modeling and simulation of permanent

8/3/2019 Modeling and Simulation of Permanent

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MODELING AND SIMULATION OF PERMANENT

MAGNET SYNCHRONOUS MOTOR DRIVE SYSTEM



Symbols

B- friction

BDCM -Brushless DC Motor

CSI -Current Source Inverter

d -Direct o polar axis

f c -crossover frequency

ia,ib,ic -Three phase currents

id -d-axis current

if -equivalent permanent magnet field current

iq -q-axis current

Im -Peak value of supply current

IGBT -Insolate Gate Bipolar Transistor

IPM -Interior Permanent Magnet

J -inertia

L -self inductance

Ld -d-axis self inductance

Lls -stator leakage inductance

Ldm -d-axis magnetizing inductance

Lqm -q-axis magnetizing inductance

Lq -q-axis self inductance

Ls -equivalent self inductance per phase

P -number of poles

PI - proportional integral

PM- Permanent Magnet

PMSM -Permanent Magnet Synchronous Motorq Quadrature or interpolar axis

Rs-stator resistance

SPM-Surface Permanent Magnet

Te develop torque

TL-load torque



Va,Vb,Vc -Three phase voltage

Vd -d-axis voltage

Vq -q-axis voltage

VSI-Voltage Source Inverter

ρ -derivative operator

λd -flux linkage due d axis

λf -PM flux linkage or Field flux linkage

λq- flux linkage due q axis

θr- rotor position

ωm- rotor speed

ωr -electrical speed

ωrated- motor rated speed



1 INTRODUCTION

Permanent magnet (PM) synchronous motors are widely used in low and mid power applications

such as computer peripheral equipments, robotics, adjustable speed drives and electric vehicles.

The growth in the market of PM motor drives has demanded the need of simulation tools capable

of handling motor drive simulations. Simulations have helped the process of developing new

systems including motor drives, by reducing cost and time. Simulation tools have the capabilities

of performing dynamic simulations of motor drives in a visual environment so as to facilitate the

development of new systems.

In this work, the simulation of a field oriented controlled PM motor drive system is developed

using Simulink. The simulation circuit will include all realistic components of the drive system.

This enables the calculation of currents and voltages in different parts of the inverter and motor

under transient and steady conditions. The losses in different parts can be calculated facilitating

the design of the inverter.

A closed loop control system with a PI controller in the speed loop has been designed to operate

in constant torque and flux weakening regions. Implementation has been done in Simulink. A

comparative study of hysteresis and PWM control schemes associated with current controllers

has been made in terms of harmonic spectrum and total harmonic distortion. Simulation results

are given for two speeds of operation, one below rated and another above rated speed.

1.1 Motivation

Modeling and simulation is usually used in designing PM drives compared to building system

prototypes because of the cost. Having selected all components, the simulation process can start

to calculate steady state and dynamic performance and losses that would have been obtained if

the drive were actually constructed. This practice reduces time, cost of building prototypes and

ensures that requirements are achieved. In works available until now ideal components have been

assumed in the inverter feeding the motor and simulations have been carried out. The voltages

and currents in different parts of the inverter have not been obtained and hence the losses and



efficiency can not be calculated. In this work, the simulation of a PM motor drive system is

developed using Simulink. The simulation circuit includes all realistic components of the drive

system. This enables the calculation of currents and voltages in different parts of the inverter and

motor under transient and steady conditions. The losses in different parts are calculated. A

comparative study associated with hysteresis and PWM control techniques in current controllers

has been made. A speed controller has also been designed for closed loop operation of the drive.

Design method for the PI controller is also given.

1.2 Previous Work

PM motor drives have been a topic of interest for the last twenty years. Different

authors have carried out modeling and simulation of such drives.

In 1986 Sebastian, T., Slemon, G. R. and Rahman, M. A. [1] reviewed permanent

magnet synchronous motor advancements and presented equivalent electric circuit models

for such motors and compared computed parameters with measured parameters.

Experimental results on laboratory motors were also given.

In 1986 Jahns, T.M., Kliman, G.B. and Neumann, T.W. [2] discussed that interior

permanent magnet (IPM) synchronous motors possessed special features for adjustable speed

operation which distinguished them from other classes of ac machines. They were robust

high power density machines capable of operating at high motor and inverter efficiencies

over wide speed ranges, including considerable range of constant power operation. The

magnet cost was minimized by the low magnet weight requirements of the IPM design. The

impact of the buried magnet configuration on the motor’s electromagnetic characteristics was

discussed. The rotor magnetic saliency preferentially increased the quadrature-axis

inductance and introduced a reluctance torque term into the IPM motor’s torque equation.

The electrical excitation requirements for the IPM synchronous motor were also discussed.

The control of the sinusoidal phase currents in magnitude and phase angle with respect to the

rotor orientation provided a means for achieving smooth responsive torque control. A basic

feed forward algorithm for executing this type of current vector torque control was discussed,

including the implications of current regulator saturation at high speeds. The key results were



illustrated using a combination of simulation and prototype IPM drive measurements.

In 1988 Pillay and Krishnan, R. [3], presented PM motor drives and classified them

into two types such as permanent magnet synchronous motor drives (PMSM) and brushless

dc motor (BDCM) drives. The PMSM has a sinusoidal back emf and requires sinusoidal

stator currents to produce constant torque while the BDCM has a trapezoidal back emf and

requires rectangular stator currents to produce constant torque. The PMSM is very similar to

the wound rotor synchronous machine except that the PMSM that is used for servo

applications tends not to have any damper windings and excitation is provided by a

permanent magnet instead of a field winding. Hence the d, q model of the PMSM can be

derived from the well known model of the synchronous machine with the equations of the

damper windings and field current dynamics removed. Equations of the PMSM are derived

in rotor reference frame and the equivalent circuit is presented without dampers. The damper

windings are not considered because the motor is designed to operate in a drive system with

field-oriented control. Because of the nonsinusoidal variation of the mutual inductances

between the stator and rotor in the BDCM, it is also shown in this paper that no particular

advantage exists in transforming the abc equations of the BCDM to the d, q frame.

As an extension of his previous work, Pillay, P. and Krishnan, R. in 1989 [4]

presented the permanent magnet synchronous motor (PMSM) which was one of several types

of permanent magnet ac motor drives available in the drives industry. The motor had a

sinusoidal flux distribution. The application of vector control as well as complete modeling,

simulation, and analysis of the drive system were given. State space models of the motor and

speed controller and real time models of the inverter switches and vector controller were

included. The machine model was derived for the PMSM from the wound rotor synchronous

motor. All the equations were derived in rotor reference frame and the equivalent circuit was

presented without dampers. The damper windings were not considered because the motor

was designed to operate in a drive system with field-oriented control. Performancedifferences due to the use of pulse width modulation (PWM) and hysteresis current

controllers were examined. Particular attention was paid to the motor torque pulsations and

speed response and experimental verification of the drive performance were given.



Morimoto, S., Tong, Y., Takeda, Y. and Hirasa, T. in 1994 [5], in their paper aimed

to improve efficiency in permanent magnet (PM) synchronous motor drives. The controllable

electrical loss which consisted of the copper loss and the iron loss could be minimized by the

optimal control of the armature current vector. The control algorithm of current vector

minimizing the electrical loss was proposed and the optimal current vector could be decided

according to the operating speed and the load conditions. The proposed control algorithm

was applied to the experimental PM motor drive system, in which one digital signal

processor was employed to execute the control algorithms, and several drive tests were

carried out. The operating characteristics controlled by the loss minimization control

algorithm were examined in detail by computer simulations and experimental results.

