CHAPTER 1

INTRODUCTION

1

Introduction

1.1 Review:

Power electronics deals with the solid state power semiconductor devices

for control of electric power. It is a branch of electrical engineering that is concerned

with the conversion and control of electrical power for various applications such as

industrial, commercial, residential, and aerospace environments. Of all the modern

power electronics converters, the voltage source inverters (VSI) is the simplest and

most widely used device with power ratings ranging from fractions of kilowatt to

megawatt level. It converts fixed DC voltage to AC voltage with controllable

frequency and magnitude. These are extensively used in motor drives, active filters,

and unified power flow controllers in power systems and uninterrupted power

supplies.

1.2 Inverters:

A device that converts dc power into ac power at desired output voltage and

frequency is called an inverter. Some industrial applications of inverters are for

adjustable speed ac drives, induction heating, stand by aircraft power supplies,

uninterruptible power supplies for computers, HVDC transmission lines etc. The dc

power input to the inverter is obtained from an existing power supply network or from

a rotating alternator through rectifier or a battery, fuel cell, photovoltaic array or

magneto hydrodynamic (MHD) generator. The configuration of ac-to-dc and dc-to-ac

inverter is called a dc-link converter. The rectification is carried out by standards

diodes or thyristor converter circuits.

As Inverters are employed to get a variable frequency supply from a dc

supply, stepped-wave inverters of figure.1.1 can be designed to behave as voltage

source or current source. Accordingly they are known as voltage source or current

source inverters. For the control of AC Motor, voltage or current should also be

controlled along with frequency. Variation in output voltage or current can be

achieved by varying the input dc voltage. This is achieved either by interposing a

chopper in between fixed voltage dc source and the inverter or the inverter may be fed

from an ac-dc converter. Output voltage and current have stepped waveform,

consequently they have substantial amount of harmonics. Variable frequency and

variable voltage ac is directly obtained from fixed voltage dc when the inverter is

controlled by Pulse Width Modulation (PWM) (figure.1.2). The Pulse Width

2

Modulation control also reduces harmonics in the output voltage. Inverters are built

using semiconductor devices such as thyristors, power transistors, IGBTs, GTOs, and

power MOSFETs. They are controlled by firing pulses obtained from a low power

control unit.

Figure: 1.1 Stepped wave Inverters

Figure: 1.2 Pulse Width Modulated Inverters

1.3 Pulse Width Modulated Inverters:

Pulse Width Modulated inverters are gradually taking over other types of

inverters in industrial applications. Pulse Width Modulation techniques are

characterized by constant amplitude pulses. The width of these pulses is, however,

modulated to obtain inverter output voltage control and to reduce its harmonic

content.

Different Pulse Width Modulation techniques are as under:

a) Single-pulse modulation

b) Multiple- pulse modulation

c) Sinusoidal- pulse modulation

In Pulse Width Modulation inverters, forced commutation is essential. The

three Pulse Width Modulation techniques listed above differ from each other in the

harmonic content in their respective output voltages. It means the choice of a

particular Pulse Width Modulation technique depends on the permissible harmonic

content in the inverter output voltage.

3

Stepped-waveSemiconductor

Inverter

Variable frequencyFixed

voltage/current ac

FixedVoltage dc

PWMSemiconductor

Inverter

Variable frequencyFixed

voltage/current ac

FixedVoltage dc

1.4 Sinusoidal Pulse Width Modulation:

The objective of sinusoidal PWM is to synthesize the motor currents as

near to a sinusoid as economically possible. The lower voltage harmonics can be

greatly attenuated leaving typically only two or four harmonics of substantial

amplitude close to the chopping or carrier frequency. The motor now tends to rotate

much more smoothly at low speed. Torque pulsations are virtually eliminated and the

extra motor losses caused by the inverter are substantially reduced. To counter

balanced these advantages the inverter control is complex, the chopping frequency is

high (typically 500-2500Hz for GTOs and up to 5000 or more for BJT transistors),

and inverter losses are higher than for the six-step mode of operation. To approximate

a sine wave, a high frequency triangular wave is compared with fundamental

frequency sine wave as shown in figure.1.3 when the low frequency sine waves are

used with 120 phase displacement; switching pattern for the six inverter devices

ensues.

Figure1.3 Illustration of sinusoidal Pulse Width Modulation

4

1.5 Organization Of Thesis

This thesis is organized as follows:

Chapter 1 Introduces the review of power electronics, introduction of inverters and

types of Pulse Width Modulated inverters.

Chapter 2 Introduces the classifications of Inverters, analysis of Induction Motor

Drives, analysis of Pulse Width Modulation and its control.

Chapter 3 Describes the technique of Space Vector Modulation, Gating signal

generation in Space Vector Modulated PWM scheme and relationship

between the Inverter leg Switching timings and the Space Vector Switching

timings.

Chapter 4 Modelling of induction motor

Chapter 5 Describes the results & discussion

Chapter 6 Describes the conclusion

5

CHAPTER 2

PULSE WIDTH

MODULATION

6

2.1Pulse Width Modulation Algorithm:

Pulse width modulation (PWM) refers to a method of carrying

information on attain of pulses, the information being encoded in the width of the

pulses, in applications to motion control, it is not exactly information we are

encoding, but a method of controlling power in motors without(significant) loss.

There are several schemes to accomplish this technique. One is to switch voltage on

and off, and let the current reticulate through diodes when the transistors have

switched off. Another technique is to switch voltage voltage polarity back and forth

with a full bridge switch arrangement, with 4 transistors. This technique may have

better linearity, since it can go right down to an effective 0% duty cycle by having the

positive and negative voltage periods precisely equal on/off techniques may have

trouble going down extremely close to 0% duty cycles, and may jitter between

minimum duty cycles of positive and negative polarity.

By using PWM algorithm, we can control the inverter output

voltage. This is done by exercising the control with in the inverter it self. By

adjusting the on and off periods of inverter, we get AC output voltage by giving a

fixed DC input voltage. In variable speed AC 2 motors, the AC output voltage from a

constant DC voltage can be obtained by using inverter. Mainly AC voltage is

dependent on two parameters, amplitude and frequency. It is essential to control the

above two parameters. In PWM techniques in order to generate the gating signals, we

compare the reference signal amplitude (Ar) with the carrier signal amplitude (Ac).

The fundamental frequency of the output voltage is determined by the reference signal

frequency. The inverter output voltage is determined by the following way when

Ar>Ac, the pole voltage is +Vdc/2. When Ar<Ac, the pole voltage is -Vdc/2. The

ratio of Ar to Ac is called modulation index. The pulse width can be varied from 0 to

180(degrees) by varying Ar from 0 to Ac.

Of the variety of modulation methods, the carrier based PWM

method is very popular due to its simplicity of implementation, known harmonic

waveform characteristics, low harmonic distortion, high switching frequency and

offer high waveform quality. The first important contribution in the carrier based

PWM area was done by schonung and stemmler in1964 with the development of the

sinusoidal PWM (SPWM) method. In this method the sinusoidal reference waveform

7

of each phase and a periodic triangular carrier wave are \compared and the

intersection points determine the commutation instants of the associated inverter leg

switches.however, this method has a poor voltage linearity range, which is at most

78.5% of the six step voltage fundamental component value, hence poor voltage

utilization.

2.2 Space Vector PWM method:

Space vector pulse width modulation is a relatively new and

popular technique in controlling motor drives. This technique is mainly based on

approximating the reference voltage instantaneously by combining the switching

states corresponding to the basis space vectors. The advantage of this technique is it

will generate less harmonics distortion in the output voltages and currents.

