Stator side
Xs Stator Reactance
Xr Rotor Reactance
Xr’ Rotor Reactance Referred to Stator side
Xm Leakage Inductance
I1 Stator Current
I2 Rotor Current
I2’ Rotor Current Referred to Stator side
Im Magnetizing Current
V0 Stator Voltage
s Slip
ωs Synchronous Speed
ωm Rotor Speed (Machine Speed)
Ωs Average Synchronous Speed (in RPM)
f Supply Frequency
p No. of Poles
Pg Air-gap Power
Pcu Copper loss in the machine
Pm Mechanical Power output of the machine
T Torque Developed by the motor
sm Slip at maximum torque
Tmax Maximum Torque
Vd DC Link Voltage
ωref Reference Speed

ωsl
ωf
Yd
Yq
Ya
Yb
Yc
Vqs
Vds
Iqs
Ids
Iqr
Idr
λds
λqs
λdr
λqr
λs
Ls
Lr
Lm
Is’
Pi
Qi
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Slip Speed
Rotor Speed at Frequency f
Space Vector in d-axis
Space Vector in q-axis
Space Vector of a-phase
Space Vector of b-phase
Space Vector of c-phase
q-axis Stator Voltage with stationary frame
d-axis Stator Voltage with stationary frame
q-axis Stator Current with stationary frame
d-axis Stator Current with stationary frame
q-axis Rotor Current with stationary frame
d-axis Rotor Current with stationary frame
d-axis Stator flux with stationary frame
q-axis Stator flux with stationary frame
d-axis Rotor flux with stationary frame
q-axis Rotor flux with stationary frame
q-axis Rotor flux with stationary frame
Stator Self-Inductance
Rotor Self-Inductance
Stator Mutual-Inductance
Complex Conjugate of Stator Current
Instantaneous Active Power
Instantaneous Reactive Power

V
CHAPTER I
INTRODUCTION
Be it domestic application or industry, motion control is required everywhere. The systems
that are employed for this purpose are called drives. Such a system, if makes use of electric
motors is known as an electrical drive. In electrical drives, use of various sensors and control
algorithms is done to control the speed of the motor using suitable speed control methods.
The basic block diagram of an electrical drive is shown below:
Figure 1.1: Block diagram of an electrical drive
Earlier only dc motors were employed for drives requiring variable speeds due to ease of
their speed control methods. The conventional methods of speed control of an induction
motor were either too expensive or too inefficient thus restricting their application to only
constant speed drives. However, modern trends and development of speed control methods of
an induction motor have increased the use of induction motors in electrical drives
extensively.
In this paper, we have studied the various methods of speed control of a 3- induction motor
and compared them using their Torque-Speed characteristics. Also the transients during the
starting of a 3- induction motor were studied using MATLAB Simulink and the effects of
various parameters such as rotor and stator resistances and inductances were analysed. Also
different control algorithms such as P, PI and PID control were studied by simulating them in
MATLAB Simulink and were compared.
SOURCE
POWER
MODULATOR
MOTOR LOAD
CONTROL
UNIT
SENSING
UNIT
INPUT
COMMAND
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CHAPTER 2
LITERATURE REVIEW
2.1 ThreephaseinductionmotorandtheirTorque-Speedanalysis
Basedon the construction of the rotor, a3- induction motor can be categorized into two
types:
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i.
ii.
Squirrel Cage Induction Motor
Wound Rotor or Slip Ring Induction Motor
The stator of both types of motors consists of a three phase balanced distributed winding with
each phase mechanically separated in space by 120 degrees from the other two phase
windings. This gives rise to a rotating magnetic field when current flows through the stator.
In squirrel cage IM, the rotor consists of longitudinal conductor bars which are shorted at
ends by circular conducting rings. Whereas, the wound rotor IM has a 3- balanced
distributed winding even on the rotor side with as many number of poles as in the stator
winding.
Considering the three phases to be balanced, the analysis of a 3- induction motor can be
done by analysing only one of the phases. The per phase equivalent circuit of an induction
motor is shown below:
Figure 2.1: Per phase equivalent circuit of a 3- induction motor
R2 and X2 are the stator referred values of rotor resistance R1 and rotor reactance X1. Slip is
defined by
s = (s – m)/ s (2.1)
where, ωm and ωs are rotor and synchronous speeds,respectively.

