Power Systems Nguyen Phung Quang and Jörg-Andreas Dittrich Vector Control of Three-Phase AC Machines Nguyen Phung Quang · Jörg-Andreas Dittrich Vector Control of Three-Phase AC Machines System Development in the Practice With 230 Figures Prof Dr Nguyen Phung Quang Dr Jörg-Andreas Dittrich Hanoi University of Technology Department of Automatic Control 01 Dai Co Viet Road Hanoi Vietnam quangnp-ac@mail.hut.edu.vn Neeserweg 31 8048 Zürich Switzerland andreas_d@swissonline.ch ISBN: 978-3-540-79028-0 e-ISBN: 978-3-540-79029-7 Power Systems ISSN: 1612-1287 Library of Congress Control Number: 2008925606 This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable for prosecution under the German Copyright Law The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Cover design: deblik, Berlin Printed on acid-free paper 987654321 springer.com Dedicated to my parents, my wife and my son Nguyen Phung Quang To my mother in grateful memory Jörg-Andreas Dittrich Formula symbols and abbreviations A B C dq E E I, E R eN f fp, fr, fs G Gfe H, h h i im imd, imq iN, iT, iF is, ir isd, isq, ird, irq is , is isu, isv, isw J K Lf g Lm, Lr, Ls Lsd, Lsq L r, L s mM, mG N pCu pv pv,fe, pFe System matrix Input matrix Output matrix Field synchronous or rotor flux orientated coordinate system Sensitivity function Imaginary, real part of the sensitivity function Vector of grid voltage General analytical vector function Pulse, rotor, stator frequency Transfer function Iron loss conductance Input matrix, input vector of discrete system General analytical vector function Vector of magnetizing current running through Lm Vector of magnetizing current dq components of the magnetizing current Vectors of grid, transformer and filter current Vector of stator, rotor current dq components of the stator, rotor current components of the stator current Stator current of phases u, v, w Torque of inertia Feedback matrix, state feedback matrix Lie derivation of the scalar function g(x) along the trajectory f(x) Mutual, rotor, stator inductance d axis, q axis inductance Rotor-side, stator-side leakage inductance Motor torque, generator torque Nonlinear coupling matrix Copper loss Total loss Iron loss VIII Formula symbols and abbreviations RI, RIN RF, RD Rfe Rr, Rs R r r s S s T Tp Tr, Ts Tsd, Tsq tD ton, toff tr t u , t v, t w u u0, u1, … , u7 UDC uN us, ur usd, usq, urd, urq us , us V w x xw z zp Zs p, / r, sd, r, Two-dimensional current controller Filter resistance, inductor resistance Iron loss resistance Rotor, stator resistance Flux controller Vector of relative difference orders Relative difference order Complex power Loss function Slip Sampling period Pulse period Rotor, stator time constant d axis, q axis time constant Protection time Turn-on, turn-off time Transfer ratio Switching time of inverter leg IGBT’s Input vector Standard voltage vectors of inverter DC link voltage Vector of grid voltage Vector of stator, rotor voltage dq components of the stator, rotor voltage components of the stator voltage Pre-filter matrix Input vector State vector Control error or control difference State vector Number of pole pair Complex resistance or impedance Stator-fixed coordinate system Transition or system matrix of discrete system Main flux linkage Vector of pole, rotor, stator flux s / s sq, Vector of rotor, stator flux in terms of Lm rd, rq dq components of the stator, rotor flux Formula symbols and abbreviations / rd , / rq , / sd , / sq i , r, , s s ϕ ADC CAPCOM DFIM DSP DTC EKF FAT GC IE IGBT IM KF MIMO MISO MRAS NFO PLL PMSM PWM SISO VFC VSI C, P / r , / r Components of / r Components of IX / s Eigen value Mechanical rotor velocity, rotor and stator circuit velocity Rotor angle, angle of flux orientated coordinate system Total leakage factor Angle between vectors of stator or grid voltage and stator current Analog to Digital Converter Capture/Compare register Doubly-Fed Induction Machine Digital Signal Processor Direct Torque Control Extended Kalman Filter Finite Adjustment Time Grid-side Converter or Front-end Converter Incremental Encoder Insulated Gate Bipolar Transistor Induction Machine Kalman Filter Multi Input – Multi Output Multi Input – Single Output Model Reference Adaptive System Natural Field Orientation Phase Locked Loop Permanent Magnet Excited Synchronous Machine Pulse Width Modulation Single Input – Single Output Voltage to Frequency