Accuracy Analysis of Frequency Converter Estimates in Diagnostic Applications

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Cosme Antonio Azuara Mar

ACCURACY ANALYSIS OF FREQUENCY CONVERTER ESTIMATES IN DIAGNOSTIC APPLICATIONS

Examiners: Professor D.Sc. Jero Ahola, D.Sc. Tero Ahonen Supervisor: D.Sc. Tero Ahonen

Cosme Antonio Azuara Mar Emmauksenkatu 6 D 13 20380 Turku, Finland Phone: +358465790168

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Abstract

Author: Cosme Antonio Azuara Mar

Title: Accuracy Analysis of Frequency Converter Estimates in Diagnostic Applications

Department: Electrical Engineering

Degree Programme: Master of Science in Electrical Engineering Major: Industrial Electronics

Place: Turku

Master’s thesis Lappeenranta University of Technology. 85 pages, 47 figures, 13 tables and 2 appendices

Year: 2013

1^st Examiner: Professor, D.Sc.(Tech.) Jero Ahola 2^nd Examiner: D.Sc.(Tech.) Tero Ahonen

Keywords: accuracy, frequency converter, estimates, diagnostics

Transportation of fluids is one of the most common and energy intensive processes in the industrial and HVAC sectors. Pumping systems are frequently subject to engineering malpractice when dimensioned, which can lead to poor operational efficiency. Moreover, pump monitoring requires dedicated measuring equipment, which imply costly investments. Inefficient pump operation and improper maintenance can increase energy costs substantially and even lead to pump failure.

A centrifugal pump is commonly driven by an induction motor. Driving the induction motor with a frequency converter can diminish energy consumption in pump drives and provide better control of a process. In addition, induction machine signals can also be estimated by modern frequency converters, dispensing with the use of sensors. If the estimates are accurate enough, a pump can be modelled and integrated into the frequency converter control scheme. This can open the possibility of joint motor and pump monitoring and diagnostics, thereby allowing the detection of reliability-reducing operating states that can lead to additional maintenance costs.

The goal of this work is to study the accuracy of rotational speed, torque and shaft power estimates calculated by a frequency converter. Laboratory tests were performed in order to observe estimate behaviour in both steady-state and transient operation. An induction machine driven by a vector-controlled frequency converter, coupled with another induction machine acting as load was used in the tests. The estimated quantities were obtained through the frequency converter’s Trend Recorder software. A high-precision, HBM T12 torque-speed transducer was used to measure the actual values of the aforementioned variables. The effect of the flux optimization energy saving feature on the estimate quality was also studied. A processing function was developed in MATLAB for comparison of the obtained data. The obtained results confirm the suitability of this particular converter to provide accurate enough estimates for pumping applications.

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Tiivistelmä

Tekijä: Cosme Antonio Azuara Mar

Nimi: Taajuusmuuttajien estimointitarkkuustutkimus diagnostiikkasovelluksissa

Osasto: Sähkötekniikan osasto

Koulutusohjelma: Sähkötekniikan diplomi-insinööritutkinto Pääaine: Teollisuuselektroniikka

Paikka: Turku

Diplomityö Lappeenrannan teknillinen yliopisto. 85 sivua, 47 kuvaa, 13 taulukkoa ja 2 liitettä

Vuosi: 2013

1. tarkastaja: Professori, TkT Jero Ahola 2. tarkastaja: TkT Tero Ahonen

Hakusanat: tarkkuus, taajuusmuuttaja, estimaatit, diagnostiikka

Nesteiden siirtyminen on yksi teollisuus- ja LVI-alojen yleisimmistä ja energiaa vaativimmista prosesseista. Matala prosessihyötysuhde johtuu usein pumppausjärjestelmien epätarkasta mitoituksesta. Lisäksi yksilöllinen mittauslaitteisto vaaditaan pumpun toiminnantarkailua varten, mikä voi edellyttää suuria investointeja. Pumppujen tehoton toiminta sekä riittämätön kunnossapito voivat aiheuttaa huomattavaa energiankustannusten kasvua ja jopa vaurioita pumpussa.

Keskipakopumppua ohjataan yleisesti oikosulkumoottorilla. Moottorin ohjaus taajuusmuuttajalla voi pienentää pumppukäytön energiankulutusta sekä parantaa prosessin säädettävyyttä. Sen lisäksi oikosulkumoottorien signaaleja voidaan estimoida nykyisillä anturittomilla taajuusmuuttajilla. Täten sekä pumpun että moottorin yhteinen valvonta ja diagnostiikka voi toteutua, mahdollistaen luotettavuutta heikentävien toimintatilojen havaitsemisen, jotka voivat aiheuttaa ylimääräisiä kunnossapitokustannuksia.

Työn tavoite on tutkia taajuusmuuttajan laskemien pyörimisnopeus-, vääntömomentti- ja akselitehoestimaattien tarkkuutta. Laboratoriomittauksia suoritettiin sekä staattisessa että transienttitiloissa estimointitarkkuuden tutkimiseksi. Testilaitteisto koostui vektorisäädetystä taajuusmuuttajasta sekä kahdesta, toisiinsä kytketystä oikosulkukoneesta, joista yksi toimi moottorina ja toinen kuormana. Estimoidut muuttujat saatiin taajuusmuuttajan Trend Recorder- ohjelmistolla. Pyörimisnopeuden ja vääntömomentin todelliset arvot mitattiin HBM:n T12-anturilla. Laboratoriomittausten aikana vuon optimoinnin vaikutusta estimaateihin tarkkailtiin myös staattisessa tilassa. Mittauksista saatujen tietojen käsittelemistä varten kehitettiin lukemisohjelma MATLAB:lla. Saadut tulokset vahvistavat, että tämä taajuusmuuttaja tuottaa tarpeeksi tarkat estimaatit pumppaussovelluksia varten.