The paper in 1997 by Wijenayake, A.H. and Schmidt, P.B. [6], described the

development of a two-axis circuit model for permanent magnet synchronous motor (PMSM)

by taking machine magnetic parameter variations and core loss into account. The circuit

model was applied to both surface mounted magnet and interior permanent magnet rotor

configurations. A method for on-line parameter identification scheme based on no-load

parameters and saturation level, to improve the model, was discussed in detail. Test schemes to

measure the equivalent circuit parameters, and to calculate saturation constants which

govern the parameter variations were also presented.



CHAPTER - 2

2 DESCRIPTION OF THE DRIVE SYSTEM

This chapter deals with the description of the different components such as permanent magnet

motors, position sensors, inverters and current controllers of the drive system. A review of

permanent magnet materials and classification of permanent magnet motors is also given.

2.1 Permanent Magnet Synchronous Motor Drive System

The motor drive consists of four main components, the PM motor, inverter, control unit and the

position sensor. The components are connected as shown in figure 2.1.



2.2 Permanent Magnet Synchronous Motor

A permanent magnet synchronous motor (PMSM) is a motor that uses permanent magnets to

produce the air gap magnetic field rather than using electromagnets. These motors have

significant advantages, attracting the interest of researchers and industry for use in manyapplications.

2.2.1 Permanent Magnet Materials

The properties of the permanent magnet material will affect directly the performance of the

motor and proper knowledge is required for the selection of the materials and for understanding

PM motors.

The earliest manufactured magnet materials were hardened steel. Magnets made from steel were

easily magnetized. However, they could hold very low energy and it was easy to demagnetize. In

recent years other magnet materials such as Aluminum Nickel and Cobalt alloys (ALNICO),

Strontium Ferrite or Barium Ferrite (Ferrite), Samarium Cobalt (First generation rare earth

magnet) (SmCo) and Neodymium Iron-Boron (Second generation rare earth magnet) (NdFeB)

have been developed and used for making permanent magnets.

The rare earth magnets are categorized into two classes: Samarium Cobalt (SmCo) magnets and

Neodymium Iron Boride (NdFeB) magnets. SmCo magnets have higher flux density levels but

they are very expensive. NdFeB magnets are the most common rare earth magnets used in

motors these days. A flux density versus magnetizing field for these magnets

is illustrated in figure 2.2.



Figure 2.2 Flux Density versus Magnetizing Field of Permanent Magnetic Materials[21]

2.2.2 Classification of Permanent Magnet Motors

2.2.2.1 Direction of field flux

PM motors are broadly classified by the direction of the field flux. The first field flux

classification is radial field motor meaning that the flux is along the radius of the motor. The

second is axial field motor meaning that the flux is perpendicular to the radius of the motor.

Radial field flux is most commonly used in motors and axial field flux have become a topic of

interest for study and used in a few applications.

2.2.2.2 Flux density distribution



PM motors are classified on the basis of the flux density distribution and the shape of current

excitation. They are PMSM and PM brushless motors (BLDC). The PMSM has a sinusoidal-

shaped back EMF and is designed to develop sinusoidal back EMF waveforms.

They have the following:

1. Sinusoidal distribution of magnet flux in the air gap

2. Sinusoidal current waveforms

3. Sinusoidal distribution of stator conductors.

BLDC has a trapezoidal-shaped back EMF and is designed to develop trapezoidal back EMF

waveforms. They have the following:

1. Rectangular distribution of magnet flux in the air gap

2. Rectangular current waveform

3. Concentrated stator windings.

2.2.2.3 Permanent magnet radial field motors

In PM motors, the magnets can be placed in two different ways on the rotor. Depending on the

placement they are called either as surface permanent magnet motor or interior permanent

magnet motor.

Surface mounted PM motors have a surface mounted permanent magnet rotor. Each of the PM is

mounted on the surface of the rotor, making it easy to build, and specially skewed poles are

easily magnetized on this surface mounted type to minimize cogging torque. This configuration

is used for low speed applications because of the limitation that the magnets will fly apart during

high-speed operations. These motors are considered to have small saliency, thus having

practically equal inductances in both axes [22]. The permeability of the permanent magnet is

almost that of the air, thus the magnetic material becoming an extension of the air gap. For a

surface permanent magnet motor Ld = Lq [8].

The rotor has an iron core that may be solid or may be made of punched laminations for

simplicity in manufacturing [23]. Thin permanent magnets are mounted on the surface of this



core using adhesives. Alternating magnets of the opposite magnetization direction produce

radially directed flux density across the air gap. This flux density then reacts with currents in

windings placed in slots on the inner surface of the stator to produce torque.

Figure 2.3 shows the placement of the magnet.

Interior PM motors have interior mounted permanent magnet rotor as shown in figure 2.4.

Each permanent magnet is mounted inside the rotor. It is not as common as the surface mounted

type but it is a good candidate for high-speed operation. There is inductance variation for this

type of rotor because the permanent magnet part is equivalent to air in the magnetic circuit

calculation. These motors are considered to have saliency with q axis inductance greater than the

d axis inductance ( Lq > Ld )[8].



2.3 Position Sensor

Operation of permanent magnet synchronous motors requires position sensors in the rotor shaft

when operated without damper winding. The need of knowing the rotor position requires thedevelopment of devices for position measurement. There are four main devices for the

measurement of position, the potentiometer, linear variable differential transformer, optical

encoder and resolvers. The ones most commonly used for motors are encoders and revolvers.

Depending on the application and performance desired by the motor a position sensor with the

required accuracy can be selected

2.3.1 Optical Encoders

The most popular type of encoder is the optical encoder as shown in figure 2.5, which consists of

a rotating disk, a light source, and a photo detector (light sensor). The disk, is mounted on the

rotating shaft, has coded patterns of opaque and transparent sectors. As the disk rotates, these

patterns interrupt the light emitted onto the photo detector, generating a digital pulse or output

signal.

Optical encoders offer the advantages of digital interface. There are two types of

optical encoders, incremental encoder and absolute encoder.



2.3.1.1 Incremental encoders

Incremental encoders have good precision and are simple to implement but they suffer from lack

of information when the motor is at rest position and in order for precise position the motor most

be stop at the starting point.

The most common type of incremental encoder uses two output channels (A and B) to sense

position. Using two code tracks with sectors positioned 90° degrees out of phase, the two output

channels of the quadrature encoder indicate both position and direction of rotation as shown in

figure 2.6. If A leads B, for example, the disk is rotating in a clockwise direction. If B leads A,

then the disk is rotating in a counter-clockwise direction. By monitoring both, the number of

pulses and the relative phase of signals A and B, it's possible to track position and direction of

rotation. Some quadrature encoders also include a third output channel, called a zero or index or

reference signal, which supplies a single pulse per revolution. This single pulse is used for

precise determination of a reference position. The precision of the encoder is fix by its code disk

but it can be increased by detecting the Up and Down transitions on both the A and B channels.

2.3.1.2 Absolute encoders

The absolute encoder, as shown in figure 2.7 captures the exact position of the rotor with a

precision directly related to the number of bits of the encoder. It can rotate indefinitely and even

if the motor stops the position can be measured or obtained.



It provides a “whole word” output with a unique code pattern representing each position. This

code is derived from independent tracks on the encoder disc (one for each “bit” of resolution)

corresponding to individual photo detectors. The output from these detectors is

HI (light) or LO (dark) depending on the code disc pattern for that particular position.

