Simulation of a Vector-Controlled PermanentMagnet Synchronous Motor Drive

ByAsish Kumar Mondal

Registration No. 210608010 of 2008-09Roll No. 160806010

Under the Supervision of

Dr. Kaushik MukherjeeDr. Mainak Sengupta

AThesis

Submitted in partial fulfillmentfor the requirements for the degree of

Master of Engineering (Electrical Engineering)

Specialization: Power Electronics and Drives

Department of Electrical EngineeringBengal Engineering and Science University,

ShibpurHowrah - 711103

West Bengal, IndiaMay 2011

Dedicated to my parents

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BENGAL ENGINEERING AND SCIENCE UNIVERSITYHOWRAH-711103

FOREWORD

We hereby forward the thesis entitled “Simulation of a Vector-ControlledPermanent Magnet Synchronous Motor Drive” submitted by Asish KumarMondal (Registration No. 210608010 of 2008-2009) as a bona-fide record of theproject work carried out by him under our supervision, in partial fulfillment of therequirements for the award of the degree of Master of Engineering in ElectricalEngineering (Specialization: Power Electronics and Drives) from this University.

Dr. Kaushik Mukherjee Dr. Mainak Sengupta(Supervisor) (Supervisor)

Dept.of Electrical Engineering Dept.of Electrical EngineeringBengal Engineering and Science University Bengal Engineering and Science University

Howrah-711103 Howrah-711103

Forwarded by:

Dr. Sukumar Chandra Konar Dr. Amit K. Das(Prof. and Head) (Prof. and Dean, FE & T)

Dept.of Electrical Engineering Bengal Engineering and Science UniversityBengal Engineering and Science University Howrah-711103

Howrah-711103

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BENGAL ENGINEERING AND SCIENCE UNIVERSITYHOWRAH-711103

CERTIFICATE OF APPROVAL

We hereby approve the thesis entitled “Simulation of a Vector-Controlled Permanent Magnet Synchronous Motor Drive” submitted byAsish Kumar Mondal (Registration No. 210608010 of 2008-2009) as a bona-fide record of the project work carried out by him under the supervision of Dr.Kaushik Mukherjee and Dr. Mainak Sengupta in partial fulfillment of therequirements for the degree of Master of Engineering in Electrical Engineering(Specialization: Power Electronics and Drives) from this University.

BOARD OF EXAMINERS

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BENGAL ENGINEERING AND SCIENCE UNIVERSITYHOWRAH-711103

ACKNOWLEDGEMENTS

I must take this opportunity to place on record my deep sense of respectand gratitude to Dr. Kaushik Mukherjee and Dr. Mainak Sengupta, who have in-troduced me in the present area of work and guided me in this work. I also wish tothank Prof. Debjani Ganguly and Dr. Prasid Syam for helping me with differentsuggestions.I am also thankful to Dr. Sukumar Chandra Konar, Prof. and Head,Dept. of EE, BESU, Shibpur, for providing the necessary support and infrastruc-ture. I am also indebted to Mr. Molay Roy, Dinesh Paswan, Chandrasekher Roy,Dipankar Debnath, Rakesh Roy, and also Mr. Avijit Ghose for helping me a lotin continuing the work. Last but not the least, I am grateful to the Almighty forpresenting me one of the best parents in this world.

(Asish Kumar Mondal)Reg. No. 210608010Roll No. 160806010

Bengal Engineering and Science University

Date:

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Abstract

This project is primarily aimed at developing a fully-operational laboratory pro-totype of a VSI-fed vector-controlled PMSM drive built around an indigenouslydeveloped motor 1kW, 48 V DC, 4-pole, 2000 rpm PMSM. A model of the motorfed through a sinusoidal pulse width modulated (SPWM) inverter system operat-ing with rotor position information, is first developed. A vector control schemeutilizing feedforward compensation technique is next proposed. The entire systemis next simulated offline in the MATLAB-SIMULINK software environment. Thesteady state and dynamic performance results have been presented. Finally, thesystem is partially simulated in real time on a Field Programmable Gates Array(FPGA) based platform built around an Altera Cyclone EP1C12Q240C8 devicewith Altera’s Quartus II software. Oscilloscopic records of the real-time simulatedsystem have been obtained also.

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Contents

1 Introduction 11.1 General discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 PMSM and brush-less versus brushed DC motors . . . . . . 21.2 Permanent Magnet Synchronous Motor Drives . . . . . . . . . . . 3

1.2.1 Permanent Magnet Synchronous Motor . . . . . . . . . . . 31.2.2 Inverter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.3 Position Sensor . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.4 Control unit . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Motivation for the Present Work . . . . . . . . . . . . . . . . . . . 61.4 Relevance and significance of the project undertaken . . . . . . . . 71.5 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . 7

2 Mathematical models and theory of vector control of PermanentMagnet Synchronous Machines 82.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Mathematical Model of a PMSM . . . . . . . . . . . . . . . . . . . 82.3 Mathematical model of three phase inverter . . . . . . . . . . . . . 122.4 SPWM in three phase inverter . . . . . . . . . . . . . . . . . . . . 142.5 Theory of Vector-Controlled PMSM . . . . . . . . . . . . . . . . . 162.6 Feed-forward compensation for achieving Vector-Control . . . . . 17

3 Design of controllers and Off-line Simulation of a Vector-ControlledPMSM Drive 20

3.0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 203.1 Design of controllers . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Simulation of the developed vector control strategy . . . . . . . . . 233.3 Simulation in MATLAB-SIMULINK platform and results . . . . . 24

4 Online or Real Time Simulation of a vector controlled PMSMdrive 334.1 Real time simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Integration Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 35

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4.2.1 Backwards Euler’s Method . . . . . . . . . . . . . . . . . . 354.3 Simulation of a series R-L-C Circuit . . . . . . . . . . . . . . . . . 36

4.3.1 FPGA design files . . . . . . . . . . . . . . . . . . . . . . . 384.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.4 Simulation of a vector controlled PMSM drive in FPGA platform . 414.5 Developments of a Model of the PMSM in FPGA . . . . . . . . . . 41

4.5.1 Normalized Equations . . . . . . . . . . . . . . . . . . . . . 414.5.2 Testing of the PMSM Block . . . . . . . . . . . . . . . . . . 42

4.6 Three-Phase Sine-PWM Inverter . . . . . . . . . . . . . . . . . . . 434.6.1 Implementation of Three-Phase Sine-PWM Inverter in FPGA 434.6.2 Carrier Generation . . . . . . . . . . . . . . . . . . . . . . . 434.6.3 Generation of Phase and Line voltage waveforms . . . . . . 44

4.7 Design of two phase to three phase Transformation in FPGA . . . 484.7.1 Testing of two phase to three phase Transformation Block . 484.7.2 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.8 Design of three phase to two phase Transformation in FPGA plat-form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.8.1 Testing of three phase to two phase transformation block . 504.8.2 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.9 PI controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5 Conclusions and Scope for future work 555.1 General Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 555.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

appendices 56

A 57A.1 Parameters and specifications of the PMSM under study . 57

B 58B.1 Block diagram and specifications of the FPGA kit . . . . . 58

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List of Figures

1.1 Permanent Magnet Synchronous Motor Drives . . . . . . . . . . . 31.2 Permanent Magnet Synchronous Motor . . . . . . . . . . . . . . . 41.3 Three phase inverter . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 PMSM model block in D-Q reference frame, denoting input andoutput variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Cross sectional view showing 3 phase winding in stator and perma-nent magnet in rotor and the rotor reference frame. . . . . . . . . . 10

2.3 D-axis equivalent circuit of the PMSM . . . . . . . . . . . . . . . 112.4 Q-axis equivalent circuit of the PMSM . . . . . . . . . . . . . . . 122.5 Mathematical model of the 3 phase 2 level voltage source inverter. 122.6 Three-Phase two level Sine-PWM Inverter with three phase load. 132.7 SPWM strategy in of a 3 phase, 2 level VSI . . . . . . . . . . . . . 152.8 Three-Phase two level Sine-PWM Inverter logic and phase and line

voltage waveform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.9 D-axis system representation of the PMSM considering feedforward

compensation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.10 Q-axis system representation of the PMSM considering feedforward

compensation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1 D-axis current control loop (system) to achieve vector control ofPMSM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Control system of the Fig3.1in simplified pole-zero form. . . . . . . 213.3 Control system of the Fig3.1 after pole-zero cancelation. . . . . . . 223.4 Model of the Vector-Controlled PMSM drive. . . . . . . . . . . . . 223.5 Q-axis current control loop (system) to achieve vector control of

PMSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.6 Simulated irds response with time, as ir∗ds (reference value of D-axis

armature current) for the Vector-Controlled PMSM drive is keptat zero all throughout but step increment in ir∗qs are given, first atstarting and then at 7 seconds, on load. Load setting are: TL = 0and f = 0.0606Nm-sec/radian.(value corresponding to about ratedtorque at rated speed.) . . . . . . . . . . . . . . . . . . . . . . . . 26