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Acknowledgements

This work was carried out at the Department of Electrical Engineering of Lappeenranta University of Technology.

I want to thank the Head of the Department of Control Engineering and Digital Systems, Jero Ahola, for the interesting research topic proposal. I wish to express my gratitude to my supervisor, Tero Ahonen, for his guidance and readiness to help me during the writing process of this thesis. I also want to thank Kyösti Tikkanen for his advice and cooperation during the tests at the Laboratory of Electrical Drives.

I am sincerely grateful to my girlfriend Heidi Saarinen for her support and understanding. Her encouragement proved to be invaluable during difficult times.

Finally, this work is dedicated to my family and especially to my parents, whose love and support have continuously driven me to pursue new goals in life.

Turku, January 2013

Cosme Antonio Azuara Mar

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Abbreviations and Symbols

Roman letters

a Unit vector

A Clarke’s transformation matrix, state matrix B Friction coefficient

B Input matrix C Output matrix

D_ Effective machine diameter [m]

fN Nominal frequency [Hz]

fs Supply frequency [Hz]

i Current space vector

IN Nominal current [A]

J Rotor moment of inertia [kgm²]

K Scaling factor in Clarke’s transformation matrix kw1 Winding factor

L Luenberger observer gain

l’ Effective core length [m]

L_m Magnetizing inductance [mH]

Lrσ Rotor leakage inductance [mH]

Lsσ Stator leakage inductance [mH]

m Number of phases

n_N Nominal speed [s^-1]

n_s Synchronous speed [s^-1]

N_s Effective number of stator coil turns p Pole-pair number

P Park’s transformation matrix

PN Nominal power [kW]

r Rotor radius [m]

RMS Root Mean Square

Rr Rotor resistance [Ω]

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Rs Stator resistance [Ω]

T d-q transformation matrix

Te Electromagnetic torque [Nm]

T_L Load torque [Nm]

T_N Nominal torque [Nm]

U Voltage space vector

U_N Nominal voltage [V]

Greek letters

 Angle

 Discrete input matrix

ef Effective air-gap length [m]

 Angle

 Permeability of vacuum [H/m]

 Pi constant

 Machine leakage factor

p Pole pitch [m]

_r Rotor time constant [s]

 Discrete state matrix

  Flux linkage space vector

 Flux linkage [Vs]

 Rotational speed [s^-1]

Subscripts

r Rotor quantity

s Stator quantity sl Slip frequency

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Superscripts

d Direct component of the rotor-flux-oriented reference frame k General reference frame

q Quadrature component of the rotor-flux-oriented reference frame s Stator coordinates

 Direct component of the reference frame fixed to the stator

 Quadrature component of the reference frame fixed to the stator

r Rotor flux linkage coordinates Symbols

 Cross product

 Del operator

 Dot product

^ Estimated variable/ peak value

 Partial derivative

Acronyms

AC Alternating Current ASD Adjustable Speed Drive CSI Current Source Inverter DC Direct Current

DTC Direct Torque Control DVC Direct Vector Control EKF Extended Kalman Filter

ELO Extended Luenberger Observer e.m.f. Electromotive Force

FOC Field Oriented Control GTO Gate Turn-off Thyristor

HVAC Heating, Ventilation and Air Conditioning

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IGBT Integrated-Gate Bipolar Transistor IGCT Integrated-Gate Turn-off Thyristor IVC Indirect Vector Control

KF Kalman Filter

LO Luenberger Observer

LUT Lappeenranta University of Technology m.m.f. Magnetomotive Force

MRAS Model Reference Adaptive System OEM Original Equipment Manufacturer PI Proportional-Integral

PWM Pulse-Width Modulation SCR Silicon Controlled Rectifier SVM Space Vector Modulation VC Vector Control

VFD Variable Frequency Drive VSD Variable Speed Drive VSI Voltage Source Inverter

VVVF Variable Voltage, Variable Frequency ZCS Zero Current Switching

ZVS Zero Voltage Switching

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Chapter 1 Introduction

1.1 Background

One of the greatest endeavours in the present age lies in the better utilization of energy resources in systems and processes. With the ever increasing world population and imminent shortage of fossil fuels, it is clear that a sustainable future requires the available energy to be used wisely. In these times, the industrial sector accounts approximately for more than 40% of the world’s electric energy consumption [1-2]; AC induction motor drive systems consume a major part of it [2].

Fig. 1.1: Left: Estimated electricity demand for all electric motors by sector; right: Estimated share of global electricity demand by end use [2].

Among motor-driven systems, pumps as well as fans and compressors are devices which constitute a major part of electricity consumption, since fluid transfer is virtually needed in all areas of industry [2-3].

Centrifugal pumps and fans are designed to have a certain operating range which must match the process characteristics. It also sets the device limits for optimal

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operation and efficiency. In the case of centrifugal pumps, such information is usually presented graphically as a curve of hydraulic head against fluid flow rate at a given speed, called the QH-curve.

