85
CHAPTER 6
REDUCTION OF TORQUE AND SPEED PULSATION
THROUGH THE PROPOSED CONTROLLER
6.1
INTRODUCTION
Variable voltage and variable frequency supply to AC drives are
invariably obtained from a three-phase voltage source inverter. A space vector
pulse width modulation (PWM) scheme is used to obtain variable voltage and
variable frequency supply. The most widely used PWM schemes for threephase voltage source inverters are carrier based sinusoidal PWM and space
vector pulse width modulation schemes. There is an increasing trend of using
SVPWM‟s because of their easier digital realization (Nayeripour et al 2010).
This Chapter focuses on step by step development of SVPWM implemented
on an induction motor using the proposed fuzzy controller. The new duty
cycle calculation of a three phase voltage source inverter is proposed here
based on space vector theory. This calculation is used to describe the
relationship between duty cycle and output voltage. Also it is used to find out
matrix converter duty cycle.
6.2
CALCULATION OF DUTY CYCLE FOR THE INVERTER
Sector representation of space vector analysis of inverter is
explained in chapter 4. At sector I, V and V are voltage vectors as shown in
Figure 6.1. Assuming V0* makes „ v‟ phase angle difference with V . This
V0* can be calculated using vector calculus by referring to Figure 6.1. The „ z„
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is switching time interval during which output voltage of the inverter is
constant. The switching time duration voltage space vectors V and V are „ 1‟
and „ 2‟. Duty cycles corresponding to
are calculated as
. If output voltages applied to stator of induction motor are equal
then its adjacent voltage active vectors are added.
V
V0 *
( 2/ z) i2
v
0
V
( 1 / z) V
Figure 6.1 Reference Voltage Vector
From Figure 6.1 it is known that reference voltage vector V0* can
be drawn as illustrated in Equation (6.1).
V0*=
(6.1)
⁄
(6.2)
⁄
(6.3)
⁄
87
where
Desired voltage transfer ratio = Mv =
Then,
√
0≤
V0*,
Under balanced input voltage condition,
(
(
(
where
Let us assume a condition for maximum current modulation ie.
.
Displacement angle
=
With reference to chapter 4 and Figure 4.4, vectors
can be identified
and
.
Mean value of the output voltages and dc link currents are
and
88
[
[
]
(
.
[
]
[
[
]
])
]
(6.4)
(6.5)
Equations (6.4) and (6.5) are voltage and current of voltage source
inverter in terms of duty cycle.
6.3
INVERTER FED INDUCTION MOTOR SIMULATION
This section describes the variation of frequency with respect to
motor speed. Induction motor supplied from a constant frequency source
admirably fulfills the requirements of substantially constant speed drives.
Many motor applications, however, require several speeds or even a
continuously adjustable range of speeds. The synchronous speed of an
induction motor depends on,


Changing the number of poles
Varying the line frequency
The salient features of speed control methods of induction motor
are based on the five possibilities, changing the number of poles, varying the
line frequency, varying the line voltage, varying the rotor resistance and
applying voltage of the appropriate frequency to rotor. The rotor is almost
always of the squirrel-cage type, which reacts by producing a rotor field
having the same number of poles as the induction stator field. Mechanical
torque developed is shown in Equation (6.6) (Fitzgerald 2003).
89
(6.6)
ω
where,
ω
ω and ω is the electrical excitation frequency of the
motor in rad/sec
Z1,eq
=
Thevenin‟s equivalent stator impedance, ohms
R1,eq
=
Thevenin‟s equivalent stator resistance, ohms
ɷs
=
Synchronous mechanical angular velocity, rad/sec
X2
=
Rotor Reactance, ohms
V1,eq
=
Equivalent stator voltage, volts
ɷe
=
Electrical excitation frequency of the induction
motor, rad/sec
∆ ɷm
=
ɷs
-
ɷm, is the difference between synchronous and
mechanical angular velocities of the motor, rad/sec
Stator voltage of the motor is shown in Equation (6.7) and its
equivalent impedance is shown in Equation (6.8).
̂
̂
(6.7)
(6.8)
90
To investigate the effect of changing frequency, it is assumed that
is negligible. Then Equation (6.7) can be written as Equation (6.9).
̂
eq
= ̂
(6.9)
And equivalent reactance with respect to stator side is shown in
Equation (6.10)
(6.10)
Let the subscript „г‟ represent rated frequency values of the
induction motor parameters. The electrical excitation frequency is varied.
Variation of stator frequency is illustrated in Equation (6.11).
ω
(
ω
г
(6.11)
Under V/f control, the equivalent source voltage as in
Equation (6.12) shall be written.
