1

Chapter 1

Vector Control

1.1 Introduction

The vector oriented contol also known as Field oriented control for induction motor was first

introduced by Blaschke in early 1970s. The main objective of this control is to independently

control the torque and the flux as in separetly excied DC machines. This is done by choosing

d-q rotating frame synchromously with the rotor flux space vector. Once the orientation is

correctly achieved, the torque is controlled by the torque producing current which is the q-

component of the stator current space vector. At the same time, the flux is controlled by the

flux producing current, which is the d-component of the stator current space vector.

The idea behind vector control is to control the AC induction motor in the similar way for DC

motor control. For a permanent magnet excitation DC motor control can be achieved by

controlling its armature current. Since the torque results from the interaction of two

perpendicular magnetic fields, which are the stator field generated by the PM excitation and

the armature field, created by the armature current. Once the flux of stator field is kept

constant, the torque can be controlled by armature current.

In the vector control method, the motor model considered is valid for transient conditions.

The idea of the vector control can be understood from the reference frame theory. The

reference frame theory is one whose real axis coincides with the rotor flux vector. This frame

is not static and does not have a constant speed during transients. The great advantage of this

non inertial frame is that for stator currents, this method allows independent flux and torque

controls as in seperately excited DC machine.

Separately excited dc motor drives are simple in control because they independently control

flux, which, when maintained constant, contributes to an independent control of torque. This

is made possible with separate control of field and armature currents, which, in turn control

the field flux and the torque independently. Moreover the dc motor control requires only the

control of the field or armature current magnitudes, providing simplicity not possible with an

ac machine. In contrast, the induction motor drive requires a coordinated control of stator

2

current magnitudes, frequencies and phase magnitude making it a complex control. As with

the dc motor drives, independent control of flux and the torque is possible in ac drives.

The stator current phasor can be resolved along the rotor flux linkages and the component

along the rotor flux linkages is the field producing current, but this requires the position of

the rotor flux linkages at every instant. If this is available then the control of the ac machines

is very similar to that of the dc drives. The requirement of phase, frequency and magnitude

control of the currents and hence the flux phasor is made possible by inverter control. The

control is achieved in field coordinates and hence it is often called the field-oriented control

or the vector control, because it relates to the phasor control of the flux linkages.

1.2 Principle Of Vector Control

The vector control of induction machine is explained by assuming that the position of the

rotor flux linkages phasor r is known. The phasor diagram of the vector control is as shown

in figure 1.1. r is at f from thee stationary reference, f is referred to as field angle. The

three-stator currents can be transformed into and axised currents in the synchronous

reference frames by using the transformation.

[ ] = [ ] * [ ] (1.1)

where [ ] is given as

[ ] = 2/3 [

] (1.2)

[

] 2/3 [

] [

] (1.3)

from which the stator current phasor is derived as

(1.4)

and the stator phase angle is given by

3

(1.5)

where and are the and the and axes currents in the synchronous reference frame

that are obtained by projecting the stator current phasor on the and axes respectively. The

current phasor magnitude remains the same regardless of the reference frame chosen as is

clear from Figure 1.1.

Fig 1.1 Phasor diagram of vector controller [12]

The stator current produces the rotor flux r and the torque Te The component of current

producing the rotor flux is in phase with r. Therefore, resolving the stator current phasor

along r reveals that the component is the field-producing component as shown in Figure

1.1. The perpendicular component is hence the torque-producing component . Thus

r (1.6)

Te r (1.7)

The components and are only dc components in steady state, because the relative speed

with respect to that of the rotor field is zero. Orientation of r amounts to consider the

synchronous reference frame and hence the flux and the torque-producing components of

currents are dc quantities, and this makes them to be used as control variables. Till now it was

4

assumed that the rotor flux position is actually available, but it has to be obtained at every

instant. This field angle can be written as,

(1.8)

where r is the rotor position and sl is the slip angle. In terms of the speeds and time, the

field angle can be written as

(1.9)

Vector control schemes thus depend upon how the instantaneous rotor flux position is

obtained and are classified as direct and the indirect vector control schemes.

The rotor flux position can also be obtained by using rotor position measurement and partial

estimation with only machine parameters, such as voltages and currents. Using this field

angle is called the indirect vector control.

1.3 Derivation Of Indirect Vector Control For Induction Motor

The dynamic equations of the induction motor in the synchronous reference frame for the

rotor taking flux as state variable is given as,

(1.10)

(1.11)

where

(1.12)

(1.13)

(1.14)

The resultant rotor flux linkage, r also known as the rotor flux linkages phasor and is

assumed to be on the direct axis to achieve field orientation. This alignment reduces the

number of variables to deal.