The pump performance at different speeds is obtained using a set of equations called the affinity laws, which correlate the pump head (H), flow rate (Q), shaft speed (n) and shaft power consumption (P):

3

nom nom nom nom

n Q H P

n Q  H  P

(1.1) Pumps have been traditionally installed with throttle control to regulate fluid flow according to the process requirements. While this method is regarded to be inefficient [4-6], a considerable amount of throttle-controlled systems still exist to this day, primarily due to their simple operation. Throttling changes the system QH-curve, intersecting with the pump QH-curve at a higher head value and decreased efficiency. On the other hand, speed control modifies the pump QH- curve, which matches the system characteristic at a lower head value.

Fig. 1.2: Flow control with two different methods in pumping systems. The two accentuated curves correspond to the system, and the finer curves to the pump. The lightly shaded area, plus the darker one correspond to the energy consumed by the pump with throttle control. The energy consumed by the pump with speed control is represented only by lightly shaded area.

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If a pump is not properly matched with the system, due to improper selection or changes in process demands, it may operate beyond its recommended flow rate range. Both high and low flow operation are problematic for the pump. As the pump is driven further outside its design limits, a number of situations can occur that pose a threat for the machine: cavitation, excessive turbulence and rotational velocity, increased vibration and heavier loadings on bearings, seals and other mechanical components [7-9].

As a consequence, the overall lifetime of these systems may decrease and their maintenance and repair costs increase. It is not uncommon for industrial pump and fan equipment to perform at sub-optimal conditions: these systems tend to be dimensioned inaccurately and the driving motors are oversized [7-9], resulting in a very low total efficiency.

1.2 Motivation for the Study

Pumps, fans and compressors consume a considerable amount of energy in e.g.

chemical, pulp & paper and petroleum industries [8], as well as in HVAC (heating, ventilation and air conditioning) applications in commercial and residential installations.

Fig. 1.3: Estimated share of global motor electricity demand by application [2].

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Energy and maintenance costs surpass greatly the initial capital investment in these machines, sometimes exceeding ten times the initial costs [8]. A considerable energy and cost savings opportunity lies in the optimal performance of pumps and fans.

Typical pumping systems in industry are driven with squirrel-cage induction motors, due to their simple construction and low maintenance requirements and costs. In contrast to other speed control methods which may require moving mechanical parts and other wear-prone elements, frequency converters utilize solid-state devices and control electronics to vary the rotational speed of an induction machine. Frequency converters thereby provide a cost-effective way of adjusting fluid flow accurately and efficiently [4-6].

Diverse control methods exist to adjust the rotational speed of a motor. Scalar control (also referred to as V/Hz control) is one of the most common methods applied to induction motor-driven pump and fan systems, since it is the simplest and most widely available alternative. At its simplest, this control method can operate without a rotational speed sensor in an open loop configuration. For low dynamics processes such as flow control, high accuracy is not critical. More demanding applications require advanced control techniques such as Vector Control (VC) or Direct Torque Control (DTC), which can employ a speed transducer in closed loop.

In an effort to simplify and integrate motor drive systems, as well as to diminish costs, sensorless estimation of motor parameters is now possible with modern control technologies [14-15], which only need phase current and voltage measurements, eliminating the need of an encoder or tachometer.

The most important pump parameters needed for condition monitoring are total head (H), flow rate (Q), shaft speed (n) and shaft power (P) [9-10]. To obtain this data, the equipment is traditionally monitored with dedicated measuring systems, such as pressure transducers and flow meters, which increase overall costs. It is therefore desired to diagnose pumps and fans with the minimum amount of

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sensors in a non-invasive way. An attractive alternative is to obtain this diagnosis via a frequency converter, as discussed in [11-13].

Pump and fan parameters can be estimated with the aid of the motor rotational speed and torque estimates, which together give also the shaft power estimate, with the aid of a sensorless control method inside the frequency converter. Pump and system models, which have been a recent topic of research at LUT [11-13], are also needed to complete a diagnosis. There are some solutions available from manufacturers that cover this topic [16-17].

However, there is little research on how these motor estimates are obtained in a commercially available frequency converter and how different aspects affect their repeatability, sensitivity to ambient conditions, dependability of estimation techniques and control methods and frequency converter components; in addition, the influence of energy saving-features such as flux-optimization [18] on motor estimates are still yet to be published.

This thesis aims to investigate the accuracy of motor rotational speed, torque and shaft power estimates performed by frequency converters and what are the factors that influence their calculation, in order to assess the converter ability to properly diagnose pump and fan operation based on these quantities.

1.3 Scope of the Thesis

In order to assess the accuracy of frequency converter estimates, a theoretical and literature study is presented and laboratory tests at different operating points are performed. The thesis is outlined as follows:

Chapter 2 reviews the basic principles of electrical drives, in particular induction motor drives, in order to provide a framework to support the concepts introduced in subsequent chapters. The induction machine and its operation principles in steady and transient states are studied. Power electronic converters, in particular

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inverters and modulation schemes are presented. Three main control schemes used in modern frequency converters are discussed.

Chapter 3 studies the different estimation schemes found in literature and their principles. The influence of a variety of factors on the estimates accuracy is investigated from a theoretical point of view.

Chapter 4 presents the methods utilized to perform tests on different frequency converters in steady-state and transient operation. A torque transducer with torque and rotational speed readings is used to record values and compare them with the theoretical estimates. The effects of flux optimization are investigated.

Chapter 5 shows the results attained by the laboratory tests.

Chapter 6 summarizes the work done and presents the conclusions reached.

Possible future work in the field is suggested. This is the final chapter of this thesis.

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Chapter 2 Principles of Induction Motor Drives

Modern industrial processes demand increased efficiency, flexibility and productivity to meet customer demands. Electronic control of electrical motors plays an important role towards achieving this goal. Motor control is implemented through what is commonly known as an electrical drive. Though definitions may vary, the electrical drive consists of electronic control systems, measurement systems, power electronic devices and often the motor itself.