Absolute encoders are used in applications where a device is inactive for long periods of time or

moves at a slow rate, such as flood gate control, telescopes, cranes, valves, etc. They are also

recommended in systems that must retain position information through a power outage.

2.3.2 Position Revolver

Position revolver as shown in figure 2.8, also called rotary transformers, works on the

transformer principle. The primary winding is placed on the rotor and depending upon the rotor

shaft angle the induced voltage at the two secondary windings of the transformer shifted by 90°

would be different. The position can be calculated using the two voltages. The resolver is

basically a rotary transformer with one rotating reference winding (Vref ) and two stator windings.

The reference winding is fixed on the rotor, and therefore, it rotates jointly with the shaft passing

the output windings, as is depicted in figure 2.8. Two stator windings are placed in quadrature

(shifted by 90°) with one another and generate the sine and cosine voltages (Vsin , Vcos)

respectively. Both windings will be further referred to as output windings. In consequence of the

excitement applied on the reference winding Vref and along with the angular movement of the

motor shaft θ, the respective voltages are generated by resolver output windings Vsin , Vcos.



The frequency of the generated voltages is identical to the reference voltage and their amplitudes

vary according to the sine and cosine of the shaft angle θ. Considering that one of the output

windings is aligned with the reference winding, then it is generated full voltage on that output

winding and zero voltage on the other output winding and vice versa. The rotor angle θ can be

extracted from these voltages shown in figure 2.9.



The shaft angle can be determined by an Inverse Tangent [8] function of the quotient of the

sampled resolver output voltages Vsin , Vcos . This determination can be expressed, in terms of

resolver output voltages, as follows:

2.4 Current Controlled Inverter

The motor is fed form a voltage source inverter with current control. The control is performed by

regulating the flow of current through the stator of the motor. Current controllers are used to

generate gate signals for the inverter. Proper selection of the inverter devices and selection of thecontrol technique will guarantee the efficacy of the drive.

2.4.1 Inverter

Voltage Source Inverters are devices that convert a DC voltage to AC voltage of variable

frequency and magnitude. They are very commonly used in adjustable speed drives and are

characterized by a well defined switched voltage wave form in the terminals [23]. Figure 2.10shows a voltage source inverter. The AC voltage frequency can be variable or constant

depending on the application.



Three phase inverters consist of six power switches connected as shown in figure 2.10 to a DC

voltage source. The inverter switches must be carefully selected based on the requirements of

operation, ratings and the application. There are several devices available today and these are

thyristors, bipolar junction transistors (BJTs), MOS field effect transistors (MOSFETs), insulated

gate bipolar transistors (IGBTs) and gate turn off thyristors (GTOs). The devices list with their

respective power switching capabilities are shown in table 2.1 MOSFETs and IGBTs are

preferred by industry because of the MOS gating permits high power gain and control

advantages. While MOSFET is considered a universal power device for low power and low

voltage applications, IGBT has wide acceptance for motor drives and other application in the low

and medium power range. The power devices when used in motor drives applications require an

inductive motor current path provided by antiparallel diodes when the switch is turned off.

Inverters with antiparallel diodes are shown in figure 2.11.



2.4.2 IGBTs

IGBTs provide high input impedance and are used for high voltage applications. The high input

impedance allows the device to switch with a small amount of energy and for high voltage

applications the device must have large blocking voltage ratings. The device behavior is

described by parameters like voltage drop or on-resistance, turn on time and turn off time. Figure

2.12 shows the characteristic plot of the device (International rectifier IGBT

(IRGIB10B60KD1) Appendix 1).

The symbolic representation and the equivalent circuit of an IGBT are shown infigure 2.13.



2.4.3 Current Control

The power converter in a high-performance motor drive used in motion control essentially

functions as a power amplifier, reproducing the low power level control signals generated in thefield orientation controller at power levels appropriate for the driven machine. High-performance

drives utilize control strategies which develop command signals for the AC machine currents.

The basic reason for the selection of current as the controlled variable is the same as for the DC

machine; the stator dynamics (effects of stator resistance, stator inductance, and induced EMF)

are eliminated. Thus, to the extent that the current regulator functions as an ideal current supply,

the order of the system under control is reduced and the complexity of the controller can be

significantly simplified.

Current regulators for AC drives are complex because an AC current regulator must control both

the amplitude and phase of the stator current. The AC drive current regulator forms the inner

loop of the overall motion controller. As such, it must have the widest bandwidth in the system

and must, by necessity, have zero or nearly zero steady-state error. Both current source inverters

(CSI) and voltage source inverters (VSI) can be operated in controlled current modes. The

current source inverter is a "natural" current supply and can readily be adapted to controlled

current operation. The voltage source inverter requires more complexity in the current regulator

but offers much higher bandwidth and elimination of current harmonics as compared to the CSI

and is almost exclusively used for motion control applications.

Current controllers can be classified into two groups, hysteresis and PWM current controllers.

Both types are discussed bellow.

2.4.3.1 PWM Current Controller

PWM current controllers are widely used. The switching frequency is usually kept constant.

They are based in the principle of comparing a triangular carrier wave of desire switching

frequency and is compared with error of the controlled signal. The error signal comes from the

sum of the reference signal generated in the controller and the negative of the actual motor



current. The comparison will result in a voltage control signal that goes to the gates of the

voltage source inverter to generate the desire output. Its control will respond according to the

error. If the error command is greater than the triangle waveform, the inverter leg is held

switched to the positive polarity (upper switch on). When the error command is less than the

triangle waveform, the inverter leg is switched to the negative polarity (lower switch on). This

will generate a PWM signal like in figure 2.14. The inverter leg is forced to switch at the

frequency of the triangle wave and produces an output voltage proportional to the current error

command. The nature of the controlled output current consists of a reproduction of the reference

current with high-frequency PWM ripple superimposed[23].

2.4.3.2 Hysteresis current controller

Hysteresis current controller can also be implemented to control the inverter currents. The

controller will generate the reference currents with the inverter within a range which is fixed by

the width of the band gap. In this controller the desired current of a given phase is summed with

the negative of the measured current. The error is fed to a comparator having a hysteresis band.

When the error crosses the lower limit of the hysteresis band, the upper switch of the inverter leg

is turned on. But when the current attempts to become less than the upper reference band, the

bottom switch is turned on. Figure 2.15 shows the hysteresis band with the actual current and the

resulting gate signals. This controller does not have a specific switching frequency and changes

continuously but it is related with the band width [4] [23].



3 MODELING OF PM DRIVE SYSTEM

This chapter deals with the detailed modeling of a permanent magnet synchronous motor. Field

oriented control of the motor in constant torque and flux-weakening regions are discussed.

Closed loop control of the motor is developed using a PI controller in the speed loop. Design of

the speed controller is discussed.

3.1 Detailed Modeling of PMSM

Detailed modeling of PM motor drive system is required for proper simulation of the system.

The d-q model has been developed on rotor reference frame as shown in figure 3.1. At any time

t, the rotating rotor d-axis makes and angle θr with the fixed stator phase axis and rotating stator

mmf makes an angle α with the rotor d-axis. Stator mmf rotates at the same speed as that of the

rotor.

The model of PMSM without damper winding has been developed on rotor reference

frame using the following assumptions:

1) Saturation is neglected.

2) The induced EMF is sinusoidal.