̂
ω
ω г
(̂
(6.12)
г
Since ̂1, eq is equal to ̂ multiplied by a reactance ratio which stays
constant with changing frequency as illustrated in Equation (6.13).
̂
ω
ω г
(̂
г
(6.13)
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Finally, the motor slip can be written
ω
ω
ω
where
ω
ω
(6.14)
ω
ω
ω
is the difference between the synchronous and
mechanical angular velocities of the induction motor. The frequency
dependence of the speed torque characteristic of an induction motor appears
only in the term R2/∆ ɷ as shown in Equation (6.15). The stator resistance R1
is negligible and electrical supply frequency of induction motor is changed.
As a result, the torque speed characteristics will simply shift along the speed
axis when ɷe is varied.
Then,
ω
(
ω
г
(
ω
(6.15)
г
Table 6.1 shows the design specifications of Induction motor for
the the matrix converter simulation.
Table 6.1 Specifications of Induction Motor Ratings for Simulation
Parameters
Ratings
Asynchronous Machine
20HP
Stator voltage
460V
Frequency
50-60Hz
Rated rpm
1760rpm
Rs(Stator Resistance)
0.1645Ω
Stator Inductance(Ls)
0.002191H
Rotor Resistance
0.1645 Ω
Rotor inductance
0.002191H
Mutual inductance
0.07614H
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The speed controller is based on the fuzzy controller that controls
the motor slip. Fuzzy controller is added to the motor speed that produces the
demanded inverter frequency. This frequency is used to generate the inverter
voltage so that the motor V/f ratio is maintained constant.
6.4
FUZZY CONTROLLER FOR VOLTAGE SOURCE
INVERTER
Over the past two decades, different types of soft computation were
used widely in VSI based electrical drives. They are,





Artificial Neural Network (ANN)
Fuzzy Logic Set (FLS)
Fuzzy Neural Network (FNN)
Genetic Algorithm Based system (GAB)
Genetic Algorithm Assisted system (GAA)
Neural networks and fuzzy logic techniques are quite different.
They have unique capabilities in information processing by specifying
mathematical relationships (Qingqing Zhou 1997).
The Fuzzification module converts the crisp values of the control
inputs into fuzzy values. A fuzzy variable has values, which are defined by
linguistic variables (fuzzy sets or subsets) such as low, medium and high.
Each variable is defined by gradually varying membership function. In fuzzy
set terminology, all the possible values can be assumed universe of discourse
(Qingqing Zhou et al). And fuzzy sets (characterized by membership
function) cover whole universe of discourse. The shape of fuzzy sets can be
triangular as we discussed in chapter 5.
The data base and the rules form the knowledge base which is used
to obtain the inference relation R. The data base contains a description of
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input and output variables using fuzzy sets. The rule base is essentially the
control strategy of the system. The proposed fuzzy based SVPWM is shown
in Figure 6.2. The speed as a feedback element is used in fuzzy controller. To
generate fuzzy Tables flux linkage current, phase angle and stator voltage are
used.
To determine a fuzzy rule from each input and output data pair, the
first step is to find the degree of each data value in every membership region
of its corresponding fuzzy domain .The variable is then assigned to the region
with the maximum degree of accuracy.
Phir Id
Flux
Iabc
Phir
Speed
Wm Teta
Iq
Iabc
Id
Teta
Iq
ABC to dq
conversion
Teta
calculation
1/z
0.98
Phir* Id*
SPACE VECTOR MODULATION
Phir*
Id*
calculation
Iabc
Iabc
Teta
Phir
Iabc*
Pulse
Iabc*
Id*
Speed
Te*
Iq*
Iq* calculation
Iq*
dq to ABC
conversion
pulses
1
Inverter
Fuzzy logic controller
I
Induction
motor
Desired speed
and torque
measurement
Figure 6.2 Proposed Fuzzy Controllers to Develop SVPWM
94
Any direct torque control scheme contains two control actions.
They are current control and speed control. The direct torque control is
subdivided into speed sensor and speed sensorless control. In this proposed
method speed sensor based direct torque control is described. Rotor speed and
three phase induction motor currents are measured. The torque is measured as
an inverse of speed. Then Id and Iq are calculated using Park transform
through Iabc. Then this vector gives the information about 600 sectors. A three
dimensional look-up-table selects the voltage vector to satisfy the torque
demands. It is detailed in Appendix 2.