In most three phases AC motor drive and utility interface

applications the neutral points are isolated and no neutral current path exists. In such

applications in the triangle intersection implementations any zero sequence signal can

be injected to the reference modulation waves. This zero-sequence waveform is used

to alter duty cycle of the inverter switches. K.G.King was the first researcher to utilize

this concept in a voltage source inverter. King realized that a three phase diode

rectifier circuit could be utilized to generate a 3 zero sequence signal. His choice of

scale, which was based on the linearity range from 78.5% to 90.7% of the six-step

voltage. This modulation method was later re-invented employing the space vector

thery.hence the method was termed “Space Vector PWM” (SVPWM). The degree of

freedom of the equal division of zero voltage vector times within a sampling period or

sub cycle is used in the space vector modulation methodology.

2.3 Analysis Of Induction Motor Fed From Non-Sinusoidal Voltage

Supply:

When motor fed from an inverter the motor terminal voltage is non-sinusoidal

but it has half-wave symmetry. A non-sinusoidal waveform can be resolved into

fundamental and its harmonic components using Fourier analysis. Because of half-

wave symmetry only odd harmonics will be present. The harmonics can be divided

into positive sequence, negative sequence and zero sequence. The harmonics, which

have the same phase sequence as that of the fundamental are known as positive

sequence harmonics. The harmonics having phase sequence opposite to fundamental

8

are called negative sequence harmonics. The harmonics, which have all the three

phase voltages in phase, are called zero sequence harmonics.

Consider the fundamental phase voltage components with the phase sequence ABC.

VAN = V1 Sin ωt

VBN = V1 Sin (ωt-2π/3)

VCN = V1 Sin (ωt-4π/3) ………………… (2.1)

The corresponding 5th and 7th harmonic voltages are:

VAN = V5 Sin 5ωt

VBN = V5 Sin 5(ωt-2π/3) = V5 Sin (5ωt-4π/3)

VCN = V5 Sin 5(ωt-4π/3) = V5 Sin (5ωt-2π/3 ) ……………… (2.2)

and

VAN = V7 Sin 7ωt

VBN = V7 Sin 7(ωt-2π/3) = V5 Sin (7ωt-2π/3)

VCN = V7 Sin 7(ωt-4π/3) = V5 Sin (7ωt-4π/3) ……… (2.3)

From the equations mentioned above it is clear that seventh harmonic

(equation 2.3) has the phase sequence ABC, which is the same as that of fundamental.

Hence it is a positive sequence harmonic. The fifth harmonic (equation 2.2) has the

phase sequence ACB, which is opposite to the fundamental. Hence it is a negative

sequence harmonic.

It can be concluded as that the harmonic voltages and currents of the order n =

6k+1 (where k is an integer) are of positive sequence and the harmonic voltages and

currents of the order n = 6k-1 are of negative sequence. Similarly the harmonics of

the order n = 3k are of zero sequence. A positive sequence harmonic n will produce a

rotating field, which moves in the same direction as the fundamental at a speed n

times that of the fundamental field. Similarly rotating field produced by a negative

sequence harmonic m will move in the direction opposite to the fundamental at m

times its speed. Zero sequence components do not produce a rotating field.

9

2.4 Classification Of Inverters:

Inverters can be broadly classified into two types:

a) Voltage Source Inverter (VSI)

b) Current Source Inverter (CSI)

Voltage Source Inverter (VSI) is one in which the dc source has small or

negligible impedance or we can say, it has stiff dc voltage source at its input

terminals. A Current Source Inverter (CSI) is fed with adjustable current from a dc

source of high impedance, i.e. from a stiff dc current source. In a CSI fed with

stiff current source, output current waves are not affected by the load.

2.5 VSI Induction Motor Drives :

Voltage source inverter allows a variable frequency supply to be obtained

from a dc supply. Figure 2.1 shows VSI employing transistors. Any other self

commutated device can be used instead of a transistor. Generally MOSFET is used in

low voltage and low power inverters, IGBT and power transistors are used up to

medium power levels and GTO is used for high power levels.

Figure 2.1 Transistor Inverter-fed Induction Motor Drive

VSI can be operated as a stepped wave inverter or a pulse width modulated

inverter. When operated as a stepped wave inverter, transistors are switched in the

sequence of their numbers with a time difference of T/6 and each transistor is kept on

for the duration T/2, where T is the time period for one cycle. Frequency of inverter

operation is varied by varying T and the output voltage of the inverter is varied by

varying dc input voltage.

10

Figure 2.2(a) VSI controlled IM drive using chopper

Figure 2.2(b) VSI controlled IM drive using controlled rectifier

Figure 2.2(c) VSI controlled IM drive using dc supply

Figure 2.2(d) VSI controlled IM drive using ac supply

When supply is dc, variable dc input voltage is obtained by connecting a

chopper between dc supply and inverter (figure 2.1(a)). When supply is ac, variable

dc input is obtained by connecting a controlled rectifier between ac supply and

11

inverter (figure 2.1(b)). A large electrolytic filter capacitor C is connected in dc link to

make inverter operation independent of rectifier or chopper and to filter out harmonics

in dc link voltage.

The rms value of fundamental voltage is given by

V = (√2 / π )Vd ………………… (2.4)

The torque for given speed can be calculated by considering only fundamental

component. Consequently, an induction motor Drive fed from a stepped wave inverter

suffers from the following drawbacks:

a) Because of low frequency harmonics, the motor losses are increased at all

speeds causing derating of the motor.

b) Motor develops pulsating torques due to fifth, seventh, eleventh and thirteenth

harmonics that cause jerky motion of the rotor at low speeds.

c) Harmonic content in motor current increases at low speeds. The machine

saturates at light loads at low speeds due to high (v/f) ratio. These two effects

overheat the machine at low speeds, thus limiting lowest speed to around 40%

of base speed.

Harmonics are reduced, low frequency harmonics are eliminated, associated

losses are reduced and smooth motion is obtained at low speeds also when inverter is

operated as a pulse width modulated inverter. Since output voltage can be controlled

by pulse width modulation no arrangement is required for the variation of input dc

voltage, hence inverter can be directly connected when the supply is dc (figure 2.2(c))

and through a diode rectifier when a supply is ac (figure 2.2(d)).

The fundamental component in the output phase voltage of a PWM inverter

operating with sinusoidal PWM is given by

V = m (Vd / 2√2) ………………… (2.5)

where m is modulation index.

The harmonics in the motor current produced torque pulsation and derate the

motor. For a given harmonic content in motor terminal voltage, the current harmonics

are reduced when the motor has higher leakage inductance; this reduces derating and

torque pulsations. Therefore, when fed from VSI, induction motors with large

(compared to when fed from sinusoidal supply) leakage inductance is used.

12

2.6 Pulse Width Modulation Control:

Output voltage from an inverter can also be adjusted by exercising a control

within the inverter itself. The most efficient method of doing this is by Pulse Width

Modulation control used within an inverter. In this method, a fixed dc input voltage is

given to the inverter and a controlled ac output voltage is obtained by adjusting the on

and off periods of the inverter components. This is the most popular method of

controlling the output voltage and this method is termed as pulse-width modulation

(PWM) control.

The advantages of Pulse Width Modulation technique are as under:

i. The output voltage control with this method can be obtained without any

additional components.

ii. With this method, lower order harmonics can be eliminated or minimized along

with its output voltage control. As higher order harmonics can be filtered easily,

the filtering requirements are minimized.

Disadvantage:

i. The main disadvantage of this method is that the SCRs are expensive as they

must possess low turn-on and turn-off times.