3) Eddy currents and hysteresis losses are negligible.

4) There are no field current dynamics.



Voltage equations are given by:

V q = Rsiq +ωr λd + ρλq 3.1

d sd r q d V = R i −ω λ + ρλ 3.2

Flux Linkages are given by



In the above equations ωr is the rotor electrical speed where as ωm is the rotor

mechanical speed.

3.1.1 Parks Transformation and Dynamic d q Modeling

The dynamic d q modeling is used for the study of motor during transient and steady state. It is

done by converting the three phase voltages an currents to dqo variables by using Parks

transformation[8].

Converting the phase voltages variables vabc to vdqo variables in rotor reference frame the

following equations are obtained



3.1.2 Equivalent Circuit of Permanent Magnet Synchronous Motor

Equivalent circuits of the motors are used for study and simulation of motors. From the d-q

modeling of the motor using the stator voltage equations the equivalent circuit of the motor can

be derived. Assuming rotor d axis flux from the permanent magnets is represented by a constant

current source as described in the following equationλ f = Ldmi f , figure 3.2 is

obtained.

3.2 PM Motor Control



Control of PM motors is performed using field oriented control for the operation of synchronous

motor as a dc motor. The stator windings of the motor are fed by an inverter that generates a

variable frequency variable voltage. Instead of controlling the inverter frequency independently,

the frequency and phase of the output wave are controlled using a position sensor as shown in

figure 3.3.

Field oriented control was invented in the beginning of 1970s and it demonstrates that an

induction motor or synchronous motor could be controlled like a separately excited dc motor by

the orientation of the stator mmf or current vector in relation to the rotor flux to achieve a desired

objective. In order for the motor to behave like DC motor, the control needs knowledge of the

position of the instantaneous rotor flux or rotor position of permanent magnet motor. This needs

a resolver or an absolute optical encoder. Knowing the position, the three phase currents can be

calculated. Its calculation using the current matrix depends on the control desired. Some control

options are constant torque and flux weakening. These options are based in the physical

limitation of the motor and the inverter. The limit is established by the rated speed of the motor,

at which speed the constant torque operation finishes and the flux weakening starts as shown in

figure 3.4.



3.2.1 Field Oriented Control of PM Motors

The PMSM control is equivalent to that of the dc motor by a decoupling control known as field

oriented control or vector control. The vector control separates the torque component of current

and flux channels in the motor through its stator excitation. The vector control of the PM

synchronous motor is derived from its dynamic model. Considering the currents as inputs, the

three currents are:

Writing equations 3.14 to 3.16 in the matrix form:



Where α is the angle between the rotor field and stator current phasor, r ω is the electrical rotor

speed.

The previous currents obtained are the stator currents that must be transformed to the rotor

reference frame with the rotor speed r ω , using Park’s transformation. The q and d axis currents

are constants in the rotor reference frames since α is a constant for a given load torque. As these

constants, they are similar to the armature and field currents in the separately excited dc

machine. The q axis current is distinctly equivalent to the armature current of the dc machine; the

d axis current is field current, but not in its entirety. It is only a partial field current; the other part

is contributed by the equivalent current source representing the permanent magnet field. For this

reason the q axis current is called the torque producing component of the stator current and the d

axis current is called the flux producing component of the stator current.Substituting equation 3.17 and 3.12 is obtain id and iq in terms of Im as follows

Using equations 3.1, 3.2, 3.8 and 3.18 the electromagnetic torque equation is obtained

as given below.



3.2.1.1 Constant torque operation

Constant torque control strategy is derived from field oriented control, where the maximum

possible torque is desired at all times like the dc motor. This is performed by making the torque

producing current iq equal to the supply current Im. That results in selecting the α angle to be 90 º

degrees according to equation 3.18. By making the id current equal to zero the torque equation

can be rewritten as:

Like the dc motor, the torque is dependent of the motor current.

3.2.1.2 Flux-weakening

Flux weakening is the process of reducing the flux in the d axis direction of the motor which

results in an increased speed range. The motor drive is operated with rated flux linkages up to a

speed where the ratio between the induced emf and stator frequency (V/f) is maintained constant.

After the base frequency, the V/f ratio is reduced due to the limit of the inverter dc voltage



source which is fixed. The weakening of the field flux is required for operation above the base

frequency.

This reduces the V/f ratio. This operation results in a reduction of the torque proportional to a

change in the frequency and the motor operates in the constant power region[22]. The rotor flux

of PMSM is generated by permanent magnet which can not be directly reduced as induction

motor. The principle of flux-weakening control of PMSM is to increase negative direct axis

current and use armature reaction to reduce air gap flux, which equivalently reduces flux and

achieves the purpose of flux-weakening control[28]. This method changes torque by altering the

angle between the stator MMF and the rotor d axis. In the flux weakening region where ωr >

ωrated angle α is controlled by proper control of id and iq for the same value of stator current. Since

iq is reduced the output torque is also reduced. The angle α can be obtained as:

Flux-weakening control realization

The realization process of equivalent flux-weakening control is as follows,

1) Measuring rotor position and speed ωr from a sensor which is set in motor rotation axis.

2) The motor at the flux weakening region with a speed loop, Te* is obtained from the PI

controller.



3.3 Speed Control of PM Motor

Many applications, such as robotics and factory automation, require precise control of speed and

position. Speed Control Systems allow one to easily set and adjust the speed of a motor. The

control system consists of a speed feedback system, a motor, an inverter, a controller and a speed

setting device. A properly designed feedback controller makes the system insensible to

disturbance and changes of the parameters.

The purpose of a motor speed controller is to take a signal representing the demanded\ speed,and to drive a motor at that speed. Closed Loop speed control systems have fast response, but

become expensive due to the need of feed back components such as speed sensors.

3.3.1 Implementation of the Speed Control Loop

For a PM motor drive system with a full speed range the system will consist of a motor, an

inverter, a controller (constant torque and flux weakening operation, generation of reference

currents and PI controller ) as shown in figure 3.5.



The operation of the controller must be according to the speed range. For operation up to rated

speed it will operate in constant torque region and for speeds above rated speed it will operate in

flux-weakening region. In this region the d-axis flux and the developed torque are reduced. The

process can be easily understood with the flow diagram in figure 3.6.



Speed controller calculates the difference between the reference speed and the actual speed

producing an error, which is fed to the PI controller. PI controllers are used widely for motion

control systems. They consist of a proportional gain that produces an output proportional to the

input error and an integration to make the steady state error zero for a step change in the input.Block diagram of the PI controller is shown in figure 3.7.



Speed control of motors mainly consist of two loops the inner loop for current and the outer loop

for speed. The order of the loops is due to their response, how fast they can be changed. This

requires a current loop at least 10 times faster than the speed loop. Since the PMSM is operated

using field oriented control, it can be modeled like a dc motor. The design begins with the

innermost current loop by drawing the block diagram. But in PMSM drive system the motor has

current controllers which make the current loop. The current control is performed by the

comparison of the reference currents with the actual motor currents.

The design of the speed loop assumes that the current loop is at least 10 times faster than speed

loop, allowing to reduce the system block diagram by considering the current loop to be of unity

gain as shown in figure 3.8.



Chapter-4

MATLAB SIMULINK INTRODUCTION

Introduction to MATLABWhat is MATLAB?

MATLAB (short for MATrix LABoratory) is a special-purpose computer program optimized

to perform engineering and scientific calculations. It is a high-performance language for

technical computing. It integrates computation, visualization, and programming in an easy-to-use

environment where problems and solutions are expressed in familiar mathematical notation.