Fig. 2.1: Basic configuration of an electrical drive.

OEMs commonly use the term frequency converter when dealing with AC drives, based on the principle of speed control through variation of electrical supply frequency; other more generic terms used in flow control applications include adjustable speed drive (ASD), variable speed drive (VSD) and variable frequency drive (VFD). In this chapter, the main components of a typical AC drive—the induction motor, the frequency converter and its control schemes—are illustrated.

2.1 The Induction Machine

For decades, the squirrel-cage induction machine has proven to be the preferred choice in the industrial sector; its simple construction guarantees high reliability,

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ruggedness and low manufacturing and maintenance costs, which are factors that favour these machines over the other types. Nevertheless, the operating principles of the induction machine exhibit a variety of nonlinearities that made it difficult to control in the past, confining its use to non-demanding applications. Thanks to the advances in semiconductor technology and modern signal processing techniques, the complexities of the induction machine can be modelled accurately and affordably enough, enabling speed, torque and position control. Thus, the dynamic requirements of most industrial applications can be met by induction machine drives.

The induction machine is a rotating, electromagnetic energy converter. Electrical current is fed to the machine via conductors winded in slots, which are located in the stator. The stator windings are connected to a three-phase AC supply in a delta or wye configuration, and are usually made of copper. The rotor is often constructed with die cast aluminium or copper conducting bars, which are short- circuited by end rings that resemble a cage, hence its name.

Fig. 2.2: Cross-section of an induction machine. The stator slots are marked in white, and the rotor bars in grey, of which a single one is shown on the right.

The stator currents develop a rotating magnetic field which induces voltages in the rotor bars. These induced voltages generate rotor currents and thus another magnetic field. The interaction of these magnetic fields results in torque production. Torque is the force necessary to turn the rotor and consequently the

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motor shaft, resulting in useful mechanical energy. These basic principles are described in more detail in the following sections.

2.1.1 Rotating Magnetic Field Production

The stator windings of an induction machine can be arranged in multiple ways, with the ultimate goal of producing a revolving magnetic field as close as a sinusoidal wave in shape as possible, in order to minimize losses. The simplest arrangement consists of three coils separated 120° from each other, going in and out of the stator through slots. This produces a pair of magnetic north and south poles.

Fig. 2.3: A single slot per pole per phase winding arrangement. p represents the pole pair, and m the number of phases [19]. The produced magnetic flux waveform is shown on the right.

The sum of the phase currents that circulate through the stator windings create a time-varying, constant amplitude magnetic field. This is best illustrated by Ampère’s circuital law:

C S

D D

H J H dl J ds

t t

 

 

    



 



   

(2.1)

where current density J creates a magnetic field intensity H around a current- carrying conductor. Due to the low frequencies involved, the displacement current term D

t



 is usually ignored. This magnetic field flows radially from the stator north pole, travels through the air gap of the machine, and reaches the rotor bars;

after that, it flows to the stator south pole, closing a magnetic circuit. We will see later that in order to create a high torque, the magnetic flux density in the air gap should be as high as possible, without reaching saturation.

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Due to the discrete nature of the stator slots, the magnetic flux density waveform in the air gap is not purely sinusoidal, but square shaped, as shown in Fig. 2.3.

This waveform has a high harmonic content, which creates extra power losses in the machine. Increasing the number of stator slots per pole per phase and distributing them around the stator has a smoothing effect on the air gap magnetic flux density waveform, creating a better approximation to a sinusoid:

Fig. 2.4: A two slot per pole per phase winding arrangement. p represents the pole pair, and m the number of phases [19]. The produced magnetic flux waveform is shown at the bottom.

Other methods to modify the waveform shape are to place the conductors in double layers, as well as short-pitching and skewing. The air gap magnetic flux density waveform travels through the air gap at synchronous speed, defined as:

(2.2)

where f_s is the supply frequency and p is the pole pair of the machine.

The magnetizing current (magnetomotive force) needed to create the air gap magnetic flux density can be quite large in induction machines and should be minimized for best efficiency, since it contributes only to reactive power. The magnetizing inductance per phase is defined as follows [19]:

60 _s

s

n f

 p

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

₁



²

2 0 ' _w _s

ef

m l k N

p

L mD 





 

(2.3)

where m is the number of machine phases, kw1 is the phase winding factor, Ns the effective number of coil turns in the stator, p the pole pair number and ef the effective air gap length. Increasing the magnetizing inductance restricts the amount of magnetizing current drawn by the machine. Therefore, a low pole count and a small air gap are typical features of industrial induction motors.

2.1.2 Torque Production

At a standstill situation, the magnetic flux density is in relative motion to the stationary rotor. Faraday’s law of induction states that an electromotive force (e.m.f.) will be induced in the rotor conductors:

C S

B B

E E dl ds

t t

 

 

    



  



  

(2.4)

Due to the presence of this e.m.f., currents are generated in the rotor bars. The current carrying bars will experience a force, given by the Lorentz force equation:

( )

dF dQ E v B  (2.5)

where F is the force over the conductor, Q is electric charge, E the electric field intensity, v the conductor speed and B the magnetic flux density. In electrical machines, the magnetic field density in this equation dominates over the electric field intensity; therefore the equation can be reworked as follows:

dF dQdl B didl B

 dt  

(2.6)

This force will therefore depend on the amount of current in the rotor conductors, the magnetic flux density and the conductor length. The magnetic flux density

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propagates perpendicularly in the rotor, producing a tangential force on its surface:

Fig. 2.5: Lorentz force acting on the rotor surface [19].

