. Induction Motor

Table of contents

Abstract……………………………………………………….…………………………………. .3

Rezumat………………………………………………………………………………………..….4

Chapter 1

Introduction………………………………………………………………………………………..5

Chapter 2

Voltage Source Inverter……………………………………………………………………………7

2.1 Three-Phase Voltage Source Inverter…………………………………………………………7

Chapter 3

Induction Motor………………………………………………………………………………….12

3.1 Physical Description…………………………………………………………………………15

3.2 Equivalent Circuit Diagrams…………………………………………………………………18

3.3 Inverse Gamma Model…………………………………………………. ……………………21

3.3.1 Stator Coordinates………………………………………………………………….21

3.3.2 Synchronous and Rotor Flux Coordinates…………………………………………22

3.3.3 Current Control…………………………………………………………………….24

3.4 Gamma Model……………………………………………………………………………….25

3.4.1 Stator Coordinates………………………………………………………………….25

3.4.2 Slip Frequency Control…………………………………………………………….27

3.5 Simulations………………………………………………………………………………………….29

Chapter 4
Field-Oriented Control (FOC)……………………………………………………………………32

4.1 Space Vector definition and projection………………………………………………………33

4.1.1 The (a,b,c(a,b) projection (Clarke transformation)…………………………….34

4.1.2 The (a,b (d,q) projection (Park transformation)……………………………….36

4.1.3 The (d,q(a,b) projection (inverse Park transformation)………………………..37

4.2 The basic scheme for the FOC………………………………………………………………37

4.3 The input for the FOC………………………………………………………………………..39

4.3.1 Current sampling……………………………………………………………………39

4.3.2 Rotor flux position…………………………………………………………………39

4.4 The PI regulator………………………………………………………………………………40

Chapter 5

Controllability Analysis………………………………………………………………………….42

5.1 Input Constraints……………………………………………………………………………..43

5.2 Polar Representation…………………………………………………………………………43

5.3 Poles and Zeros………………………………………………………………………………46

5.4 Relative Gain Array (RGA)………………………………………………………………….48

5.5 Singular Values Revisited……………………………………………………………………49

Chapter 6

Conclusions ………………………………………………………………………………………52

List of symbols……………………………………………………………………………………54

List of Acronyms…………………………………………………………………………………55

Appendix A………………………………………………………………………………………56

Appendix B………………………………………………………………………………………57

Appendix C………………………………………………………………………………………58

Appendix D………………………………………………………………………………………64

Bibliography……………………………………………………………………………………..66

Copyright…………………………………………………………………………………………67

Abstract

This paper considers robust control of an induction motor drive, consisting of an input filter and a voltage source inverter. It is shown how parameter errors influence the torque loop and the conclusion is that the motor leakage inductance should not be overestimated, especially not with a large desired control bandwidth. Based on the closed-loop model, expressions for the controller parameters are derived to obtain required stability margins.

The behavior of an induction motor is accurately described by well known mathematical models. For practical use, the motor parameters must be estimated and adapted to the operating conditions. As high-performance control methods are model based, the closed-loop performance is influenced by the accuracy of the estimated motor parameter. It is valuable to know which motor parameters are most critical and how model errors affect performance. These issues have been explored for the so-called classical field-oriented control or vector control. From models including these non-ideal properties of the control system, explicit expressions for stabilization controllers to use with Field-oriented control are derived. This is valuable as stabilization often is designed through costly and time-consuming manual tuning. Due to the invention of Field Oriented Control the control of the induction motor looks like the control of a direct current motor. In this way reasonable stability margins are obtained, while minimizing the negative effects on torque control.

In this paper has been implemented a graphical user interface of the induction motor drive which simulates the torque of the drive according to the rotor speed. On the plot is shown the maximum torque that cam be achieved by the drive.

It have been studied the controllability an stability of the induction motor. The stability of the drive was represented in the pole-zero map of the induction motor.

Rezumat

Sub îndrumarea Domnului Prof. Dr. Ing. Danuț Ilea, eu, Mihai Pomană am ales această lucrare deoarece am vrut să cercetez mai în amănunt controlul mașinii de inducție.

Motorul de inducție a fost studiat în amănunt, au fost analizate schemele echivalente ale acestuia, s-a studiat controlabilitatea și stabilitatea acestuia. Tot în acest capitol a fost implementată o interfața grafică a motorului de inducție ce calculează raportul dintre turația și cuplul acestuia, după schema echivalentă a circuitului. Deasemenea tot pe acest grafic a fost evidențiat și cuplul maxim pe care îl atinge motorul la o anumită turație.

Controlul motorului de inducție este compus dintr-un filtru de intrare și un invertor. Filtrul de intrare, este direct conectat la linia de tensiune, este folosit ca rezervor de energie pentru a controla armonicele generate de invertor. In timpul realizării comenzii/controlului, filtrul de intrare este adesea neglijat.

In aplicații de tracțiune, condițiile de operare ca viteza, sarcina și temperatura variază. In practică toate procesele de semnal folosite nu sunt adaptate corespunzător sau masurate și de aceea nu toți parametrii sunt implementați în practică. Pentru a garanta o performanță satisfăcătoare, se apelează la algoritmi de control robust. Comportamentul motorului de inducție este în detaliu descris în numeroase modele matematice. In practică, parametrii motorului trebuie adaptați și estimați conform condițiilor de operare. Este important de știut ce parametrii ai motorului sunt critici și cum erorile afectează performanțele. Aceste probleme au fost dezbătute în așa numitul “Field-oriented control”(FOC) sau controlul vectorilor, datorită căruia controlul unui motor de inducție se apropie de cel al unui motor de curent continuu.A fost explicată și schema de bază după care se realizează acest control.

Simulările efectuate în această lucrare au fost realizate în Mathlab și evidențiază într-o manieră practică rezultatele teoretice obținute.

Chapter 1

Introduction

This work treats the robust control of an induction motor drive used for traction applications. An induction motor drive control consists of an input filter and a voltage source inverter. The input filter, which is directly connected to the line, is needed as an energy reservoir and to suppress switching harmonics generated by the inverter. During control design, the input filter is often neglected. To compensate for a varying filter voltage in a practical application, the inverter modulation ratio is modified to ensure that the desired stator voltages always are applied to the motors. With a resonant input filter, this kind of suppression of voltage variations may lead to power oscillations between the inverter and the input filter. Hence there is a trade-off between efficient disturbance rejection and stability of the drive. In a traction application, the operating conditions, such as speed, load and temperature, vary over a wide range. Furthermore, the electrical environment of the drive depends on the number of surrounding trains and feeder stations, and the distances between them. In practice all process signals needed for parameter adaptation are not feasibly measurable and all parameter variations are not practical to implement. To always guarantee satisfying performance, robust control algorithms are therefore required.

The behavior of an induction motor is accurately described by well known mathematical models. For practical use, the motor parameters must be estimated and adapted to the operating conditions. As high performance control methods are model based, the closed-loop performance is influenced by the accuracy of the estimated motor parameter. It is valuable to know which motor parameters are most critical and how model errors affect performance. These issues have been explored for the so-called classical field-oriented control or vector control. In a classic way, torque control of an induction motor drive is designed while neglecting the dynamics of the input filter. In connection with a poorly damped input filter, this property of the controlled inverter may destabilize the drive. These stability problems are not unique for induction motor drives but have been treated also for other kinds of switching converters, e.g. DC-DC converters. For these applications, the stability problem has been solved by adding resistance to the input filter. For high power drives, this solution is not feasible due to the large additional power losses. The behavior of the inverter is then changed through control to emulate an increased resistance. In a torque controlled drive, this may be achieved by modifying the torque reference as a function of

variations of the voltage.

Chapter 2

Voltage Source Inverter

Inverters [1] are used in a wide range of applications, from small switched power supplies for a computer to large electric utility applications to transport bulk power. There are many different power circuit topologies and control strategies used in inverter designs. Different design approaches are used to address various issues that may be more or less important depending on the way that the inverter is intended to be used.

2.1 Three-Phase Voltage Source Inverter

Three-phase inverters are used for variable-frequency drive applications and for high power applications such as HVDC (High-voltage direct current) power transmission. A basic three-phase inverter consists of three single-phase inverter switches each connected to one of the three load terminals. For the most basic control scheme, the operation of the three switches is coordinated so that one switch operates at each 60 degree point of the fundamental output waveform. This creates a line-to-line output waveform that has six steps. The six-step waveform has a zero-voltage step between the positive and negative sections of the square-wave such that the harmonics that are multiples of three are eliminated as described above. When carrier-based PWM techniques are applied to six-step waveforms, the basic overall shape, or envelope, of the waveform is retained so that the 3rd harmonic and its multiples are cancelled.

The control input to the induction motor is the three-phase stator voltage, which is generated through the voltage source inverter in Figure 3-1. A three phase voltage source inverter is depicted in Figure 2-1, consisting of three legs, one for each phase. The legs contain IGBTs (Insulated Gate Bipolar Transistor) in parallel with diodes.

Figure 2-1: Three-phase voltage source inverter.