Typical uses include:

Math and computation

Algorithm development

Modelling, simulation and prototyping

Data analysis, exploration and visualization

Scientific and engineering graphics

Application development, including Graphical User Interface (GUI) building

MATLAB is an interactive system whose basic data element is an array that does not require

dimensioning. This allows you to solve many technical computing problems, especially those

with matrix and vector formulations, in a fraction of the time it would take to write a program in

a scalar non-interactive language such as C, C++ or Fortran.

MATLAB has evolved over a period of years with input from many users. In university

environments, it is the standard instructional tool for introductory and advanced courses in

mathematics, engineering and science. In industry, MATLAB is the tool of choice for high-

productivity research, development and analysis.

MATLAB features a family of application-specific solution called Toolboxes. Very important to

most users of MATLAB, toolboxes allow you to learn and apply specialized technology.

Toolboxes are comprehensive collections of MATLAB function (m-files) that extend the

MATLAB environment to solve particular classes of problems. Areas in which toolboxes are

available include signal processing, control systems, neural networks, fuzzy logic, wavelets,

image processing, simulation and many others.



MATLAB System

The MATLAB system consists of five main parts:

1. The MATLAB language. This is a high-level matrix/array language with control flow

statements, functions, data structures, input/output, and object-oriented programming

features. It allows both “programming in the small” to rapidly create quick and dirty

throw-away programs, and “programming in the large” to create complete large and

complex application programs.

2. The MATLAB working environment. This is the set of tools and facilities that you work

with as the MATLAB user or programmer. It includes facilities for managing the

variables in your workspace and importing and exporting data. It also includes tools for

developing, managing, debugging, and profiling M-files, MATLAB’s applications.

3. Handle Graphics. This is the MATLAB graphics system. It includes high-level

commands for two-dimensional and three-dimensional data visualization, image

processing, animation, and presentation graphics. It also includes low-level commands

that allow you to fully customize the appearance of graphics as well as to build complete

Graphical User Interfaces on your MATLAB applications.

4. The MATLAB mathematical function library. This is a vast collection of computational

algorithms ranging from elementary functions like sum, sine, cosine, and complex

arithmetic, to more sophisticated functions like matrix inverse, matrix eigenvalues,

Bessel functions, and fast Fourier transforms.

5. The MATLAB Application Program Interface (API). This is a library that allows you

to write C and Fortran programs that interact with MATLAB. It include facilities for

calling routines from MATLAB (dynamic linking), calling MATLAB as a computational

engine, and for reading and writing MAT-files.

The Advantages of MATLAB

MATLAB has many advantages compared to conventional computer languages for technical

problem solving. Among them are:

1. Ease of Use. MATLAB is an interpreted language. Program may be easily written and

modified with the built-in integrated development environment and debugged with the



MATLAB debugger. Because the language is so easy to use, it is ideal for the rapid

prototyping of new programs.

2. Platform Independence. MATLAB is supported on many different computer systems,

providing a large measure of platform independence. At the time of this writing, the

language is supported on Windows NT/2000/XP, Linux, several versions of UNIX and

the Macintosh.

3. Predefined Function. MATLAB comes complete with an extensive library of predefined

functions that provide tested and pre-packaged solutions to many basic technical tasks.

For examples, the arithmetic mean, standard deviation, median, etc. these and hundreds

of other functions are built right into the MATLAB language, making your job much

easier. In addition to the large library of function built into the basic MATLAB language,

there are many special-purpose toolboxes available to help solve complex problems in

specific areas. There is also an extensive collection of free user-contributed MATLAB

programs that are shared through the MATLAB Web site.

4. Device-Independent Plotting. Unlike most other computer languages, MATLAB has

many integral plotting and imaging commands. The plots and images can be displayed on

any graphical output device supported by the computer on which MATLAB is running.

5. Graphical User Interface. MATLAB includes tools that allow a programmer to

interactively construct a graphical user interface, (GUI) for his or her program. With this

capability, the programmer can design sophisticated data-analysis programs that can be

operated by relatively inexperienced users.

6. MATLAB Compiler. MATLAB’s flexibility and platform independence is achieved by

compiling MATLAB programs into a device-independent p-code, and then interpreting

the p-code instructions at runtime. Unfortunately, the resulting programs can sometimes

execute slowly because the MATLAB code is interpreted rather than compiled.

Disadvantages of MATLAB

MATLAB has two principal disadvantages. The first is that it is an interpreted language and

therefore can execute more slowly than compiled languages. This problem can be mitigated by



properly structuring the MATLAB program, and by the use of the MATLAB compiler to

compile the final MATLAB program before distribution and general use.

The second disadvantage is cost: a full copy of MATLAB is five to ten times more expensive

than a conventional C or Fortran compiler. This relatively high cost is more than offset by then

reduced time required for an engineer or scientist to create a working program, so MATLAB is

cost-effective for businesses. However, it is too expensive for most individuals to consider

purchasing. Fortunately, there is also an inexpensive Student Edition of MATLAB, which is a

great tool for students wishing to learn the language. The Student Edition of MATLAB is

essentially identical to the full edition.

Getting Started

Starting MATLAB

You can start MATLAB by double-clicking on the MATLAB icon or invoking the application

from the Start menu of Windows. The main MATLAB window, called the MATLAB Desktop,

will then pop-up and it will look like this:

Figure 2.1: The Default MATLAB desktop

When MATLAB executes, it can display several types of windows that accept commands or

display information. It integrates many tools for managing files, variables and applications



within the MATLAB environment. The major tools within or accessible from the MATLAB

desktop are:

1. The Current Directory Browser

2. The Workspace Window

3. The Command Window

4. The Command History Window

5. The Start Button

6. The Help Window

If desired, this arrangement can be modified by selecting an alternate choice from [View]

[Desktop Layout]. By default, most MATLAB tools are “docked” to the desktop, so that they

appear inside the desktop window. However, you can choose to “undock” any or all tools,

making them appear in windows separate from the desktop.

The Command Window

The Command Window is where the command line prompt for interactive commands is

located. Once started, you will notice that inside the MATLAB command window is the

text:

Click in the command window to make it active. When a window becomes active, its titlebar

darkens. The “>>” is called the Command Prompt, and there will be a blinking cursor right after

it waiting for you to type something. You can enter interactive commands at the command

prompt (>>) and they will be executed on the spot.



As an example, let’s enter a simple MATLAB command, which is the date command. Click

the mouse where the blinking cursor is and then type date and press the ENTER key.

MATLAB should then return something like this:

Where the current date should be returned to you instead of 01-Sep-2006. Congratulation! You

have just successfully executed your first MATLAB command!

The Command History Window

Figure 2.3: The Command History Window

The Command History Window, contains a log of commands that have been executed within

the command window. This is a convenient feature for tracking when developing or debugging

programs or to confirm that commands were executed in a particular sequence during a multi-

step calculation from the command line.

The Current Directory Browser



Figure 2.4: The Directory Browser

The Current Directory Browser displays a current directory with a listing of its contents. There

is navigation capability for resetting the current directory to any directory among those set in the

path. This window is useful for finding the location of particular files and scripts so that they can

be edited, moved, renamed or deleted. The default directory is the Work subdirectory of the

original MATLAB installation directory.