Assuming ideal sinusoidal flux density and current distributions, and the flux direction to be perpendicular to the conductor, the electromagnetic torque equation can be obtained by integrating the Lorentz force equation around the rotor periphery and multiplying by the rotor radius [19]:



 

ˆ cos

2 ˆ

0

p

e r F rAB

T 





(2.7) where A^ˆis the peak current density in the conductor, B^ˆthe peak magnetic flux density,_pthe pole pitch and r the rotor radius. This expression is valid for a two- pole machine. Developing the expression further, the electromagnetic torque can be presented in vector form, particularly useful while performing advanced control tasks:

 

3

e 2 m s

T  p  i (2.8)

where i_srepresents the stator current space vector, p the pole number and _mthe flux linkage space vector of the air gap. In order for the rotor to turn, this generated torque has to be higher than the inertia of the rotor, and additionally higher than the opposing torque of a possible load connected to the shaft.

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Newton’s second law of motion can be used to describe the mechanics of the rotor shaft:

r r

L

e B

dt J d T

T  







(2.9)

where T_e is the generated electromagnetic torque, T_L the torque produced by the load, J the moment of inertia of the rotor, _r the rotor speed and B a friction coefficient. It is important to stress that the machine torque is dependent on the relative motion between the stator magnetic flux density and the rotor. If they were to rotate at the same speed, no relative motion would exist and no currents would be induced in the rotor. Consequently, even though the magnetic field of the rotor rotates as fast as the stator magnetic field, the rotor itself cannot attain synchronous speed.

The necessary relative speed difference between the rotor and stator field is defined as slip, which is usually represented as a fractional quantity:

s s

n n s n 



(2.10) where n_s is the synchronous speed and n is the rotor speed. The induced e.m.f., rotor resistance and frequency of the rotor currents are all dependant on the slip.

2.1.3 Torque-Speed Characteristic

The torque-speed characteristic illustrates the behaviour of the induction machine at different speeds. At standstill, the relative motion between stator flux and rotor is at its maximum (slip value is 1) and hence a large e.m.f. and currents are induced in the rotor. At this moment, the rotor impedance is predominantly inductive, as the rotor frequency is the same as the stator flux. Very high stator currents are drawn from the mains. Soft-starters and frequency converters can be

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used to prevent the mechanical and electrical stresses on the machine and on the grid associated with these large currents.

After start-up, the machine quickly reaches the operating point, determined by the load torque. This point must be located between the synchronous speed and the speed at breakdown torque for stable operation. In this region, there is a quasi- linear relationship between torque and speed since the rotor frequency relative to the stator is quite small. This results in a resistive circuit where currents and voltages are in phase with the stator flux. When the slip is close to zero, the induced e.m.f. is very small and the rotor resistance is high, resulting in very low rotor currents and therefore low torque. With the exception of wind power systems and other specific applications, the induction machine is seldom used as a generator. The characteristics and features concerning the generating region are therefore beyond the scope of this work.

Fig. 2.6: Torque vs. speed characteristic of an induction machine.

2.1.4 Dynamic Motor Model

An induction machine is a complex dynamical system. The equivalent circuit of the machine commonly used in steady-state analysis is not applicable during transient operation. Instead, the governing equations used to describe the motor become non-linear differential equations, with time-varying parameters. The

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intricate nature of induction machine transients was once the reason for DC machine ubiquity in the field of servo drives and other dynamically demanding applications.

However, the emergence of novel research during the 70s jumpstarted the development of dynamic AC machine control. Due to the additional expenses of an equivalent-sized DC machine, AC machines have since gained ground in almost all technical sectors.

Nowadays, modern frequency converters often include a motor model in order to achieve satisfactory transient performance and precise control of position, speed and torque. The nature of motor models is based in the reference frame and space vector theories, used in VC and DTC.

Space Vector and Reference Frame Theory

The space vector theory is a tool that can be used to develop a framework for the machine model during transients. It assumes a magnetic flux density distributed sinusoidally in the air gap (ignoring space and time harmonics) and neglects magnetic saturation as well as temperature related nonlinearities. The rotor of the machine is also assumed to be smooth: asymmetries and slotting effects are discarded. Assuming a balanced system, any given time-varying, three-phase quantity can be expressed as a space vector:

0 1 2

( ) 2 ( ) ( ) ( )

3 ^u ^v ^w

x t  a x t a x t a x t 

(2.11) where

2 j3

a e



 , which is the unit vector that separates each quantity by 120°, and x is the desired quantity to be converted. This space vector is a rotating complex variable representation of a three-phase value. Since the handling of vectors is not suitable for most microprocessors, a two-phase equivalent system must be employed. This means decomposing each vector into its real and imaginary components. For this purpose, a linear transformation called Clarke’s transformation (also  transformation) can be employed [25]. If zero-sequence

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quantities are not present in the system, only the and components are taken into account. The transformation takes the following form:

2 1 1

( ) 3 3 3 ( )

1 1 ( )

( ) 0 ( )

3 3

u v w

x t x t

K x t

x t x t





     

 

    

      A

(2.12)

1

1 0

( ) 1 3 ( )

( ) 2 2 ( )

( ) 1 3

2 2

u v w

x t x t

x t K

x t x t







 

 

   

 

   

     

  

 

 

A (2.13)

where K represents a scaling factor. The factor could be chosen so that the transformed quantity is power invariant



^K ^ ^{3 / 2}



or equal either to the peak value



^K^¹



or the RMS value



^K ^^{1/ 2}



of the original phase-quantity [20].