(2.2)Further, s = 120f/p rpm
Where f and p are supply frequency and number of poles, respectively.
Since, stator impedance drop is generally negligible compared to terminal voltage V, the
equivalent circuit can be simplified to that shown below:
Figure 2.2: Per phase approximate equivalent circuit of a 3- induction motor
Rotor current
( )
(2.3)
Power transferred to rotor (or air-gap power)
(2.4)
Rotor copper loss is
(2.5)
Electrical power converted into mechanical power
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(2.6)

Torque developed by motor
(2.7)
Thus,
(2.8)
Substituting the value of I2 into the above equation, we get,
( )
(2.9)
Differentiating T with respect to s and equating to zero gives the slip for maximum torque
(2.10)
√
Substituting Sm in T gives the value of maximum torque, thus
[ √( )]
(2.11)
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CHAPTER 3
TRANSIENTS DURING STARTING OF
A 3- INDUCTION MOTOR
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A model of a 3- induction motor was setup in MATLAB SIMULINK and the rotor and
stator currents, speed, electromagnetic torque and the Torque-Speed characteristics were
observed with different values of rotor and stator resistances and impedances.
The SIMULINK model is shown below.
Figure 3.1: SIMULINK model of a 3- Induction motor
The different machine details followed by their corresponding outcomes are shown in this
chapter.
It should be noted that all the simulations were made for Zero Load Torque. However,
the inertia and friction were taken into consideration.

3.1 Lowstatorinductance(~0.05mH)
Figure 3.2: Parameters of 3- induction motors (Low stator impedance)
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3.2Mediumstatorinductance(~0.7 mH)
Figure 3.8: Parameters of 3- induction motors (Medium stator inductance)
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3.3Highstatorinductance(~2 mH)
Figure 3.14: Parameters of 3- induction motors (High stator inductance)
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3.4LowRotorResistance(~0.1 )
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Figure 3.20: Parameters of 3- induction motors (Low Rotor Resistance)

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0.5003
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3.5HighRotorResistance(~0.5)
Figure 3.26: Parameters of 3- induction motors (High Rotor Resistance)

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Simulation and Speed Control of Induction Motor Drives 2012
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3.6LowStatorResistance(~0.16)
Figure 3.32: Parameters of 3- induction motors (Low Stator Resistance)
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3.7HighStatorResistance(~0.8 )
Figure 3.38: Parameters of 3- induction motors (High Stator Resistance)
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On the basis of the above outcomes, the following observations were made:
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i. On increasing the motor inductance (either rotor or stator), the transients lasted for
longer period i.e., the machine took longer time to achieve its steady state speed,
current and torque. Also the start was a bit jerky.
ii. On increasing the rotor resistance, there was no effect on the steady state time but
the machine started with lesser jerks, i.e., the fluctuations in the transient period
were reduced. Also the maximum torque occurred at a lower speed.
iii. On increasing the stator resistance, the steady state time increased as well as the
machine started with more jerks. Thus the stator resistance must be kept as low as
possible.

CHAPTER 4
ANALYSIS OF VARIOUS METHODS
FOR SPEED CONTROL OF IM
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The various methods of speed control of 3- Induction motor are as under:
1. Pole Changing
2. Variable Supply Frequency
3. Variable rotor resistance control
4. Variable supply voltage control
5. Constant V/f control
6. Slip recovery
7. Vector Control
However, we shall not be analyzing the pole changing and the variable supply frequency
methods as these are very rarely used. This chapter deals with the basic theory behind the
several methods of speed control. Hereafter, they are discussed one after the other.