The Workspace Window



Figure 2.5: The Workspace Window

The Workspace Window provides an inventory of all the items in the workspace that are

currently defined, either by assignment or calculation in the Command Window or by importing

with a load or similar command from the MATLAB command line prompt. These items consist

of the set of arrays whose elements are variables or constants and which have been constructed

or loaded during the current MATLAB session and have remained stored in memory. Those

which have been cleared and no longer are in memory will not be included. The Workspace

Window shows the name of each variable, its value, its array size, its size in bytes, and the class.

Values of a variable or constant can be edited in an Array Editor which is launched by double

clicking its icon in the Workspace Window.

The Help Window



Figure 2.6: The Help Window

You can access the online help in one of several ways. Typing help at the command prompt will

reveal a long list of topics on which help is available. Just to illustrate, try typing help general.

Now you see a long list of “general purpose” MATLAB commands. In addition, you can also get

help on the certain command. For example, date command. The output of help also refers to

other functions that are related. In this example the help also tells you, See also NOW, CLOCK,

DATENUM. You can now get help on these functions using the three different commands as

well.

There is a much more user-friendly way to access the online help, namely via the MATLAB

Help Browser. Separate from the main desktop layout is a Help desktop with its own layout.

This utility can be launched by selecting [Help] [MATLAB Help] from the Help pull down

menu. This Help desktop has a right side which contains links to help with functions, help with

graphics, and tutorial type documentation.

The Start Button

The Start Button (see figure 2.1) allows a user to access MATLAB tools, desktop tools, help

files, etc. it works just like the Start button on a Windows desktop. To start a particular tool, just

click on the Start Button and select the tool from the appropriate sub-menu.

Interrupting Calculations

If MATLAB is hung up in a calculation, or is just taking too long to perform an operation,

you can usually abort it by typing [CTRL + C] (that is, hold down the key labeled CTRL, and

press C).

Ending a Session

One final note, when you are all done with your MATLAB session you need to exit MATLAB.

To exit MATLAB, simply type quit or exit at the prompt. You can also click on the special

symbol that closes your windows (usually an × in the upper right-hand corner). Another way to



exit is by selecting [File] [Exit MATLAB]. Before you exit MATLAB, you should be sure to

save any variables, print any graphics or other files you need, and in general clean up after

yourself.

MATLAB Basics

Variables and Arrays

The fundamental unit of data in any MATLAB program is the array. An array is a collection

of data values organized into rows and columns and known by a single name. Individual data

values within an array are accessed by including the name of the array followed by subscripts in

parentheses that identify the row and column of the particular value. Even scalars are treated as

arrays by MATLAB – they are simply arrays with only one row and one column. There are three

fundamental concepts in MATLAB, and in linear algebra, are scalars, vectors and matrices.

1. A scalar is simply just a fancy word for a number (a single value).

2. A vector is an ordered list of numbers (one-dimensional). In MATLAB they can be

represented as a row-vector or a column-vector.

3. A matrix is a rectangular array of numbers (multi-dimensional). In MATLAB, a two-

dimensional matrix is defined by its number of rows and columns.

In MATLAB matricies are defined inside a pair of square braces ([]). Punctuation marks of a

comma (,), and semicolon (;) are used as a row separator and column separator, respectfully.



You can also use a space as a row separator, and a carriage return (the ENTER key) as a column

separator as well.

Examples 3.1

Below are examples of how a scalar, vector and matrix can be created in MATLAB.

Indexing into an Array

Once a vector or a matrix is created you might needed to extract only a subset of the data, and

this is done through indexing.

Figure 3.1: An array is a collection of data values organized into rows and columns



Individual elements in an array are addressed by the array name followed by the row and

column of the particular element. If the array is a row or column vector, then only one subscript

is required. For example, according to the example 3.1:

o my_vector2(2) is 5

o my_matrix(3,2) is 4 or my_matrix(7) is 4

The Colon Operator

The colon (:) is one of MATLAB’s most important operators. It occurs in several different

forms.

Examples 3.2

1. To create an incremental or a decrement vector

∗ NOTES: If the colon is used by itself within subscript, it refers to all the elements in a row or

column of a matrix!



Concatenating Matrices

Matrix concatenation is the process of joining one or more matrices to make a new matrix. The

expression C = [A B] horizontally concatenates matrices A and B. The expression C = [A; B]

vertically concatenates them.

Examples 3.3

Reshaping a Matrix

Here are a few examples to illustrate some of the ways you can reshape matrices.

Examples 3.4

Reshape 3-by-4 matrix A to have dimensions 2-by-6.

Examples 3.5

Transpose A so that the row elements become columns or vice versa. You can use either the

transpose function or the transpose operator (’). To do this:



General Function for Matrix and Vector

There are many MATLAB features which cannot be included in these introductory notes.

Listed below are some of the MATLAB functions regard to matrix and vector.

Basic Vector Function

MATLAB includes a number of built-in functions that you can use to determine a number of

characteristics of a vector. The following are the most commonly used such functions.

Examples 3.6

The following example demonstrates the use some of these functions.

Functions to Create a Matrix



This following section summarizes the principal functions used in creating and handling

matrices. Most of these functions work on multi-dimensional arrays as well.

Examples 3.7

The following example demonstrates the use some of these functions.

Arithmetic Operations

MATLAB can be used to evaluate simple and complex mathematical expressions. When we

move from scalars to vectors (and matrices), some confusion arises when performing arithmetic

operations because we can perform some operations either on an element-by-element (array

operation) or matrices as whole entities (matrix operation). Expressions use familiar arithmetic

operators:

Array Operators



Examples 3.8

The following example demonstrates the use some of these operations.

Matrix Operators



Examples 3.9

Built-in Variables

MATLAB uses a small number of names for built-in variables. An example is the ans variable,

which is automatically created whenever a mathematical expression is not assigned to another

variable. Table below lists the built-in variables and their meanings. Although you can reassign

the values of these built-in variables, it is not a good idea to do so, because they are used by the

built-in functions.

Examples 3.10



Common MATLAB Functions

A few of the most common and useful MATLAB functions are shown in table below. These

functions will be used in many times. It really helps you when one needs to manage variables

and workspace and to perform an elementary mathematical operation.

Managing Variables and Workspace

Examples 3.11

Built-in Function of Elementary Math



Examples 3.12

Plotting and Visualization

Plotting in MATLAB



MATLAB has extensive facilities for displaying vectors and matrices as graphs, as well as

annotating and printing these graphs. This section describes a few of the most important graphics

functions and provides examples of some typical applications.

Creating a Plot

The plot function has different forms, depending on the input arguments. If y is a vector,

plot(y) produces a piecewise linear graph of the elements of y versus the index of the elements of

y. If you specify two vectors as arguments, plot(x,y) produces a graph of y versus x.

For example, to plot the value of the sine function from zero to 2π, use:

This is the basic command of plotting a graph. Besides that, MATLAB has commands which

will let you add titles and labels and others in order to make your figures more readable.

However, you need to keep the figure window open while executing these commands.



The current limits of this plot can be determined from the basic axis function. So, in order to

modify the x-axis within [0 2π], you need to run this function:

Annotating Plots

You can adjust the axis tick-mark locations and the labels appearing at each tick mark. For

example, this plot of the sine function relabels the x-axis with more meaningful values.



Creating a Semilogarithmic Plot

Semilogarithmic plot is another type of figuring a graph by rescaling if the new data falls outside

the range of the previous axis limits.

Specifying the Color and Size of Lines

You can control a number of line style characteristics by specifying values for line properties.