The choice of the scaling factor is not consistent in the literature and thus differs according to the source.

This two-phase system is fixed to the stator. In order to further simplify the machine equations, a different reference frame is chosen. The choice of reference frame can yield several advantages depending on the type of control approach selected or machine type characteristics. A second transformation is needed to rotate the system to the desired reference frame.

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Fig. 2.7: A given flux space vector (rotor, stator or magnetizing) decomposed in a fixed and a synchronously rotating reference frame. The coordinate system terminology adopted here is:

for stator coordinates and d-q for the desired reference frame coordinates.

In principle, the desired reference frame can be static or moving at any given speed , as illustrated in Fig. 2.7. The chosen reference frame can be fixed to a certain flux space vector. As an example, the following reference frame is assumed to be fixed to the rotor flux linkage space vector. This reference frame rotates at the same electrical frequency as the magnetic field of the rotor, and it is expressed here as the d-q coordinate system; however, notation may vary according to the source. The transformation generally takes the following form:

( ) cos sin ( )

( ) sin cos ( )

d q

x t x t





 

    

    

T (2.14)

1 ( ) cos sin ( )

( )

( ) sin cos

d q

x t x t





 

      

    

 

   

T (2.15)

This reference frame has the important characteristic of transforming the alternating nature of a stator reference frame quantity into a DC quantity (time- invariant) [22].

A more compact representation can be obtained by combining both transformations into one. This transformation bears the name of d-q

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transformation or Park’s transformation [28]. If the zero-sequence components are not taken into account, the transformation takes the following form:

2 2

cos cos cos ( )

( ) 3 3

( ) 2 2 ( )

sin sin sin ( )

3 3

u d

v q

w

x t x t

K x t

x t x t

 

  

 

  

       

 

        

          

P (2.16)

1

cos sin

( ) 2 2 ( )

( ) cos sin

3 3 ( )

( ) 2 2

cos sin

3 3

u

d v

q w

x t x t

x t K

x t x t

 

 

 

 

 



 

 

 

    

 

          

P (2.17)

The induction machine can be compactly represented by a space vector equivalent circuit that illustrates its main parameters. For the sake of clarity, and due to the variety of choices for a reference frame, this circuit is presented in a general reference frame rotating at speed _k, which may be selected arbitrarily. Any other reference frame can be reached from the general reference frame by selecting the appropriate speed of rotation [20]

Fig. 2.8: Space vector model of the induction machine in a general reference frame. The superscript k indicates a general reference frame.

From the space vector model, the following fundamental machine equations can be obtained:

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k s k k

k s s s k

s j

dt i d

R

U  ^ _





(2.18)



k r



r^k k

r k

r r k

r j

dt i d

R

U     







(2.19)

 

k k k k k k

s L is_ s Lm ir is L is s L im r

      (2.20)

 

k k k k k k

r L ir_ r Lm is ir L ir r L im s

      (2.21)

 

k k k k

m Lm is ir L im m

    (2.22)

where Lm stands for magnetizing inductance, Ls for stator inductance and Lr for rotor inductance; L_sand L_rstand for the leakage inductances of the stator and rotor respectively. From these equations, so-called voltage and current models can be obtained to calculate flux linkages, an important part of DTC and vector control theory. The voltage equation of the stator contains the ohmic losses of the stator resistance, the stator e.m.f. and a voltage term induced due to the relative motion between the general reference frame and the stator. In the rotor voltage equation, the extra voltage term is generated also due to the relative motion between the general reference frame and the rotor.

2.2 Power Electronic Converters

As discussed before, speed control is the most efficient method in flow applications due to its versatility and ease of implementation. As seen in equation (2.2), the rotational speed of an induction machine depends on the supply frequency and pole-pair number. Mechanical pole-pair changing techniques only allow for a discrete number of speed values and it is therefore greatly limited compared to frequency control. Since the AC power available from the mains ought to maintain constant voltage and frequency at all times, a power converter is required to modify the power flow appropriately. Commercial products typically include a rectifier unit, an intermediate circuit with an energy storing element (or DC-link) and an inverter unit.

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Fig. 2.9: Basic power processing blocks used in frequency converters.

The energy storage element can be either an inductor or a capacitor, creating a current source or a voltage source respectively. Even though the inverter stage is usually at the end of the power processing chain, it is commonplace to name the whole system an inverter. When the energy storage element is an inductor, the unit is called a Current Source Inverter (CSI); in the case of a capacitor, the term Voltage Source Inverter (VSI) is used instead.

To maintain as constant as possible DC power, the energy storage element has to be dimensioned sufficiently large, which can be costly. Direct AC-AC alternatives, such as cycloconverters, exhibit limitations in the frequency output, and the large inductor and load requirements in CSI’s restrict the technology to specific, high power applications [22]. However, at the expense of higher harmonic content and control requirements, the size of a capacitor of VSI can be reduced significantly [23]. A VSI can also be suitable for multi-motor applications and a wide degree of design variations can be found. Therefore, a VSI yields a degree of versatility not commonly found in other configurations, making it one of the most widely used converters in industry.

Due to the high power involved in most industrial drives, the semiconductor devices used in VSI’s are typically used as switches instead of operating in their linear region. Ideally this yields a lossless device, however, and especially in inductive circuits such as motor drives, this is not true due to hard switching. This means that a large amount of voltage and current are present simultaneously in the device during the switching instant, producing losses [31]. Zero-current and zero- voltage switching (ZCS and ZVS respectively) are soft switching methods aimed

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to improve the situation, however, the extra complexity added to the system diminishes reliability and increases costs [22].