4.1 VariableRotorResistance
This method is applicable only to the wound rotor motor as external resistance can be added
to it through the slip rings.
A MATLAB code was developed to observe the variation in Torque-Speed characteristics of
a 3- induction motor with variable rotor resistance. The MATLAB code is given in
Appendix 1 and the output Torque-Speed characteristics are shown in Figure 4.1 below. The
machine details used for the code execution are shown in Table 1 at the end of the chapter.
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Figure 4.1: Torque-Speed characteristics of a 3- IM withVariable rotor resistance
External resistances can be connected in the rotor circuit during starting. This increases the
starting torque (Equation 2.9 with s=1) and reduces the starting current (Equation 2.3). By
making use of appropriate value of resistors, the maximum torque can be made to appear
during starting. This can be used in applications requiring high starting torque. Once the
motor is started, the external resistance can be cut out to obtain high torque throughout the
accelerating range. As external resistances are connected, most of the I2
R loss is dissipated
through them thus the rotor temperature rise during starting is limited.
4.2VariableStatorVoltage
As can be seen from Equation 2.9, the torque developed by an induction motor varies as
square of the voltage applied to its stator terminals. Thus by varying the applied voltage, the
electromagnetic torque developed by the motor can be varied. This method is generally used
for small squirrel-cage motors where cost is an important criterion and efficiency is not.
However, this method has rather limited range of speed control.

A MATLAB code was developed to observe the variation in torque-speed characteristics of a
3- induction motor with variable stator voltage. The MATLAB code is given in Appendix 2
and the output Torque-Speed characteristics are shown in Figure 4.2 below. The machine
details used for the code execution are shown in Table 1 at the end of the chapter.
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Figure 4.2: Torque-Speed characteristics of a 3- IM with Variable stator voltage
As the supply voltage is decreased, the value of maximum torque also decreases (Equation
2.11). However it still occurs at the same slip as earlier (Equation 2.10). Even the starting
torque and the overall torque reduce (Equation 2.9). Thus the machine is highly underutilized.
Thus this method of speed control has very limited applications.
4.3ConstantV/f Control
We vary the stator voltage in such a way that the flux remains constant by simultaneously
varying the supply frequency such that the ratio V/f remains constant.
A MATLAB code was developed to observe the variation in torque-speed characteristics of a
3- induction motor with constant V/f. The MATLAB code is given in Appendix 3 and the

output Torque-Speed characteristics are shown in Figure 4.3 below. The machine details used
for the code execution are shown in Table 1 at the end of the chapter.
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Figure 4.3: Torque-Speed characteristics of a 3- IM with constant V/f ratio
The AC supply is rectified and then applied to a PWM inverter to obtain a variable
frequency, variable magnitude 3- AC supply.
The electromagnetic torque developed by the motor is directly proportional to the magnetic
field produced by the stator and the flux produced by the stator is proportional to the ratio of
applied voltage and frequency of supply. Therefore, by varying the voltage and frequency by
the same ratio, flux and hence, the torque can be kept constant throughout the speed range.
This makes constant V/f method the most common speed control method of an induction
motor.
4.3.1ClosedLoopV/fspeedcontrolmethod
The basis of constant V/f speed control of induction motor is to apply a variable magnitude
and variable frequency voltage to the motor. Both the voltage source inverter and current
source inverters are used in adjustable speed ac drives. The following block diagram shows
the closed loop V/f control using a VSI.

Figure 4.4: Block diagram for closed loop V/f control on a 3- IM
A speed sensor or a shaft position encoder is used to obtain the actual speed of the motor. It is
then compared to a reference speed. The difference between the two generates an error and
the error so obtained is processed in a Proportional controller and its output sets the inverter
frequency. The synchronous speed, obtained by adding actual speed ωf and the slip speed ωsl,
determines the inverter frequency. The reference signal for the closed-loop control of the
machine terminal voltage Vs is generated from frequency.
A MATLAB code was developed to observe the variation of frequency as well as the
operating zone of the motor with the variation of load torque. Along with the frequency, the
voltage was also varied to make V/f ratio constant so that the air gap flux and the maximum
torque remain constant. The MATLAB code is given in Appendix 4 and the output Torque-
Speed characteristics are shown in Figure 4.5 below. The machine details used for the code
execution are shown in Table 1 at the end of the chapter.
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Figure 4.5: Input Data (Machine details) for Closed loop Constant V/f Speed Control Method

Rotor Speed(rpm)
Speed 1200 rpm
Torque
(N-m)
Rotor Speed(rpm)
Figure 4.6 Torque-Speed Characteristics with Starting Load Torque 1.5 Nm and Reference
Speed 500 rpm
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Torque
(N-m)