2.7 Applications Of Pulse Width Modulations:

a) PWM can be used to reduce the total amount of power delivered to a load

without losses normally incurred when a power source is limited by resistive

element. This is because the average power delivered is proportional to the

modulation duty cycle.

b) High frequency PWM power control systems are easily realizable with

semiconductor switches. The discrete on/off states of the modulation are used

to control the state of the switches which correspondingly control the voltage

across or the current through the load.

c) PWM is also often used to control the supply of electrical power to another

device such as in speed control of electric motors, volume control of Class D

audio amplifiers or brightness control of light sources and many other power

electronics applications.

13

d) PWM is also used in efficient voltage regulators. By switching voltage to the

load with the appropriate duty cycle, the output will approximate a voltage at

the desired level. The switching noise is usually filtered with an inductor and a

capacitor.

e) PWM is sometimes used in audio effects and amplification like sound

synthesis, in particular subtractive synthesis as it gives sound effect similar to

chorus or slightly detuned oscillator played together. The ratio between the

high and low level is typically modulated with low frequency oscillator.

2.8 Types Of Modulations:

2.8.1 Linear Modulation:

The simplest modulation to interpret is where the average ON time of the

pulses varies proportionally with the modulating signal. The advantage of linear

processing for this application lies in the ease of de-modulation. The modulating

signal can be recovered from the PWM by low pass filtering. For a single low

frequency sine wave as modulating signal modulating the width of a fixed frequency

(fs) pulse train the spectra is as shown in Figure 2.3 Clearly a low pass filter can

extract the modulating component fm. 

Figure 2.3 Spectra of PWM

14

2.8.2 Saw Tooth PWM:

The simplest analog form of generating fixed frequency PWM is by

comparison with a linear slope waveform such as a sawtooth. As seen in Figure 2.4

the output signal goes high when the sine wave is higher than the sawtooth. This is

implemented using a comparator whose output voltage goes to logic HIGH when the

input is greater than the other. 

Figure 2.4 Sine Saw tooth PWM

Other signals with straight edges can be used for modulation a rising ramp carrier will

generate PWM with Trailing Edge Modulation. 

Figure 2.5 Trailing Edge Modulation

It is easier to have an integrator with a reset to generate the ramp in Figure 2.5 but the

modulation is inferior to double edge modulation.

15

2.8.3 Regular Sampled PWM:

The scheme illustrated above generates a switching edge at the instant of crossing of

the sine wave and the triangle. This is an easy scheme to implement using analog

electronics but suffers the imprecision and drift of all analog computation as well as

having difficulties of generating multiple edges when the signal has even a small

added noise. Many modulators are now implemented digitally but there is difficulty is

computing the precise intercept of the modulating wave and the carrier. Regular

sampled PWM makes the width of the pulse proportional to the value of the

modulating signal at the beginning of the carrier period. In Figure 2.6 the intercept of

the sample values with the triangle determine the edges of the Pulses. For a saw tooth

wave of frequency fs the samples are at 2fs.

Figure 2.6 Regular Sampled PWM

There are many ways to generate a Pulse Width Modulated signal other than

fixed frequency sine saw tooth. For three phase systems the modulation of a Voltage

Source Inverter can generate a PWM signal for each phase leg by comparison of the

desired output voltage waveform for each phase with the same saw tooth. One

alternative which is easier to implement in a computer and gives a larger

MODULATION DEPTH is using SPACE VECTOR MODULATION.

 

16

CHAPTER 3

SPACE VECTOR

MODULATION

17

3.1 Introduction:

Power Electronics has evolved as a major branch in Electrical Engineering

over the past three decades. Power Electronics is primarily concerned with efficient

and optimal use of the resources of electrical energy. It offers intelligent and tangible

solutions to achieve the objectives of flexible and efficient control of various

electrical apparatus and systems.

Inverters form an important class of power electronic circuits, which convert

DC power to AC power. With the advent of power semiconductor technology,

modern power devices such as BJTs, MOSFETs and IGBTs replaced SCRs at low and

medium power level, as these devices do not require the complex commutation

circuitry to turn off them. Of these devices, IGBTs have aroused a particular interest

in recent times as these devices inherit the simplicity of control from MOSFETs and

superior conduction characteristics from the BJTs.

A classical sinusoidal modulation limits the phase duty cycle signal to the

inner circle. The space vector modulation schemes extend this limit to the hexagon by

injecting the signal third harmonic. The result is about 10% (2/1.73 x 100%) higher

phase voltage signal at the inverter output. The PWM modulation chops alternatively

two adjacent phase voltage and zero voltage signals in a certain pattern producing the

switching impulses for the inverter Sa, Sb and Sc. Various SVM modulation schemes

have been proposed in literature and some recent analyzes show that there is a trade-

off between the switching loses and the harmonic content, so-called THD, produced

by the SVM modulation.

3.2 Two-Level Inverters And Modulation Schemes:

Inverters built with the power devices have become very popular and were

accepted by the industry owing to their simplicity and ruggedness. With the

advancements in the Pulse Width Modulated (PWM) control schemes, the harmonic

spectrum of the output voltage can be maneuver to contain a pronounced fundamental

component and to transfer the harmonic energy to the components of higher

frequency. This is desirable, as it is relatively easier to filter out the components of

18

higher frequency compared to the components of the lower frequency. A typical two-

level inverter is shown in Figure3.1.

Figure 3.1 Conventional Induction motor drive using a two-level inverter

Sinusoidal Pulse Width Modulation (SPWM) is one of the most popular

schemes devised for the control of a two-level inverter. In SPWM, a modulating sine

wave corresponding to the fundamental frequency of the output voltage is compared

with a triangular carrier wave of high frequency, which corresponds to the switching

frequency of the devices. Each leg of the two-level inverter is controlled by the

corresponding modulating wave. The modulating waves for the individual legs are

displaced by 1200 with respect to each other as shown in the top trace of Figure 3.2

Figure3.2 Modulating and carrier signals in SPWM for a two-level inverter (Top)

and pole voltage (bottom) showing two levels

19

Vdc

Time

0

Thus, the inverter employed in the system shown in Figure2.1 is a two-level

inverter because any pole voltage e.g. VAO assumes one of the two possible values

namely 0 (when S4 is turned on) or Vdc (when S1 is turned on) as shown in Figure3.2

The ratio of the peak value of the modulating signal and the peak value of the

carrier signal is defined as the amplitude modulation ratio (also called modulation

index) and is denoted as ma. The ratio of the frequencies of the carrier wave and the

modulating wave is defined as the frequency modulation ratio and is denoted as mf.

In the range of linear modulation, 0 < <1. For the situation depicted in Figure.3.2,

= 0.8 and = 15.

The pole voltage waveform contains significant amount of common mode

voltage. The common mode voltages, also called the zero-sequence voltage, are

comprised of the triplen harmonic components in the pole voltages. In the circuit

depicted in Figure.3.1, these are dropped between the points ‘O’ and ‘N’.

Consequently the load phase voltages do not possess the zero sequence voltage.

The waveform of output voltage (the motor phase voltage waveform ) and

the motor phase current when the inverter is operated in the range of linear-

modulation are shown in Figure.3.3.

Figure.3.3 Typical waveforms of phase voltage (Top) and phase current

(Bottom) of a two-level inverter in the range of linear modulation

20

A two-level inverter is capable of being operated in the six-step mode in which the

inverter displays the maximum voltage capability. Typical phase voltage and phase

current when a two-level inverter is operated in this mode is shown in Figure.3.4. The

typical harmonic spectra of the phase voltage when the inverter is operated in the

range of linear modulation and in the six-step mode are shown in Figure.3.5.