Fig. 2.10: Power losses during switching transients in power devices.

One of the most typical configurations used in induction motor drives is comprised of a diode bridge, a DC-link capacitor and an inverter unit with IGBT’s and freewheeling diodes. A degree of versatility can be gained by the use of Silicon Controlled Rectifiers (SCR) in order to control the power stored in the DC-link; however, their extra control requirements and protection devices make the use of silicon diodes a more cost-effective option for most applications. For high power applications, thyristor devices such as Gate Turn-off Thyristors (GTO) and Integrated Gate Turn-off Thyristors (IGCT) are used instead of IGBT’s as inverter switches.

Fig. 2.11: Typical VSI configuration for induction motor drives.

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2.2.1 PWM Techniques

In order to produce the required three-phase voltage and current output required by the motor, the inverter switches have to be controlled in a specific order. The switching sequence defines the order in which the switches are turned on and off, thus modifying the amount of power supplied to the motor. Even though the output voltage waveform of the inverter is a stream of pulses, the motor current should be as sinusoidal as possible in order to prevent power losses. One way to achieve this is through Pulse-Width Modulation (PWM).

Fig. 2.12: Classification of PWM methods.

In this section, two of the most popular continuous methods for induction motor drives are discussed: carrier-based modulation and space vector modulation.

Carrier-Based Modulation

One of the simplest methods to implement PWM is the sine-triangle comparison method. Sinusoidal PWM (SPWM) uses a saw-tooth carrier wave to modulate the pulse train to be produced by the switches, therefore producing a current

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waveform whose fundamental frequency is the same as the reference wave. The reference waveforms are ideal, three-phase voltages.

Fig. 2.13: Sine-triangle comparison of phase-voltages Uu,, Uv, and Uw. Uuv represents the pulsed output command, along with its fundamental component.

The simplest case of PWM is the square-wave operation, where the amplitude of the reference waves is made much larger than that of the carrier wave. This creates a long continuous pulse that will change polarity when the reference waves do, in essence behaving as a square-wave. Although this gives the highest output voltage, the signals also carry rich harmonic content. This will create a heavily distorted current waveform at the motor terminals. Thus, the switching frequency has to be high enough in order to transfer the harmonic content alongside it. This is achieved while operating in the linear region. Operation in the overmodulation region might be desirable depending on the application.

In addition to sinusoidal PWM, other methods listed in Fig. 2.12 such as 3^rd harmonic injection and discontinuous PWM can offer advantages such as diminished harmonic distortion or reduced switch stress. A more in-depth treatment of PWM methods can be found in [22, 31, 32].

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Fig. 2.14: Different modulation regions of a Sinusoidal-PWM. A modulation index ma = 1 corresponds to the end of the linear region and the start of the overmodulation region. A modulation index ma = 3.24 marks the start of the square wave region.

Space Vector Modulation

Space Vector Modulation (SVM) is a widely used method suitable for handling space vectors. A three-phase frequency converter with two-level switches produces eight different switching states, two of which are so-called zero-states—

all three phases on the same potential—and six of which are so-called active switching states.

Fig. 2.15: Different switching states in a two-level inverter. The upper leg is connected to +UDC (1) and the lower leg is connected to the neutral point (0).

The active switching states represent voltage vectors, which are plotted in a complex plane. The area in between each voltage vector is defined as a sector.

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The basic idea of SVM is that a space vector can be defined on any given angle by alternating a sector’s adjacent switching states and zero states with correct switching times.

Fig. 2.16: Space vector modulation sectors (I to VI) and voltage vectors (U1 to U6). The zero vectors are U0 and U7. The overmodulation region is represented by the shaded area.

The reference vector is a voltage space vector, decomposed into its two components. With the reference vector and the DC-link voltage, a modulation index is calculated. This is the main parameter to decide if the SVM is working in the linear or overmodulation range. The overmodulation range operates in a similar fashion as the square-wave region of a PWM approach, in that it allows for a higher voltage output at the expense of extra losses and harmonic content.

In order to generate a switching sequence, first it is needed to determine in which sector of the modulation plane the reference voltage vector is. This is done by obtaining the angle between the two components of said vector. After this, and with the value of the modulation index, the adequate voltage vectors are selected by calculating their on-times. In case of linear modulation operation, the zero vectors are employed; they are not used while operating in the overmodulation region. After having calculated the on-times of the voltage vectors, the amplitude of a triangular signal is compared with them in order to create the switching sequence. The frequency of the triangular signal corresponds to the switching frequency, and its amplitude has the same value as the switching period.

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2.3 Control Strategies

In motor drive systems, it is usually desired to control torque and rotational speed.

In more demanding applications, such as machine tools, servo mechanisms and robotic arms, precise position control might be required as well. A traditional approach is to arrange the control variables in a closed-loop cascade configuration, with dedicated sensors for feedback. Controllers of the Proportional-Integral type (PI) are the most common in motion control systems.

Routinely, torque and flux control are located in the innermost loop of the cascade approach, as a fast current loop is usually required. By far, rotational speed is the most common control variable in modern industrial drives. Pumps and fans fall in the category of speed-controlled machinery.

Fig. 2.17: Cascade control loop in motor drives.