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3.7 Simulated irqs response with time, for the Vector-Controlled PMSMdrive, with ir∗ds is kept at zero all throughout but step increment of1Amps. in ir∗qs is given, first at starting and then again a further in-crement of 1 Amps. (finally to 2 Amps.) is introduced at 7 seconds,on load. Load setting are same as mentioned in Fig.3.6 . . . . . . . 27

3.8 Same irds (top) and irqs (bottom) response as given in Fig.3.6 andFig.3.7 with time scale expanded in and around 7 seconds (instantat which ir∗qs is given a step increment to 2Amps. finally from 1Amps.) 28

3.9 Simulated Te (instantaneous electromagnetic torque) response withtime for the Vector-Controlled PMSM drive, with operating condi-tions remaining same as mentioned in Fig. 3.6 . . . . . . . . . . . 29

3.10 Same Te waveform with time as that of Fig.3.9, but with time scaleexpanded in and around 7 seconds (instant at which ir∗qs is given astep increment to 2Amps. finally from 1Amps.) . . . . . . . . . . . 30

3.11 Simulated ωr (electrical Speed in rad/sec) response with time for theVector-Controlled PMSM drive for operating conditions as detailedunder the caption of Fig.3.6 . . . . . . . . . . . . . . . . . . . . . 31

3.12 Simulated phase current waveforms of the PMSM’s armature (ia,ib, ic) during steady state. . . . . . . . . . . . . . . . . . . . . . . 32

4.1 Triggering Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2 Block Diagram of Backwards Euler’s Method . . . . . . . . . . . . 354.3 Series RLC Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.4 FPGA design file of a series RLC circuit . . . . . . . . . . . . . . 384.5 Transient waveforms of circuit current and input applied voltage for

Vg = 100V , R = 10Ω, L = 20mH and C is forced to a value veryvery high, i.e. representing a series R-L circuit case. . . . . . . . . 39

4.6 Transient waveforms of the real-time simulated capacitor voltageand input applied voltage for Vg = 100V , R = 10Ω, L = 20mH,C = 4uF in a series R-L-C circuit. . . . . . . . . . . . . . . . . . . 40

4.7 Transient waveform of iqin matlab simulink with no load and vd=20Vvq=150V. Peak value of the waveform is 18 Amps or 1.8 pu . . . . 43

4.8 Simulated transient waveform of iqin FPGA with no load and 0.04pe-runit vd and 0.3perunit vq Peak value of the waveform is 1.8 pu(5volt is one pu) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.9 Transient waveform of idin matlab simulink with no load and vd=20Vand vq=150V. Peak value of the waveform is 9 Amps or 0.9 pu andsteady state value is 0.43pu . . . . . . . . . . . . . . . . . . . . . . 45

4.10 Simulated transient waveform of idin FPGA with no load and 0.04pe-runit vd and 0.3perunit vq. Peak value of the waveform is 0.9 pu (5volt is one pu) and steady state value is 0.43pu . . . . . . . . . . . 46

4.11 Transient waveform of electrical speed(in rad/sec ) in matlab simulinkwith no load and vd=20V and vq=150V . . . . . . . . . . . . . . . 47

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4.12 Simulated transient waveform of electrical speed(in rad/sec )in FPGAwith no load and 0.04 per unit vd and 0.3 per unit vq . . . . . . . . 48

4.13 Block diagram of carrier generation . . . . . . . . . . . . . . . . . 494.14 Triangle Carrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.15 A phase voltage of a sine PWM inverter with 50Hz reference signal

and 1pu dc link voltage. . . . . . . . . . . . . . . . . . . . . . . . . 504.16 Gate pulse for the switching device T1. . . . . . . . . . . . . . . . . 514.17 FFT analysis of the signal in Fig 4.16. in Tektronix oscilloscope(TDS-

1001). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.18 Output of the two phase to three phase transformation block. ‘A’

phase in channel 1 and ‘B’ phase in channel 2 . . . . . . . . . . . . 534.19 Output result of the three phase to two phase transformation block.

It is connected with a two phase to three phase transformationblock. In put of the two phase to three phase transformation blockv(d)=2.5v and v(q)=2.5v. . . . . . . . . . . . . . . . . . . . . . . . 54

B.1 Block diagram of the FPGA kit. . . . . . . . . . . . . . . . . . . . 58

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Chapter 1

Introduction

1.1 General discussions

The availability of high energy density permanent magnets (PM) has usheredin the era of high-performance drives using PM motors. The drawbacks of thecommutator-brush assembly of the conventional DC motors has narrowed downthe option to the use of Permanent Magnet Synchronous Motors (PMSM) andBrushless DC motors (BLDC).BLDC motors and PMSM can potentially be deployed in any area currently ful-filled by brushed DC motors. Cost and control complexity prevent PMSM andBLDC motors from replacing brushed motors in most common areas of use. Nev-ertheless, BLDC motors have come to dominate many applications: Consumerdevices such as computer hard drives, CD/DVD players, and PC cooling fans useBLDC motors exclusively. Low speed, low power brushless DC motors are usedin direct-drive turntables for “analog” audio records to name a few applications.Application of high power BLDC motors are found in electric vehicles, hybridvehicles and some industrial machinery. The Segway Scooter, a two-wheeled, self-balancing electric vehicle, is an example. An established commercial product inthe form of the Vectrix Maxi-Scooter also uses BLDC technology. A number ofelectric bicycles use BLDC motors that are sometimes built right into the wheelhub itself. As elsewhere, researchers in the Department of Electrical Engineering,BESU, Shibpur, also initiated efforts towards developing prototypes of PMSMand BLDC drive started around the year 2004. A complete set-up is in operationaround a motor that was also fully designed and fabricated in the laboratory. Thepresent project is a part of a journey forward.

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1.1.1 PMSM and brush-less versus brushed DC mo-tors

Brushed DC motors have been in commercial use since 1886. BLDC motors,however have only been commercially possible since 1962. Limitations of brushedDC motors overcome by BLDC motors include lower efficiency and susceptibility ofthe commutator assembly to mechanical wear and consequent need for servicing,at the cost of potentially less rugged and more complex and expensive controlelectronics [1].

In the BLDC motor, the electromagnets do not move; instead, the permanentmagnets rotate and the armature remains static. This gets around the prob-lem of how to transfer current to a moving armature. In order to do this, thebrush-system/commutator assembly is replaced by an electronic controller. Thecontroller performs the same timed power distribution found in a brushed DCmotor, but using a solid-state circuit rather than a commutator/brush system [1].

Because of induction of the windings, power requirements, and temperaturemanagement, some interface circuitry is necessary between digital controller andmotor. The multiple transitions between high and low voltage levels are crude ap-proximations to a trapezoid or (ideally) a sinusoid; they reduce harmonic content[1]. BLDC motors offer several advantages over brushed DC motors, includinghigher efficiency and reliability, reduced noise, longer lifetime (no brush and com-mutator erosion), elimination of ionizing sparks from the commutator, more power,and overall reduction of electromagnetic interference (EMI) [1]. With no windingson the rotor, they are not subjected to centrifugal forces, and because the wind-ings are supported by the housing, they can be cooled by conduction, requiring noairflow inside the motor for cooling. This in turn means that the motor’s internalscan be entirely enclosed and protected from dirt or other foreign matter.

The maximum power that can be applied to a PMSM motor is exception-ally high, limited almost exclusively by heat, which can weaken the magnets(Neodymium-iron-boron magnets typically have Curie Temperatures of 3100C).A PMSM motor’s main disadvantage is higher cost, which arises from two issues.First, PMSM motors require complex electronic speed controllers to run. BrushedDC motors can be regulated by a comparatively simple controller, such as a rheo-stat (variable resistor). However, this reduces efficiency because power is wastedin the rheostat. Second, some practical uses have not been well developed in thecommercial sector. For example, in the Radio Control (RC) hobby, even commer-cial brushless motors are often hand-wound while brushed motors use armaturecoils which can be inexpensively machine-wound [1].

PMSM motors are often more efficient at converting electricity into mechanicalpower than brushed DC motors. This improvement is largely due to the absenceof electrical and friction losses due to brushes. The enhanced efficiency is greatestin the no-load and low-load region of the motor’s performance curve. Under highmechanical loads, PMSM motors and high-quality brushed motors are comparable

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in efficiency.Vector drives are DC controllers that take the extra step of converting back

to AC for the motor; they are sophisticated inverters. The DC-to-AC conversioncircuitry is usually expensive and less efficient, but these have the advantage ofbeing able to run smoothly at very low speeds or completely stop (and providetorque) in a position not directly aligned with a pole. Motors used with a vectordrive are typically called AC motors. When running at low speeds and under load,they don’t cool themselves significantly; temperature rise has to be allowed for.