Figure.3.4 Typical waveforms of phase voltage (Top) and phase current (Bottom) of a

two-level inverter in the six-step mode of operation

Figure.3.5 Typical normalized harmonic spectra of the phase voltage when the

inverter is operated in the range of linear modulation with m a = 0.8 and m f =

15 (Top) and in the six-step mode (Bottom)

21

From the harmonic spectra presented in Figure.3.5, it is clear that in the range

of linear- modulation, the predominant harmonics are pushed to the order of the

switching frequency. In the six-step mode of operation, the harmonic order is given

by (n = 1, 2, 3…).

These spectra explain as to why SPWM control for two-level inverters has

become popular. In the range of linear-modulation, not only a smooth control over the

fundamental component of the output voltage is obtained, but also the harmonic

spectrum is acceptable. By increasing the switching frequency, one may push

significant harmonics further up. However, for high power applications, this is not

attempted as the switching losses in the power semiconductor devices also increase.

Another possibility of reducing harmonic distortion is to eliminate certain

specific harmonics. This modulation scheme is known as the Selective Harmonic

Elimination (SHE). It is possible to suppress one harmonic component by each

commutation per quarter period. Each commutation notch per quarter period provides

one degree of freedom. With an appropriate selection of the degrees of freedom, it

is possible to control the fundamental component and to eliminate harmonics.

The advantage with this scheme is that, it allows the undesirable lower order

harmonics to be eliminated, without making the switching frequency very high.

However, this scheme involves the numerical solution of nonlinear equations and is

difficult to implement for a large value of .

Another popular modulation scheme for the control of two-level inverters is the Space

Vector Modulation.

The space vector constituted by the pole voltages , and is

defined as:

………….

(3.1)

The relationship between the phase voltages, VAN, VBN, VCN and the pole

voltages, VAO, VBO and VCO is given by:

VAO = VA N + VN O;

VBO = VB N + VN O;

VCO = VCN + VN O; …………(3.2)

22

Since VAN + VBN + VCN = 0,

VNO = (VAO + VBO + VCO) / 3 ……….. (3.3)

Where, is the common mode voltage.

From equation.3.1 and equationn.3.2 it is clear that the phase voltages VAN, VBN

and VCN also result in the same space vector VS.

The space vector VS can also be resolved into two rectangular components namely Vd

and Vq. It is customary to place the -axis along the A-phase axis of the induction

motor. Hence:

VS = Vd + Vq …….………….. (3.4)

The relationship between (Vd, Vq) and the instantaneous phase voltages

(VAN,, VBN,, VCN) is given by the conventional ABC- transformation as below:

…………… (3.5)

Each pole in a two-level inverter can independently assume two values namely

0 and . Therefore, the total number of states a two-level inverter can assume is 8

(i.e. 23).

These states are graphically illustrated in Figure.3.6a through Figure.3.6h. In

these diagrams, the symbols ‘1’ and ‘0’ respectively indicate that the top switch and

the bottom switch in a given phase leg are turned on. In any given phase leg, the top

switch and the bottom switch are turned on complimentarily.

The switching states from 1 to 6 are known as “Active” states and the states 7

and 8 are known as “Null” or “Zero” or “Passive” state.

23

TABLE 3.1:

Switching States Of Two-

Level Inverter In All Sectors

The following example illustrates the method of determination of the space vector

location for a given state. When the inverter assumes a state of ‘2’ (+ + -) as shown in

Figure. 3.6c, then the pole voltages are:

VAO = (Vdc/2); VBO = (Vdc/2); VC0 = - (Vdc/2) …………….. (3.6)

Hence the space vector for this state is given by from eqn.1.1,

Vs = (Vdc/2) + (Vdc/2). exp [j (2π/3)] - (Vdc/2). exp [j (4π/3)]

= (Vdc/2) + (Vdc/2). [-(1/2) + j (√3/2)] - (Vdc/2). [-(1/2) - j √3/2)]

= Vdc. [(1/2) + j (√3/2)]

= Vdc at 600 …………………….. (3.7)

The space vector locations for the rest of the states may similarly be evaluated.

Figure.3.6a State 8 (0 0 0)

24

Switching

StateSequence Result

1 (1 0 0 ) Vdc at 00

2 (1 1 0 ) Vdc at 600

3 (0 1 0) Vdc at 1200

4 (0 1 1) Vdc at 1800

5 (0 0 1) Vdc at 2400

6 (1 0 1) Vdc at 3000

7 (1 1 1) 0

8 (0 0 0) 0

Figure.3.6b State 1 (1 0 0)

Figure.3.6c State 2 (1 1 0)

Figure.3.6d State 3 (0 1 0)

Figure.3.6e State 4 (0 1 1 )

25

Figure.3.6f State 5 (0 0 1)

Figure.3.6g State 6 (1 0 1)

Figure.3.6h State 7 (1 1 1)

26

The space vector locations for a two-level inverter form the vertices of a regular

hexagon, forming 6 sectors as shown in Figure.3.7. For the states 8 (0 0 0) and 7(1 1

1) the motor phases are short-circuited and therefore are not connected to the source.

These states are called the zero states or null states during which there is no power

flow from the source to the motor. Hence, by controlling the duration of these zero

state intervals, we can control the output voltage magnitude. It is worth noting that in

six-state mode of operation, such intervals of zero state switching do not exist.

Consequently, the output voltage magnitude in an inverter operating in a square wave

mode must be controlled by controlling the input DC link voltage. The rest of the

vectors 1 through 6 are called the active vectors.

3.3 Space Vector locations in a Hexagon

Figure.3.7 Space vector locations for a two-level inverter

The vector OT’ in Figure.3.7 represents the reference voltage space vector

corresponding to the desired value of the fundamental components for the output

phase voltages. It is obtained by substituting the instantaneous values of the reference

phase voltages, sampled at regular time intervals, in eqution.3.1. It may be noted that

there is no direct way to generate the sample. It can be reproduced in the average

sense by switching amongst the inverter states situated at the vertices, which are in the

closest proximity to it. For the situation depicted in Figure.3.7, the sample can be

realized by switching among the inverter states situated at the vertices O, A and B

following a certain sequence. The vectors OT’, AT’ and BT’ respectively denote the

deviation of the sample when the inverter states situated at O, A, and B are switched

to construct the sample in the average sense. Therefore, one may conclude that the

24

realization of the sample in the average sense produces switching ripple

corresponding to these vectors (OT’, AT’ and BT’).

By decreasing the sampling time interval (it can be achieved by increasing the

sampling frequency), we can closely track the reference vector OT’. But a higher

sampling frequency results in a higher switching frequency of the power devices.

Therefore, high sampling frequencies are generally abstained in high power

applications.

The symbols and respectively denote the time periods for which the

active vector along the leading edge and the lagging edge are switched for the

realization of the reference voltage space vector in a given sampling time period. The

sampling time period is denoted by the symbol Ts.

It can be shown that

………. (3.8)

In equation.3.8, denotes the amplitude of the reference vector and ‘ ’

represents the position of the reference vector with respect to the beginning of the

sector in which the tip of the reference vector is situated.

In order to minimize the switching of the power semiconductor devices in the

inverter, it is desirable that switching should take place in one phase of the

inverter only for a transition from one state to another. For the situation depicted

in Figure.3.7 (i.e. when the sample is situated in sector-1), this objective is met if

the switching sequence (8-1-2-7-2-1-8-1-2-7…) is used. Therefore, the zero-

interval is divided into two equal halves of length . These half-intervals

are placed at the beginning and end of every sampling interval . If the half at the

beginning is realized with the state ‘8’(0 0 0), then the state at the end is realized

with the state ‘7’ (1 1 1) and vice-versa. Figure.3.8 depicts a typical switching

sequence when the sample is situated in sector-1 (Figure.3.7). By extending this

procedure, the gating signals can be generated when the sample is situated in the

other sectors.