Converter Voltage Source Inverter Microcontroller, microprocessor Table of Contents A Basic Problems Principles of vector orientation and vector orientated control structures for systems using three-phase AC machines Formation of the space vectors and its vector orientated philosophy Basic structures with field-orientated control for three-phase AC drives Basic structures of grid voltage orientated control for DFIM generators References 1.1 1.2 1.3 1.4 2.1 2.2 2.3 2.3.1 2.3.2 2.3.3 2.4 2.4.1 2.4.2 2.4.3 2.5 2.5.1 2.5.2 2.5.3 2.6 Inverter control with space vector modulation Principle of vector modulation Calculation and output of the switching times Restrictions of the procedure Actually utilizable vector space Synchronization between modulation and signal processing Consequences of the protection time and its compensation Realization examples Modulation with microcontroller SAB 80C166 Modulation with digital signal processor TMS 320C20/C25 Modulation with double processor configuration Special modulation procedures Modulation with two legs Synchronous modulation Stochastic modulation References 1 11 15 17 18 23 26 26 28 29 31 33 37 45 49 49 51 53 58 XII 3.1 3.1.1 3.1.2 3.2 3.2.1 3.2.2 3.3 3.3.1 3.3.2 3.4 3.4.1 3.4.2 3.5 3.6 3.6.1 3.6.2 3.6.3 3.6.4 3.7 4.1 4.2 4.3 4.3.1 4.3.2 4.4 4.4.1 4.4.2 4.4.3 4.5 Table of Contents Machine models as prerequisite to design the controllers and observers General issues of state space representation Continuous state space representation Discontinuous state space representation Induction machine with squirrel-cage rotor (IM) Continuous state space models of the IM in stator-fixed and field-synchronous coordinate systems Discrete state space models of the IM Permanent magnet excited synchronous machine (PMSM) Continuous state space model of the PMSM in the field synchronous coordinate system Discrete state model of the PMSM Doubly-fed induction machine (DFIM) Continuous state space model of the DFIM in the grid synchronous coordinate system Discrete state model of the DFIM Generalized current process model for the two machine types IM and PMSM Nonlinear properties of the machine models and the way to nonlinear controllers Idea of the exact linearization Nonlinearities of the IM model Nonlinearities of the DFIM model Nonlinearities of the PMSM model References Problems of actual-value measurement and vector orientation Acquisition of the current Acquisition of the speed Possibilities for sensor-less acquisition of the speed Example for the speed sensor-less control of an IM drive Example for the speed sensor-less control of a PMSM drive Field orientation and its problems Principle and rotor flux estimation for IM drives Calculation of current set points Problems of the sampling operation of the control system References 61 61 61 63 69 70 78 85 85 88 90 90 93 95 97 98 100 102 103 105 107 108 110 116 118 125 127 128 134 135 139 12 Appendices 12.1 Normalizing - the important step towards preparation for programming So far, the algorithms with their control variables and parameters are given in the originally derived physical form An implementation or a programming in this form is mainly impossible Before this practical implementation can start, all algorithms have to be normalized and scaled if necessary, e.g for fixed point and partly also for floating point processors The purpose of the normalization consists of transferring these variables and parameters into a unity-less form and thus to prepare them for programming The scaling is primarily necessary to increase the numerical accuracy which is of great importance for the use of fixed point processors This important step towards preparation for programming is demonstrated on two examples a) Example 1: The first equation of (3.55), which can be written in detail as follows, serves as the first example: isd (k + 1) = 11 isd (k ) + 12 isq (k ) + 13 / rd (k ) + h11 usd (k ) isq (k + 1) = 12 isd (k ) + 11 isq (k ) 14 / rd (k ) + h11 usq (k ) (12.1) From (12.1) the following can be noticed: • The variables like currents isd , isq , / rd and voltages usd , usq have to be normalized • From the parameters only Φ11, Φ13 are already unity-less All others have to be normalized For the normalization of the currents, the maximum inverter current Imax is often chosen For the normalization of the voltage, the maximum value, 326 Appendices