The IGBT is a transistor, which can either conduct or block current in the direction of the arrow in the figure (in the other direction it always blocks). In each inverter leg, only one of the IGBTs is turned on at a time to prevent a short circuit of the link. If for example the upper IGBT in phase A is turned on and the lower is turned off, a positive current iA(t) must pass through

the upper IGBT, whereas a negative current must go through the upper diode. If we neglect voltage drops of the IGBTs and diodes, the potential at point A then in both cases equals the higher potential of the capacitor in Figure 2-1. For the other case, with the upper IGBT turned off and the lower turned on, the potential at point A equals the lower potential of the capacitor. Hence, simplified we may consider the legs as switches as shown in Figure 2-2. Shown in this figure are also the three stator windings of the induction motor and the three stator voltages (or phase voltages) denoted uA(t), uB(t) and uC(t). Note that stator voltages refer to voltages across the stator windings and not to the inverter ter

Three-phase inverters are used for variable-frequency drive applications and for high power applications such as HVDC (High-voltage direct current) power transmission. A basic three-phase inverter consists of three single-phase inverter switches each connected to one of the three load terminals. For the most basic control scheme, the operation of the three switches is coordinated so that one switch operates at each 60 degree point of the fundamental output waveform. This creates a line-to-line output waveform that has six steps. The six-step waveform has a zero-voltage step between the positive and negative sections of the square-wave such that the harmonics that are multiples of three are eliminated as described above. When carrier-based PWM techniques are applied to six-step waveforms, the basic overall shape, or envelope, of the waveform is retained so that the 3rd harmonic and its multiples are cancelled.

The control input to the induction motor is the three-phase stator voltage, which is generated through the voltage source inverter in Figure 3-1. A three phase voltage source inverter is depicted in Figure 2-1, consisting of three legs, one for each phase. The legs contain IGBTs (Insulated Gate Bipolar Transistor) in parallel with diodes.

Figure 2-1: Three-phase voltage source inverter.

The IGBT is a transistor, which can either conduct or block current in the direction of the arrow in the figure (in the other direction it always blocks). In each inverter leg, only one of the IGBTs is turned on at a time to prevent a short circuit of the link. If for example the upper IGBT in phase A is turned on and the lower is turned off, a positive current iA(t) must pass through

the upper IGBT, whereas a negative current must go through the upper diode. If we neglect voltage drops of the IGBTs and diodes, the potential at point A then in both cases equals the higher potential of the capacitor in Figure 2-1. For the other case, with the upper IGBT turned off and the lower turned on, the potential at point A equals the lower potential of the capacitor. Hence, simplified we may consider the legs as switches as shown in Figure 2-2. Shown in this figure are also the three stator windings of the induction motor and the three stator voltages (or phase voltages) denoted uA(t), uB(t) and uC(t). Note that stator voltages refer to voltages across the stator windings and not to the inverter output voltages (i.e., the differences in potentials between the terminals and the zero point of the inverter).

Figure 2-2: Simplified model of a three-phase voltage source inverter connected to the stator

windings of an induction motor. Shown here are also the three stator voltages uA, uB and uC.

Through the switches in Figure 2-2, the potentials at the points A, B and C can each be set to two different values. If we assign potential zero to the point 0 in the figure, these two values are ±Ud(t)/2. We will denote the potentials vA(t), vB(t) and vC(t), and use the following representation

(2.1)

where kA(t) only can take the two values ±0.5. By also introducing the notation vstar(t) for the potential at the star point of the motor, the stator voltages in Figure 2-2 may be expressed as

(2.2)

The vector k(t) with elements kA(t), kB(t) and kC(t) in (2.2) is the coupling vector, which we have used previously without a formal definition.

Through (2.2), the stator voltages have been expressed using three equations (three-dimensional vectors), one per phase. If the motor is Y-coupled and the star point is not connected, as in Figure 2-2, it follows from Kickoff’s current law that the three stator currents always add to zero. This implies that also the sum of the stator voltages vanishes, i.e.,

(2.3)

The constraint (2.3) implies that the three stator voltages (as well as the currents) are not independent and all information about the stator voltages may be captured by a two-dimensional quantity. For this purpose, the so called space vector representation has been introduced. Space vectors are complex numbers and we have to transform three-dimensional quantities into complex-valued. Due to the constraint (2.3), all information about the three-phase stator voltage in (2.2) is contained in the space vector of the stator voltage, which is given by

(2.4)

where space vectors are denoted in bold face. By comparing the space vector equation (2.4) to the corresponding three-phase expression (2.2), we note that the term containing the potential at the star point has disappeared in (2.4). This is because the space vector transformation maps equal offsets to all three phase quantities, so called zero sequences, to zero. To motivate why zero sequences have no influence, we note that the voltages across the stator impedances in Figure 2-2 (the stator voltages) are not affected by adding or subtracting the same amount to all potentials at the points A, B and C. This also means that the zero sequence of the three-phase coupling vector does not affect the three-phase stator voltage (note that k(t) does not in general satisfy the constraint (2.3)).

With two possible positions of each of the three switches, the inverter in Figure 2-2 may be put in eight different states. Two of these inverter states result in zero stator voltage. These are the combinations with all switches in the upper position or all switches in the lower position. The corresponding (zero) space vectors are sometimes referred to as the zero voltages. The remaining six non-zero stator voltage space vectors are shown in Figure 2-3, where the length of each vector is 2/3Ud

Figure 2-3: The vectors us1 – us6 represent the six non-zero stator voltage space vectors that can be generated by the three-phase voltage source inverter.

The three (physical) stator voltages uA(t), uB(t) and uC(t) can be obtained from a space vector by projecting it onto unit vectors in the directions of us1, us3 and us5 in Figure 2-3. For example this means that the real part of a space vector corresponds to the A-component of the three-phase quantity.

Chapter 3

Induction Motor

In 1882, Nikola Tesla identified the rotating magnetic field principle, and pioneered the use of a rotary field of force to operate machines. He exploited the principle to design a unique two-phase induction motor in 1883. In 1885, Galileo Ferraris independently researched the concept. In 1888, Ferraris published his research in a paper to the Royal Academy of Sciences in Turin. Introduction of Tesla's motor from 1888 onwards initiated what is sometimes referred to as the Second Industrial Revolution, making possible the efficient generation and long distance distribution of electrical energy using the alternating current transmission system, also of Tesla's invention (1888) [1]. Before the invention of the rotating magnetic field, motors operated by continually passing a conductor through a stationary magnetic field (as in homopolar motors). The three-phase induction motor was invented already in the 19th century. Compared to DC motors, induction motors have higher power densities and are mechanically more robust, which make them the ideal motor in many applications. On the other hand, they require AC power supplies and more sophisticated control. First with the invention of field-oriented control (or vector control) in the late 1960’s, [5], control of an induction motor could be compared to that of a separately excited DC motor. The need of advanced power electronics further delayed the widespread use of induction motors in industrial and traction applications till the 1980’s. However, squirrel-cage induction motors fed voltage source inverters (VSI) is standard in traction applications. Where a polyphase electrical supply is available, the three-phase (or polyphase) AC induction motor is commonly used, especially for higher-powered motors. The phase differences between the three phases of the polyphase electrical supply create a rotating electromagnetic field in the motor.

Through electromagnetic induction, the rotating magnetic field induces a current in the conductors in the rotor, which in turn sets up a counterbalancing magnetic field that causes the rotor to turn in the direction the field is rotating. The rotor must always rotate slower than the rotating magnetic field produced by the polyphase electrical supply; otherwise, no counterbalancing field will be produced in the rotor. Induction motors are the workhorses of industry and motors up to about 500 kW (670 horsepower) in output are produced in highly standardized frame sizes, making them nearly completely interchangeable between manufacturers (although European and North American standard dimensions are different). Very large induction motors are capable of tens of thousands of kW in output, for pipeline compressors, wind-tunnel drives and overland conveyor systems.

Figure 3-1: Model of an induction motor drive.

In traction applications, space and weight constraints tend to keep the capacitance C of the input filter relatively small. To meet regulations on harmonics suppression, or to assure a certain required input impedance of the drive, the inductance L has to be relatively large. In

combination with a small resistance R to keep the power losses acceptable, the resulting input filters get poorly damped with high resonance peaks at the frequency ω0= 1/√(LC). This is illustrated in Figure 3-2, showing the Bode plot of the transfer function from link current id(t) to line current i(t), which is given by the following equation

(3.1)

Here p is the differential operator and the filter data are taken from Appendix A (input filter with 35.6 Hz resonance frequency). The large resonance peak is hence a consequence of a desired small gain of the filter at high frequencies to suppress switching frequency harmonics. To quantify the damping properties of the input filter, we use the damping factor ζ, which is defined by

(3.2)

The damping factor for the filter used to generate Figure 3-2 is only 0.02.

Figure 3-2: Bode plot of input filter transfer function.

The simulation was made in Mathlab and the code for the Bode plot ca be found in Apendix B.

3.1 Physical Description

The induction motor [6] consists of a stationary part, the stator which have coils supplied with AC current to produce a rotating magnetic field, and a rotating part, the rotor which is attached to the output shaft that is given a torque by the rotating field. A stator with a three-phase distributed stator winding is shown in Figure 3-3.

Figure 3-3: Illustration of the stator of an induction motor.

The rotor often is of the squirrel cage type, which simply consists of a number of rotor bars connected through two end rings as shown in Figure 3-4. Just as with the stator, the electrical effect of the rotor may be modeled as an equivalent three-phase winding.