25

Figure.3.8 Gating signal generation in Space Vector Modulated PWM

scheme

In Figure.3.8, the symbols , and respectively denote the time duration

for which the top switch in each phase leg is turned on. It can be seen from

Figure.3.8 that the chopping frequency of each phase of the inverter is equal to

half of the sampling frequency. Table-3.2 depicts the switching sequence for all

the sectors.

Inner sector

number

On-sequence in an

equivalent single inverter

drive

Off-sequence in an equivalent

single inverter drive

1 8-1-2-7 7-2-1-8

2 8-3-2-7 7-2-3-8

3 8-3-4-7 7-4-3-8

4 8-5-4-7 7-4-5-8

5 8-5-6-7 7-6-5-8

6 8-1-6-7 7-6-1-8

Table - 3.2 Switching sequences for two-level Inverter in all the sectors

for Space Vector Modulation

26

One of the important advantages of the Space Vector PWM over the sine-

triangle PWM is that it gives nearly 15% more output voltage compared to the latter,

while still remaining in modulation. Space vector modulation can also be regarded as

a carrier based PWM technique with the modification that, the reference waveform

has triplen harmonics in addition to the fundamental.

The conventional implementation of the space vector PWM involves the following

steps:

1. Sector identification

2. Calculation of the active vector switching time periods T1 and T2 using

equation.3.8.

3. Translation of the active vector switching time periods and into the

inverter leg switching timings , and .

4. Generation of the gating signals for the individual power devices using the

inverter leg switching timings , and .

Therefore, owing to this basic approaching manner of the conventional space

vector PWM method, the overall process of this algorithm is complex and the

implementation is formidable.

This code given in the Appendix is based on the concept of “Effective time”

which is the time duration for which the load is connected to the supply. This

algorithm reduces the execution time by more than 25% while the memory

requirement is reduced to 15% compared to the conventional PWM method. The task

of generating the gating signals is accomplished naturally and automatically with this

algorithm. This algorithm is extended for all the dual inverter schemes.

3.4 Relationship between the Inverter leg Switching Timings and the Space Vector

Switching Timings

Whenever the tip of the reference voltage space vector OT (Figure.3.7) is

situated in one of the outer sectors in a dual-inverter scheme, OT’ is the transformed

reference voltage vector corresponding the actual reference voltage vector OT.

The inverter leg switching timings (denoted by , and ) are defined

as the respective time duration for which the phases of the inverter in an equivalent

27

conventional scheme (a single 2-level inverter driving a normal motor) are connected

to the positive terminal of the DC power supply to realize OT’. The DC-link voltage

of the equivalent conventional scheme to realize OT’ is equal to the side-length of the

inner hexagon.

The inverter leg switching timings ( , and ) to realize OT’ using an

equivalent conventional drive are determined (Figure 3.9).

Figure 3.9 Realization of OT’ in the inner hexagon

The inverter leg switching timings are then translated into the space vector

combinational switching timings ( , and ) as there exists an explicit

relationship between the inverter leg switching timings ( , and ) and the

space vector combinational switching timings ( , and ) depending upon which

inner sector the tip of OT’ is situated in.

28

Case i) Tip of OT’ is situated in Sector-1 of the inner hexagon:

Figure.3.10 The Gating pulses for the A, B and C phases

when the tip of OT’ is situated in sector-1 in the inner hexagon

From Figure 3.10 it is evident that in this case:

………. (3.9)

29

Case ii) Tip of OT’ is situated in Sector-2 of the inner hexagon:

Figure.3.11 The Gating pulses for the A, B and C phases

when the tip of OT’ is situated in sector-2 in the inner hexagon

From Figure 3.11 it is evident that in this case:

……………… (3.10)

30

Case iii) Tip of OT’ is situated in Sector-3 of the inner hexagon:

Figure 3.12 The Gating pulses for the A, B and C phases

when the tip of OT’ is situated in sector-3 in the inner hexagon

From Figure 3.12 it is evident that in this case:

……………. (3.11)

31

Case iv) Tip of OT’ is situated in Sector-4 of the inner hexagon:

Figure 3.13The Gating pulses for the A, B and C phases

when the tip of OT’ is situated in sector-4 in the inner hexagon

From Figure 3.13 it is evident that in this case:

…………. (3.12)

32

Case v) Tip of OT’ is situated in Sector-5 of the inner hexagon

Figure.3.14 The Gating pulses for the A, B and C phases

when the tip of OT’ is situated in sector-5 in the inner hexagon

From Figure.3.14 it is evident that in this case:

……………….. (3.13)

33

Case vi) Tip of OT’ is situated in Sector-6 of the inner hexagon

Figure.3.15 The Gating pulses for the A, B and C phases

when the tip of OT’ is situated in sector-6 in the inner hexagon

From Figure.3.15 it is evident that in this case:

…………….. (3.14)

34

Thus, the relationship between the inverter leg switching timings ( , and )

and the space vector switching timings ( , and ) is obtained graphically in all

the inner sectors. These relationships are presented in Table C.2 to enable a quick

reference.

Sector in which

OT’ is situated

1 - -

2 - -

3 - -

4 - -

5 - -

6 - -

Table.3.3: Space vector Switching Timings

These switching timings ( , and ) are then used to switch amongst the

closest vertices around the tip of the actual reference voltage space vector OT to

realize it in the average sense.

Translation of the active Space Vector Switching time periods and into

the Inverter leg Switching Timings , and are derived and given below in the

table.

35

Table 3.4 Inverter leg Switching Timing

36

Sector in

which OT’ is

situated

1

2

3

4

5

6

CHAPTER 4

MODELLING OF

INDUCTION MOTOR

37

4.1 Introduction

Before going to analyse the any motor or generator it is very much

important to obtain the machine in terms of equivalent mathematical equations.

Traditional per phase equivalent circuit has been widely used in steady state

analysis and design of induction motor, but it is not appreciated to predict the

dynamic performance of the motor. The dynamic of considers the instantaneous

effects of varying voltage/currents, stator frequency, and torque disturbance. The

dynamic model of the induction motor is derived by using a two-phase motor in

direct and quadrature axes. This approach is desirable because of the conceptual

simplicity obtained with two sets of windings, one on the stator and the other in

the rotor. The equivalence between the three phase and two phase machine

models is derived from simple observation, and this approach is suitable for

extending it to model an n-phase machine by means of a two phase machine.

The concept of power invariance is introduced; the power must be equal

in the three-phase machine and its equivalent two-phase model. Derivations for

electromagnetic torque involving the currents and flux linkages are given. The

differential equations describing the induction motor are non-linear. For stability

and controller design studies, it is important to linearize the machine equations

around a steady state operating point to obtain small signal equations. In or

adjustable speed drive, the machine normally constituted as element within a

feedback loop, and therefore its transient behaviour has to be taken into

consideration. The dynamic performance of an ac machine is somewhat complex

because the three phase rotor windings move with respect to the three phase

stator windings.

An induction motor can be looked on as a transformer with a moving

secondary, where the coupling coefficients between the stator and rotor phases

change continuously with the change of rotor position θr. The machine model can

be described by differential equations with time varying mutual inductances, but

such a model tends to be very complex. Hence, to reduce complexity it is

necessary to transform the three-phase machine into equivalent two-phase

machine beside, high performance drive control such as vector control, is based

38

on the dynamic d-q model of the machine. Therefore, to understand vector

control principled, a good understanding of d-q model is mandatory.