which is 2UDC1)/3, is chosen The quantity UDC is itself variable and, with respect to the hardware, has to be normalized by Umax while measuring The equation (12.1) is totally identical with the following: isd (k + 1) I max = isd (k ) 11 I max +h11 isq (k + 1) I max = + isq (k ) 12 I max + / rd 13 (k ) I max 2U max U DC (k ) usd (k ) 3I max U max U DC isd (k ) 12 I max + isq (k ) 11 / rd 14 I max (k ) (12.2) I max 2U max U DC (k ) usq (k ) 3I max U max U DC In the equation (12.2) new symbols are introduced and replaced: / isd , sq / N usd , sq N N ; rd = rd ; usd , sq = isd , sq = 2U DC I max I max +h11 N h11 N = ku U DC ; ku 2U U N = h11 max ; U DC = DC 3I max U max (12.3) Superscripts N: normalized quantities The parameters Φ12 , Φ14 are frequency dependent The value fmax is used for the normalization of the frequencies Using (3.54) it can be written then: fs = k f f sN 12 = sT = f sT = ( f max T ) f max 14 = T= fT= f maxT = kf f N In (12.4) the symbols mean: 1) UDC: DC link voltage of the inverter f f max (12.4) Normalizing - the important step towards preparation for programming f sN = fs f max ;fN= f f max ; k f = f maxT ; k f = 327 f maxT (12.5) The equation (12.1) can now be rewritten in the normalized form, in which the constants ku, kf1 and kf2 as well as the constant parameters Φ11, Φ13 have to be calculated only at the beginning, i.e at the initialization of the system N isd (k + 1) = N 11 isd (k ) + N 12 isq (k ) + 13 /N rd (k ) N 11 isq (k ) 14 /N rd (k ) N N +h11 usd (k ) N isq (k + 1) = N 12 isd (k ) + (12.6) N N +h11 usq (k ) The original equation (12.1) exists now in programmable form without loss of its physical meaning The voltages represent the degree of modulation in this normalized form in which the variable DC link voltage N UDC is considered by the parameter h11 , which has to be updated on-line To achieve the most possible numerical accuracy with fixed point or integer arithmetic, the normalized quantities (represented in hexadecimal form) are shifted to the left (multiplied with 2) as much as possible without producing overflow For normalized currents, voltages and frequencies which accept only values smaller than one, the multiplication factor can be e.g 215 for 16 bit fixed point processors This process is commonly described as the scaling The multiplication factor of 215 is the scaling factor which at the same time means the number of digits behind the comma For parameters, which by their nature are already greater than one, the scaling has to be carried out in a way that on the one hand the maximum word length is used, but on the other hand overflow is avoided simultaneously In principle, this problem does not exist any more with the use of floating point processors and only appears for conversions between data types again, e.g between integer numbers and signed floating point numbers b) Example 2: A further typical example is shown by normalizing and scaling of the quantities within the equation (12.6) for the calculation of the angular velocity ω 328 Appendices (k + 1) = (k ) + (k )T = (k ) + f (k )T (12.7) If the values 2π and fmax are chosen as normalizing quantities for the angle and frequency, and it is considered that: • the frequency has to be signed (e.g positive for right and negative for left rotation), and • the angle has to be unsigned (i.e only forwards counting 0, π, 2π, 3π, 4π ) then e.g 215 is possible to be used as scaling factor for the frequency at 16 bit word length, and 216 for the angle From (12.7) it will be obtained: (k + 1) (k ) f = 216 + 215 216 (12.8)a (2 f maxT ) f max 2 or 216 N (k + 1) = 216 N (k ) + 215 f N (k ) k f (12.8)b In the equation (12.8)b it means: ã 216 ì N the unsigned integer calculation quantity for the angle, • 215 × f N the signed calculation quantity for the frequency 12.2 Example for the model discretization in the section 3.1.2 A system of second order following (3.5) is given with: a b A= B= C= (12.9) a b a) Method 1: Series expansion with truncation after the linear term (Euler) The use of (3.14) provides: + aT T b ; H =T Φ= (12.10) T + aT b b) Method 2: Series expansion with truncation after the quadratic term The use of (3.14) provides again: + aT + (aT ) Φ= ( T) T (1 + aT ) T (1 + aT ) + aT + (aT ) 2 ( T) (12.11)a Example for the model discretization in the section 3.1.2 H = bT + aT 329 T (12.11)b T + aT c) Method 3: Euler