A squirrel cage rotor is the rotating part commonly used in an AC induction motor. An electric motor with a squirrel cage rotor is sometimes called a squirrel cage motor. In overall shape it is a cylinder mounted on a shaft. Internally it contains longitudinal conductive bars of aluminum or copper set into grooves and connected together at both ends by shorting rings forming a cage-like shape. The name is derived from the similarity between this rings-and-bars winding and a hamster wheel (presumably similar wheels exist for pet squirrels).The core of the rotor is built of a stack of iron laminations.

Figure 3-4: Squirrel cage rotor with rotor bars connection by two end rings.

The drawing shows only three laminations of the stack but many more are used. The field windings in the stator of an induction motor set up a rotating magnetic field around the rotor. The relative motion between this field and the rotation of the rotor induces electrical current flow in the conductive bars. In turn these currents flowing lengthwise in the conductors react with the magnetic field of the motor to produce force acting at a tangent to the rotor, resulting in torque to turn the shaft. In effect the rotor is carried around with the magnetic field but at a slightly slower rate of rotation. The difference in speed is called “slip” and increases with load. The conductors are often skewed slightly along the length of the rotor to reduce noise and smooth out torque fluctuations that might result at some speeds due to interactions with the pole pieces of the stator. The number of bars on the squirrel cage determines to what extent the induced currents are fed back to the stator coils and hence the current through them. The constructions that offer the least feedback employ prime numbers of bars. The iron core serves to carry the magnetic field across the motor. In structure and material it is designed to minimize losses. The thin laminations, separated by varnish insulation, reduce stray circulating currents that would result in eddy current loss. The material is a low carbon but high silicon iron with several times the resistance of pure iron, further reducing eddy-current loss. The low carbon content makes it a magnetically soft material with low hysteresis loss. Applying a sinusoidal three-phase voltage to the terminals of an induction motor results in a rotating flux in the stator. This is schematically illustrated with the large rotating magnet in Figure 3-5. Most of the flux propagates to the rotor, where the rotor bars hence see a magnetic flux passing by. Consequently, voltages across the rotor bars are induced, which in turn give rise to rotor currents. These rotor currents generate flux and the rotor is therefore represented as a small magnet in Figure 3-5. Together with the original flux, the additional flux produced by the rotor winding generates a torque, which strives to align the two fluxes. The torque hence forces the rotor to follow the applied stator flux. Note, however, that a non-zero torque requires that the stator flux rotates asynchronously to the rotor (as opposed to the case with synchronous motors). Otherwise no currents are induced, as the flux seen by the rotor bars then is constant. This fact also motivates the name asynchronous motor, which also is used for this type of motor. The frequency of the induced electrical rotor quantities equals the frequency of the flux seen by the rotor bars, which is called the slip frequency and is denoted by ω2. The slip frequency hence corresponds to the difference between the frequency of the rotating flux and the electrical rotational frequency of the rotor. By electrical rotor frequency we mean np times the mechanical rotor speed ωm, where np is the number of pole pairs of the motor. The phase windings of the stator are usually arranged to give multiple poles, see [9]. With np pole pairs, one physical rotation of the rotor corresponds to np rotations for the electrical quantities of the rotor. The relevant rotor frequency in this case therefore is npωm.

Figure 3-5: Schematic picture of an induction motor. The magnetic field produced by the stator is represented by the large rotating magnet. The varying flux induces currents and hence also flux in the rotor, which is illustrated by the small magnet in the figure. Torque is generated to align the stator and rotor fluxes.

Although torque is the main control variable, i.e., the main quantity we want to control, usually also the flux in the motor is controlled. The induction motor is designed to operate at a certain nominal flux and running with for example larger fluxes drives the motor into saturation, which produces large currents in the stator windings. Using a too small flux leads to a reduced maximum torque. In practical applications, the flux may be varied as a function of load as to minimize the total stator current.

The schematically shown controller in Figure 3-6 controls the operation of the inverter through the coupling vector k(t). Based on measured motor, the coupling vector is determined to make the motor torque follow a reference, which is denoted by Tref in Figure 3-6. Given that the induction motor is a non-linear multi-input multi-output (MIMO) system with varying parameters, control of these systems is a challenging task that has received rather much attention in the literature over the last decades, see e.g. [7]. In most of these publications, the influence of the power supply is neglected. The controlled drive may then be represented by the block diagram in Figure 3-10, where the torque is influenced by the reference signal via the system Gc and through a disturbance voltage through a transfer function Gd. The influence of the motor speed is not explicitly shown in the figure as we often will consider the speed to be constant in this paper.

Figure 3-6: Model of a controlled induction motor. Here the dynamics of the input filter are not

considered.

3.2 Equivalent Circuit Diagrams

Mathematically, the induction motor can be compactly described by using complex-valued space vectors. Magnetically linear motors are assumed with inductances varying sinusoidally with the rotor position. It should be noted that different models exist that correctly describe the motor as seen from the stator terminals (at least if iron losses are neglected). Depending on the purpose, a certain model, or equivalent circuit diagram (ECD), may be preferred in a certain situation.

Here is an example of an exact equivalent circuit diagram.

Figure 3-7: Equivalent circuit of an induction motor.

Where, R1 is the stator resistance per phase, X1 is the stator reactance per phase, R is the equivalent rotor resistance referred to stator per phase, X is the equivalent rotor reactance referred to stator per phase, Rc is the resistance representing core losses, Xm is the magnetizing reactance per phase, V1 is the per phase supply voltage to the stator, s is the slip of the motor.

For example, with a control scheme oriented to the rotor flux, such as the field-oriented control (FOC), an ECD with simple rotor equations may be preferred. Such an ECD is the so called inverse Γ-model, which is shown in Figure 3-6. The ECD consists of a stator mesh and a rotor mesh, corresponding to the two three-phase windings of the motor. The two meshes are connected by the magnetizing inductance Lm and the rotor mesh also contains the so called rotor EMF as a voltage source. Each winding contains resistance, where Rs denotes stator resistance and Rr denotes rotor resistance. In practice, each winding also contains leakage inductance as not all flux produced in one winding passes through to the other winding. However, in the inverse Γ-model all leakage inductance has been put in the stator mesh, where the total leakage inductance is denoted Lσ. This is not really physically correct, but simplifies the rotor equations. Seen from the stator terminal, the representation is correct, although for example the internal rotor current ir is scaled compared to the current that would be measured on a real motor. Finally, , and in the ECD represent the stator current, the stator flux and the rotor flux.

Figure 3-8: Inverse Γ-model ECD of the induction motor.

For the control method Indirect Self Control (ISC), the stator flux is the central quantity. In this case the Γ-model shown in Figure 3-9 is suitable, where all leakage inductance is put in the rotor mesh to simplify the expressions for the stator equations. The term “Γ- model” refers to the location of the inductances in the ECD in Figure 3-9 (which also motivates the name “inverse Γ-model” for the ECD in Figure 3-8). Although rotor quantities and parameters in Figure 3-6 are not equal to those in Figure 3-9, the same notation is still used for simplicity. It should be clear from the context which model is used in a certain chapter. In connection with ISC, all quantities refer to the Γ-model and in connection to FOC all quantities refer to the inverse Γ-model. The model in Figure 3-8, as well as the model in Figure 3-9, is given in stator fixed coordinates, where space vectors are expressed in a coordinate system attached to the stator. However, when working with FOC in Chapter 4, we will use a description in rotor flux coordinates. In order to distinguish between quantities in the two coordinate systems, quantities of the inverse Γ-model in stator fixed coordinates are written with superscripts s.

Figure 3-9: Γ-model ECD of the induction motor.

3.3 Inverse Gamma Model

3.3.1 Stator Coordinates

From the ECD in Figure 3-8 it follows that the induction motor is described by the following equations in stator coordinates

(3.3)

(3.4)

where ES is the so called back EMF. The torque is not directly given in Figure 3-8 but can be calculated from rotor flux and stator current as [9].

(3.5)

where the asterisk denotes complex conjugate and α and β denote the real and imaginary parts of the (stator-fixed) space vectors.

3.3.2 Synchronous and Rotor Flux Coordinates

The equations in Subsection 3.3.1 are said to be in stator coordinates. This means that the space vectors are represented in a coordinate system that is attached to the stator (does not rotate). Also the rotor quantities are expressed in this fixed coordinate system. The real and imaginary parts of a general space vector QS(t) in stator coordinates are denoted Qα and Qβ, respectively.

Figure 3-10: Illustration of synchronous coordinates. The synchronous coordinate system with

axes denoted by d and q is displaced the angle θ1 relative to the stator fixed coordinate system

with axes denoted α and β.

We now introduce a change of coordinates as

(3.6)

where θ1(t) is the angle of a rotating coordinate system, see the illustration of the coordinate transformation in Figure 3-10. The new quantity Q(t) is said to be in synchronous coordinates and the d-component is referred to as the direct component and the q-components is referred to as the quadrature component of the space vector.

