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

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Appendices U U U ra rb rc = I = I = I ra rb rc dΨ ra Rr + dt dΨ rb Rr + dt dΨ rc Rr + dt And flux-currents equations: Ψ Ψ Ψ SA SB SC = L S = L S = L S I I I SA SB SC + L + L M M + L M I I I ra rb rc (A.1. 13) (A.1. 14) Ψ = L I ra rb rc r Ψ = L I r Ψ = L I r ra rb rc + L + L + L M M M I I I SA SB SC Matrix form of above equations: (A.1. 15) dΨSK U SK = RSI SK + (A.1. 16) dt dΨ rK U rK = RrI rK + (A.1. 17) dt Ψ = L I + L I (A.1. 18) SK rK S r SK rK M M RK Ψ = L I + L I (A.1. 19) SK In order to reduce the set of equations (A.1.12)-(A.1.15) from 12 to 4 the complex space vectors are used. Additionally, to eliminate the dependence of mutual inductance on the angle of rotor rotation relative to the stator, transformation into a common rotating coordinate system with and an arbitrary angular speed needed. Therefore, following equations can be written: U U dΨ dt Ω K is SK SK = RSI SK + + jΩKΨSK (A.1. 20) R I dΨ dt j ( Ω K − pbΩm ) ΨrK rK rK = r rK + + (A.1. 21) Ψ = L I + MI (A.1. 22) SK rK S r SK rK rK Ψ = L I + MI (A.1. 23) Where the d rK ( L I + MI ) S transformation, while j ( L I + MI ) [63]. SK dt K S rK SK represents the inner voltage (EMF) of the Ω is the inner voltage of the motor rotation rK 144
Appendices A.2. Coordinate Transformation A.2.1. Three-Phase to Two-Phase Conversion (ABC/αβ ) ⎡k ⎢ ⎣k α β ⎡ ⎤ ⎢ ⎥ = ⎦ ⎢ ⎢⎣ 1 1 3 0 2 3 0 ⎤⎡k ⎥⎢ ⎥⎢ k 0 ⎥⎦ ⎢⎣ k A B C ⎤ ⎥ ⎥ ⎥⎦ (A.2. 1) A.2.2. Two-Phase to Three-Phase Conversion (αβ /ABC) ⎡k ⎢ ⎢ k ⎢⎣ k A B C ⎡ ⎢ 1 ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ − ⎥⎦ ⎢ ⎢ ⎢ − ⎣ 1 2 1 2 − 0 3 2 3 2 ⎤ 0 ⎥ ⎥ ⎥⎡k 0 ⎥⎢ ⎥⎣k 0 ⎥ ⎥⎦ α β ⎤ ⎥ ⎦ (A.2. 2) A.2.3. Rectangular to Rectangular Coordinate Conversion (αβ /xy) and (xy/αβ ) With the assumption that the rectangular components k , k of space vector k in α β stationary coordinatesαβ are known. Knowing from space vector theory that: k = ke and : k jγ ke - space vector in stationary αβ coordinates, ( γ −γ K ) − jγ K j K = = k angular speed Ω (see Fig. A. 1. 1). K Therefore the components e - space vector in rotating xy coordinates with overall k , k of the xy rotating system with angular speed x y Ω K can be written: ke = − jγ K ( k + jk )( cosγ − j sinγ ) ( k cosγ + k sinγ + j( − k sinγ + k cosγ ) α = α K β β K Rewritten into matrix form: ⎡kx ⎤ ⎡cosγ γ ⎤⎡k K sin K α ⎤ ⎢ ⎥ = ⎢ ⎥⎢ ⎥ ⎣k y ⎦ ⎣- sinγ γ ⎦⎣k K cos K β ⎦ In an analogous way the inverse transformation can be found: k = jγ Ke K ( k + jk )( cosγ + j sinγ ) K ( k cosγ − k sinγ + j( k sinγ + k cosγ ) x = Rewritten into matrix form: K x y y K K x α K K K = K = y β K K (A.2. 3) (A.2. 4) (A.2. 5) 145
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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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Page 133 and 134: References [18] G. Buja, M. P. Kazm
Page 135 and 136: References [58] J. G. Kassakian, M.
Page 137 and 138: References [96] H. Mao, D. Boroyevi
Page 139 and 140: References [133] A. Tripathi, A. M.
Page 141 and 142: References [167] M. Jasiński, D.
Page 143 and 144: Symbols Employed I L - I LA I r - r
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