Rotor Speed(rpm)
Speed 1500 rpm
Initially the motor was started at a reference speed and a constant load torque. Then the load
torque was varied and according to that the speed of operation shifted. The speed was taken
as a feedback and the error in the speed was calculated and the frequency was adjusted
accordingly by proportional controller. This corrected frequency is given to the voltage
source inverter. Simultaneously the voltage is varied such that the V/f ratio remains constant
for any value of frequency. Hence we can an approximate constant speed for different loads.
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Torque
(N-m)

4.3.2OpenLoopV/fspeedcontrolmethodusingPIcontroller
A SIMULINK block was created to analyse the open loop constant V/f control method using
PI controller and the Stator current (Figure 4.10), DC voltage (Figure 4.11), Electromagnetic
torque (Figure 4.12) and Rotor speed (Figure 4.13) were plotted against time. The
SIMULINK block is given below followed by the outcomes.
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Figure 4.9: SIMULINK block of open loop constant V/f speed control using PI controller
Figure 4.10: Variation of Stator current of a 3- in case of open loop PI control for constant
V/f control method

Figure 4.11: Variation of DC bus voltage of a 3- in case of open loop PI control for constant
V/f control method
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Figure 4.12: Variation of Torque of a 3- in case of open loop PI control for constant V/f
control method

Figure 4.13: Variation of Rotor Speed of a 3- in case of open loop PI control for constant
V/f control method
4.3.3ClosedLoopV/fspeedcontrolmethodusingPIcontroller
A SIMULINK block was created to analyse the close loop constant V/f control method using
PI controller and the Stator current (Figure 4.15), DC voltage (Figure 4.16), Electromagnetic
torque (Figure 4.17) and Rotor speed (Figure 4.18) were plotted against time. The
SIMULINK block is given below followed by the outcomes.
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Figure 4.14: SIMULINK block of close loop constant V/f speed control using PI controller

Figure 4.15: Variation of Stator current of a 3- in case of closed loop PI control for constant
V/f control method
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Figure 4.16: Variation of DC Bus Voltage of a 3- in case of closed loop PI control for
constant V/f control method

Figure 4.17: Variation of Torque of a 3- in case of closed loop PI control for constant V/f
control method
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Figure 4.18: Variation of Rotor Speed of a 3- in case of closed loop PI control for constant
V/f control method

4.4VectorControl Method
The induction motor is the most widely used electrical motor due to its rugged structure, low
cost and reliability. However, the nonlinearity in the Torque-Voltage relationship of an IM
makes its analysis difficult. Also it is a fifth order system making its dynamic response poor.
Development of Vector Control analysis has enabled us to get as good dynamic performance
from an IM as a dc motor. The torque and the flux components can be controlled
independently using vector control just like in a dc motor.
In order to analyse vector control, we need to develop a dynamic model of the IM. This is
done by converting the 3- quantities into 2-axes system called the d-axis and the q-axis.
Such a conversion is called axes transformation. The d-q axes can be chosen to be stationary
or rotating. Further, the rotating frame can either be the rotor oriented or magnetizing flux
oriented. However, synchronous reference frame in which the d-axis is aligned with the rotor
flux is found to be the most convenient from analysis point of view.
A major disadvantage of the per phase equivalent circuit analysis is that it is valid only if the
three phase system is balanced. Any imbalance in the system leads to erroneous analysis.
Even this problem is eradicated if we use the d-q model.
4.4.1 d-qEquivalent Circuit
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. For any space vector Y, let
us define two real quantities Sq and Sd as,
S = Sq - j Sd (4.1)
In other words,
Sq = Re (S) and Sd = - Im (S)
Figure 4.9 illustrates the relationship between d-q axis and stationary a-b-c frame. It should
be noted that d- and q-axes are defined on a rotating reference frame at the speed of ωa with
respect to fixed a-b-c frame.
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4.4.2 Axes Transformation
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Figure 4.19: Angular relationships between reference axes
(4.2)
Substituting ωa = 0, the above equation can be written as below: (This is called stationary
reference frame)
(4.3)
Sometimes vector control includes calculation in rotor reference frame (frame is attached to
the rotor rotating at ωo). In this case, ωa = ωo in equation 4.2. Hence the matrix will be
changed as