In order to express the motor equations from the previous subsection in synchronous coordinates, we need to know how derivates are affected by the change of coordinates. For that purpose we note that the derivative of a space vector in stator coordinates can be expressed as

(3.7)

That is,

(3.8)

Where

(3.9)

From the definition of a space vector in synchronous coordinates in (3.6) and the expression for the derivatives in (3.7), it follows that the two motor equations (3.3) and (3.4) are transformed to

(3.10)

(3.11)

The torque equation (3.5), finally, may now be represented as

(3.12)

The synchronous coordinates are often chosen as rotor flux coordinates, i.e., θ1 is chosen as the angle of the rotor flux space vector. In rotor flux coordinates, the q-component of the flux therefore is zero by definition. The equation for the stator current space vector (3.3) then becomes

(3.13)

where we used the notation Ψr = Ψrd. The corresponding equation for the rotor flux space vector can be separated into one equation for the rotor flux magnitude and one equation for the stator frequency as

(3.14)

(3.15)

where ω1 is the frequency of the rotor flux space vector, see (3.9). Finally, the torque can be calculated from rotor flux and stator current as

(3.16)

which follows from (3.12).

3.3.3 Current Control

For a separately excited DC motor, the torque is proportional to the product of flux and armature current. By keeping the flux constant (through a constant magnetization current), the torque may be controlled by varying the armature current. For an induction motor, the torque equation (3.5) is a little bit more complicated. However, in rotor flux coordinates, the situation resembles that of the DC motor, see the torque equation (3.16). Here the torque is given as the product of the rotor flux magnitude Ψr and the quadrature current component isq. From the rotor flux equation (3.14), we see that the rotor flux Ψr is only influenced by the direct current component isd. The control strategy with FOC therefore is to use the direct current component to regulate the rotor flux to a constant value and then to use the quadrature component to control the torque (the current components can be modified independently). By working in rotor flux coordinates, we hence can reuse ideas from DC-motor control.

3.4 Gamma Model

With ISC, the stator flux is a central quantity and for that purpose the Γ-model, with its simple stator equations is used when modeling the induction motor. The motor equations of the Γ-model in stator coordinates are given in Subsection 3.4.1, where also the idea behind ISC (and other DTC techniques) is presented. In Subsection 3.4.2, steady-state relations between the slip frequency and the torque are derived.

3.4.1 Stator Coordinates

By using the ECD in Figure 3-9, the fluxes of the induction motor can be described in stator coordinates.

(3.17)

(3.18)

Note that the equations are in stator coordinates, although no superscripts s are used. Further, the stator current is given by

(3.19)

and the torque may be calculated as

(3.20)

where α and β denote the real and imaginary components of the space vectors. By inserting the expression for the stator current (3.19) into the torque equation (3.20), we may also compute the torque as

(3.21)

By using a polar representation of the space vectors as

(3.22)

the torque in (3.21) can be written as

(3.23)

where the load angle δ(t) was defined as the difference between the angles of the stator and rotor flux space vectors. The load angle is illustrated in Figure 3-11.

Figure 3-11: Definition of load angle δ(t), which is the angle between the stator and rotor fluxes.

With DTC methods, torque is controlled by increasing or decreasing the load angle.

The idea with DTC methods is to keep the stator flux magnitude constant and affect the torque via the load angle δ. As the rotor flux varies relatively slowly compared to the stator flux, an increase in load angle transiently gives an increase in torque. The load angle is affected by varying the frequency of the stator flux space vector. In the following subsection, steady-state relations between the load angle, the torque and the slip frequency are derived.

3.4.2 Slip Frequency Control

In this subsection we derive a steady-state relation between the torque and the slip frequency (with constant stator flux), which will be used in connection with ISC. We also derive an expression for the so called pull-out torque, which is the maximum torque that can be generated with a constant stator flux magnitude.

With the polar representation (3.22), the rotor equation (3.18) can be represented as

(3.24)

By separating Equation (3.22) into real and imaginary parts, it follows that

(3.23)

(3.24)

where the rotor leakage time constant T σ = L σ /Rr was introduced. At steady state, the rotor flux magnitude is constant, which means that (3.23) reduces to

(3.25)

where subscripts 0 denote steady-state values. By using (3.25), Equation (3.24) gives the following expression for the steady state slip frequency

(3.26)

We may now insert the two steady state relations (3.25) and (3.26) into the torque equation (3.21), to write the steady-state torque as

(3.27)

Equation (3.27) shows that if the rotor flux is constant, the torque is proportional to the slip frequency. If instead of regulating the rotor flux, the stator flux is kept constant, the rotor flux magnitude varies with the load and expression (3.27) gets a little more complicated. By again using the steady state relation (3.25) and the following trigonometric identity

(3.28)

the torque equation (3.27) can be written as

(3.29)

The steady-state torque according to (3.29) is plotted in Figure 3-12 as a function of slip frequency with constant stator flux magnitude. Here we see that around the origin, the steady state torque depends almost linearly on slip frequency but eventually reaches a maximum for positive slip frequencies and a minimum for negative slip frequencies. The maximum (or minimum) steady state torque is called the pull-out torque or break-down torque and occurs at

(3.30)

The pull-out torque is given by

(3.31)

Increasing the slip frequency above the value (3.30) results in a decreasing torque. The increased slip frequency, however, increases losses in the rotor, which may damage the motor. The steady state slip frequency should therefore always be upper limited by (3.30).

Figure 3-12: Steady state torque as a function of steady state slip frequency. Here it is seen that the torque approximately varies linearly with the slip frequency for small slip frequencies. Above the so called pull-out slip frequency (for positive slip frequencies) the torque decreases with increasing slip frequency, while losses increase in the motor. A control algorithm therefore should limit the steady state slip to values below the pull-out slip frequency.

3.5 Simulations

The next simulations were made in Mathlab . I made an graphical user interface shown in Figure 3-13 for the equivalent circuit diagram to give a note of originality to the paper. The code for this simulation can be found in Appendix C. The values have been adjusted to obtain the desired results.

Figure 3-13: Graphical user interface. Here are presented the parameters of the drive and operation conditions.

Figure 3-14: Torque diagram. This figure evidentiates the starting torque and the maximum torque that can be achieved by the induction motor.

Chapter 4

Field-Oriented Control (FOC)

A major achievement in the area of induction motor control was taken with the invention of field-oriented control (FOC) or vector control in the late 1960s. Until then, induction motors had been controlled using so called scalar control methods, like the volt-hertz control. Here the magnitude and frequency of the stator voltage are determined from steady-state properties of the motor, which leads to poor dynamic performance. On the other hand, FOC uses a vector model of the drive that is valid also during transients, which facilitates faster control. The idea with field orientation is to mimic control of DC motors also for induction motors. To accomplish this, the motor equations are described in a coordinate system oriented to the (rotor) flux in the motor. The torque and flux can then be controlled through different components of the stator current. In the rotating coordinate system, all motor quantities are constant at steady state and PI controllers can hence be used to ensure zero steady-state errors. It has later been shown that the classical field-oriented controller can be interpreted as asymptotic exact linearization [8]. Field oriented control was developed by a research group lead by Prof. Werner Leonard in Brauschweig, Germany.

Field-oriented (vector) controls, which enable AC motors to behave like DC motors, constitute a major application. The brush and commutator assembly in a dc motor ensures that the field (stator) current is always at right angles to the armature (rotor) current. Known as field orientation, this condition allows the rotor to generate the maximum torque for which it is rated.

The Field Orientated Control (FOC) [13] consists of controlling the stator currents

represented by a vector. This control is based on projections which transform a threephase time and speed dependent system into a two co-ordinate ( d and q co-ordinates) time invariant system. These projections lead to a structure similar to that of a DC machine control. Field orientated controlled machines need two constants as input references: the torque component (aligned with the q co-ordinate) and the flux component (aligned with d co-ordinate). As Field Orientated Control is simply based on projections the control structure handles instantaneous electrical quantities. This makes the control accurate in every working operation (steady state and transient) and independent of the limited bandwidth mathematical model. The FOC thus solves the classic scheme problems, the ease of reaching constant reference (torque component and flux component of the stator current) and the ease of applying direct torque control because in the ( d,q) reference frame.

By maintaining the amplitude of the rotor flux () at a fixed value we have a linear relationship between torque and torque component (iSq). We can then control the torque by controlling the torque component of stator current vector.

4.1 Space Vector definition and projection

The three-phase voltages, currents and fluxes of AC-motors can be analyzed in terms of

complex space vectors. With regard to the currents, the space vector can be defined as follows. Assuming that ia, ib, ic are the instantaneous currents in the stator phases, then the complex stator current vector is defined by:

(4.1)

where and , represent the spatial operators. The following diagram

shows the stator current complex space vector:

Figure 4-1: Stator current space vector and its component in (a,b,c)

where (a,b,c) are the three phase system axes. This current space vector depicts the three phase sinusoidal system. It still needs to be transformed into a two time invariant co-ordinate system. This transformation can be split into two steps: Firs step: (a,b,c)(a,b) (the Clarke transformation) which outputs a two co-ordinate time variant system and second step: (a,b)( d,q) (the Park transformation) which outputs a two co-ordinate time invariant system.

4.1.1 The (a,b,c)(a,b) projection (Clarke transformation)

The space vector can be reported in another reference frame with only two orthogonal

axis called (a,b). Assuming that the axis a and the axis a are in the same direction we

have the following vector diagram:

Figure 4-2: Stator current space vector and its components in (a,b)

The projection that modifies the three phase system into the (a,b) two dimension

orthogonal system is presented below.

(4.2)

for a TMS320F240 software implementation refer to report (BPRA048). We obtain a two

co-ordinate system that still depends on time and speed.