4.2 Reference Frames:

The required transformation in voltages, currents, or flux linkages is

derived in a generalized way. The reference frames are chosen to be arbitrary and

particular cases, such as stationary, rotor and synchronous reference frames are

simple instances of the general case. R.H. Park, in the 1920s, proposed a new

theory of electrical machine analysis to represent the machine in d – q model. He

transformed the stator variables to a synchronously rotating reference frame fixed

in the rotor, which is called Park’s transformation. He showed that all the time

varying inductances that occur due to an electric circuit in relative motion and

electric circuits with varying magnetic reluctances could be eliminated. In 1930s,

H.C Stanley showed that time varying Inductances in the voltage equations of an

induction machine due to electric circuits in relative motion can be eliminated by

transforming the rotor variables to a stationary reference frame fixed on the

stator. Later, G. Kron proposed a transformation of both stator and rotor variables

to a synchronously rotating reference that moves with the rotating magnetic field.

4.3Axes Transformation (3φ To 2φ)

We know that per phase equivalent circuit of the induction motor is only

valid in steady state condition. Nevertheless, it is not holds good while dealing

with the transient response of the motor. In transient response condition the

voltages and currents in three phases are not in balance condition. It is too much

difficult to study the machine performance of the machine by analyzing with

three phases. In order to reduce this complexity the transformation of axes from 3

– Φ to 2 – Φ is necessary. An other reason for transformation is to analyze any

machine of n number of phases, an equivalent model is adopted universally, that

is ‘d – q’ model.

39

Fig 4.1:3- to 2- Transformation

Consider a symmetrical three-phase induction machine with stationary as-

bs-cs axis at 2/3 angle apart. Our goal is to transform the three-phase stationary

reference frame (as-bs-cs) variables into two-phase stationary reference frame

(ds-qs) variables. Assure that ds-qs over are oriented at angle as shown in fig: the

voltages can be resolved into as-bs-cs components and can be

represented in matrix from as,

……………4.1

The corresponding inverse relation is

……………4.2

Here is zero-sequence convenient to set = 0 so that qs axis is aligned with

as-axis. Therefore ignoring zero-sequence component, this can be simplified as-

……………………… 4.3

…………………………4.4

4.4 MOTOR MODEL

40

Fig 4.2 Two-phase equivalent diagram of induction motor

The two-phase equivalent diagram of three-phase induction motor with

stator and rotor windings referred to d – q axes are shown in Fig 2.2. The

winding are spaced by 90o electrical and rotor winding , is at an angle θr from

the stator d-axis. It is assumed that the d axis is leading the q axis for clockwise

direction of rotation of the rotor. If the clockwise phase sequence is d-q, the

rotating magnetic field will be revolving at the angular speed of the supply

frequency but counter to the phase sequence of the stator supply. Therefore the

rotor is pulled in the direction of the rotating magnetic field i.e. counter

clockwise, in this case. The currents and voltages of the stator and rotor windings

are marked in figure [2.2]. The number of turns per phase in the stator and rotor

respectively are T1 and T2. A pair of poles is assumed for this figure. But it is

applicable with slight modification for any number of pairs of poles if it is drawn

in terms of electrical degrees. Note that r is the electrical rotor position at any

instant, obtained by multiplying the mechanical rotor position by pairs of

electrical poles. The terminal voltages of the stator and rotor windings can be

expressed as the sum of the voltage drops in resistances, and rate of change of

flux linkages, which are the products of currents and inductances.

From the above figure the terminal voltages are as follows,

41

Vqs = Rqiqs + p (Lqqiqs) + p (Lqdids) + p (Lq i ) + p (Lqi)

Vds = p (Ldqiqs) + Rdids + p (Lddids) + p (Ldi) + p (Ldi)

V = p (Lqiqs) + p (Ldids)+ Ri + p (Li) + p (Li)

V = p (Lqiqs) + p (Ldids) + p (Li) + R i + p (Li)

Where p is the differential operator d/dt, and vqs, vds are the terminal

voltages of the stator q axis and d axis. V, V are the voltages of rotor and

windings, respectively. iqs and ids are the stator q axis and d axis currents,

respectively. i and i are the rotor and windings currents, respectively. Lqq,

Ldd, L and L are the stator q and d axis winding and rotor and winding

self-inductances, respectively.

The following are the assumptions made in order to simplify the equation (2.5).

i. Uniform air-gap

ii. Balanced rotor and stator winding with sinusoidal distributed mmf.

iii. Inductance in rotor position is sinusoidal and

iv. Saturation and parameter changes are neglected

From the above assumptions the equation (2.5) modified as

vqs = (Rs + Ls p)iqs + Lsr p(i sinr) – Lsr p(i cosr)

vds = (Rs + Ls p)ids + Lsr p(i cosr ) + Lsr p(i sinr)

v = Lsr p (iqs sinr) + Lsr p (ids cosr) + (Rrr + Lrrp) i

v = - Lsrp (iqscosr) + Lsr p (ids sinr) + (Rrr + Lrrp) i

Where

42

4.5

4.6

4.7

By applying Transformation to the α and β rotor winding currents and voltages

the equation 2.6 will be written as

4.8

The rotor equations in above equation 2.8 refereed to stator side as in the case of

transformer equivalent circuit. From this, the physical isolation between stator

and rotor d-q axis use eliminated.

Derivative of r, a = transformer ratio = (stator turns)/(rotor turns)

;

; 4.9

vqr = avqrr ; vdr = avdrr

Magnetizing and control inductances are

4.10

Magnetizing inductance of the stator is 4.11

From equations 2.9, 2.10 &2.11 the equation 2.8 is modified as

4.12

Where θor = r = d/dt and p= d/dt

The dynamic equations of the induction motor in any reference frame can

be represented by using flux linkages as variables. This involves the reduction of

a number of variables in the dynamic equations. Even when the voltages and

currents are discontinuous the flux linkages are continuous [12]. The stator and

rotor flux linkages in the stator reference frame are defined as

4.13

43

4.13.a

From (2.12) and (2.13) we get

4.14

Since the rotor windings are short circuited, the rotor voltages are zero. Therefore

4.15

From (2.15), we have

4.16

By solving the equations (4.13), (4.14), (4.15) and (4.16) we get the following

equations

4.17

4.18

4.19

4.20

4.21

4.22

The electromagnetic torque of the induction motor in stator reference frame is

given by

44

) 4.23

or

4.24

The electro-mechanical equation of the induction motor drive is given by

4.25

By using the equations from (4.17) to (4.25), the induction motor

model is developed in stator reference frame

In this thesis Induction machine is modeled in stationary

reference frame ( =0). The terms corresponding to speed are eliminated. Eqns

(4.14), (4.17), (4.24) and (4.25) are used to model the machine. Simulink model

of Induction machine is presented in Chapter 4.

CHAPTER 5

45

SIMULINK AND

IMPLEMENTATION OF

INDUCTION MACHINE

5.1. Induction Motor Model:

The induction machine d-q or dynamic equivalent

circuit is shown in fig. One of the most popular induction motor models

derived from this equivalent circuit is Krause's model. According to his

model, the modeling equations in flux linkages form are as follows

46

Then, the modeling equations of a squirrel cage induction motor in state-space

47

5.2 Simulation:

The inputs of a squirrel cage induction machine are the three-phase

voltages, their fundamental frequency, and the load torque. The outputs,

on the other hand, are the three phase currents, the electrical torque, and

the rotor speed.

48

The d-q model requires that all the three-phase variables have to be

transformed to the two-phase synchronously rotating frame. Consequently

the induction machine model will have the blocks transforming the three-

phase voltages d-q frame and d-q currents back to three-phase currents.

The induction machine model implemented in this work is shown in

fig.4.1.