Multiple Plots

Often, it is desirable to place more than one plot in a single figure window. This is achieved

by three ways:

The subplot Function

The subplot Function breaks the figure window into an m-by-n matrix of small subplots and

selects the ith subplot for the current plot. The plots are numbered along the top row of the figure

window, then the second row, and so forth.



Multiple plots

You can assign different line styles to each data set by passing line style identifier strings to plotand placing a legend on the plot to identify curves drawn with different symbol and line types.

The hold Function

The hold command will add new plots on top of previously existing plots.



Line Plots in Three-Dimensions

Now, the three-dimension analog of the plot function is plot3. if x, y and z are three vectors of the

same length, plot3(x,y,z) generates a line in 3-D through the points whose coordinates are the

elements of x, y and z and then produces a 2-D projection of that line on the screen.

Three-Dimensional Surface Mesh Plots

The first step in displaying a function of two variables, z = f(x,y), is to generate X and Y matrices

consisting of repeated rows and columns, respectively, over the domain of the function. Then use

these matrices to evaluate and graph the function. Meshgrid function transforms the domain

specified by two-vectors, x and y, into matrices X and Y .

Programming

Data Types

There are many different types of data that you can work with in MATLAB. You can build

matrices and arrays of floating point and integer data, characters and strings, logical true and

false states, etc. you can also develop your own data types using MATLAB classes. Two of the



MATLAB data types, structures and cell arrays, provide a way to store dissimilar types of data in

the same array.

There are 15 fundamental data types (or classes) in MATLAB. Each of these data types is in the

form of an array. This array is a minimum of 0-by-0 in size and can grow to an n-dimensional

array of any size. Two-dimensional versions of these arrays are called matrices. All of the

fundamental data types are shown in lowercase text in the diagram below. Additional data types

are user-defined, object-oriented user classes (a subclass of structure), and java classes, that you

can use with the MATLAB interface to Java. Matrices of type double and logical may be either

full or sparse. For matrices having a small number of nonzero elements, a sparse matrix requires

a fraction of the storage space required for an equivalent full matrix. Sparse matrices invoke

special methods especially tailored to solve sparse problems.



M-File Programming

MATLAB provides a full programming language that enables you to write a series of MATLAB

statements into a file and then execute them with a single command. You write your program in

an ordinary text file, giving the file a name of filename.m. The term you use for filename

becomes the new command that MATLAB associates with the program. The file extension of .m

makes this a MATLAB M-file. M-files can be scripts that simply execute a series of MATLAB

statements, or they can be functions that also accept arguments and produce output. You create



M-files using a text editor, then use them as you would any other MATLAB function or

command. The process looks like this:

Kinds of M-files

There are two kinds of M-files

Functions

Functions are M-files that accept input arguments and return output arguments. They operate on

variables within their own workspace. This is separate from the workspace you access at the

MATLAB command prompt

Example 5.2



If you would like, try entering these commands in an M-file called average.m. The average

function accepts a single input argument and returns a single output argument. To call the

average function, enter

The Function Definition Line

The function definition line informs MATLAB that the M-file contains a function, and specifies

the argument calling sequence of the function. The function definition line for the average

function is

All MATLAB functions have a function definition line that follows this pattern.



The Function Name - MATLAB function names have the same constraints as variable names.

The name must begin with a letter, which may be followed by any combination of letters, digits,

and underscores. Making all letters in the name lowercase is recommended as it makes your M-

files portable between platforms.

Flow Control

MATLAB has several flow control constructs:

1. if

2. continue

3. break

4. switch and case

5. for

6. while

If

The if statement evaluates a logical expression and executes a group of statements when the

expression is true. The optional elseif and else keywords provide for the execution of alternate

groups of statements. An end keyword, which matches the if, terminates the last group of

statements. The groups of statements are delineated by the four keywords--no braces or brackets

are involved

Continue

The continue statement passes control to the next iteration of the for or while loop in which it

appears, skipping any remaining statements in the body of the loop. In nested loops, continue

passes control to the next iteration of the for or while loop enclosing it.



Break

The break statement lets you exit early from a for or while loop. In nested loops, break exits from

the innermost loop only.

Switch and Case

The switch statement executes groups of statements based on the value of a variable or

expression. The keywords case and otherwise delineate the groups. Only the first matching case

is executed. There must always be an end to match the switch.

For

The for loop repeats a group of statements a fixed, predetermined number of times. A matching

end delineates the statements.

While

The while loop repeats a group of statements an indefinite number of times under control of alogical condition. A matching end delineates the statements.



SIMULINK

Simulink is a graphical extension to MATLAB for the modeling and simulation of

systems. In Simulink, systems are drawn on screen as block diagrams. Many elements of block

diagrams are available (such as transfer functions, summing junctions, etc.), as well as virtual

input devices (such as function generators) and output devices (such as oscilloscopes). Simulink

is integrated with MATLAB and data can be easily transferred between the programs. In this

tutorial, we will introduce the basics of using Simulink to model and simulate a system.

Simulink is supported on Unix, Macintosh, and Windows environments, and it is

included in the student version of MATLAB for personal computers. For more information on

Simulink, contact the MathWorks.

The idea behind these tutorials is that you can view them in one window while running Simulink

in another window. Do not confuse the windows, icons, and menus in the tutorials for your

actual Simulink windows. Most images in these tutorials are not live - they simply display what

you should see in your own Simulink windows. All Simulink operations should be done in your

Simulink windows.

STARTING SIMULINK

Simulink is started from the MATLAB command prompt by entering the following command:

simulink

Alternatively, you can click on the "Simulink Library Browser" button at the top of the

MATLAB

command window as shown below:



The Simulink Library Browser window should now appear on the screen. Most of the blocks

needed for modeling basic systems can be found in the subfolders of the main "Simulink" folder

(opened by clicking on the "+" in front of "Simulink"). Once the "Simulink" folder has been

opened, the Library Browser window should look like:



Basic Elements

There are two major classes of elements in Simulink: blocks and lines. Blocks are used to

generate, modify, combine, output, and display signals. Lines are used to transfer signals from

one block to another.

Blocks

The subfolders underneath the "Simulink" folder indicate the general classes of blocks available

for us to use:

· Continuous: Linear, continuous-time system elements (integrators, transfer functions,

state-space models, etc.)

· Discrete: Linear, discrete-time system elements (integrators, transfer functions, statespace

models, etc.)

· Functions & Tables: User-defined functions and tables for interpolating function values



· Math: Mathematical operators (sum, gain, dot product, etc.)

· Nonlinear: Nonlinear operators (coulomb/viscous friction, switches, relays, etc.)

· Signals & Systems: Blocks for controlling/monitoring signal(s) and for creating

subsystems

· Sinks: Used to output or display signals (displays, scopes, graphs, etc.)

· Sources: Used to generate various signals (step, ramp, sinusoidal, etc.)

Blocks have zero to several input terminals and zero to several output terminals. Unused input

terminals are indicated by a small open triangle. Unused output terminals are indicated by a

small triangular point. The block shown below has an unused input terminal on the left and an

unused output terminal on the right.

Lines

Lines transmit signals in the direction indicated by the arrow. Lines must always transmit signals

from the output terminal of one block to the input terminal of another block. One exception to

this is that a line can tap off of another line. This sends the original signal to each of two (or

more) destination blocks, as shown below:

Lines can never inject a signal into another line; lines must be combined through the use of a

block such as a summing junction.