4.1.2 The (a,b)->(d,q) projection (Park transformation)

This is the most important transformation in the FOC. In fact, this projection modifies a

two phase orthogonal system (a,b) in the d,q rotating reference frame. If we consider the

d axis aligned with the rotor flux, the next diagram shows, for the current vector, the

relationship from the two reference frame:

Figure 4-3: Stator current space vector and its component in (a,b) and in the d,q

rotating reference frame

where is the rotor flux position. The flux and torque components of the current vector

are determined by the following equations:

(4.3)

These components depend on the current vector (a,b) components and on the rotor flux

position; if we know the right rotor flux position then, by this projection, the d,q component becomes a constant. For TMS320F240 software implementation refer to report (BPRA048). We obtain a two co-ordinate system with the following characteristics: it is a two co-ordinate time invariant system and with iSd (flux component) and iSq (torque component) the direct torque control is possible and easy.

4.1.3 The (d,q)(a,b) projection (inverse Park transformation)

Here, we introduce from this voltage transformation only the equation that modifies the

voltages in d,q rotating reference frame in a two phase orthogonal system:

(4.4)

The outputs of this block are the components of the reference vector that we call ;

is the voltage space vector to be applied to the motor phases. For TMS320F240

software implementation refer to report (BPRA048).

4.2 The basic scheme for the FOC

The following diagram summarizes the basic scheme of torque control with FOC

Figure 4-4: Basic scheme of FOC for AC-motor

Two motor phase currents are measured. These measurements feed the Clarke transformation module. The outputs of this projection are designated iSa and iSb. These two components of the current are the inputs of the Park transformation that gives the current in the d,q rotating reference frame. The iSd and iSq components are compared to the references iSdref (the flux reference) and iSqref (the torque reference). At this point, this control structure shows an interesting advantage: it can be used to control either synchronous or induction machines by simply changing the flux reference and obtaining rotor flux position. As in synchronous permanent magnet motors, the rotor flux is fixed (determined by the magnets) there is no need to create one. Hence, when controlling a PMSM, iSdref should be set to zero. As induction motors need a rotor flux creation in order to operate, the flux reference must not be zero. This conveniently solves one of the major drawbacks of the “classic” control structures: the portability from asynchronous to synchronous drives. The torque command iSqref could be the output of the speed regulator when we use a speed FOC. The outputs of the current regulators are vSdref and vSqref; they are applied to the inverse Park transformation. The outputs of this projection are vSaref and vSbref which are the components of the stator vector voltage in the a,b stationary orthogonal reference frame. These are the inputs of the Space Vector PWM. The outputs of this block are the signals that drive the inverter. Note that both Park and inverse Park transformations need the rotor flux position. Obtaining this rotor flux position depends on the AC machine type (synchronous or asynchronous machine). Rotor flux position considerations are made in a following paragraph.

4.3 The input for the FOC

Fundamental requirements for the FOC are a knowledge of two phase currents (as the

motor is star-connected, the third phase current is also known, since ia+ib+ic=0) and

the rotor flux position.

4.3.1 Current sampling

The measured phase currents ia and ib are sampled and converted by an A/D converter.

The correct working of the FOC depends on the true measurement of these currents.

4.3.2 Rotor flux position

Knowledge of the rotor flux position is the core of the FOC. In fact if there is an error

in this variable the rotor flux is not aligned with d-axis and iSd and iSq are incorrect flux and

torque components of the stator current. The following diagram shows the (a,b,c), (a,b)

and ( d,q) reference frames, and the correct position of the rotor flux, the stator current

and stator voltage space vector that rotates with d,q reference at synchronous speed.

Figure 4-5: Current, voltage and rotor flux space vectors in the d,q rotating reference frame and their relationship with a,b,c and a,b stationary reference frame

In the induction machine the rotor speed is not equal to the rotor flux speed (there is a

slip speed), then it needs a particular method to calculate . The basic method is the

use of the current model which needs two equations of the motor model in d,q

reference frame.

4.4 The PI regulator

An electrical drive based on the Field Orientated Control needs two constants as control

parameters: the torque component reference ISqref and the flux component reference ISdrefef.

The classic numerical PI (Proportional and Integral) regulator is well suited to regulating

the torque and flux feedback to the desired values as it is able to reach constant references, by correctly setting both the P term (Kpi) and the I term (Ki) which are respectively responsible for the error sensibility and for the steady state error. The numerical expression of the PI regulator is as follows:

(4.5)

Which can be represented by the following figure:

Figure 4-6: Classical Numerical PI Regulator Structure

Stator flux magnitude and torque are controlled through PI controllers.

Chapter 5

Controllability Analysis

In this chapter we perform an input-output controllability analysis of the inverter fed induction motor to reveal its relevant properties from a control point of view. This analysis includes dynamic properties, but also consequences of constraints on the inputs are considered. The controllability analysis is performed at zero torque and nominal flux at three different stator

frequencies, namely 10% (OP1), 50% (OP2) and 90% (OP3) of base speed ωbase. It will be shown that the induction motor has a sharp resonance peak at the operating point stator frequency, where the height of the resonance peak increases with the motor speed. It is also shown that coupling between the inputs and outputs may give large disturbances in torque at pure flux control at higher speeds with input uncertainty. Further, the upper limitation of the

voltage magnitude reduces the possibilities to suppress voltage disturbances at higher speeds. As the voltage excites a high gain direction of the plant, such disturbances have a large effect on the outputs.

Remark: The controllability an alysis is only carried out for rotor speeds below base speed. Above base speed, the stator voltage magnitude saturates and only the frequency of the stator voltage can be modified. Here the induction motor hence represents a system with one input and two outputs. However, disturbances still affect the magnitude of the stator voltage.

5.1 Input Constraints

In this paper we consider induction motors fed by a voltage source inverter. The fundamental stator voltage magnitude therefore is limited by

(5.1)

where Ud is the voltage. This is a constraint on the stator voltage amplitude but we also should put restrictions on the stator voltage frequency to prevent the steady state slip frequency, ω20, from exceeding the pull-out slip frequency (3.30), i.e.,

(5.2)

where ω10 is the steady state stator frequency. Note that with constant rotor speed, the constraint on the deviation of the (steady-state) stator frequency around an operating point with zero slip frequency becomes

(5.3)

In the sequel we will use linearized models where deviations of signals around stationary operating points are modeled. A delta notation as in (5.3) could then have been used. However, to simplify notation, the deltas will be omitted.

5.2 Polar Representation

The outputs of the motor are here considered to be the torque and the stator flux magnitude. Further, from the input constraints given by (5.1) and (5.3), it makes sense to consider the stator voltage magnitude and frequency as inputs. We therefore derive a linear model with the mentioned inputs and outputs. To facilitate this, we use the polar notation of the space vectors introduced by (3.22). With the polar representation, the torque can be expressed by (3.23), i.e.,

(5.4)

To obtain a linear model of the process, the motor equations (3.17) and (3.18) are rewritten in the following way

(5.5)

(5.6)

(5.7)

(5.8)

where δ is the load angle, see (3.23), and

(5.9)

Equations (5.4) and (5.5)-(5.8) now form a (real-valued) non-linear state space model of the induction motor. Inputs are the stator voltage magnitude and frequency and outputs are the stator flux magnitude and the torque (via (5.4)). Acting on the system is also the rotor speed, which is considered as a disturbance.

The non-linear model can be linearized around stationary operating points. From (5.5)-(5.8) it follows that at an operating point with torque, stator flux and mechanical rotor speed specified by T0, Ψs0 and ωm0, the stationary states are given by

(5.10)

and the stationary inputs by

(5.11)

By linearizing the equations (5.4) – (5.8) we now obtain a linear model of the plant, which we represent as

(5.12)

where G is a 2×2 transfer matrix representing the dynamics from inputs to outputs and Gv ω is a 2×1 matrix representing the dynamics from the rotor speed to the outputs. Further, the notation was introduced.

In addition to the rotor speed, also the link voltage is considered a disturbance. The influence of the voltage on the three-phase stator voltage was discussed in connection with PWM, where also a method to suppress the influence of a varying voltage was introduced. Here we neglect this compensation and thus model the stator voltage as

(5.13)

Equation (5.13) can be linearized and represented in polar form as

(5.14)

Note that the voltage only affects the magnitude of the stator voltage space vector in (5.14). We also see that the voltage adds to the input of the plant. If we move the disturbance to the output instead, the linearized model of the induction motor will be represented by

(5.15)

where the disturbance vector and the disturbance transfer function Gv were introduced. The model (5.15) is visualized in Figure 5-1.

Figure 5-1: Linear induction motor model. The input to the motor, which here is represented by

the stator voltage magnitude and stator voltage frequency, affects the outputs, i.e., the stator flux

magnitude and torque via the transfer function G. The outputs are also affected by the disturbances DC-link voltage and motor speed via the disturbance transfer function Gv.

5.3 Poles and Zeros

The poles and zeros of the linear process model G derived in the previous section are shown in Figure 5-2 at zero torque and varying stator frequency.

Figure 5-2: Pole-zero map of the induction motor. x’s and o’s denote poles and zeros at the three

operating points OP1, OP2 and OP3. Here it is seen that the poles are in the left half plane, hich

means that the motor model is stable. One poorly damped pole pair moves along the imaginary

axis with approximately the operating point stator frequency. The poles here are shown at zero

operating point torque. At non-zero torque the locations of the poles are only slightly changed.