(4.4)
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For dynamic simulation of induction motors, equation 4.3 or equation 4.4 may be used.
Also we have
(4.5)
(4.6)
(4.7)
Vqs = Rs Iqs + pqs +s ds
Vds = Rs Ids + p ds - s qs
0 = Rr Iqr + pqr + r dr
0 = Rr Idr + p dr - r qr (4.8)
where flux linkage variables are defined by
λqs = Ls Iqs + Lm Iqr (4.9)
λds = Ls Ids + Lm Idr (4.10)
λqr = Lm Iqs + Lr Iqr (4.11)
λdr = Lm Ids + Lr Idr (4.12)
3- quantities converted into d-q quantities can be expressed as:
(4.13)Sqs = (2/3) Re{exp(-ja) (Sa + Sb + 2
Sc)}
Sds = - (2/3) Im{exp(-ja) (Sa +Sb +2
Sc)} (4.14)
(4.15)

and its inverse transform is given by
(4.16)
For any frame of reference, in terms of space vector instantaneous input power can be written
as,
Pi = (3/2) Re(Vs Is’ ) (4.17)
or
Pi = (3/2) [ Vds Ids + Vqs Iqs] (4.18)
The reactive power Qi can also be defined as
Qi = (3/2) Im(Vs Is’ ) (4.19)
or
Qi = (3/2) [ Vqs Ids - Vds Iqs] (4.20)
Torque in terms of d-q parameters is given by,
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(4.21)

A MATLAB code was developed to observe the variations in q-axis and d-axis stator currents
with change in stator voltage for a three phase induction motor. The MATLAB code is given
in Appendix 5 and the machine parameters are given in table 2 at the end of the chapter.
Following graphs were obtained:
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Figure 4.20: Variation of q-axis stator current with change in stator voltage
Figure 4.21: Variation of d-axis stator current with change in stator voltage

Table 1: Machine details used in MATLAB codes execution for variable
rotor resistance, variable stator voltage and constant V/f control
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RMS value of supply voltage (line-to-line) 415 Volts*
Number of poles 4
Stator resistance 0.075 ohm
Rotor resistance 0.1 ohm**
Frequency 50 Hz***
Stator leakage reactance at 50 Hz frequency 0.45 ohm
Rotor leakage reactance at 50 Hz frequency 0.45 ohm
V/f ratio (ONLY FOR CONTANT V/f CONTROL) 8.3
* For Variable stator voltage and constant V/f methods, different values of supply voltage are
given in the respective graphs (Figure 4.2 and Figure 4.3, respectively).
** For Variable rotor resistance method, different values of rotor resistance are given in the
graph (Figure 4.1).
*** For constant V/f method, different values of supply frequency were used such that V/f
ratio remained constant at 8.3.

Table 2: Motor rating and parameters used in MATLAB code execution for
Vector control method
53 | Page
Rated Power 12 KW
Rated Stator Voltage (line to line) 230 V RMS
Operation Frequency 50 Hz
Number of Poles 4
Stator Resistance 0.095 ohms
Rotor Resistance 0.2 ohms
Stator Reactance 0.68 ohms
Rotor Reactance 0.672 ohms
Magnetizing Reactance 18.7 ohms

CHAPTER 5
CONCLUSIONS
Torque-Speed characteristics for different methods of speed control of an IM were obtained
and analysed by developing MATLAB codes.
In rotor resistance control method the starting torque can be varied with the variation of rotor
resistance. The maximum torque however, remains unaffected. Thus for operations requiring
high starting torque, the rotor resistance can be varied to even obtain the maximum torque
during starting. But simultaneously the copper losses will increase due to increase of
resistance. So this method is highly inefficient and cannot be used throughout the operation.
In variable supply voltage control method of speed control, the maximum torque decreases
with the decrease of supply voltage and thus the motor remains underutilized. So even this
method cannot be used for good performance.
In constant control, by use of rectifier and PWM inverter, we can vary the supply voltage as
well as the supply frequency such that the ratio remains constant so that the flux remains
54 | Page
constant too. So we can get different operating zone for various speeds and torques and also
we can get different synchronous speed with almost same maximum torque. Thus the motor
is completely utilized and also we have a good range of speed control.
Also from the SIMULINK model for the starting of an induction motor with varying
parameters, it was deduced that the stator resistance must be kept as low as possible so as to
reduce the steady state time during starting and also to obtain a smoother start. Increasing the
rotor resistance leads to increase in the starting torque (maximum torque occurs at a lesser
speed) however, it also leads to a jerky start. Decreasing the inductance (either rotor or stator)
lets the machine achieve its steady state quicker with slightly lesser jerks.
The traditional per phase equivalent circuit analysis of an induction motor has the
disadvantage that it is valid only if the system is a balanced one. Any imbalance in the system
leads to erroneous analysis. Also the dynamic response of the motor cannot be obtained from
the per phase equivalent circuit. The vector control method or the d-q axes model leads to a
simpler analysis of an induction motor. A d-q axes model with the d-axis aligned along the
synchronously rotating rotor frame, leads to the decoupled analysis where the torque and the
flux components can be independently controlled just like in case of a dc motor.