A motor can be optimized for AC (i.e. vector control) or it can be optimized forDC (i.e. block commutation). A motor which is optimized for block commutationwill typically generate trapezoidal EMF. One can easily observe the shape of theEMF by connecting the motor wires (at least two of them) to a ’scope and thenhand-cranking/spinning the shaft.

1.2 Permanent Magnet Synchronous Motor

Drives

Permanent Magnet Synchronous Motor Drives consists of four main components,the Permanent Magnet (PM) motor, inverter, control unit and the position sensor.

Figure 1.1: Permanent Magnet Synchronous Motor Drives

1.2.1 Permanent Magnet Synchronous Motor

A permanent magnet synchronous motor (PMSM) is a motor that uses permanentmagnets to produce the air gap magnetic field rather than using electromagnets.

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These motors have significant advantages, attracting the interest of researchersand industry for use in many applications.

Figure 1.2: Permanent Magnet Synchronous Motor

1.2.2 Inverter

Voltage Source Inverters are power electronics converters where the average powerflows from the DC voltage side to AC voltage side with variable frequency andmagnitude. They are very commonly used in adjustable speed drives and are char-acterized by a well defined switched voltage waveform in the terminals. The ACvoltage frequency can be variable or constant depending on the application. Three

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Figure 1.3: Three phase inverter

phase inverters consist of six power switches connected to a DC voltage source.The inverter switches must be carefully selected based on the requirements of op-eration, ratings and the application. There are several devices available today andthese are thyristors, bipolar junction transistors (BJTs), Metal Oxide Semicon-ductor Field Fffect Transistors (MOSFETs), Insulated Gate Bipolar Transistors(IGBTs), Gate Turn Off Thyristors (GTOs) and Integrated Gate CommutatedThyristors (IGCT). MOSFETs and IGBTs are preferred by industry because ofthe MOS gating permits high power gain and control advantages. While MOSFETis 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 andmedium voltage and power ranges. The power devices when used in motor drivesapplications require an inductive current path provided by antiparallel (freewheel-ing) diodes when the switch is turned off. Inverters with antiparallel diodes areshown in fig. 1.3.

1.2.3 Position Sensor

Operation of permanent magnet synchronous motors requires position sensors inthe rotor shaft when operated without damper winding. The need of knowingthe rotor position requires the development 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 onesmost commonly used for motors are encoders and revolvers. Depending on the ap-plication and performance desired by the motor a position sensor with the requiredaccuracy can be selected.

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Optical Encoders

The most popular type of encoder is the optical encoder, which consists of a rotat-ing disk, a light source, and a photo detector (light sensor). The disk, is mountedon the rotating shaft, has coded patterns of opaque and transparent sectors. Asthe disk rotates, these patterns interrupt the light emitted onto the photo detector,generating a digital pulse or output signal. Optical encoders offer the advantagesof digital interface. There are two types of optical encoders, incremental encoderand absolute encoder.

Incremental encoders

Incremental encoders have good precision and are simple to implement but theysuffer from lack of information when the motor is at rest position and in orderfor precise position the motor most be stop at the starting point. The mostcommon type of incremental encoder uses two output channels (A and B) to senseposition. Using two code tracks with sectors positioned 900 degrees out of phase,the two output channels of the quadrature encoder indicate both position anddirection of rotation. If A leads B, for example, the disk is rotating in a clockwisedirection. If B leads A, then the disk is rotating in a counter-clockwise direction.It can be noted that for vector control with sinusoidal pulse width modulated(SPWM) inverter or Spcae Vector Pulse Width Modulated (SVPWM) inverter,rotor position information at very high resolutions and with great accuracy is amust. This is often achieved by incremental encoders.

1.2.4 Control unit

The controller is the heart of the drives system. In modern days micro-controllers,microprocessors, digital signal processors (DSP) and FPGA based processors areused to provide the intelligence of the drive system. Control unit senses the presentstate of the system and take the necessary action according to the control require-ment.

1.3 Motivation for the Present Work

The present work is a continuation of efforts to develop high-performance PMSMand BLDC drives in the Department of Electrical Engineering of this University.It may be mentioned here that the prototype motor used in this work has beenfully designed and fabricated here as a part of a previous postgraduate work.The motor thus developed was later on used in another subsequent post-graduatework to successfully develop a PMSM drive system. However, in that work theinverter was run in the 120O conduction mode with inverter switching obviouslycontrolled through rotor-position feedback. The next target set was to develop a

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high-performance PMSM drive with vector control. Of course, the inverter thenhas to operate with an appropriate PWM logic. Also, voltage and current sensingand feedback need to be implemented. The first step towards development ofsuch a drive may be to verify its projected performance through both off-line andrealtime simulation. Hence this project.

1.4 Relevance and significance of the project

undertaken

As outlined in the abstract, the vector control scheme presented in this thesis,rests on a feed-forward compensation technique. It will be seen later that theresults thus obtained are extremely encouraging and remove certain short-comingsof the scheme presented in the previous work. This can be looked upon as acontribution of this thesis. It is conjectured that when this scheme will be triedon the practical set-up accurate results will be obtained having both very goodsteady-state performance and excellent dynamic response.

1.5 Organization of the thesis

The thesis is organized in five chapters.

After the present introductory chapter, Chapter 2 deals with the modellingof the PMSM drive system consisting of a model of the machine in Park’s D-Qreference frame [4] and model of the inverter operating with sinusoidal PWM logic.Finally a brief theory of vector-control vis-a-vis the PMSM drive is presented.

Chapter 3 deals with off-line simulation of the PMSM drive system in MAT-LAB Simulink software. The design of the PI-controllers is also presented.

Chapter 4 presents modelling of the PMSM drive in discrete domain leadingto a real time simulation of the same drive system. This exercise in done on anFPGA platform built around an Altera Cyclone EPIC12Q device and Quartus IIsoftware.

Chapter 5 presents general concluding comments on the present work done.It also discusses the scope for future work in the related field.

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Chapter 2

Mathematical models andtheory of vector control ofPermanent MagnetSynchronous Machines

2.1 Introduction

The Permanent Magnet Synchronous Machine (PMSM) is 3 phase synchronousmachine with a balanced 3 phase distributed winding in the stator and a permanentmagnet in the rotor [2]. As no electrical power is to be fed into the rotor (forabsence of winding in rotor), no slipring-brushes are required. This improves theruggedness of the machine and all problem related to presence of slip-ring-brushassembly are therefore absent in this machine. For this reason, sometimes, in someliteratures, PMSM’s are also described as Brushless DC (BLDC) machine. Theconstruction of a BLDC machine is also similar to that of a PMSM with the onlyexception in the design that a BLDC machine exhibits a near-trapezoidal inducedEMF distribution with time, whereas, a PMSM exhibits a near-sinusoidal inducedEMF distribution [3]. Mathematical model for electromechanical analysis of aPMSM can be formulated in the standard a-b-c frame [4] or in the Park’s D-Qrotor reference frame [4]. In the present model, the Park’s D-Q rotor referenceframe has been used and is hence described next.

2.2 Mathematical Model of a PMSM

The stator of the PMSM and the wound rotor synchronous motor (SM) with ar-mature in stator are similar. In addition there is no difference between the backemf produced by a permanent magnet in a PMSM and that produced by an ex-

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cited coil in a SM. Hence the mathematical model of a PMSM is similar to that ofthe wound rotor SM. The rotor frame of reference is chosen because the PMSM 3phase armature winding is fed from a 3 phase voltage source inverter (VSI), whichis switched in synchronism with the rotor position information of the PMSM.Hence the frequency of the voltage or current in the PMSM armature winding atall instants is same as the electrical speed of the machine; electrical speed beingrelated to mechanical speed through the no. of poles of the machine. Making asynchronous machine run in this way is the “self-synchronous” mode of operation,which was initially proposed by Sato et al [5]. The following assumptions are madewhich deriving the D-Q model of the PMSM in rotor reference frame :1. Saturation is neglected.2. The back emf is sinusoidal.3. Eddy currents and hysteresis losses are negligible.The mathematical model is presented as a block in Fig. 2.1, where the three

Figure 2.1: PMSM model block in D-Q reference frame, denoting input andoutput variables

armature phase voltages (machine assumed to be star connected), load torque pa-rameters are input variables to the motor; and the armature current, electromag-netic torque, electrical speed, mechanical speed and rotor position are consideredoutput variables. The rotor position is fed back as an input variable to the motormodel. With the assumptions, the important equations of the PMSM in the rotorreference frame are [4]:

vrqs = (Rs + pLq)irqs + ωrLdi

rds + ωrψ

0 (2.1)

vrds = (Rs + pLd)irds − ωrLqi

rqs (2.2)

Te = (32)(

P

2)[ψ0irqs + (Ld − Lq)irqsi

rds] (2.3)

Te = J(2P

)d

dtωr + f(

2P

)ωr + TL (2.4)

9

Figure 2.2: Cross sectional view showing 3 phase winding in stator andpermanent magnet in rotor and the rotor reference frame.