The locations of the poles and zeros at the three operating points defined above are marked with x’s and o’s, respectively. The system is stable at all operating points with one LHP zero and four LHP poles. As the stator frequency increases, two poles approach the zero, whereas the remaining two poles move along the imaginary axis with the stator frequency. Consequently,

the system gets less damped with increasing rotor speed. The smaller the stator resistance, the closer the poles get to the imaginary axis.

5.4 Relative Gain Array (RGA)

The relative gain array (RGA) was introduced by Bristol in 1966 as a measure of interactions for decentralized control [2]. The RGA of a non-singular square matrix G, denoted RGA(G) or Λ(G) is a square matrix defined by

(5.16)

where the operation x denotes element by element multiplication (Hadamard or Schur product).

However, the RGA also is an indicator of sensitivity to uncertainty. Large RGA elements mean poor robustness for inverse based controllers in presence of independent input uncertainty, [3]. By large is here meant values above 10, [3]. On the other hand, large RGA elements also

mean that there is strong coupling and even nominal performance with a diagonal controller may not be satisfactory. Hence, in general, large RGA elements indicate a plant that is inherently difficult to control.

The maximum RGA elements for the induction motor at the three operating points are plotted in Figure 5-3 as solid, dashed and dotted lines. As seen, there are peaks in the RGA elements corresponding to the operating point stator frequency and the heights increase with the stator frequency. Most problematic are large RGA elements around the desired bandwidth. One

possibility to avoid potential problems is then to set the bandwidth well above the operating point stator frequency, where the RGA elements are small. Due to bandwidth constraints, for example due to time delays, this may not be possible at higher stator frequencies (where low switching frequencies give long time delays) and the RGA indicator limits the achievable bandwidth at those operating points. With a bandwidth independent of the operating point, it follows that it probably has to be less than the stator frequency at OP3.

Remark: Many control algorithms aim at decoupling the control of torque and flux, for example the ISC. Such controllers are inverse based and are consequently sensitive to diagonal input uncertainty.

Figure 5-3: Maximum RGA elements at the three considered operating points. The solid curve

shows the RGA elements at OP1, the dashed curve shows the RGA elements at OP2 and the

dotted curve shows the RGA elements at OP3. We note that the peaks of the RGA elements

appear at the operating point stator frequencies and increase with higher speed. Large RGA

elements for example indicate sensitivity to independent input uncertainty.

5.5 Singular Values Revisited

In this section we try to physically motivate the resonance peak of the maximum singular value at the operating point stator frequency. We will do this by examining the gain of the motor from stator voltages to stator currents. This gain has been studied in for example [4]. At constant speed we can form the Laplace transforms of the motor equations (3.3) and (3.4) to obtain the following transfer function representation

(5.17)

Where

(5.18)

Note that the transfer function in (5.18) has complex-valued coefficients and is therefore typed in bold face (in equivalence to space vectors). For systems that can be represented on this form, that the singular values of the corresponding real-valued representation of the system can be calculated as

(5.19)

The inputs giving maximum gains are hence given by the so called positive and negative sequences, i.e.,

(5.20)

The gains (5.19) are shown at the speeds ωm = 0, 10 and 50 Hz in Figure 5-4.We see that the gain is largest at low frequencies. This may be physically motivated as only the winding resistances limit the current at low frequencies, whereas the current is also limited by the inductances at higher frequencies.

(A) (B)

(C)

Figure 5-4: Maximum and minimum gains of the induction motor in stator coordinates at (A) 0

Hz, (B) 10 Hz and (C) 50 Hz mechanical speed. Maximum gains appear at low frequencies.

Stationary currents are only limited by the winding resistances, whereas inductances also limit

sinusoidal currents.

The simulations were made in Mathlab and the code from which it were implemented is presented in Appendix D.

Chapter 6

Conclusions

This thesis has considered robust control of an induction motor drive. A controllability analysis of the inverter fed induction motor was first performed to better understand fundamental properties and limitations of the plant from a control point of view. The controllability analysis showed that the motor has large RGA elements for large operating point stator frequencies, which lead to high sensitivity to model errors with a decoupling controller, and the variations in the voltage affect the magnitude of the stator voltage, which acts in a high gain direction around the resonance frequency of the motor. Due to input constraints and limited control bandwidth, disturbances in the voltage therefore have a large effect of the motor outputs at high stator frequencies.

We have examined properties of an inverter fed induction motor. Usually, the induction motor equations are expressed using space vectors in stator or synchronous coordinates. In this paper we have used a linear model of the motor on polar form with the stator voltage magnitude and frequency as inputs. This way the input constraint of the stator voltage magnitude is easily analyzed. Outputs of the linear model are the torque and the stator flux magnitude. It has been shown that the linear motor model is stable at all examined operating points. The dynamics of the motor strongly vary with the motor speed with a poorly damped pole pair that moves along the imaginary axis. These poles give the maximum singular value of the motor a sharp resonance peak at the operating point stator frequency. The height of the peak increases with the speed of the motor. Here we have extended the analysis of the linea r motor model to also consider the relative gain array (RGA) and consequences of input limitations. It was shown that also the RGA elements show a resonance peak at the operating point stator frequency, where the height increases with the speed. Large RGA elements mean that the cross coupling of the plant is significant. For good nominal performance, a decoupling controller is required. However, a decoupling controller can lead to poor robust performance in presence of independent input uncertainty. We then conclude that it may be difficult to achieve both good nominal and robust performance of the induction motor at higher speeds. Inputs with perfect flux control act in a low gain direction of the induction motor, which means that large inputs are needed for fast flux control. The coupling of the motor then means that model errors may cause pure flux control to accidentally excite high gain directions instead of the desired ideal flux direction.

In this paper we dealt with the Field Orientated Control of three-phase AC machines. Following a description of common major drawbacks of classic control structures it has been shown how the Field Orientated Control overcomes these deficiencies and what kind of benefits Field Orientated Controlled AC drives can bring. By explaining in detail each of the FOC modules necessary, it presents a clear introduction to efficient vector control of AC drives.

List of symbols

iA(t), iB(t), iC(t) current trough each phase.

uA(t), uB(t), uC(t) phase voltages

vstar(t) potential at the star point of the motor

kA(t), kB(t), kC(t) coupling vectors

C capacitance

L inductance

R resistance

ω0 frequency

ζ damping factor

ω2 slip frequency

ωm rotor speed

np the number of pole pairs of the motor

Tref reference torque

R1 stator resistance per phase

X1 stator reactance per phase

R equivalent rotor resistance referred to stator per phase

X equivalent rotor reactance referred to stator per phase

Rc resistance representing core losses

Xm magnetizing reactance per phase

V1 phase supply voltage to the stator

s slip of the motor

Rs stator resistance

Rr rotor resistance

Lσ leakage inductance

stator current

stator flux

rotor flux.

θ1(t) angle of a rotating coordinate system

ω1 frequency of the rotor flux space vector

δ(t) load angle

Tσ rotor leakage time constant

ω20 steady state slip frequency

ω10 steady state stator frequency

List of Acronyms

DTC Direct Torque Control

ECD Equivalent Circuit Diagram

FOC Field-Oriented Control

IGBT Insulated Gate Bipolar Transistor

ISC Indirect Self Control

RGA Relative Gain Array

PWM Pulse Width Modulation

VSI Voltage Source Inverter

DC Direct Current

AC Alternative Current

HVDC High-voltage direct current

MIMO Multi Input Multi Output

Appendix A

Drive Data Set

Stator resistance R1 = 0.24 Ω

Rotor resistance R2 = 0.3 Ω

Stator reactance per phase X1=0.6 Ω

Equivalent rotor reactance referred to stator per phase X2=1.2 Ω

Resistance representing core losses Rc=300 Ω

Magnetizing reactance per phase Xm=50 Ω

Power absorbed by the rotor Prot = 200W

Efficiency η=88.27

Supply voltage to the stator V1=380V

Slip of the motor s = 0.16

Stator inductance Lm = 6.2 mH

Leakage inductance Lσ = 0.79 mH

Number of pole pairs np = 2

Base speed ωbase = 528 rad/s

Rated flux Ψ0 = 0.9 Vs

Input Filter Data

Input filter with 35.6 Hz resonance frequency:

Filter inductance: L = 5 mH

Filter capacitance: C = 4 mF

Filter resistance: R = 40 mΩ

Appendix B

The code for Bode plot:

R=40*10^(-3);

L=5*10^(-3);

C=4*10^(-3);

num=[1]

den=[C*L C*R 1]

w=logspace(-1,3)

[mag,phase]=bode(num,den,w)

subplot(2,1,1)

loglog(w,mag)

title('Bode Diagram');

ylabel('Magnitude [abs]');

grid;

subplot(2,1,2)

semilogx(w,phase)

ylabel('Phase [deg]');

xlabel('Frequency [Hz]');

grid;

Apendix C

The code for graphical user interface and the torque plot.

function varargout =Transformer2wImp(varargin)

if nargin == 0

fig = openfig(mfilename,'reuse');

handles = guihandles(fig);

guidata(fig, handles);

ax4=axes('Position',[0.7 .72 .2 .2]);

[x,map] = imread('IMPic1','jpg');

image(x)

set(gca,'visible','off')

ax1=axes('Position',[0 .48 .9 .23]);