REFERENCES
55 | Page
1 Gopal K. Dubey, “Fundamental Of Electrical Drives”, Narosa Publication House,
Second Edition, 2011
2 A. E. Fitzgerald, Charles Kingsley, Jr. And Stephan D. Umans, “Electrical Machinery”,
McGraw-Hills Publications, Year 2002
3 “IEEE Standard Test Procedure for Polyphase Induction Motors and Generators”, volume
112, issue 1996 of IEEE, by IEEE Power Engineering Society
4 Scott Wade, Matthew W. Dunnigan, and Barry W. Williams, “Modelling and Simulation
of Induction Machine Vector Control with Rotor Resistance Identiﬁcation”, IEEE
transactions on power electronics, vol. 12, no. 3, may 1997.
5 D.W. Novotney, et al (editor), “Introduction to Field Orientation and High Performance
AC drives”, IEEE IAS tutorial course, 1986.
6 Ramon Blasco Blasco Gimenez, “High Performance Sensorless Vector Control of
Induction Motor Drives”, The University of Nottingham, December 1995.

APPENDICES
Appendix1: MATLABCodeforSpeedControlof3- Inductionmotor using
VariableRotorResistance
function out = inductionvarRr()
Vl1=input('Enter the Suppy Voltage (line to line) RMS value: ');
P=input('Enter the number of poles: ');
Rs=input('Stator Resistance: ');
Rr1=input('Enter the first Rotor Resistance: ');
Rr2=input('Enter the second Rotor Resistance: ');
Rr3=input('Enter the third Rotor Resistance: ');
Rr4=input('Enter the fourth Rotor Resistance: ');
Rr5=input('Enter the fifth Rotor Resistance: ');
Xs=input('Stator Leakage Reactance @ 50 Hz frequecny: ');
Xr=input('Rotor Leakage Reactance @ 50 Hz frequecny: ');
Ls=Xs/(2*pi*50);
Lr=Xr/(2*pi*50);
Wsync1=4*pi*50/P;
Tmf2=zeros(Wsync1*500+1,1);
m=1;
for Wrotor1=0:0.002:Wsync1
Tmf1(m)=(3*(((Vl1^2)*Rr1/((Wsync1-Wrotor1)/Wsync1))/((Rs+Rr1/((Wsync1-
Wrotor1)/Wsync1))^2+(2*pi*50*Ls+2*pi*50*Lr)^2))/Wsync1); %star connected
m=m+1;
end
m=1;
Wrotor1)/Wsync1))^2+(2*pi*50*Ls+2*pi*50*Lr)^2))/Wsync1);
m=m+1;
end
m=1;
m=m+1;
end
56 | Page

m=1;
m=m+1;
end
m=1;
m=m+1;
end
plot(Tmf1);
hold on;
plot(Tmf2);
plot(Tmf3);
plot(Tmf4);
plot(Tmf5);
hold off;
ylabel('Torque(N-m)');
xlabel('Rotor Speed(Rad/s)');
end
57 | Page