0-n

unit vector

abc - syn induction motor d-q model syn - abc

Te

Tl

wr

we

ia

ic

ib

currents ia , ib , ic

Torque

angular speed

5

43

2

1

iqs

ids

sin theta

cos theta

ia

ib

ic

we

sin theta

cos theta

vqs

vds

we

Tl

iqs

ids

Te

wr

idr

iqr

Van

Vbn

Vcn

sin theta

cos theta

vqs

vds

va 0

vb 0

vc 0

Van

Vbn

Vcn

Tl

we

Fig: 5.1 induction machine d-q model

49

It consists of five major blocks. The o-n Conversion, abc-syn conversion,

syn-abc conversion unit, vector calculation and induction machine d-q

model blocks.

The following subsections will explain each block.

A. O-n conversion block:

This is implemented in Simulink by passing the input voltages through a

Simulink “Matrix Gain" block, which contains the above transformation

matrix.

This block is required for an isolated neutral system, otherwise it can be

bypassed. The transformation done by this block can be represented as

follows:

50

Fig5.2:o-n conversion block

B. Unit vector calculation block

Unit vectors cosθ, sinθ are used in vector rotation blocks, "abc-syn

conversion block” and "syn-abc conversion block ".The angle θ is

calculated directly by integrating the frequency of the three-phase

voltages .

The unit vectors are obtained simply by taking the sine and cosine of

Theta. In this block is also the rotor position can be inserted, it Simulink

"Integrator” blocks.

51

C. abc-syn conversion block

To convert three-phase voltages to voltages in the-phase rotating frame

and, they are first converted to two phase stationary frame and then from

stationary frame to synchronously rotating frame. Where the super

script’s’ refers to stationary frame.

Equation (20) is implemented similar to eq(18) because it is a simple

matrix transformation. Eq. (21). However, contains the unit vector,

therefore a simple matrix transformation cannot be used. Instead, Vqs and

Vds are calculated using basic Simulink "Sum" and "Product" blocks.

vqs '

vds '

vds

2

vqs

1

-K-

cos theta

5

sin theta

4

Vcn

3

Vbn

2Van

1

52

Fig5.3: abc-syn conversion block

D. syn-abc conversion block

This block does the opposite of abc-syn conversion block for current

variables using (22) & (23) and the same implementation technique

as below.

sin

iqs '

ids ' ic

3

ib

2

ia

1

-K-

-K-

cos theta

4

sin theta

3

ids

2

iqs

1

Fig5.4: syn-abc conversion block

Simulation Tip 1 Do not use derivatives. Some signals will have

discontinuities and or ripple that would in spikes when the differentiated.

Instead try to integrals to and basic erythematic.

Simulation Tip 2 Beware of the algebraic loops. Algebraic loops will

appear when there is feedback loop in your system, when the calculation

of present value of a variable requires its present value when simulink

notices an algebraic loop, it will try to solve it.

Simulink "Memory" block, which delays its input signal by one sampling

time however this might affect the system. If possible avoid algebraic

loops.

For example, fig 4.1(given) shows the utilization file for a 30kw induction

machine. Before the simulation, this file has to be executed at the Mat lab

prompt; otherwise simulink will display an error message.

The modelled induction machine is simulated with the parameters given

and applying 220v, three-phase ac voltage at 60Hz with just an inertia

53

load.

The below one is example for Mat lab initiation of simulink variable

initialisations

5.3 Machine Initialising File

% initialisation

rr=0.39; % rotor resistance (ohm)

rs=0.19; % stator resistance (ohm)

lls=0.21e-3; % stator inductance (H)

llr=0.6e-3; % rotor inductance (H)

lm=4e-3; % magnatising inductance (H)

fb=100; % base frequency (Hz)

p=4; % number of poles

j=0.0226; % moment of inertia

% Impedance and angular speed calculations

lr=llr+lm;

tr=lr/rr;

wb=2*pi*fb; % base speed

xls=wb*lls; % stator impedance

xlr=wb*llr; % rotor impedance

xm=wb*lm; % magnatising impedance

xmlstar=1/((1/xls)+(1/xm)+(1/xlr));

% User defined values

we=2*pi*fb; % stator angular electrical frequency

Tl=0; % Load torque (N-mt)

5.4 Program:

function A=csqn(par) v=par(1); r=par(2)+(2*pi); t=par(3); T=(1/10000); Vdc=500; M=(1.5)*(v/Vdc);

54

r=mod(r,(2*pi)); nsect=fix(r/(pi/3))+1; r1=mod(r,(pi/3)); Ta=M*T*sin((pi/3)-r1)/sin(pi/3); Tb=M*T*(sin(r1))/sin(pi/3); if mod(nsect,2)==1 T1=Ta; T2=Tb; rf=r1; else T1=Tb; T2=Ta; rf=pi/3-r1; end Tz=T-T1-T2; s=[1 -1 -1;1 1 -1;-1 1 -1;-1 1 1;-1 -1 1;1 -1 1;1 1 1;-1 -1 -1]; sq=[8 1 2 7 2 1 8;8 3 2 7 2 3 8;8 3 4 7 4 3 8;8 5 4 7 4 5 8;8 5 6 7 6 5 8;8 1 6 7 6 1 8]; tt=mod(t,2*T); if tt<(Tz/2) A(1)=s(sq(nsect,1),1); A(2)=s(sq(nsect,1),2); A(3)=s(sq(nsect,1),3); elseif tt<(Tz/2)+T1 A(1)=s(sq(nsect,2),1); A(2)=s(sq(nsect,2),2); A(3)=s(sq(nsect,2),3); elseif tt<(Tz/2)+T1+T2 A(1)=s(sq(nsect,3),1); A(2)=s(sq(nsect,3),2); A (1)=s(sq(nsect,4),1); A(2)=s(sq(nsect,4),2); A(3)=s(sq(nsect,4),3); elseif tt<(Tz/2)+T1+T2+Tz+T2 A(1)=s(sq(nsect,5),1); A(2)=s(sq(nsect,5),2); A(3)=s(sq(nsect,5),3); elseif tt<(Tz/2)+T1+T2+Tz+T2+T1 A(1)=s(sq(nsect,6),1); A(2)=s(sq(nsect,6),2); A(3)=s(sq(nsect,6),3); else A(1)=s(sq(nsect,7),1); A(2)=s(sq(nsect,7),2); A(3)=s(sq(nsect,7),3); end end

5.5 Simulation Results

5.5.1 Induction d-q model block:

In this subdivision each equation from the induction machine model is

55

implemented in a different block. First consider the flux linkage state

equations because flux is required to calculate all the other variables.

These equations could be implemented using simulink “state-space”

block, but to have the access to each point of the model, implementation

using discrete blocks is preferred. The results for the simulink

implementation of induction machine are shown in figure 4.1.7 tracing the

speed, torque as well as phase currents with improved transient stability

as well as dynamic stability.