θr =∫ t

0ωr(t)dt (2.5)

Where,vrqs is the Q-axis armature voltage of the motor in rotor reference frame.

vrds is the D-axis armature voltage of the motor in rotor reference frame.

Rs is the stator resistance and Ld, Lq are the D-axis and Q-axis synchronous in-ductances respectively.irqs is the Q-axis armature current of the motor in rotor reference frame.irds is the D-axis armature current of the motor in rotor reference frame.ωr is the electrical speed of the motor in radian per second.Te is instantaneous electromagnetic torque of the motor.ψ0 is the peak value of per phase flux linkage of the permanent magnet rotor,referred to stator, in rotor reference frame.J is the moment of inertia of the motor with load.f is the viscous damping coefficient of the motor. and P is number of poles.θr, the instantaneous rotor position, has been described in Fig. 2.2

vrds, vr

qs are obtained from the three armature phase voltages (van,vbn and vcn)

10

through the Park’s transformation defined below:

vrqs

vrds

v0

=

23

cos(θr) cos(θr − 1200 cos(θr + 1200

sin(θr) sin(θr − 1200) sin(θr + 1200)12

12

12

van

vbn

vcn

(2.6)

abc variables are obtained from dq variables through inverse Park transform definedbelow:

van

vbn

vcn

=

cos(θr) sinθr

cos(θr − 1200) sin(θr − 1200)cos(θr + 1200) sin(θr + 1200)

[vrqs

vrds

](2.7)

The same Park’s transformation matrix and its inverse are not only applicableto relate the three phase armature voltages and the corresponding D-Q voltages,but they are also applicable for the corresponding currents and flux linkages.

The total input power to the machine in terms of abc variables isPower = vania + vbnib+vcnic (ia, ib, ic are the PMSM armature phase currentsand output power in motoring mode, motoring torque are assumed positive)while, in terms of d, q variablesPower = 3(vr

dsirds + vr

qsirqs)/2

The zero sequence variables are not considered as armature of motor is assumedstar connected and balanced operation would occur. With the help of equations2.1, 2.2 and the laid-down assumptions, d-axis and q-axis equivalent circuits ofPMSM can be developed which are shown in Fig.s 2.3 and 2.3.

Figure 2.3: D-axis equivalent circuit of the PMSM

11

Figure 2.4: Q-axis equivalent circuit of the PMSM

2.3 Mathematical model of three phase in-

verter

In application such as uninterruptible ac power supplies and ac motor drives,three-phase inverters are commonly used to supply three phase load. The PMSMarmature winding is to be supplied from a 3 phase VSI whose power electronicsdevices (switches) would be switched according to the rotor position informationfor achieving Vector-Control. The power circuit of a typical 3 phase, 2 level VSI

Figure 2.5: Mathematical model of the 3 phase 2 level voltage source inverter.

catering to a 3 phase armature winding of a 3 phase AC motor is shown in Fig.2.6.The inverter devices marked as T1, T2, T3, T4, T5, T6 are to be switched to achieveVector-Control as per a Sinusoidal Pulse Width Modulation (SPWM) strategy.For such a strategy, at any point of time, in Fig.2.6, either two top devices andone bottom devices (not belonging to the same leg) are gated on or, two bottomdevices and one top devices (not belonging to the same leg) are gated on. No twodevices of one leg of the inverter should be gated on at the same time as this would

12

short circuit the DC bus voltage Vdc. To derive the mathematical model of such aninverter, three Boolean variables Sa, Sb and Sc are conceived here; each denotingthe state of conducting device (switch) of a particular leg (i.e. a or b or c accord-ingly). The Boolean variable Sa can assume a value of either ‘0’ or ‘1’. The stateSa=0 will mean that bottom device (T4) for inverter leg ‘a’ would be conducting,and, Sa=1 will mean that top device (T1) for the same leg ‘a’ would be conduct-ing. Same logic holds good for the Boolean variables ‘Sb’, and ‘Sc’ denoting theswitching states of inverter legs ‘b’ and ‘c’ respectively. The mathematical modelof the 3 phase, 2 level VSI, as shown in Fig2.5 as a block, should have the DC linkvoltage (Vdc), the 3 switching functions (Boolean variables) Sa, Sb and Sc as inputvariables and should have the 3 phase voltages van, vbn and vcn as output variables.The output variables of the inverter will form as the input phase voltages to befed to the PMSM armature winding (Star connected). From Fig.2.6, the 3 phase

Figure 2.6: Three-Phase two level Sine-PWM Inverter with three phase load.

13

voltages vao, vbo and vco may be represented in terms of the switching functions as:

vao = VdcSa

vbo = VdcSb

vco = VdcSc

(2.8)

where, vao is the voltage of point ‘a’ with respect to -ve DC link bus. Sim-ilar nomenclature is also applicable for other two phases. The 3 phase voltageimpressed on the star connected armature winding of PMSM (these are outputvoltage of the inverter) can be represent as Now,

van = vao − vno

vbn = vbo − vno

vcn = vco − vno

(2.9)

Where vno= The voltage of the neutral point ‘n’ with respect to the point ‘o’ ofthe DC bus. van + vbn + vcn = vao + vbo + vco − 3vno assuming that the machinebeing balanced, van + vbn + vcn = 0so vno = vao+vbo+vco

3Hence inverter phase voltages can be expressed as:

van = vao − vao + vbo + vco

3=

2vao − vbo − vco

3=

2Sa − (Sb + Sc)3

(2.10)

similarly

vbn =2Sb − (Sc + Sa)

3(2.11)

and

vcn =2Sc − (Sa + Sb)

3(2.12)

The above equation form the mathematical model of the 3 phase SPWM VSI,which will be utilized later.

2.4 SPWM in three phase inverter

In Sinusoidal Pulse Width Modulation (SPWM) technique a sinusoidal controlsignal at the desired frequency is compared with a triangular wave form. Thefrequency of the triangular waveform decides the inverter switching frequency andit is generally kept constant along with its amplitude. The objective of pulse

14

Figure 2.7: SPWM strategy in of a 3 phase, 2 level VSI

width modulation is to control the out put voltages in magnitude and frequencywith a constant input dc link voltage. To obtain balanced three phase outputvoltages from a three phase PWM inverter, the same triangular voltage waveformis compared with the three sinusoidal reference voltages (i.e. 1200 out of phase),as shown in the Fig.s 2.7 and 2.8.

15

Figure 2.8: Three-Phase two level Sine-PWM Inverter logic and phase andline voltage waveform.

In the same figures, vtri is the triangular waveform and vcontrol,a, vcontrol,b andvcontrol,c are three reference sinusoids (each 1200 out of phase). There are threecomparators, one for each phase. In a comparator of a particular phase, the cor-responding phase reference sinusoid and the triangular wave are compared. Eachcomparator output is taken and also complimented and ultimately these outputsform as switching signals of the respective power electronics devices (switches) ofthe inverter. Due to such a switching strategy, the representative pole voltages(vao, vbo) of legs a and b and the resultant line voltage vab are shown in Fig.2.8.

2.5 Theory of Vector-Controlled PMSM

An separately excited DC motors, flux and armature current can independentlycontrolled by controlling separate controlled voltage source in the two separatecircuit. The electromagnetic torque in such motors can be written as, Te = Ktφia.There is an inherent decoupling in the operating mechanism of a conventionalcompensated DC motor because the armature MMF are orthogonal in space [7].

16

Once rated flux is established in the separately excited DC motor by setting thefield current, a sudden change caused in armature current causes the electromag-netic torque to change extremely fast. Such fast dynamic torque response is notpossible in any AC motors with conventional low-cost controllers. Vector-Controlalso synonymous with field oriented control, when applied to PMSM’s will resultin similar fast dynamic torque response, optimum torque at steady state and lesstorque ripple. However cost increased to develop the controller is more as strategyis not simple.If irds in a PMSM is controlled to remain zero at each instant, then, from equation2.3,we find

Te =32

P

2ψ0i

rqs (2.13)

Because ψ0 is constant, depending on the design of permanent magnet in a partic-ular PMSM, the developed torque varies directly with irqs. Equation2.13 resemblesthe torque equation of a conventional DC motor (with permanent magnet field)and irqs can be imagined to be similar to the armature current of a conventionalDC motor. The faster irqs can be changed, the faster Te changes. In practicalPMSMs, irqs can be changed very fast and hence developed torque can be changedvery fast, just as in a conventional DC motor.As evident from equation 2.1, a dynamics in irqs is not only dictated by changingirqs but also by irds. Similarly, equation 2.2 suggests that a dynamics in irds is alsodictated by irqs. In order to ensure that irqs and irds vary independently in order toensure fast transient response a feed-forward compensation technique is discussednext, which has been employed in this work.