5

The alignment of the d-axis with rotor flux phasor yields

(1.15)

(1.16)

(1.17)

Substituting Equations 1.13 to 1.16 in Equations 1.10 and 1.11 causes the new rotor equations

to be,

(1.18)

(1.19)

Thus from Equations 1.13 and 1.14 the rotor currents are derived as

(1.20)

(1.21)

From equations 1.18 and 1.19 the following equation can be obtained

(1.22)

From Equation (1.18),

(1.23)

where

(1.24)

Equation 1.22 resembles the field equation in a separately excited dc machine whose time

constant is usually in the order of seconds. Substituting the rotor currents, the torque

expression can be obtained as

( )

( ) (1.25)

6

From Equation 1.25 it can be observed that torque is proportional to the product of the rotor

flux linkages and the stator q-axis current. This resembles the air gap torque expression of the

dc motor, which is proportional to the product of the field flux linkages and the armature

current. If the rotor flux linkage is maintained constant then the torque is simply proportional

to the torque-producing component of the stator current as in the case of the separately

excited dc motor. Similar to the dc machine time constant, which is of the order of few

milliseconds, the time constant of the torque current is also of the same order.

Equations 1.22 and 1.25 complete the transformation of the induction machine into an

equivalent separately excited dc motor from control point of view. The stator current phasor

is the sum of the d and the q axes stator currents in any reference frame given as

(1.26)

and the axes to phase current relationship is obtained from Equation 5.4.

1.4 Implementation Of Indirect Vector Control

The bloch diagram shows the indirect vector control of an induction motor. The control

reference frame is fixed to the rotor flux position which is defined by the rotor mechanical

angle . The rotor position is used in the conversion from reference frame.

7

Fig 1.2 Block diagram of vector control

The induction motor is fed by the current controled voltage source inverter. The stator

currents are regulated by hysteresis cureent contoller which genertaes gate pulses for

switching devices in the inverter.

The motor torque is controlled by the quadrature axis component of the stator current . The

rotor flux is controlled by the direct axis compnent . The motor speed is regulated by a

8

control loop which produces the torque control signal . The

and current references

are converted into phase current references

for current regultors.

1.5 Vector control model

Fig 1.3 Vector control model

This is the complete model for vector contol of induction motor drive. The motor parametrs

such as stator current and rotor speed is taken as input. As we can see in the vector contol

block the output is the gate pulses which is fed to the inverter. The output voltage of inverter

is fed back to motor to from a closed loop. Reference speed is set to the desired speed which

we intend to run our motor.

Specification of motor can be given as

Line Voltage = 440 volts

Power = 50 hp

Stator Resistance= .087 ohm

Stator leakage Impedence=0.8 mH

Rotor resisatnce= 0.228 ohm

Rotor leakage Impedence= 0.8 mH

Magnetising Impedence= 34.7 mH

9

Inertia= 0.662 Kg/m2

Niumber of poles= 4

Friction Coefficent =0.01 N-m/sec

Frequency =50 Hz

Pulses generated by vector controlled is modelled as per given figure which is the model

inside subsystem named Vector Control

Fig 1.4 Subsystems of vector control model

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1.6 Expalnation of various subsytems

Fig 1.5 ABC to dq transformation

This block transforms the three pahse stator current into two phase using transformation

as discribed earlier. From the equation the formula to calculate and can be given as:

( (

) (

)) (1.27)

( (

) (

)) (1.28)

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Fig 1.6 Rotor Flux calculation

From the equation and it follows that

(

) (

) (1.29)

So after laplace transform rotor flux can be calculated as

(

) (1.30)

Fig 1.7 Theta calculation

Rotor angle calculation can be done from the equation

(1.31)

12

Where is the mechanical speed converted to electrical speed by multiplying with the

factor

.

Fig 1.8 Torque Estimation

The PI controller is used to estimate the reference torque by taking rotor speed and refeence

speed as input. The saturation block is used to limit the value of torque to a safe level.

Fig 1.9 calculation

References flux remains constant to make motor operate within its rated speed. To take motor

above the rated speed a flux weaking is required. In our project our scope is limited upto

rated speed only and hence it is taken as constant rated motor flux i.e 0.7 and the references

direct axis stator current is calculated as:

(1.32)

13

Fig 1.10 calculation

From the equation it follows that the quadrature references value can be given as :

(

)

(1.33)

Fig 1.11 transformation

The estimated reference direct and quadrature axis stator current is transformed into three

phase stator current by the inverse of transformation which can be given as :

[ ] = * [ ] (1.34)

14

Where,

[

(

) (

)

(

) (

) ]

(1.35)

So we can find the value for the stator current in three phase according to this transformation

and it comes out to be

(1.36)

(

)

(

) (1.37)

(

)

(

) (1.38)

Fig 1.12 PWM implementation

15

References current and motor actual current is compared in a hysteresis band to generate a

gate pulse. As the band is made smaller in size the resulatant wave approaches the sine

pattern. The gate pulse switches the device in inverter and the output voltage is fed back to

the motor to make a closed loop.

