Direct Power and Torque Control of AC/DC/AC Converter-Fed ...

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Symbols Employed Indices c – dq – m – min – – xy command value, reference value, … space vector in synchronous rotating dq coordinates, maximum, peak value, minimum value, … space vector in synchronous rotating xy coordinates, Mathematical symbols * U ⋅ – scalar product = Re { I } = ( U I + U I ) L I L L I L U , L L Lα Lα Lβ Lβ * U × – vector product = Im{ I } = ( U I −U I ) U , L L Lβ Lα Lα Lβ * I L – conjugate complex number, Abbreviations EMI – electromagnetic interference, IM – induction motor, MO– modulus optimum, PI – proportional-integral controller, PWM – pulse width modulation, SO– symmetry optimum, UPF – unity power factor, VM – virtual machine, VSC – voltage source converter, VSI – voltage source inverter, VSR – voltage source rectifier, SVM – space vector modulator, 140
A. Appendices A.1. Space vector in coordinate systems An advantage of space vectors is possibility their representation in various systems of rectangular coordinates. Therefore, space vectors are very convenient mathematical tool described three phase systems. Spatial projection of space vector k on the phase axes ( A ,B,C ) are expressed as (see [63]): 2 [ k] , k ( t ) = Re[ a k] , k ( t ) Re[ ak] k A ( t ) = Re B C = (A.1. 1) Assuming special steady-state case (phase waveforms are sinusoidal with frequency Ω = 2πf ) an important relation between space vector and positivesequence system can be described: jϕ jΩ t k = K1 = K A = kme e (A.1. 2) Where: ϕ - angle of initial phase. A.1.1. Fixed System of Coordinates - αβ Introducing a fixed system of rectangular coordinates the following space vector can be defined: k = k α + jk β (A.1. 3) Where α is the real axis and β the imaginary one: α Re[ k] , k = Im[ k] k = β (A.1. 4) A.1.2. Rotating System of Coordinates Let consider that space vector in fixed system of coordinates αβ is known. It will be transformed into a rotating system of coordinates with angular frequency d w γ Ω w = , where γ w is an angle between real axis α of the fixed system of dt coordinates and real axis d m of rotating one. Then, taking into account rotating system of coordinates the following expressions could be written: − j w j w k = k , and opposite k w e γ w Where: e γ k = (A.1. 5)
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POLITECHNIKA WARSZAWSKA WARSAW UNIV
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Acknowledgements The work presented
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Contents 4.2.2. Energy of the DC-li
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Chapter 1 1. Introduction 1.1. AC/D
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1. Introduction Tab. 1. 1. Current
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1. Introduction DC-link voltage to
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1. Introduction then nominal speed
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1. Introduction [69]. From that tim
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Chapter 2 2. Voltage Source Convert
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2. Voltage Source Converters - VSC
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Chapter 3 3. Vector Control Methods
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3. Vector Control Methods of AC/DC/
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Chapter 4 4. Direct Power and Torqu
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4. Direct Power and Torque Control
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