2.6 Feed-forward compensation for achieving

Vector-Control

The three phase inverter to be used to feed the 3 phase PMSM armature wouldbe a VSI. We conceptually convert the three phase voltages to its equivalent twophase voltages by applying the equation 2.6. and then apply these two voltagesto the PMSM D-Q model. From equation 2.4 we find that, if we want to changethe torque of the machine then we have to change the irqs or irds or both of themotor. irds has to be maintained zero to achieve vector control and hence irqs isto be changed to vary the developed torque. Equation 2.2 and 2.1 show that theq-axis voltage equation contains q-axis and as well as d-axis current quantities.Also d-axis voltage equation contains q-axis current quantities. So, we rewrite theequation (2.1) in the form shown below

vrqs − ωrLdi

rds − ωrψ

0 = (Rs + pLq)irqs

and assume

vrqs − ωrLdi

rds − ωrψ

0 = v′qs (2.14)

17

where v′qs is assumed to be another conceptual Q-axis voltage such that

v′qs = (Rs + pLq)irqs (2.15)

From equation 2.15 it can be said that the conceptual Q-axis voltage, v′qs , now en-

tirely and independently dictates the irqs dynamics. vrqs would therefore be related

to v′qs by the equation,

vrqs = v

′qs + ωrLdi

rds + ωrψ

0 (2.16)

Similarly if we rewrite the equation (2.2) in the form shown below

vrds + ωrLqi

rqs = (Rs + pLd)irds

and if we assume

vrds + ωrLqi

rqs = v

′ds (2.17)

so we can say , v′ds = (Rs + pLd)irds

By similar logic, we can say now that a conceptual D-axis voltage, v′ds , now entirely

and independently dictates the irds dynamics. vrqs would therefore be related to v

′qs

by the equation,vrds = v

′ds − ωrLqi

rqs (2.18)

Now, two Linear Time Invariant(LTI) single input single output (SISO) systemscan now be thought about, which together, would describe the PMSM. One ofthis two would have v

′ds as input, irds as output (Fig2.9) and the other would have

v′qs as input and irqs as output(Fig2.10) equation and together form the basis for

feed-forward commentation which would be use in order to achieve vector controlof a PMSM drive reported in this work. The design of controllers for achievingvector control together with the feed-forward compensation technique is describedin the subsequent chapter.

Figure 2.9: D-axis system representation of the PMSM considering feedfor-ward compensation.

18

Figure 2.10: Q-axis system representation of the PMSM considering feedfor-ward compensation.

19

Chapter 3

Design of controllers andOff-line Simulation of aVector-Controlled PMSMDrive

3.0.1 Introduction

This chapter discusses the design of controllers along with the control stratgy toimplement vector control on a particular Permanent Magnet Synchronous Motor(PMSM) through a 3 phase Sinusoidal Pulse Width Modulated (SPWM) VoltageSource Inverter (VSI). Subsequently, the chapter discusses an off-line simulatedstudy of a vector control implementation of the PMSM drive performed throughMATLAB-SIMULINK software platform. The rating and the parameters of thechosen PMSM are given in the appendix A.1. Motor rated input voltage is 400Volts (line to line) which will be delivered from a three phase SPWM inverter.The relation between line voltage and the DC link voltage can be expressed [3] as

VLL1 = 0.612maVdc (3.1)

where, VLL1 is the fundamental component of the line voltage, Vdc is the DC linkvoltage and ma is the modulation index. We want to have linear modulation(i.e. ma ≤ 1) through out the operating zone of the vector-controlled drive toensure that current fed to the machine have low contents of lower order harmonicsin order to have as low torque ripple as possible. We would like to restrict themaximum value of ma to 0.9. Required Vdc to produce the rated voltage for thePMSM is found to be 726.21 Volts. We have finally chosen Vdc to be 750 Volts.Considering 750 Volts DC link voltage and the power rating of the PMSM, a twolevel inverter with Insulated Gate Bipolar Transistors (IGBT) should be opted forimplementation. Hence, we decided the switching frequency to be 5kHz. for the 3

20

phase SPWM inverter.

3.1 Design of controllers

For vector control strategy we ideally want to have irds = 0 at each instant oftime. In order to achieve this, we should have a reference input ir∗ds set at zero.The block diagram of Fig.2.9 suggest that v

′ds should be controlled in a proper

way to control irds. The nature of transfer function relating irds(S) and v′ds(S),

‘S’ being the Laplace operator, suggests that, if v′ds(S) is made the output of a

Proportional-Integral(PI) controller, whose input should be the error between ir∗ds

and irds, then at steady state, irds can be made to track ir∗ds, i.e. after the dynamicsis over (and it can be made quick), irds should remain zero. The complete D-axiscontrol system is illustrated in Fig. 3.1.

Figure 3.1: D-axis current control loop (system) to achieve vector control ofPMSM.

Figure 3.2: Control system of the Fig3.1in simplified pole-zero form.

We now wish to have pole-zero cancelation in the control system of Fig. 3.1

21

Figure 3.3: Control system of the Fig3.1 after pole-zero cancelation.

Figure 3.4: Model of the Vector-Controlled PMSM drive.

(simplified form shown in Fig. 3.2). Hence,

(s +Kid

Kpd) = (s +

Rs

Ld) (3.2)

hence,Kid

Kpd=

Rs

Ld(3.3)

After pole-zero cancelation, the system of Fig. 3.2 reduces to that shown in Fig.3.3.

As the switching frequency of the inverter is considered 5kHz, as the band-width of the current loop is taken to be one order less, i.e. 500Hz. so,

Ld

Kpd=

12π × 500

(3.4)

As the machine parameters (Rs, Ld) are known, equations 3.3 and 3.4 can be solvedfor two unknowns, Kpd and Kid. Thus, the parameters of the D-axis current PI

22

controller of the PMSM under study are found to be:

Kpd = 309.82 (3.5)

Kid = 17269.79 (3.6)

Figure 3.5: Q-axis current control loop (system) to achieve vector control ofPMSM

Fig. 3.5 suggests that the mathematical relationship between v′qs and irqs is

in similar form as that between v′ds and irds. With irds maintained at zero (con-

trolled by the D-axis current control loop), the developed torque of the PMSMdrive should be controlled by varying irqs, as required. Hence, we must conceiveof a ir∗qs (irqs reference), and we would like the irqs (actual Q-axis current) to trackir∗qs. Therefore, the Q-axis current control loop should look similar to the D-axiscurrent control loop and is shown in Fig. 3.5. Following design procedures exactlysimilar to those followed for designing the D-axis current controller, setting thebandwidth of this loop also at 500 Hz, the Q-axis current controller gains Kpq andKiq have been found as:

Kpq = 201.24 (3.7)

Kiq = 17210.64 (3.8)

3.2 Simulation of the developed vector con-

trol strategy

Fig. 3.4 shows the complete model of the developed vector controlled PMSMdrive. The input ir∗qs stands for the reference value of the Q-axis current, whichwould decide the value of the developed torque. The actual irqs of the PMSM issensed, fed back and is compared with ir∗qs. The error is generated and is fed tothe Q-axis current controller. This controller is a PI controller and development ofthis controller has been already discussed in section 3.1. The PI controller outputforms v

′qs and employing the feed forward technique presented in the section 2.6,

23

vr∗qs is computed with the help of of sensed value of ωr and irds. ir∗ds, another input

to the drive, is the reference value of the D-axis current, which should be zero forachieving the desired vector control. The actual irds of the PMSM is sensed, fedback and is compared with ir∗ds . The error in D-axis current is generated and isfed to the D-axis current controller. The design of this PI controller is alreadydetailed in section 3.1. The D-axis current controller output forms v

′ds. Employing

the feed forward commentation technique discussed in section 2.6, vr∗ds is computed

with the help of of sensed value of ωr and irds. The two inputs vr∗ds and vr∗

qs , thusformed, in ideal case, should be the actual vr

ds and vrqs of the PMSM. This is

ensured through the 3 phase VSI controlled by the SPWM technique, describedearlier. The “ dq to abc transformation ” block of Fig. 3.4 accepts vr∗

ds and vr∗qs

and θr of the PMSM as the three inputs and computes the corresponding a-b-cvariables v∗control,a, v∗control,b and v∗control,c, which would be required for implement-ing the SPWM strategy, discussed in section 2.4. The “SPWM inverter ” block ofFig.3.4 accepts those three voltages. The DC link voltage value of 750 volts anda triangular wave Vtri of 5kHz. frequency are also fed to this block. Followingthe SPWM strategy, as discussed during the model development of the SPWM in-verter in section 2.4. van, vbn and vcn are computed. These subsequently serve asthe 3 input stator terminal voltages of the PMSM. The “ PMSM” block of Fig.3.4receives those three voltages as its input along with θr, the rotor position (whichis one of its output and is fed back) of the PMSM. The mathematical model ofthe “ PMSM” block is already discussed in the section 2.2. This block computesits output variables irds, irqs (actual D-axis and Q-axis values of PMSM armaturecurrents respectively), θr, ωr and Te. These output variables are fed back to theother blocks of Fig.3.4 as required for complete performance analysis.