The DC voltage to be fed to inverter is decided by the voltage rating of motor. If is the

rated line voltage of motor then the DC voltage is given by the formula

(1.39)

In our model the is fixed to 415 volts, so the is taken as approx 750 volt.

Below are waveforms for different desired speed. The first waveform is when the reference

speed is fixed to 200 rad/sec.

The second waveform is for reference speed fixed to 170 rad/sec.

From all these waveforms we see that the motor speed is controlled according to the desired

speed given by user.

Fig 1.13 Speed waveform when reference speed set to 200 rad/sec

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Fig 1.14 Speed waveform when reference speed set to 170 rad/sec

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Chapter 2

Space Vector Pulse Width Modulation

2.1 Introduction

With advances in solid-state power electronic devices and microprocessors, various pulse-

width-modulation (PWM) techniques have been developed for industrial applications. Pulse-

width modulation (PWM) is a technique where the duty ratio of a pulsating waveform is

controlled by another input waveform. The intersections between the reference voltage

waveform and the carrier waveform give the opening and closing times of the switches.

PWM is commonly used in applications like motor speed control, converters, audio

amplifiers, etc. For example, it is used to reduce the total power delivered to a load without

losses, which normally occurs when a power source is limited by a resistive element. PWM is

used to adjust the voltage applied to the motor. Changing the duty ratio of the switches

changes the speed of the motor. The longer the pulse is closed compared to the opened

periods, the higher the power supplied to the load is. The change of state between closing

(ON) and opening (OFF) is rapid, so that the average power dissipation is very low compared

to the power being delivered. PWM amplifiers are more efficient and less bulky than linear

power amplifiers. In addition, linear amplifiers that deliver energy continuously rather than

through pulses have lower maximum power ratings than PWM amplifiers.

There is no single PWM method that is the best suited for all applications and with advances

in solid-state power electronic devices and microprocessors, various pulse-width modulation

(PWM) techniques have been developed for industrial applications. Space vector PWM

(SVPWM) is also one of these techniques.

In the SVPWM technique, the duty cycles are computed. The SVPWM technique can

increase the fundamental component by up to 27.39% that of SPWM. The fundamental

voltage can be increased up to a square wave mode where a modulation index of unity is

reached. SVPWM is accomplished by rotating a reference vector around the state diagram,

which is composed of six basic non-zero vectors forming a hexagon. A circle can be

inscribed inside the state map and corresponds to sinusoidal operation.

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2.2 Implementation Procedure of a Two-Level Space Vector PWM

Fig 2.1 Flow diagram of SVPWM implementation

A simplified flow diagram for the implementation of the SVPWM algorithm is shown in the

Fig 2.1. The procedure for implementing a two-level space vector PWM is as follows:

1. Calculate the angle and reference voltage vector V ref based on the input voltage

components.

2. Find the sector in which V ref lies, and the adjacent space vectors of V k and V k+1 based on

the sector angle .

3. Find the time intervals Ta and Tb and T0 based on Ts, and the angle .

4. Determine the modulation times for the different switching states.

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2.2.1Angle and Reference Voltage Vector

Voltage vector a b c V V Vector

V0 0 0 0 0 0 0

V1 1 0 0

0 V0

V2 1 1 0

V60

V3 0 1 0

V120

V4 0 1 1

0 V180

V5 0 0 1

V240

V6 1 0 1

V300

V7 1 1 1 0 0 0

Table 2.1 Voltage Vectors, Switching Vector, and

In the Space Vector PWM, the three-phase output voltage vector is represented by a reference

vector that rotates at an angular speed of = 2f. The Space Vector PWM uses the

combinations of switching states to approximate the reference vector V ref . A reference

voltage vector V ref that rotates with angular speed in the plane represents three

sinusoidal waveforms with angular frequency in the coordinate system. Each output

voltage combination in Table 2.1 corresponds to a different voltage space vector. Three

sinusoidal and balanced voltages are given by the relations:

(2.1)

(

) (2.2)

(

) (2.3)

For any three-phase system with three wires and equal load impedances, we have

(2.4)