Appendix2:MATLABCodeforSpeedControlof3- Inductionmotor using
VariableStatorVoltage
function out = inductionvarV()
Vl1=input('Enter the first Suppy Voltage (line to line) RMS value: ');
Vl2=input('Enter the second Suppy Voltage (line to line) RMS value: ');
Vl3=input('Enter the third Suppy Voltage (line to line) RMS value: ');
Vl4=input('Enter the fourth Suppy Voltage (line to line) RMS value: ');
Vl5=input('Enter the fifth Suppy Voltage (line to line) RMS value: ');
Rr=input('Rotor Resistance: ');
Ls=Xs/(2*pi*50);
Lr=Xr/(2*pi*50);
Wsync1=4*pi*50/P;
m=1;
Tmf1(m)=(3*(((Vl1^2)*Rr/((Wsync1-Wrotor1)/Wsync1))/((Rs+Rr/((Wsync1-
m=m+1;
end
m=1;
m=m+1;
end
m=1;
m=m+1;
end
m=1;
58 | Page

m=m+1;
end
m=1;
m=m+1;
end
plot(Tmf1);
hold on;
plot(Tmf2);
plot(Tmf3);
plot(Tmf4);
plot(Tmf5);
hold off;
xlabel('Rotor Speed(Rad/s)');
end
59 | Page

Appendix3:MATLABCodeforSpeedControlof3- Inductionmotor using
ConstantV/f control
60 | Page
function out = inductionconstVf()
Vll=input('Suppy Voltage (line to line) RMS value @ 50 Hz: ');
f2=input('Enter the second frequency: ');
f3=input('Enter the third frequency: ');
f4=input('Enter the fourth frequency: ');
f5=input('Enter the fifth frequency: ');
Rr=input('Rotor Resistance: ');
Ls=Xs/(2*pi*50);
Lr=Xr/(2*pi*50);
Vlnf1=Vll/(3^0.5);
Vlnf2=Vlnf1*f2/50;
Vlnf3=Vlnf1*f3/50;
Vlnf4=Vlnf1*f4/50;
Vlnf5=Vlnf1*f5/50;
Wsync1=4*pi*50/P;
Wsync2=4*pi*f2/P;
Wsync3=4*pi*f3/P;
Wsync4=4*pi*f4/P;
Wsync5=4*pi*f5/P;
m=1;
Tmf1(m)=(3*(((Vlnf1^2)*Rr/((Wsync1-Wrotor1)/Wsync1))/((Rs+Rr/((Wsync1-
m=m+1;
end
m=1;
Wrotor2)/Wsync2))^2+(2*pi*f2*Ls+2*pi*f2*Lr)^2))/Wsync2);
m=m+1;
end
m=1;

m=m+1;
end
m=1;
m=m+1;
end
m=1;
m=m+1;
end
plot(Tmf1);
hold on;
plot(Tmf2);
plot(Tmf3);
plot(Tmf4);
plot(Tmf5);
hold off;
xlabel('Rotor Speed(Rad/s) * 100');
end
61 | Page

Appendix4:MATLABCodeforClosedLoopSpeedControlof3- Induction
motorusingConstantV/f
function out = inductionconstVfclosed()
Input1;
Tmm=[];
Wrotormat=[];
Ls=Xs/(2*pi*50);
Lr=Xr/(2*pi*50);
Vfratio=Vratedph/200; %Constant V/f ratio = Rated Voltage/Maximum frequency
that is applied (taken as 200 Hz)
%Find the value of Frequency at which the motot shall be started so that
the given operating point (Tlstarting,Wref) lies in the stable zone
if Tlstarting==0
Wsync=Wref;
f=Wsync*P/120;
V=Vfratio*f;
else
for f=1:0.001:200
Wsync=120*f/P;
s=(Wsync-Wref)/Wsync;
sm=(Rr/((Rs^2+(2*pi*f*Ls+2*pi*f*Lr)^2)^0.5));
if Wref<Wsync && s<sm %to make sure that the operating point lies
in the stable zone
f=f;
V=Vfratio*f;
if Tlstarting>=(0.95*(3*(((V^2)*Rr/((Wsync-
Wref)/Wsync))/((Rs+Rr/((Wsync-
Wref)/Wsync))^2+(2*pi*f*Ls+2*pi*f*Lr)^2))/Wsync)) &&
Tlstarting<=(1.05*(3*(((V^2)*Rr/((Wsync-Wref)/Wsync))/((Rs+Rr/((Wsync-
Wref)/Wsync))^2+(2*pi*f*Ls+2*pi*f*Lr)^2))/Wsync))
break;
end
end
end
end
Tm=zeros(100001,1);
Wrot=zeros(100001,1);
m=1;
%Store the values of torque at different rotor speeds to plot the Torque-
Speed characteristics
for Wrotor=0:Wsync/100000:Wsync
62 | Page