% This program draws the induction motor equivalent circuit

ths = 0:.0025:pi;

xs1 =1.45+0.075*cos(ths);

ys1 = 1.2 +0.075*sin(ths);

xs2 =1.6+0.075*cos(ths);

xs3 =1.75+0.075*cos(ths);

xs4 =1.9+0.075*cos(ths);

x5 = [0.75 0.75 .8];

y5 =[1.2 1.0 1.0];

x6 = [1.1 1.3 1.3];

y6 = [1.0 1.0 0];

xw = [0.65 0.65 1.4 1.4 0.65];

yw = [0.95 1.35 1.35 0.95 0.95];

% R_c

xr=[.5 .5 .3 .3 .35 .25 .35 .25 .35 .25 .35 .25 .3 .3 .5 .5];

yr=[1.2 1 1 .775 .75 .7 .65 .60 .55 .50 .45 .40 .375 .2 .2 0];

% X_m

thm=-pi/2:.0025:pi/2;

xm=0.7+0.05*cos(thm);

ym=.05*sin(thm);

xxm=[xm xm xm xm] ;

yxm=[.4+ym 0.5+ym 0.6+ym 0.7+ym];

x1=[0.4 .65 .7 .80 0.9 1.0 1.1 1.2 1.25 1.575 ];

y1=[1.2 1.2 1.3 1.1 1.3 1.1 1.3 1.1 1.2 1.2];

plot(x1-1.2, y1,'b', 'erasemode','none')

hold on

axis('equal')

axis off

plot(xs1-1,ys1,'b', xs2-1,ys1, 'b', xs3-1,ys1,'b', xs4-1, ys1,'b')

xv1=[.3 .7 .7];

yv1=[1 1 .75];

xv2=[.5 .5 .7 .7 ];

yv2=[0 .2 .2 .35];

plot(xr+.8, yr, 'b')

plot(xxm+.8, yxm, 'b')

plot(xv1+.8, yv1,'b', xv2+.8, yv2, 'b')

x21=[3.2-.025 3.6 3.6];

y21=[1.2 1.2 .9];

x22=[-.8 3.6 3.6];

y22=[0 0 .3];

xL2=[3.55 3.55 3.65 3.65 3.55];

yL2=[.3 .9 .9 .3 .3];

plot(x21, y21, 'b',x22, y22,'b')

plot(xL2, yL2,'b')

x2=1.0+[0.6 .65 .7 .80 0.9 1.0 1.1 1.2 1.25 1.575 ];

y2=[1.2 1.2 1.3 1.1 1.3 1.1 1.3 1.1 1.2 1.2];

plot(x2, y2,'b')

plot(1.2+xs1,ys1,'b', 1.2+xs2,ys1, 'b', 1.2+xs3,ys1,'b', 1.2+xs4, ys1,'b')

xd1=[.9750 1.6];

yd1=[1.2 1.2];

plot(xd1, yd1, 'b')

text(-.835,1.1625, '\circ', 'color', [0 0 1])

text(-.835,-.0375, '\circ', 'color', [0 0 1])

text(-.8, .6, 'V_{1\phi}', 'Fontsize', 8, 'color', [0 0 .675])

text(1.6, .35, 'X_{m}', 'Fontsize', 8, 'color', [0 0 .675])

text(.9, .35, 'R_{c}', 'Fontsize', 8, 'color', [0 0 .675])

text(-.34, .975, 'R_1', 'Fontsize', 8, 'color', [0 0 .675])

text(.6, 1.05, 'X_1', 'Fontsize', 8, 'color', [0 0 .675])

text(1.85, 1, 'R_2^\prime', 'Fontsize', 8, 'color', [0 0 .675])

text(2.9, 1.05, 'X_2\prime', 'Fontsize', 8, 'color', [0 0 .675])

text(3.7, .6, 'R_L^\prime= {R_2}^\prime(1-s)/s', 'Fontsize', 8, 'color', [0 0 .675])

%text(5.5, .6, 'V_2', 'Fontsize', 8, 'color', [0 0 .675])

%text(8.75, .6, 'S_{Load}', 'Fontsize', 8, 'color', [0 0 .675])

text(-.2, -.1, 'Equivalent Circuit per phase referred to the Stator', 'color',[0 0 .675])

%%%%%%%%

hold off

%%%%%%%%%%%%

%Squirrel cage

ax2=axes('Position',[0.3 .2 .25 .25]);

R=.5; N=60;

[x,y,z]= cylinder(R,N);

Rotor=mesh(x,y,z); axis off, axis('square')

set(Rotor, 'erasemode', 'background');

rotation_increment=.2; % degree

rotation_axis=[0 0 1];

rotation_origin=[0 0 0];

incr=30/rotation_increment;

for loop =.1:incr

rotate(Rotor, rotation_axis, incr, rotation_origin);

drawnow

end

%hold off

%%%%%%%%%%%%

if nargout > 0

varargout{1} = fig;

end

elseif ischar(varargin{1}) % INVOKE NAMED SUBFUNCTION OR CALLBACK

try

if (nargout)

[varargout{1:nargout}] = feval(varargin{:}); % FEVAL switchyard

else

feval(varargin{:}); % FEVAL switchyard

end

catch

disp(lasterr);

end

end

% –––––––––––––––––––––––

function varargout = V_LText_Callback(h, eventdata, handles, varargin)

% –––––––––––––––––––––––

function varargout = f_sText_Callback(h, eventdata, handles, varargin)

% –––––––––––––––––––––––

function varargout = P_nText_Callback(h, eventdata, handles, varargin)

%–––––––––––––––––––––––

function varargout = R_1Text_Callback(h, eventdata, handles, varargin)

% –––––––––––––––––––––––

function varargout = X_1Text_Callback(h, eventdata, handles, varargin)

% –––––––––––––––––––––––

function varargout = R_2Text_Callback(h, eventdata, handles, varargin)

% –––––––––––––––––––––––

function varargout = X_2Text_Callback(h, eventdata, handles, varargin)

% –––––––––––––––––––––––

function varargout = R_cText_Callback(h, eventdata, handles, varargin)

% –––––––––––––––––––––––

function varargout = X_mText_Callback(h, eventdata, handles, varargin)

% –––––––––––––––––––––––

function varargout = SlipText_Callback(h, eventdata, handles, varargin)

% –––––––––––––––––––––––

function varargout = ProtText_Callback(h, eventdata, handles, varargin)

% –––––––––––––––––––––––

function varargout = pushbuttonSolve_Callback(h, eventdata, handles, varargin)

%function [Rclv, Xmlv, Rchv, Xmhv, Zelv, Zehv, Ic, Io, Re, Zemag, Error] = OcScTest;

Error = 0;

V_LHandle=findobj(gcbf, 'Tag','V_LText');

Vm=eval(get(V_LHandle,'String'));

f_sHandle=findobj(gcbf, 'Tag','f_sText');

f=eval(get(f_sHandle,'String'));

P_nHandle=findobj(gcbf, 'Tag','P_nText');

P=eval(get(P_nHandle,'String'));

R_1Handle=findobj(gcbf, 'Tag','R_1Text');

R1=eval(get(R_1Handle,'String'));

X_1Handle=findobj(gcbf, 'Tag','X_1Text');

X1=eval(get(X_1Handle,'String'));

R_2Handle=findobj(gcbf, 'Tag','R_2Text');

R2=eval(get(R_2Handle,'String'));

X_2Handle=findobj(gcbf, 'Tag','X_2Text');

X2=eval(get(X_2Handle,'String'));

R_cHandle=findobj(gcbf, 'Tag','R_cText');

Rc=eval(get(R_cHandle,'String'));

X_mHandle=findobj(gcbf, 'Tag','X_mText');

Xm=eval(get(X_mHandle,'String'));

SlipHandle=findobj(gcbf, 'Tag','SlipText');

S=eval(get(SlipHandle,'String'));

ProtHandle=findobj(gcbf, 'Tag', 'ProtText');

Prot=eval(get(ProtHandle,'String'));

j = sqrt(-1);

Z1 = R1 + j*abs(X1);

Z2s= R2 +j*abs(X2);

if Rc ==0 |Xm==0

Error = 1;

plot(0, 0), axis off

text(-1, .6,'Data error: R_c an R_m cannot be zero, 0 < R_c \leq inf & 0 < X_m \leq inf', 'color', [1 0 0], 'erasemode','xor')

text(-1, .25, 'Enter the correct value for R_c or X_m and try again. If neglected enter inf','color',[1 0 0], 'erasemode','xor');

return, end

if S > 1 | S <= 0

Error=2;

plot(0, 0), axis off

text(-1, .5,'Data error: s must be in per unit in the following range', 'color', [1 0 0], 'erasemode','xor')

text(-1, .25, '0 < s \leq 1. Enter the correct value for s and try again.','color',[1 0 0], 'erasemode','xor');

return, end

if R2 ==0

Error = 3;

plot(0, 0), axis off

text(-.6, .4,'Data error: R_2 cannot be zeo, enter the correct value and try again', 'color', [1 0 0], 'erasemode','xor')

return, end

%%%%%%%

Vmp=Vm/sqrt(3);

V= Vmp+j*0;

Ns=120.*f/P;

Nr=(1.-S)*Ns;

Ws=2.*pi*Ns/60;

Wr=(1.-S)*Ws;

Zrd=R2+j*abs(X2);

if S~= 0

RLd= R2*(1-S)/S;