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The space vector with magnitude Vref rotates in a circular direction at an angular velocity of

where the direction of rotation depends on the phase sequence of the voltages. If it has a

positive phase sequence, then it rotates in the counter clockwise direction. Otherwise, in the

clockwise direction with a negative phase sequence. The three-phase voltages could be

described as, and , in a two-dimensional plane. The magnitude of each active vector is

. The active vectors are 60

o apart and describe a hexagon boundary. The locus of the

circle projected by the space reference vector Vref depends on V 0, V 1, V 2, V 3, V 4, V 5, V 6,

V 7

The magnitude of the reference vector is:

| V re f | = (V2

+ V2

) (2.5)

The phase angle is evaluated from

=

(2.6)

Where [0,2]

2.2.2 Sector Determination:

It is necessary to know in which sector the reference output lies in order to determine the

switching time and sequence. The phase voltages correspond to eight switching states: six

non-zero vectors and two zero vectors at the origin. Depending on the reference voltages V

and V , the angle of the reference vector can be used to determine the sector as per Table 2.2.

Table 2.2 Sector Determination

Sector Degrees

1 0 < 60

2 60 < 120

3 120 < 180

4 180 < 240

5 240< 300

6 300 < 360

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2.2.3 Time Durations Ta, Tb, T0

The duty cycle computation is done for each triangular sector formed by two state vectors.

The magnitude of each switching state vector is

and the magnitude of a vector to the

midpoint of the hexagon line from one vertex to another is

. The reference space vector

rotates and moves through different sectors of the complex plane as time increases. In each

PWM cycle, the reference vector Vref is sampled at a fixed input sampling frequency fs.

During this time, the sector is determined and the modulation vector Vref is mapped onto two

adjacent vectors. The non-zero vectors can be represented by

(2.7)

for k =1, 2, 3, 4, 5, 6

Therefore, the non-zero vectors for V k and V k+1 become

*

+ (2.8)

*

+ (2.9)

Due to symmetry in the patterns in the six sectors, the integration

(2.10)

can be carried out for only half of the pulse-width modulation period

. Zero voltages are

applied during the null state times:

(2.11)

Now,

(2.12)

Thus, the product of the reference voltage vector V ref and Ts/2 equals the sum of the voltage

multiplied by the time interval of the chosen space vectors. The reference voltage vector V ref

can be expressed as function of Vk and Vk+1 as

22

(2.13)

V ref = V + j V (2.14)

Where Ta and Tb denote the required on-time of the active-state vectors Vk and V k+1 during

each sample period and k is the sector number denoting the reference location. The calculated

times Ta and Tb are applied to the switches to produce space vector PWM switching patterns

based on each sector. The switching time is arranged according to the first half of the

switching period while the other half is a reflection forming a symmetrical pattern. T0 and T7

are the times of the null state vectors.

If Vref lies exactly in the middle between two vectors, for example between V1 and V2 with an

angle of , Ta for V1 will be equal to Tb for V2. If V ref is closer to V2 than V1, it means that

Tb will be greater than Ta. If Vref coincides with V2, then Ta will be equal to zero. If the

reference keeps making a circular trajectory inside the hexagon, then T0 is greater than zero,

the output voltage will be a sinusoidal waveform in the under-modulation region.

Assuming that the reference voltage and the voltage vectors Vk and Vk+1 are constant during

each pulse-width modulation period Ts and splitting the reference voltage Vref into its real and

imaginary components (V and V) gives the following result:

[

]

( *

+ *

+)

Or

[

]

*

+ [

] (2.15)

These equations require computations involving trigonometric functions. Ta and Tb can be

calculated as:

23

[

]

*

+ [

] (2.16)

2.3 Modelling of SVPWM

Fig 2.2 Subsystem of SVPWM

This is complete model for SVPWM in an open loop.

Fig 2.3 Reference Vector calculation

24

Three phase voltage is converted to coordinate by applying following formula:

(

) (2.17)

(

) (2.18)

Fig 2.4 Sector identification

The value of is calculated by the equations 1.16 and the is given by

(2.19)

25

Fig 2.5 Calculation of , , .

26

Conclusion

It has been shown vector control techniques can be applied successfully to standard squirrel

cage induction motors. The performance is superior to that obtained with thyristor D.C drives

and these motors are also becoming a more economical approach to variable speed. The

verification of the proposed control is performed by simulation results at the different speed.

The SVPWM is used for controlling the switching of the machine side converter. Advantages

of this method include a higher modulation index, lower switching losses, and less harmonic

distortion. However, the SVPWM technique is complex in implementation.

27

References

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