Separately excited DC motors were once the preferred choice for speed and torque control in industry and are still used today for specific applications. The DC machine behaviour is described by the following equations:

m

A f

E k  (2.23)

A

T kfI (2.24)

( )

f f If

  (2.25)

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where EA is the induced voltage, IA the armature current, T the electromagnetic torque, the magnetizing flux and k a constant that summarizes the details of machine construction. These simple relations allow for simple control techniques and excellent dynamic properties in a DC machine, as the separation between field and armature circuits provide actual torque and flux decoupling. Consequently, once the magnetizing flux reaches its nominal value, speed can be adjusted by varying the armature voltage; likewise, torque can be controlled by modifying the armature current. This inherent decoupling makes the DC machine quite a flexible device from a control point of view.

2.3.1 Scalar Control

Among the simplest methods to control induction motor speed lies a variety of techniques collectively named scalar control, as they are based on linear, steady- state relationships that vary only in magnitude. Through circuit analysis, it can be shown that the developed electromagnetic torque is a quadratic function of supply voltage and it is proportional to slip when supply frequency is kept constant [22].

Developing equation (2.4) further, and assuming a sinusoidal flux wave, the machine e.m.f. can be described as follows:

ˆsin( )

ˆ

ˆ_s _s ^s _s _scos( _s )

d t

e N t

dt

 

  

 

 

    (2.26)

where ˆe_s is the peak stator e.m.f., _s the stator angular frequency and ˆ_s the peak stator flux linkage. If the ohmic voltage drop in the stator is assumed to be negligible, the stator e.m.f. and the supply voltage are of similar magnitude. In the case of constant load torque, it follows that by only diminishing the supply voltage, an additional amount of magnetizing current is needed to compensate the fall in magnetic flux and sustain the required torque. This is essentially a slip control strategy, which greatly reduces efficiency and limits the speed control range to certain types of motor designs. On the other hand, varying only the supply frequency and keeping voltage constant saturates the magnetic circuit and

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increases the magnetizing current, creating extra losses and heating issues become a concern.

In order to avoid these shortcomings, it is preferable to relocate the whole torque- speed characteristic of the machine to the desired operating point, that is, adjust the synchronous speed. As it can be inferred from equation (2.2), this can be accomplished by varying the supply frequency. If the magnetizing flux is kept constant, nominal torque can be obtained even during start-up if required and magnetizing current intake is thus minimized.

Equation (2.27) suggests that voltage can be varied along with frequency up to base speed:

2

S

S S m

e U U

 f

    (2.27)

As the ratio of supply voltage and frequency should be maintained constant, the variable-voltage, variable-frequency (VVVF) method of speed control is commonly called Volts per Hertz (V/Hz) control. V/Hz control is a very popular control strategy in industrial pump and fan drives, as it is highly efficient and flexible, covering a very wide range of operating regions with satisfactory performance and reliability. It is also one of the cheapest methods for induction machine speed regulation, especially when used in an open-loop configuration, as there is no need for rotational speed feedback.

The possibility of going beyond synchronous speed is also possible, which is shown in Fig. 2.18:

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Fig. 2.18: Operating regions of an induction machine in one quadrant. From standstill to base speed, stator voltage is increased linearly while flux linkage is maintained constant (constant torque region). Beyond base speed, flux linkage is decreased and supply voltage is left at nominal value (constant power region). When approaching the breakdown torque of the machine, flux linkage must be readjusted in order to avoid malfunction (high speed region).

The constant torque region is where the constant ratio V/Hz is applied in order to achieve maximum torque. At low frequencies in the order of 10 Hz, the stator e.m.f. becomes so small that it is no longer approximately equal to the supply voltage, due to the increased effect of the ohmic voltage drop. This lowers the V/Hz ratio, hence magnetizing flux and torque values decline. In this situation, modern frequency converters implement a voltage boost (which can be seen as a torque boost) in order to keep flux at its rated value and deliver the desired torque if the load requires it. This feature is also sometimes referred to as IR compensation, as the outcome of an increased load is reflected on a higher voltage drop in the stator circuit, which must be compensated.

Due to concerns of overvoltage in the machine insulation, it is recommended to keep voltage constant after reaching base speed. However, if higher speed is desired past this point, increasing frequency would result in diminished magnetizing flux. This is the area of field weakening, where torque decreases proportional to frequency while nominal power is maintained. V/Hz drives are also capable of four quadrant operation and high rotational speeds, nonetheless, these features are seldom required for pump and fan applications and therefore will not be discussed in this work.

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Commercial scalar drives are often implemented in an open-loop configuration.

Fig. 2.19 demonstrates one possibility. The input command is often a frequency or speed value and a user-defined ramp function determines its rate of change. The resulting signal is then fed to the V/Hz regulator, which determines the V/Hz ratio according to the operating region of the motor. An optional torque boost is present which may be user-defined or a factory preset. This voltage output is the reference value to a PWM or SVM modulator alongside the frequency command, which creates a switching sequence.

Fig. 2.19: Open-loop V/Hz scheme with slip compensation and current limiting.

Slip compensation is a mechanism based on the fact that load changes are followed by current changes. As stator current measuring is commonplace due to safety reasons, this information can be used to create a slip command proportional to changes in current (and hence torque) to prevent the slowdown produced by a load change. In this way, the slip compensator produces a frequency value that can be added to or subtracted from the input command and allows the drive to keep the previous operating speed. If the maximum tolerated value for the stator current is reached, the input frequency is not allowed to increase anymore until the current falls back to a safe level.

Accuracy Analysis of Frequency Converter Estimates in Diagnostic Applications

Abstract

Tiivistelmä

Acknowledgements

Contents

Abbreviations and Symbols

Chapter 1

Introduction

Chapter 2

Principles of Induction Motor Drives















 

















 

 

 