56

Te

3

Wr

2

Iabc1

Subsystem5

Scids

iqs

Sciqs

ids

wr

Te

Subsystem4

Ids

Iqs

Iabc

Subsystem3

scids

sciqs

scidr

sciqr

ids

iqs

idr

iqr

Subsystem2

Vds

Vqs

Ids

Iqs

Wr

Idr

Iqr

Scids

Sciqs

Scidr

Sciqr

Subsystem1

Vas

Vbs

Vcs

Vds

Vqs

[Iqr ]

[Idr ]

[Iqs]

[Ids]

Gain 4

1/2

Gain

-K-

From 4

[Iqr ]

From 3

[Idr ]

From 2

[Iqs ]

From 1

[Ids ]

Vcs3

Vbs

2

Vas

1

Figure 5.5 induction machine d-q model

57

Figure 5.6 results for induction machine d-q model

58

inverter

Pulse a

Pulse b

Pulse c

Van

Vbn

Vcn

Vabc to Vdq Calculation

Vas

Vbs

Vcs

Vds

Vqs

Scope 7

Scope 3Scope 1

SECTOR

vab sector

RoundingFunction

floor

Radiansto Degrees

R2D

Phase3

Phase2

Phase1

Motor

Vas

Vbs

Vcs

Iabc

Wr

Te

MATLAB Fcn 1

MATLABFunction

MATLAB Fcn

MATLABFunction

Display 3

Display 2

Display 1

Display

Clock

Cartesian toPolar

Fig.5.7 SVPWM FEEDING INDUCTION MOTOR

59

Figure 5.8 Results for svpwm feeding induction motor

60

Vao

Vbo

VcoVcn

3

Vbn

2

Van

1

Switch 3

Switch 2

Switch 1

Subtract 3

Subtract 2

Subtract 1

Gain 6

1/3

Gain 5

2/3

Gain 4

1/3

Gain 3

2/3

Gain 2

1/3

Gain 1

2/3

Constant 2

-0.5*Vd

Constant 1

0.5*Vd

Pulse c

3

Pulse b

2

Pulse a

1

FIG:5.13 main in SVPWM / SVPWM controller / subsystem / linearmodulation

FIG:5.14 main in SVPWM / SVPWM controller / subsystem / over modulation

61

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-200

0

200

Vco

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-200

0

200

Vbo

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-200

0

200

t

Vco

FIG:5.15 inverter model

FIG: 5.16 3 phase to 2 phase converter

62

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-500

0

500V

ab

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-500

0

500

Vbc

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-500

0

500

t

Vca

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-200

0

200

Van

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-200

0

200

Vbn

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-200

0

200

t

Vcn

5.16 Pole Voltage of Two–level Inverter

Figure 5.19 Line voltage of Two-level Inverter

63

Figure 5.20 Phase voltage of Two-level Invert

CONCLUSION

64

6. CONCLUSION

In the conventional svpwm algorithm, the angle and sector information is used to

calculate the switching times of the devices. Hence, the complexity involved is more.

Hence to reduce the computational burden involved in conventional approach, this

thesis presents a unified pwm approach using the concept of offset and effective

times. First, the offset time and effective times are calculated using the concept of

imaginary switching times. Then, by changing the offset time various continuous and

discontinuous pwm methods have been generated from the unified pwm algorithm.

To validate the proposed pwm algorithms, simulation studies and

experimental tests have been carried out using matlab and dspace ds1004 kit. From

the results it can be observed that, in the continuous pwm algorithm, the pulse pattern

is continuous and hence the switching losses are more. However, in discontinuous

pwm algorithms, the modulating wave is clamped to either positive dc bus or

negative dc bus for a period of 180 degrees in every fundamental cycle. Hence, the

switching frequency and switching losses can be reduced by 33.33% compared with

continuous pwm algorithms.

65

FUTURE SCOPE

66

7. SCOPE FOR FUTURE WORK

This thesis can be extended for the speed control of Induction Motor as we are

getting varying voltage levels depending upon the switching timings of the Inverter

and also can be used for harmonic analysis of higher order. This can be further

implemented using Dpwm for more-level Inverters.

67

APPENDIX

The relationship between the phase voltages , , and the pole

voltages , and is given by:

VAO = VA N + VN O;

VBO = VB N + VN O;

VCO = VCN + VN O;

The space vector constituted by the pole voltages VAO, VBO and VCO, is

defined as:

Vs = VAO + VBO .exp [j (2π/3)] + VCO .exp [j (4π/3)]

Case (i):

For Switching State –1 the sequence is (+ - -), then the expression for Space

vector will be as below:

VAO = (Vdc/2); VBO = - (Vdc/2); VC0 = - (Vdc/2)

Vs = (Vdc/2) - (Vdc/2). exp [j (2π/3)] - (Vdc/2). exp [j (4π/3)]

= (Vdc/2) - (Vdc/2). [-(1/2) + j (√3/2)] - (Vdc/2). [-(1/2) - j √3/2)]

= Vdc

Case (ii):

For Switching State –2 the sequence is (+ + -), then the expression for Space

vector will be as below:

VAO = (Vdc/2); VBO = (Vdc/2); VC0 = - (Vdc/2)

Vs = (Vdc/2) + (Vdc/2). exp [j (2π/3)] - (Vdc/2). exp [j (4π/3)]

= (Vdc/2) + (Vdc/2). [-(1/2) + j (√3/2)] - (Vdc/2). [-(1/2) - j √3/2)]

= Vdc. [(1/2) + j (√3/2)]

= Vdc at 600

68

Case (iii):

For Switching State –3 the sequence is (- + -), then the expression for Space

vector will be as below:

VAO = - (Vdc/2); VBO = (Vdc/2); VC0 = - (Vdc/2)

Vs = - (Vdc/2) + (Vdc/2). exp [j (2π/3)] - (Vdc/2). exp [j (4π/3)]

= - (Vdc/2) + (Vdc/2). [-(1/2) + j (√3/2)] - (Vdc/2). [-(1/2) - j √3/2)]

= Vdc. [- (1/2) + j (√3/2)]

= Vdc at 1200

Case (iv):

For Switching State –4 the sequence is (- + +), then the expression for Space

vector will be as below:

VAO = - (Vdc/2); VBO = (Vdc/2); VC0 = (Vdc/2)

Vs = - (Vdc/2) + (Vdc/2). exp [j (2π/3)] + (Vdc/2). exp [j (4π/3)]

= (Vdc/2) - (Vdc/2). [-(1/2) + j (√3/2)] - (Vdc/2). [-(1/2) - j √3/2)]

= (-1) Vdc

= Vdc at 1800

Case (v):

For Switching State –5 the sequence is (- - +), then the expression for Space

vector will be as below:

VAO = - (Vdc/2); VBO = - (Vdc/2); VC0 = (Vdc/2)

Vs = - (Vdc/2) - (Vdc/2). exp [j (2π/3)] + (Vdc/2). exp [j (4π/3)]

= - (Vdc/2) - (Vdc/2). [-(1/2) + j (√3/2)] + (Vdc/2). [-(1/2) - j √3/2)]

= Vdc. [- (1/2) - j (√3/2)]

= Vdc at 2400 or Vdc at (-1200)

Case (vi):

For Switching State –6 the sequence is (+ - +), then the expression for Space

vector will be as below:

VAO = (Vdc/2); VBO = - (Vdc/2); VC0 = (Vdc/2)

69

Vs = (Vdc/2) - (Vdc/2). exp [j (2π/3)] + (Vdc/2). exp [j (4π/3)]

= (Vdc/2) - (Vdc/2). [-(1/2) + j (√3/2)] + (Vdc/2). [-(1/2) - j √3/2)]

= Vdc. [(1/2) - j (√3/2)]

= Vdc at 3000 or Vdc at (-600)

Case (vii):

For Switching State –7 the sequence is (+ + +), then the expression for Space

vector will be as below:

VAO = (Vdc/2); VBO = (Vdc/2); VC0 = (Vdc/2)

Vs = (Vdc/2) + (Vdc/2). exp [j (2π/3)] + (Vdc/2). exp [j (4π/3)]

= (Vdc/2) + (Vdc/2). [-(1/2) + j (√3/2)] + (Vdc/2). [-(1/2) - j √3/2)]

= 0

Case (viii):

For Switching State –8 the sequence is (- - -), then the expression for Space

vector will be as below:

VAO = - (Vdc/2); VBO = - (Vdc/2); VC0 = - (Vdc/2)

Vs = - (Vdc/2) - (Vdc/2). exp [j (2π/3)] - (Vdc/2). exp [j (4π/3)]

= - (Vdc/2) - (Vdc/2). [-(1/2) + j (√3/2)] - (Vdc/2). [-(1/2) - j √3/2)]

= 0

70

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