3.3 Simulation in MATLAB-SIMULINK plat-

form and results

The control system of Fig.3.4, as explained in section 3.2, has been realized throughMATLAB-SIMULINK software and the results are presented in Fig.3.6, 3.7, 3.8,3.9, 3.10, 3.11, 3.12.

ir∗ds, i.e. the reference value of the D-axis current is continuously maintained atzero. Initially a step signal of 1A is given as ir∗qs and is maintained till 7 seconds.The speed invariant component of the load torque (TL) setting is fixed at zero. andthe value of f is set at 0.0606 Nm−sec

radian all throughout. These settings, representingthe total load on the machine, have been kept constant in view the rating of thechosen PMSM under study. At 7 sec. another step change is introduced in ir∗qs

value. It is increased to 2A and then again maintained constant till the end ofsimulation time, i.e. 15 seconds.

Fig. 3.6 shows the actual D-axis component of stator current of the PMSMi.e. irds, with time. Its mean value has been found to be remain almost constant at

24

zero with the PWM ripples riding on the mean value. It almost instantaneouslylatches on to its reference value (almost with fraction of a millisecond). It hasbeen found to remain almost undisturbed even un and around 7 seconds when ir∗qs

has been suddenly increased. This proves the effectiveness of the vector controlstrategy implemented through this simulation.

Fig. 3.7 shows the actual Q-axis component of stator current of the PMSM i.e.irqs, with time. Its mean value has also been found to be latch on to its referencevalue (ir∗qs) of 1 Amps within fraction of a millisecond and remain undisturbedthereafter. A high frequency PWM ripple rides on the mean value of 1 Amps.At 7 seconds, when ir∗qs is again step changed to 2 Amps. and the mean value ofthe actual Q-axis current ir∗qs has been again found to track its reference value atsteady-state. Fig. 3.8 provides a zoomed-in view of irqs when it increases form1Amps and finally settles at 2Amps within a duration of about 0.0016 seconds.This is as per the designed bandwidth of the Q-axis current control loop which hasbeen chosen as 500Hz. (after pole zero cancelation, described earlier, the forwardpath transfer function of the Q-axis current loop has become Kpq

SLq, i.e. 1

S∗0.00032

and 5× 0.00032=0.0016 second.). Fig. 3.8 also shows that within the same timeinterval, the mean value of irds has changed insignificantly, proving the effectivenessof the decoupled control achieved.

Fig. 3.9 shows the simulated waveform of the electromagnetic torque withtime. The mean value is found to change extremely fast in response to the ir∗qs

command. Fig.3.10 shows the time-expanded version of the same electromagnetictorque waveform in and around 7 seconds (when ir∗qs command is suddenly changed)and the mean value of this is also found to settle within a duration of 0.0016 sec-ond, which is the settling time of irqs. This again proves the effectiveness of thevector control strategy implemented.

Fig.3.11 has the speed response of the PMSM drive with time and Fig.3.12presents the three line current waveforms of the PMSM at steady state.

25

Figure 3.6: Simulated irds response with time, as ir∗ds (reference value of D-axis armature current) for the Vector-Controlled PMSM drive is kept atzero all throughout but step increment in ir∗qs are given, first at starting andthen at 7 seconds, on load. Load setting are: TL = 0 and f = 0.0606Nm-sec/radian.(value corresponding to about rated torque at rated speed.)

26

Figure 3.7: Simulated irqs response with time, for the Vector-ControlledPMSM drive, with ir∗ds is kept at zero all throughout but step incrementof 1Amps. in ir∗qs is given, first at starting and then again a further incrementof 1 Amps. (finally to 2 Amps.) is introduced at 7 seconds, on load. Loadsetting are same as mentioned in Fig.3.6

27

Figure 3.8: Same irds (top) and irqs (bottom) response as given in Fig.3.6 andFig.3.7 with time scale expanded in and around 7 seconds (instant at whichir∗qs is given a step increment to 2Amps. finally from 1Amps.)

28

Figure 3.9: Simulated Te (instantaneous electromagnetic torque) responsewith time for the Vector-Controlled PMSM drive, with operating conditionsremaining same as mentioned in Fig. 3.6

29

Figure 3.10: Same Te waveform with time as that of Fig.3.9, but with timescale expanded in and around 7 seconds (instant at which ir∗qs is given a stepincrement to 2Amps. finally from 1Amps.)

30

Figure 3.11: Simulated ωr (electrical Speed in rad/sec) response with timefor the Vector-Controlled PMSM drive for operating conditions as detailedunder the caption of Fig.3.6

31

Figure 3.12: Simulated phase current waveforms of the PMSM’s armature(ia, ib, ic) during steady state.

32

Chapter 4

Online or Real TimeSimulation of a vectorcontrolled PMSM drive

4.1 Real time simulation

Real-time Simulation refers to simulating the equations describing the mathemati-cal model of a physical system, which can execute at the same rate as actual time.In other words, the model runs at the same rate as the actual physical system.For example, if a motor starts from rest and settles to a steady speed after ‘t1’seconds due to application of a particular input voltage at a particular loading;the real-time simulation of the system in a processor would comprise of solvingthe mathematical equations governing the same system such that the solved out-puts will be available to the user after ‘t1’ seconds only. An off-line simulation(MATLAB-SIMULATION based simulation) would typically takes much longertime to solve the same equations to yield results.

In off-line simulation, the delay depends on the complexity and simulationparameters. This is because the actual time involved in the calculation of thevariables is more. On the other hand, in Real-Time Simulation, the results areproduced almost instantly. This is possible if the system model is implementedby an electronic circuit. For real-time simulation of comparatively less-complexsystems, a less costly alternative is required. A Field Programmable Gates Array(FPGA) is a suitable platform for implementing such systems. The basic advantageof an FPGA is that any module can be implemented on FPGA by its equivalentcircuit model.

This equivalent model is a combination of sequential logic elements and com-binational logic element. The outputs of the combinational elements change statewhenever the inputs change, whereas the sequential output only changes state withthe transition of clock. During each clock cycle, the present states of the system

33

are calculated. The calculations are split into two stages as shown in Fig 4.1. Inthe first stage, the present states of the system are calculated using the previousstates of the system. In the second stage, the value are updated. For the purposeof real time simulation, a programming device is required that can handle heavymathematical operations in very short time. The basic advantage of FPGA is thatit can be programmed in parallel. Thus the implementation of network equationson an FPGA results in very short execution time.

Figure 4.1: Triggering Timing

34

4.2 Integration Methods

The dynamic systems may be represented generally in the form of Ordinary Dif-ferential Equation.

dyi

dt= ei(xj , yk, t) (4.1)

Where xj(t) are the independent forcing functions, yk(t) are the state variablesand t the time as variable of integration (independent variable). The procedure forsolving a system of equations simply involves applying the one-step technique forevery equation at each step before proceeding to the next step. There are severalintegration methods to solve the differential equations viz. Back Euler’s Method,Heun’s Method, Runge-Kutta Method etc. These methods differ in complexity.Each integration method can be implemented by digital logic elements in FPGA.Because of its simplicity, Backwards Euler’s algorithm is widely used and is usedin the present work.

4.2.1 Backwards Euler’s Method

In this method, the time axis is subdivided into several intervals. In each interval,ei is approximated by a constant representing the average of ei in that interval.A new value of yi is predicted using the slope (equal to the first derivative at theprevious value) to extrapolate linearly over the step size 4 t.

yi(n) = yi(n− 1) + ei(n− 1)* 4t

Figure 4.2: Block Diagram of Backwards Euler’s Method

35

4.3 Simulation of a series R-L-C Circuit

Figure 4.3: Series RLC Circuit

36

An electrical series R-L-C circuit is shown in Fig 4.3. A transient currentand Voltages are established in the circuit when the switch is suddenly closed.Equations that describe the transient behavior of the circuit 4.3 are The differentialequations of a series R− L− C circuit are as shown in equations 4.2 and 4.3.