A signal can be either a scalar signal or a vector signal. For Single-Input, Single-Output systems,

scalar signals are generally used. For Multi-Input, Multi-Output systems, vector signals are often

used, consisting of two or more scalar signals. The lines used to transmit scalar and vector



signals are identical. The type of signal carried by a line is determined by the blocks on either

end of the line.

Building a System

To demonstrate how a system is represented using Simulink, we will build the block diagram for

a simple model consisting of a sinusoidal input multiplied by a constant gain, which is shown

below:

This model will consist of three blocks: Sine Wave, Gain, and Scope. The Sine Wave is a Source

Block from which a sinusoidal input signal originates. This signal is transferred through a line in

the direction indicated by the arrow to the Gain Math Block. The Gain block modifies its input

signal (multiplies it by a constant value) and outputs a new signal through a line to the Scope

block. The Scope is a Sink Block used to display a signal (much like an oscilloscope). We begin

building our system by bringing up a new model window in which to create the block diagram.

This is done by clicking on the "New Model" button in the toolbar of the Simulink Library

Browser (looks like a blank page).

Building the system model is then accomplished through a series of steps:

1. The necessary blocks are gathered from the Library Browser and placed in the model

window.

2. The parameters of the blocks are then modified to correspond with the system we are

modelling.



3. Finally, the blocks are connected with lines to complete the model.

Each of these steps will be explained in detail using our example system. Once a system is built,

simulations are run to analyze its behavior.

Gathering Blocks

Each of the blocks we will use in our example model will be taken from the Simulink Library

Browser. To place the Sine Wave block into the model window, follow these steps:

1. Click on the "+" in front of "Sources" (this is a subfolder beneath the "Simulink" folder) to

display the various source blocks available for us to use.

2. Scroll down until you see the "Sine Wave" block. Clicking on this will display a short

explanation of what that block does in the space below the folder list:

3. To insert a Sine Wave block into your model window, click on it in the Library Browser

and drag the block into your workspace.

The same method can be used to place the Gain and Scope blocks in the model window. The

"Gain" block can be found in the "Math" subfolder and the "Scope" block is located in the "Sink"



subfolder. Arrange the three blocks in the workspace (done by selecting and dragging an

individual block to a new location) so that they look similar to the following:

Modifying the Blocks

Simulink allows us to modify the blocks in our model so that they accurately reflect the

characteristics of the system we are analyzing. For example, we can modify the Sine Wave block

by double-clicking on it. Doing so will cause the following window to appear:

This window allows us to adjust the amplitude, frequency, and phase shift of the sinusoidal

input. The "Sample time" value indicates the time interval between successive readings of the

signal. Setting this value to 0 indicates the signal is sampled continuously.



Let us assume that our system's sinusoidal input has:

· Amplitude = 2

· Frequency = pi

· Phase = pi/2

Enter these values into the appropriate fields (leave the "Sample time" set to 0) and click "OK" to

accept them and exit the window. Note that the frequency and phase for our system contain 'pi'

(3.1415...). These values can be entered into Simulink just as they have been shown. Next, we

modify the Gain block by double-clicking on it in the model window. The following window

will then appear:

Note that Simulink gives a brief explanation of the block's function in the top portion of this

window. In the case of the Gain block, the signal input to the block (u) is multiplied by a

constant (k) to create the block's output signal (y). Changing the "Gain" parameter in this

window changes the value of k.

For our system, we will let k = 5. Enter this value in the "Gain" field, and click "OK" to close

the window.

The Scope block simply plots its input signal as a function of time, and thus there are no system

parameters that we can change for it. We will look at the Scope block in more detail after we

have run our simulation.



Connecting the Blocks

For a block diagram to accurately reflect the system we are modeling, the Simulink blocks must

be properly connected. In our example system, the signal output by the Sine Wave block is

transmitted to the Gain block. The Gain block amplifies this signal and outputs its new value to

the Scope block, which graphs the signal as a function of time. Thus, we need to draw lines from

the output of the Sine Wave block to the input of the Gain block, and from the output of the Gain

block to the input of the Scope block.

Lines are drawn by dragging the mouse from where a signal starts (output terminal of a

block) to where it ends (input terminal of another block). When drawing lines, it is important to

make sure that the signal reaches each of its intended terminals. Simulink will turn the mouse

pointer into a crosshair when it is close enough to an output terminal to begin drawing a line, and

the pointer will change into a double crosshair when it is close enough to snap to an input

terminal. A signal is properly connected if its arrowhead is filled in. If the arrowhead is open, it

means the signal is not connected to both blocks. To fix an open signal, you can treat the open

arrowhead as an output terminal and continue drawing the line to an input terminal in the same

manner as explained below

Properly Connected Signal Open Signal

When drawing lines, you do not need to worry about the path you follow. The lines will route

themselves automatically. Once blocks are connected, they can be repositioned for a neater

appearance. This is done by clicking on and dragging each block to its desired location (signals

will stay properly connected and will re-route themselves).

After drawing in the lines and repositioning the blocks, the example system model should look

like:



In some models, it will be necessary to branch a signal so that it is transmitted to two or more

different input terminals. This is done by first placing the mouse cursor at the location where the

signal is to branch. Then, using either the CTRL key in conjunction with the left mouse buttonor just the right mouse button, drag the new line to its intended destination. This method was

used to construct the branch in the Sine Wave output signal shown below:

The routing of lines and the location of branches can be changed by dragging them to their

desired new position. To delete an incorrectly drawn line, simply click on it to select it, and hit

the DELETE key.



Running Simulations

Now that our model has been constructed, we are ready to simulate the system. To do this, go to

the Simulation menu and click on Start, or just click on the "Start/Pause Simulation" button in

the model window toolbar (looks like the "Play" button on a VCR). Because our example is a

relatively simple model, its simulation runs almost instantaneously. With more complicated

systems, however, you will be able to see the progress of the simulation by observing its running

time in the the lower box of the model window. Double-click the Scope block to view the output

of the Gain block for the simulation as a function of time. Once the Scope window appears, click

the "Autoscale" button in its toolbar (looks like a pair of binoculars) to scale the graph to better

fit the window. Having done this, you should see the following:

Note that the output of our system appears as a cosine curve with a period of 2 seconds and

amplitude equal to 10. Does this result agree with the system parameters we set? Its amplitude

makes sense when we consider that the amplitude of the input signal was 2 and the constant gain

of the system was 5 (2 x 5 = 10). The output's period should be the same as that of the input

signal, and this value is a function of the frequency we entered for the Sine Wave block (which

was set equal to pi). Finally, the output's shape as a cosine curve is due to the phase value of pi/2we set for the input (sine and cosine graphs differ by a phase shift of pi/2).

What if we were to modify the gain of the system to be 0.5? How would this affect the output of

the Gain block as observed by the Scope? Make this change by double-clicking on the Gain

block and changing the gain value to 0.5. Then, re-run the simulation and view the Scope (the



Scope graph will not change unless the simulation is re-run, even though the gain value has been

modified). The Scope graph should now look like the following:

Note that the only difference between this output and the one from our original system is the

amplitude of the cosine curve. In the second case, the amplitude is equal to 1, or 1/10th of 10,

which is a result of the gain value being 1/10th as large as it originally was.



CHAPTER-5

SIMULATION RESULTS

SIMULINK BLOCK DIAGRAM OF PERMANENT MAGNET SYNCHRONOUS MACHINE



SPEED CONTROL OF PMSM



ELECTROMAGNETIC TORQUE OF PMSM

modeling and simulation of permanent

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