R2ds=R2/S;

else RLd =inf; R2ds=inf; end

Z2=R2ds + j*abs(X2);

if Rc == inf | Rc ==0, Gc = 0; else Gc = 1/Rc; end

if Xm == inf | Xm==0, Bm = 0; else Bm = 1/abs(Xm); end

Y2=1/Z2 ;

Ye=Gc-j*Bm;

if (Gc == 0 & Bm == 0) Ze = inf+j*inf; else Ze = 1/Ye; end

Yf=Y2+Ye;

Zf=1/Yf; Rf = real(Zf); Xf = imag(Zf);

Zin=Z1+Zf; Rin = real(Zin); Xin = imag(Zin);

if (Gc ~= 0 | Bm) ~= 0

Zth= Z1*Ze/(Z1+Ze);

Vthc=V*Ze/(Z1 +Ze);

else

Zth=Z1 ;

Vthc=V ;

end

Vth=abs(Vthc); Tth = angle(Vthc)*180/pi;

Rth=real(Zth);

Xth=imag(Zth);

Rdd=sqrt(Rth^2 + (Xth+X2)^2);

Smax=R2/Rdd;

Pmax=1.5*Vth^2./(Rth+Rdd);

Tmax=Pmax/Ws;

I1=V/Zin;

I1m=abs(I1);

T1=angle(I1)*180./pi;

%if real(Ze) == inf & imag(Ze) == inf I2 =I1; else

if Rc==inf & Xm==inf, I2=I1; else

I2= Ze*I1/(Ze+Z2); end

T2=angle(I2)*180/pi;

I2m=abs(I2);

Io=I1-I2; Iom=abs(Io); Ao=angle(Io)*180/pi;

Pcore=3*real(Ze)*Iom^2;

S1=3.*V*conj(I1);

SCL=3*R1*I1m^2;

Pin=real(S1);

Pg=3.* R2ds*I2m^2;

RCL=3.*R2*I2m^2;

Pm=Pg-RCL;

Td=Pg/Ws;

I2st = Vth/(Zth+Z2s);

Pgst = 3.*R2*abs(I2st)^2;

Tst = Pgst/Ws;

if S ~= 1

Po=Pm-Prot; hp = Po/746;

To=Po/Wr;

Ef=100*Po/Pin;

else, end

h0 = figure('Units','normalized', …

'PaperUnits','points', …

'Position',[0.15 0.09 0.8 0.8], …

'Tag','Fig2');

% Torque Speed characteristics

slip1 = Smax:-.005:.0001;

if Smax > 1, Smax =1;else,end

slip2 = 1:-.005:Smax;

slip=[slip2, slip1];

np1 = length(slip1);

np2 = length(slip2);

R2d1 = R2./slip1;

R2d2 = R2./slip2;

Z2d1=R2d1+ j*X2*ones(1,np1);

Z2d2=R2d2+ j*X2*ones(1,np2);

I2d1 = Vth./(Zth*ones(1,np1)+Z2d1);

I2d2 = Vth./(Zth*ones(1,np2)+Z2d2);

Pgd1 = 3*R2d1.*abs(I2d1).^2;

Pgd2 = 3*R2d2.*abs(I2d2).^2;

Tdev1 = Pgd1/Ws;

Tdev2 = Pgd2/Ws;

nr1 = Ns*(ones(1,np1)-slip1);

nr2 = Ns*(ones(1,np2)-slip2);

if length(nr2) > 4

x1= 1.7*nr2(4);

else x1 = 1.7*nr1(2); end

y1 = 0.86*Tst;

x2 = 0.96*(1- Smax)*Ns; y2 = 0.89*Tmax;

x3 = 0.95*Nr; y3 =0.93*Td;

fy = 0:Td/20:Td; nf=length(fy);

fx = ones(1,nf)*Nr;

plot(nr1, Tdev1, 'm', nr2, Tdev2, 'r'),grid

xlabel('Rotor speed, rpm', 'color', [0 0 0.675])

ylabel('Developed Torque, N-m' , 'color', [0 0 0.675])

title('Motor torque-speed curve', 'color', [0 0 0.675])

text(x1, y1, 'T_{st}', 'color',[0 0 0.675])

text(x2, y2, 'T_{max}', 'color', [0 0 0.675])

if S~=1

if S >= Smax

text(x1,.75*y1, ['s_{max} = ',num2str(Smax), ', s \geq s_{max}'], 'color', [1 0 0])

text(x1, .5*y1,'Motor cannot operate in this region.', 'color', [1 0 0])

else, end

hold off,

else, end

% –––––––––––––––––––––––

function varargout = pushbuttonExit_Callback(h, eventdata, handles, varargin)

close all

Apendix D

The code for Figure 5-4 (A)

%wm=0

Rs=18.5e-3;

Rr=17.3e-3;

Lm=6.2e-3;

Ls=0.79e-3;

np=2;

wm=0;

w=logspace(-1,4);

num1=[1 Rr/Lm-j*np*wm];

den1=[Ls Rs+Rr+Ls*(Rr/Lm-j*np*wm) Rs*(Rr/Lm-j*np*wm)];

g1=freqs(num1, den1, w);

g2=freqs(num1,den1,-w);

mag1=abs(g1);

mag2=abs(g2);

loglog(w,mag1,w,mag2)

xlabel('Frequency [Hz]')

title('Singular values of G')

The code for Figure 5-4 (B)

%wm=10

Rr=17.3e-3;

Lm=6.2e-3;

np=2;

wm=10;

Ls=0.79e-3;

Rs=18.5e-3;

w=logspace(-1,4);

num1=[1 Rr/Lm-j*np*wm];

den1=[Ls Rs+Rr+Ls*(Rr/Lm-j*np*wm) Rs*(Rr/Lm-j*np*wm)];

g1=freqs(num1, den1, w);

g2=freqs(num1,den1,-w);

mag1=abs(g1);

mag2=abs(g2);

loglog(w,mag1,w,mag2)

xlabel('Frequency [Hz]')

title('Singular values of G')

The code for Figure 5-4 (C)

%wm=50

Rr=17.3e-3;

Lm=6.2e-3;

np=2;

wm=50;

Ls=0.79e-3;

Rs=18.5e-3;

w=logspace(-1,4);

num1=[1 Rr/Lm-j*np*wm];

den1=[Ls Rs+Rr+Ls*(Rr/Lm-j*np*wm) Rs*(Rr/Lm-j*np*wm)];

g1=freqs(num1, den1, w);

g2=freqs(num1,den1,-w);

mag1=abs(g1);

mag2=abs(g2);

loglog(w,mag1,w,mag2)

xlabel('Frequency [Hz]')

title('Singular values of G')

Biblography

[1] http://en.wikipedia.org/wiki/Inverter_(electrical)

http://en.wikipedia.org/wiki/Induction_motor

[2] E. H. Bristol. On a new measure of interaction for multiovariable process control, IEEE Transactions on Automatic Control, AC-11, 1966.

[3] S. Skogestad and I. Postlethwaite. Multivariable feedback control. John Wiley & Sons, 1996.

[4] L. Harnefors. On analysis, control and estimation of variable-speed drives, Ph D thesis, KTH, Stockholm, 1997.

[5] F. Blaschke. The principle of field orientation as applied to the new transvector closed-loop control system for rotating machines. Siemens Review, vol 39, 217-220, 1972.

[6] Henrik Mosskull: Robust Control of an Induction Motor Drive, April 2006.

[7] J.A. Santisteban and R.M. Stephan. Vector control methods for induction machines: An Overview. IEEE Transactions on Education, 2001.

[8] R. Marino. S. Peresada and P. Valigi. Adaptive input-output linearizating control of induction motors. IEEE Trans. Automatic Control, vol. 38, Feb. 1993.

[9] K. P. Kovacs. Transient phenomena in electrical machines. Elsevier, 1984.

[10] L. Harnefors, K. Pietilainen, L. Gertmar. Torque-maximizing fieldweakening control: design, analysis, and parameter selection, IEEE Transactions on Industrial Electronics, Vol. 48, Issue 1, Feb., 2001.

[11] P. A. S. De Wit, R. Ortega, and I. Mareels. Indirect field-oriented control of induction motors is robustly globally stable, Automatica, Vol. 32, No. 2, 1996.

[12] M. Jänecke, R. Kremer and G. Steuerwald. Direct self control, a novel method of controlling asynchronous machines in traction applications. In: Record of the EPE-Conference, Aachen, 1989.

[13] Texas Instruments Europe. Field Orientated Control of 3-Phase AC-Motors, Literature Number: BPRA073, February 1998

Declarația autorului

Subsemnatul declar prin prezenta că ideile, partea de proiectare și experimentală, rezultatele, analizele și concluziile cuprinse în această lucrare de diplomă constituie efortul meu propriu, mai puțin acele elemente ce nu-mi aparțin, pe care le indic în bibliografie și le recunosc ca atare.

Declar de asemenea, că după știința mea, lucrarea în această formă este originală și nu a mai fost niciodată prezentată sau depusă în alte locuri sau altor instituții decât cele indicate în mod expres de mine.

În calitate de autor, cedez toate drepturile de utilizare și modificare a lucrării de diplomă către Universitatea Transilvania din Brașov.

ABSOLVENT,

Pomană Mihai

Brașov,

10 iunie 2006