6.5
FUZZY CONTROLLER FOR TORQUE MESUREMENT
Fuzzy logic based rule approach is defined using „if-then‟
statements. Fuzzy rule is stored in the fuzzy rule base. Each input and output
data is processed and rules are generated. Fuzzy rule base is based on
knowledge base. It is in the form of two dimensional tables. Fuzzy controller
utilizes speed error and change in speed error. The speed error is calculated
using reference speed and speed signal feedback. Speed error and change in
speed error are fuzzy inputs. These two inputs are normalized using two
triangular membership functions. Fuzzy output is speed of induction motor.
This is used to measure torque. This measurement involves I
*
d
. Equation
(6.16) is used to measure I d* and I q*. Equation (6.17) is used to measure Tref.
This is based on speed difference of induction motor for different loads.
(6.16)
where
T
=
Lm =
Motor torque, Nm
mutual inductance between rotor and stator magnetic
circuits, Hendry
Lr
=
rotor inductance, Hendry
95
Id *
=
direct axis current, Amperes
Iq *
=
quadrature axis current, Amperes
This speed difference can be found using fuzzy controller based on
reference speed and speed of the motor. The calculated I d* and I q* are used to
measure Vd* and Vq*. Equation (6.17) is used to measure torque and it can be
used to develop space vector pulse width modulation.
ω
where
6.6
ω
ω
(6.17)
J =
Moment of Inertia
B =
Friction coefficient
P =
Number of poles
T =
Sampling Period
FUZZY MEMBERSHIP FUNCTION ASSOCIATED WITH
SPACE VECTOR PULSE WIDTH MODULATION
Fuzzy variables associated with speed control of induction motors
are selected and normalized between -1 and 1. These fuzzy variables can be
multiplied to accommodate the desired variable by selecting suitable scale
factor. Following sets are defined to get desired speed of the Induction motor.
Measured Speed
=
{NB, NM, NS, ZE, PS, PM, PB}
Change in speed is stated by the following set,
Change in speed
=
{NB, NM, NS, ZE, PS, PM, PB}
Desired speed represented by the following sets
=
{NB, NM, NS, ZE, PS, PM, PB}
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Based on experience we have selected membership function.
Figures 6.3 to 6.6 show the membership function of input and output
variables.
Figure 6.3 Membership Function for Error Speed
Figure 6.4 Membership Function for Change in Error Speed
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Figure 6.5 Membership Function for Output Set Speed
Figure 6.6 Surface View of Rule Base to Get Set Speed
6.7
FUZZY RULE BASE DEVELOPMENT
Fuzzy rules are combined using the following connectives AND
and ALSO. Fuzzy variables are expressed by natural language. For example
NB
= Negative Big
=
Low speed level 3
NM
= Negative Medium
=
Low speed level 2
NS
= Negative Small
=
Low speed level 1
ZE
= Zero
=
desired speed
PS
= Positive Small
=
High speed level 1
98
PM
= Positive Medium
=
High speed level 2
PB
= Positive Big
=
High speed level 3
Fuzzy set operations are Union, Intersection and Complement. Let
X and Y are two fuzzy sets in U with membership function of X and Y. U is
universe of discourse. This U is a collection of fuzzy variables {}. The
membership function (XUY) of the union XUY is max {X (),Y ()}. The
membership function  (X∩Y) of the union X∩Y is min {X (), Y ()}
The membership function X is the complement of a fuzzy set X is
1 - X ().
Table 6.2 Fuzzy Rule Base for Inverter Fed Drive
Error
Change
NB
NM
NS
Z
PS
PM
PB
NB
NB
NB
NB
NB
NM
NS
Z
NM
NB
NB
NB
NM
NS
Z
PS
NS
NB
NB
NM
NS
Z
PS
PM
Error
Z
NB
NM
NS
Z
PS
PM
PB
PS
NM
NS
Z
PS
PM
PB
PB
PM
NS
Z
PS
PM
PB
PB
PB
PB
Z
PS
PM
PB
PB
PB
PB
From Table 6.2 there are 49 rules. These rules are composed in the
following manner.
If
Induction motor speed is (.) and
Change in Induction motor speed is (.) Then Desired speed is (.).
Based on that, torque will be evaluated which is used to measure Id
and I q.
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6.8
CONCLUSION
In this chapter fuzzy based SVPWM generation is proposed.
Calculation of space vector pulse width modulation duty cycle for inverter
with respect to output voltage and output current is described. Apart from that
the following discussions are made about proposed controller and results are
discussed in chapter 9.






Relationship between voltage vectors and duty cycle is
illustrated.
Inverter fed induction motor simulation details are provided.
Fuzzy controller design for VSI is illustrated.
Fuzzy controller for torque measurement is given.
Fuzzy membership is associated with SVPWM.
Fuzzy rule base is developed.