Vg = Ri + Lpi + vc (4.2)

i = Cpvc (4.3)

Where, Vg, R, L, C, i and vc are the applied voltage, series resistance, seriesinductance, series capacitance and capacitor voltages respectively.The equations are first normalized with the help of arbitrary values Vb, Rb, whereib = Vb/Rb. this two first-order linear differential equations that can be solvedusing Euler’s numerical methods. The equations are first normalized with thehelp of arbitrary values

Vg

Vb=

R

Rb

i

ib+

L

Rbp

i

ib+

vc

Vb(4.4)

i

ib= CRbp

vc

Vb(4.5)

With the following abbreviations, Vg

Vb= V ∗

g , iib

= i∗, RRb

= R∗, vcVb

= v∗c ,LRb

= τLR,CRb = τCR, a non-dimensional equation results:

[τLRpi∗

τCRpvc

]=

(−R∗ −1

1 0

) (i∗

v∗c

)+

(10

)V ∗

g (4.6)

Say, the parameters of the circuit are: Vg = 100V , R = 10Ω, L = 20mH, C = 4uFSo, the base values of the quantities are shown below:

Voltage(Vb) 100VCurrent(Ib) 10A

R∗ 100/10 = 10ΩτLR 2e−3

τCR 40e−6

Step time(dT ) 25.6us

The p.u. (per unit) values chosen are shown below and the negative value aretaken as the one’s complement of its corresponding positive value.pu value Equivalent digital Value Equivalent decimal value

2pu 7FFFh 32767d

1pu 3FFFh 16383d

0pu 000h 0d

-1pu C000h 49152d

-2pu 8000h 32768d

37

4.3.1 FPGA design files

In the laboratory, an FPGA kit (Altera FPGA chip based) consisting of an FPGAboard, an interface card containing buffers, Analog to Digital converter converters(ADC), Digital to Analog converters (DAC), is present. The detailed specificationsof this kit are furnished in Appendix B.1. The per-unitized equations describedin section 4.3 are programmed in this kit using the QUARTUS-II design tool ofAltera Corporation, which is a free software, downloadable from the website ofAltera Corporation. Fig.4.4 shows the ‘printscreen’ image of a typical FPGAdesign file. The FPGA digital outputs are converted into analog voltages throughthe DAC of the FPGA kit.

The equation 4.6 are implemented in the said FPGA platform

Figure 4.4: FPGA design file of a series RLC circuit

38

4.3.2 Results

The results of the variables are fed to DAC and seen in the oscilloscope as shownin Fig.s 4.5 and 4.6:

Figure 4.5: Transient waveforms of circuit current and input applied voltagefor Vg = 100V , R = 10Ω, L = 20mH and C is forced to a value very veryhigh, i.e. representing a series R-L circuit case.

39

Figure 4.6: Transient waveforms of the real-time simulated capacitor voltageand input applied voltage for Vg = 100V , R = 10Ω, L = 20mH, C = 4uF ina series R-L-C circuit.

40

4.4 Simulation of a vector controlled PMSM

drive in FPGA platform

Same dynamic model of the PMSM as mentioned in the chapter 2 will be usedto derive the vector-control algorithm in FPGA platform to decouple the d-axis,q-axis quantities in the drive system. So we have to build different blocks of thedrive system for simulating the vector-control scheme in FPGA. We have developedthose blocks as mentioned in Fig. 3.4.

4.5 Developments of a Model of the PMSM

in FPGA

Mathematical model of the PMSM has been described in the section 2.1. Equations2.1, 2.2, 2.3, 2.4 and 2.5 have to be normalized to developed a PMSM model inFPGA. Parameters of the motor has been given in the Appendix A.1.

4.5.1 Normalized Equations

The step time of the system is chosen as 25.6 micro sec. The equation 2.1 is re-peated here.

vrqs = (Rs + pLq)irqs + ωrLdi

rds + ωrψ

0

Dividing the above equation by Vb (base value of the Voltage) and rearrangingit, we can get

Rspirqsp + Lq

Rbddt i

rqsp = vr

qsp − ωbRbωrpLdi

rdsp − ωb

V bωrpψ0(suffix ‘p’ implies a per-unitized quantity)if eqsp = vr

qsp − ωbRbωrpLdi

rdsp − ωb

V bωrpψ0

so, irqsp = RbLq

∫[eqsp −Rspi

rqsp]dt

Using the Euler’s Method of integration, the equation can be written asirqsp(n) = irqsp(n− 1) + 0.02vr

qsp − 0.0123ωrp(n− 1)irdsp(n− 1)− 0.0161ωrp(n− 1)−0.0022irqsp(n− 1)

Note that, the sampling time Ts for the implementation is chosen as 25.6 microsecsimilarly other equations 2.2, 2.3, 2.4 and 2.5 can also be normalized in the formshown bellow.irdsp(n) = irdsp(n−1)+0.013vr

dsp +0.00522ωrp(n−1)irqsp(n−1)−0.00143irdsp(n−1)

41

Tep(n) = 0.4022irqsp(n) + 0.1086irqsp(n)irdsp(n)

ωrp(n) = ωrp(n−1)+[0.000476irqsp(n−1)−0.0000299ωrp(n−1)−0.00273Tlb(n−1) + 0.0002irqsp(n− 1)irdsp(n− 1)]

θep(n) = θep(n− 1) + 0.001708ωep(n− 1)

All the operations are basically arithmetic operations. The digital realizationfor the above equations requires adders, subtractors, multipliers and dividers. Allthese entities are available in the library of Quartus-II tool. Apart from thesearithmetic logic entities, D-Flip Flops are also needed for storing previous datavalues, in the case of performing integration.The base values of the quantities are shown below:

Voltage(Vb) 500VCurrent(Ib) 10A

Rb 500/10 = 50Ωωb 314.16rad/sec

Step time(dT ) 25.6us

pu value Equivalent digital Value Equivalent decimal value2pu 7FFFh 32767d

1pu 3FFFh 16383d

0pu 000h 0d

-1pu C000h 49152d

-2pu 8000h 32768d

4.5.2 Testing of the PMSM Block

The MATLAB-SIMULINK based model of the PMSM block has already beendeveloped and discussed. To that block, inputs vd, vq values are set at 20V and150V respectively. So in FPGA per-unitized value of V d and V q will be 655 (0.04per unit) and 4914 (0.3 per unit) respectively. We compare the output resultsas obtained from the MATLAB-SIMULINK based developed simulation and thatobtained from real-time simulation through the FPGA kit.

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Figure 4.7: Transient waveform of iqin matlab simulink with no load andvd=20V vq=150V. Peak value of the waveform is 18 Amps or 1.8 pu

We have thus verified that the on-line and off-line simulation results matchclosely with each other.

4.6 Three-Phase Sine-PWM Inverter

4.6.1 Implementation of Three-Phase Sine-PWM In-verter in FPGA

The steps can be summarized as follows: 1.Generation of Triangle carrier2.PWM generation

4.6.2 Carrier Generation

The carrier used for Sine-Triangle Modulation is a triangular waveform. In theFPGA, it is generated digitally by using a counter. The triangular carrier, in

43

Figure 4.8: Simulated transient waveform of iqin FPGA with no load and0.04perunit vd and 0.3perunit vq Peak value of the waveform is 1.8 pu(5 voltis one pu)

analog methods is usually a bipolar signal. In digital implementation both themodulating signal and the carrier used are unipolar in nature. Hence the peak ofthe carrier is kept at 2 p.u. A binary up-down counter is configured as shown inFig.4.13 for generating the required carrier. 1 p.u is set as 3FFH. Since the carrierpeak is set at 2 p.u, the maximum count is 7FFH. The p.u value for the amplitudeof carrier is chosen based on the switching frequency required. In the FPGAcontroller board used, the clock frequency is 20MHz. This means for countingup to 7FFH, it takes 102.4us. The frequency of carrier and hence the switchingfrequency used here is nearly 5 kHz. The period of the MSB of the 12-bit counteris 204.8 us.

4.6.3 Generation of Phase and Line voltage waveforms

It can be concluded that the switching signals are having correct phase relationwith the phase-A control signals.Now, to test the switching signals further, one real-time inverter is prepared in-side FPGA and that real-time-inverter is switched with these generated switchingsignals. The generated phase voltages of that real-time-inverter is shown in theoscilloscope to check their relative phase relation and shape.