Tm(m)=(3*(((V^2)*Rr/((Wsync-Wrotor)/Wsync))/((Rs+Rr/((Wsync-
Wrotor)/Wsync))^2+(2*pi*f*Ls+2*pi*f*Lr)^2))/Wsync);
if Tm(m)<0
Tm(m)=0;
end
Wrot(m)=Wrotor;
m=m+1;
end
Tmax=(3*(V^2/(Rs+(Rs^2+(2*pi*f*Ls+2*pi*f*Lr)^2)^0.5))/(2*Wsync)); % Maximum
Torque the given motor can deliver for the current values of V and f
% Vary the Load Torque From 0.1 to Tmax with a step of Tmax/10 and apply
the closed loop P control to maintain the motor speed at Wref for any
permissible values of load torque
for Tl=0.1:Tmax/10:Tmax
Tmm=[Tmm Tm];
Wrotormat=[Wrotormat Wrot];
for a=100001:-1:1
if (Tm(a)*0.95)<=Tl && Tl<=(Tm(a)*1.05)
W=Wrot(a);
break;
end
end
Werror=Wref-W; % Error in speed to be corrected
n=1;
f=f+(Werror*P/120); % The frequency and hence the Speed Corrected
Proportionately (ie P controller used)
Wsync=120*f/P;
V=Vfratio*f;
% Store the values of torque at different rotor speeds to plot the Torque-
Speed characteristics
for Wrotor=0:Wsync/100000:Wsync
Tm(n)=(3*(((V^2)*Rr/((Wsync-Wrotor)/Wsync))/((Rs+Rr/((Wsync-
Wrotor)/Wsync))^2+(2*pi*f*Ls+2*pi*f*Lr)^2))/Wsync); %Star connected
if Tm(n)<0
Tm(n)=0;
end
Wrot(n)=Wrotor;
n=n+1;
end
end
[c,d]=size(Tmm);
[e,h]=size(Wrotormat);
63 | Page

figure;
% Plot the Torque-Speed Characteristics of the motor for various values of
Load Torque
hold on;
for g=1:1:d
plot(Wrotormat(:,g),Tmm(:,g));
end
hold off;
end
64 | Page

Appendix5:MATLABCodetoobservethevariationsinq-axisandd-axis
statorcurrentswithchangeinstatorvoltagefora3- inductionmotor
function out = vectorcontrol( )
clc;
clear all;
%Motor detalis (Rated)
P = 12*10^-3;
V = 230;
Va = 230/sqrt(3) ;
fe = 50;
We = 2*pi*fe;
Lambda = sqrt(2)*Va/We;
I= P/(sqrt(3)*V) ;
Ipeakbase = sqrt(2)*I;
poles=4 ;
%Paramteres for 50-Hz motor
VI0 = V/sqrt(3) ;
X10 = 0.67;
X20 = 0.67;
Xm0 = 18.0;
R1 = 0.1;
R2 = 0.2;
%d-q parameters
Lm = Xm0/We;
LS = Lm + X10/We;
LR = Lm + X20/We;
Ra = R1;
RaR = R2 ;
% Operating point
Wm = 2*n*pi/60;
Wme = (poles/2)*Wm;
Pmech = 9.7*10-3;
Tmech = Pmech/Wm;
Te=zeros(1001,1);
for n = 1:1:41
lambda_D_R = (0.8 + (n-1)*0.4/40)*Lambda;
iQ(n) = (2/3) * (2/poles) * (LR/Lm) * (Tmech/lambdaDR) ;
iD(n) = (lambdaDR/Lm) ;
iQm(n) = - (Lm/LR)*iQ(n);
Ia(n) =sqrt((iD(n)^2 + iQ(n)^2)/2) ;
We(n) = Wme - (RaR/LR)*(iQ(n)/iD(n)) ;
fe(n) = We(n)*poles/120 ;
Varms(n) = sqrt( ((Ra*iD(n)-We(n)*(LS-Lm^2/LR)*iQ(n)) ^2 +(Ra*iQ(n)+
We(n)*LS*iD(n))^2) /2) ;
end
plot (Varms,iQ)
figure
plot(Varms,iD)
end
65 | Page

Simulation and speed control of induction motor drives

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Simulation and speed control of induction motor drives