44

Figure 4.9: Transient waveform of idin matlab simulink with no load andvd=20V and vq=150V. Peak value of the waveform is 9 Amps or 0.9 pu andsteady state value is 0.43pu

Now, if Fig 2.6is noted,vao = VdcSa

vbo = VdcSb

vco = VdcSc

(4.7)

where, vao is the voltage of point ‘a’ with respect to -ve DC link bus. Similarnomenclature is also applicable for other two phases. Now, Sa, Sb, Sc are switchingfunctions of the respective phases and also are the switching signals of T1, T3, T5respectively. So, Sa =1 when, T1 is ON and, Sa = 0 when, T4 is ON. Similar logicis applicable for other two phases. Now,

van = vao − vno

vbn = vbo − vno

vcn = vco − vno

van + vbn + vcn = 0

(4.8)

45

Figure 4.10: Simulated transient waveform of idin FPGA with no load and0.04perunit vd and 0.3perunit vq. Peak value of the waveform is 0.9 pu (5volt is one pu) and steady state value is 0.43pu

So, it can be calculated that

vno =vao + vbo + vco

3(4.9)

Hence inverter phase voltages can be expressed as: van = vao − vao+vbo+vco

3 =2vao−vbo−vco

3 = vab−vac

3So, in terms of line voltages, the phase voltages can be written as:

van = vab−vac

3vbn = vbc−vba

3vcn = vca−vcb

3

(4.10)

46

Figure 4.11: Transient waveform of electrical speed(in rad/sec ) in matlabsimulink with no load and vd=20V and vq=150V

Here the fundamental waveform amplitude in nearly 11 db. The amplitude isdisplayed in dB, where 0dB is equal to 1 volts RMS. From the manual of TektronixTDS-1001 oscilloscope, we find that

dBV RMS = 20logVRMS

1VRMS(4.11)

so, Vrms = 101120

=3.54so the peak value of the output signal=

√2× 3.54 = 4.96

For the unipolar voltage switching peak value of vao=Vdc if ma=1[3].1 Per unit value of Vdc is equal to 5 volts and the peak value of this signal is 4.96volt which is nearly equal.

47

Figure 4.12: Simulated transient waveform of electrical speed(in rad/sec )inFPGA with no load and 0.04 per unit vd and 0.3 per unit vq

4.7 Design of two phase to three phase Trans-

formation in FPGA

It is done by converting the three phase voltages and currents to dqo variables byusing Parks transformation. As shown in equation 2.7. For sine and cosine wavegeneration 1024 samples per cycle are used. The phase shifted sine and cosinevalues can be generated by a C or MATLAB program and it can be and storedin the ROM configured in FPGA. The sine and cosine wave amplitude is chosenas 1 p.u (3FFFh) six ROM are used to generate six phase shifted sin and cosinewave.those sin and cosine wave are generated depending upon theta.

4.7.1 Testing of two phase to three phase Transforma-tion Block

For testing this block we have taken vd=2.5v and vq=2.5v. and the frequency ofthe theta is 38Hz. So the output v(a), v(b), and v(c) will be three 1200 phaseshifted wave form with positive peak value 3.5v

4.7.2 Output

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Figure 4.13: Block diagram of carrier generation

Figure 4.14: Triangle Carrier

After getting satisfactory output, we make this block as a top level entity for futurework.

4.8 Design of three phase to two phase Trans-

formation in FPGA platform

Three phase to two phase transformation block has made with the help of park’stransformation

vq

vd

v0

=

23

cos(θr) cos(θr − 1200 cos(θr + 1200

sin(θr) sin(θr − 1200) sin(θr + 1200)12

12

12

va

vb

vc

(4.12)

49

Figure 4.15: A phase voltage of a sine PWM inverter with 50Hz referencesignal and 1pu dc link voltage.

4.8.1 Testing of three phase to two phase transforma-tion block

For testing of this block we connect the output of the two phase to three phasetransformation block, developed earlier. If this block is modeled properly, thenoutput of this block has to match with the input of the two phase to three phaseTransformation Block. We have found the same result to be true.

4.8.2 Output

50

Figure 4.16: Gate pulse for the switching device T1.

This block therefore will form a top level entity for future work.

4.9 PI controller

The design of d-axis and q-axis Current PI controller are done based on certainconsiderations. The current loop is much faster that mechanical time constant (i.e.mechanical inertia, J is assumed to be very heavy). Design of the PI controller hasbeen described in Chapter 2. Input of the d-axis and q-axis current PI controller iscurrent but output is voltage so we can write q-axis current PI controller equationas follows.vr′qs = Kpqieq + Kiq

∫ieqdt

ieq is difference between q-axis reference and actual current, if divide both side byVb

vr′qsp = Kpq

Rbieqp + Kiq

Rb

∫ieqpdt

similarly d-axis per-unitized PI controller equation will bevr′dsp = Kpd

Rbiedp + Kid

Rb

∫iedpdt

51

Figure 4.17: FFT analysis of the signal in Fig 4.16. in Tektronixoscilloscope(TDS-1001).

52

Figure 4.18: Output of the two phase to three phase transformation block.‘A’ phase in channel 1 and ‘B’ phase in channel 2

53

Figure 4.19: Output result of the three phase to two phase transformationblock. It is connected with a two phase to three phase transformation block.In put of the two phase to three phase transformation block v(d)=2.5v andv(q)=2.5v.

54

Chapter 5

Conclusions and Scope forfuture work

5.1 General Conclusions

This thesis is a record of simulation work, both off-line and in real-time, carriedout on a proposed closed loop PMSM drive to be implemented on an existing set-up fully designed and fabricated in the laboratory. The fabrication and runningof the same with position feedback (so that the inverter devices can appropriatelyswitched) was already achieved by a previous post-graduate student with a 1200

conduction mode inverter. It was intended that before implementing the vectorcontrol scheme on the actual setup the same be checked through simulation bothoff-line and real-time. The same was done due to certain short-coming in theefforts of the present author and due to unforeseen circumstances, the practicalimplementation of vector control could not be done. However, the hardware isready in all aspects, though not tested. Thus speed position encoders and hallcurrent sensors and voltage sensors have been installed.As for the simulation work done first the system model was developed. Dynamicsof the machine (electro-mechanical)and the power converter (electrical) were rep-resented through appropriate equations. Once the model was developed, off-linesimulation of a vector controlled PMSM drive was done in MATLAB-SIMULINKenvironment. The real-time simulation was done on FPGA platform as mentionedearlier.The present simulation work may help future researchers in their tests on thepractical set-up.

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5.2 Future work

The following activities related to the future development aspects of the PMSMdrive, which may be taken up in future.

• Complete real time simulation of the entire Vector-Controlled PMSM drivewithin the FPGA platform.

• Experimental hardware implementation of the Vector-Controlled PMSMdrive.

• Successful running of a position sensorless scheme of the PMSM drive inFPGA platform.

• Estimating the torque and torque ripples of the PMSM at dynamic conditionby finite element methods (FEM) based techniques using ANSY S software

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Appendix A

A.1 Parameters and specifications of the PMSM

under study

The ratings and parameters of the machine are shown below:4-pole, 1.5 KW, 400 Volts (L-L), 2.17 A, 1500rpm

stator-Armature: 3-phase, star-connected, per phase synchronous inductance, Ld

=0.09867, Lq=0.06409 per phase, resistance(rs)=5.5Ω.

Rotor-Field: permanent magnet with peak value of per phase flux linkage (ψ0)=1.2814wb−turnes

J = 0.0581Kg −m2, fnl = 0.0606Nm− sec/radian.

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Appendix B

B.1 Block diagram and specifications of the

FPGA kit

Figure B.1: Block diagram of the FPGA kit.

Cyclone FPGA: This is the heart of the board in which all the algorithms areimplemented. The Cyclone IC details are given below:

58

Name FPGAPart No EP1C12Q240C8

Manufacturer ALTERANo of Pins 240Package PQFP

No of Logic elements 12,080No of PLL 2

Maximum clock frequency using PLL 275 MHzPower supply required for core 1.5VPower supply required for I/O 3.3V

Power supply required for PLL circuit 1.5VAnalog to digital converter(ADC): Analog to digital converter is used to convertAnalog signals to digital signals. In the FPGA kit we have used a Program forinterfacing the DAC to use its 4 channels. For this Program conversion timeincreases to 10 micro seconds.The details of ADC are shown in table.

Name ADCPart No AD7864 AS-1

Manufacturer ANALOG DEVICESNo of pins 44Package MQFP

Power supply 5VInput analog voltage levels -10V to +10V

Output analog voltage levels -10V to +10VNo of channels 4

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Bibliography

[1] R. Krishnan, “Electric Motor Drives: Modeling, Analysis, and Control”,2001Prentice Hall,inc.

[2] J. R. Hendershot and T. J. E. Miller, “Design of Brushless Permanent-Magnet Motors (Monographs in Electrical and Electronic Engineering)”,Magna Physics Publications, Oxford Science Publications, 1995

[3] N. Mohan, T. M. Undeland, W. P. Robbins, “Power Electronics Converters,Applications and Design”, Second Edition, John Wiley and Sons, 1996

[4] P. C. Krause, “Analysis of Electric Machinery”, McGraw Hill, New York,1996.

[5] N. Sato, “A study of commutatorless motor”, Electrical Engineering in Japan,vol. 84, pp. 42-52, August 1964.

[6] Pragasan Pillay, R. Krishnan, “Modeling of permanent megnet motor drives”,IEEE Tran. On Industrial electronics , Vol. 34, No. 4 pp. 537-541, NOV. 1988

[7] P.S. Bimbhra, “Generalized theory of electrical machines”, 4th edition ,Khanna Publisher, 1987

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