discretization in a suitable coordinate system The method is applicable if the system matrix A owns the symmetry properties of the special block diagonal structure (12.9) which is often the case at the modeling of three-phase machines In this case, state and input quantities can be understood as complex variables x = xx + j x y (12.12) • x (t ) = (a j ) x (t ) + b u (t ) At first, the continuous system is viewed in a coordinate system which circulates with the frequency -ω with respect to the target coordinate system, i.e x = x e j t (to the topic “transformation of coordinates” cf chapter 1) Considering the product rule, the following is obtained for the time derivative: • (12.13) x (t ) = a x (t ) + b u (t ) The discretization using Euler method leads to the following discrete state equation: x (k + 1) = (1 + aT ) x (k ) + bT u (k ) (12.14) For the inverse transformation into the target coordinate system, the equation (12.14) has to be subjected to the counter-rotation, i.e x (k ) = x (k )e j (k ) ,x (k + 1) = x (k + 1)e j (k +1) (12.15) Thereat, the discrete transformation angle ϑ results by Euler discretization as follows: (12.16) (k + 1) = (k ) + T The state equation is now in the target coordinate system: x (k + 1) = e j T (1 + aT ) x (k ) + bT u (k ) or resolved with discrete transfer matrices: cos T sin T cos T ; H = bT Φ = (1 + aT ) sin T cos T sin T (12.17) sin T cos T (12.18) d) Method 4: Substitute function using the Sylvester-Lagrange substitute polynomials The eigen values of the continuous system matrix A will be: 1,2 =a± j 330 Appendices It follows then for equations (3.19) to (3.22): M ( )=( )( 2) M1 = ( m1 = ( R( ), M2 =( ) , m2 = ( 1 )= 1e 2T 2e 1T ) 1) 2 + e (12.19) 1T e 2T Therewith the substitute function R(A) can be given as: 1T e 2T e 1T e 2T 2e R (A) = Φ = I+ A (12.20) and finally the system matrix Φ: cos T sin T Φ = e aT sin T cos T (12.21) The input matrix H is calculated by direct integration of Φ according to (3.12): H= eaT (a cos T + b a + sin T ) a eaT (a sin T cos T ) + eaT (a sin T cos T ) + e aT (a cos T + sin T ) a (12.22) 12.3 Application of the method of the least squares regression The method of the least squares regression is often used for the optimization of control loops or the identification of the system parameters The goal is normally to find an approximate function y(x) in the form of a polynomial of nth order y ( x ) = a0 + a1 x + a2 x + + an x n (12.23) from a set of m experimental measurement pairs [yi, xi], (i = 1,2,3, ,m) and by the prerequisite, that the loss function (cf [Rojiani 1996]): S= m i=1 yi y ( xi ) (12.24) is minimized A typical application example is the off-line identification of the main inductance Ls (cf section 6.4.4, figure 6.18) in dependence on Application of the method of the least squares regression 331 the magnetizing current iμ As an approach for L(i)1) a polynomial of 4th order, thus n = 4, is very suitable L (i ) = a0 + a1 i + a2 i + a3 i + a4 i (12.25) The task is now to determine the coefficients a0, a1, a2, a3 and a4 from m pairs [Li, ii] with (i=1,2,3, ,m) To minimize the loss function, at first (12.25) has to be inserted into (12.24), and then the partial derivations S S S S S ; ; ; ; (12.26) a0 a1 a2 a3 a4 have to be set to zero Thereby a system with (n+1)=5 linear equations results (cf [Rojiani 1996]): m m m i=1 m m i=1 m i=1 m i=1 m i=1 ii ii2 ii3 ii4 i=1 m i=1 m i=1 m i=1 ii ii2 ii3 ii4 ii5 i=1 m i=1 m i=1 m i=1 m i=1 ii2 ii3 ii4 ii5 ii6 m i=1 m i=1 m i=1 m i=1 m i=1 ii3 ii4 ii5 ii6 ii7 m i=1 m i=1 m i=1 m i=1 m i=1 m ii4 ii5 a0 ii6 a1 a2 = a3 ii7 a4 ii8 i=1 m i=1 m i=1 m i=1 m i=1 Li ii Li ii2 Li (12.27) ii3 Li ii4 Li The system (12.27) can be merged into the following form: A [5,5]* a[5] = b [5] (12.28) thereat A [5,5] and b [5] are given by the measurement pairs, following (12.27) If C is used as programming language, then the calculation can be realized by the following program section as an example, where A [5,5] and b [5] are summarized in a matrix A [5,6] with b [5] as the 6th column In this example it is assumed, that „MeasNum“ is the number of the measurement pairs and „PolyOrder“ the order of the approached polynomial Here it holds: MeasNum = 10; PolyOrder = During measuring the current is increased by 0,1×ImNominal step by step from 0,1×ImNominal to ImNominal (e.g nominal magnetizing 1) For simplification Ls is replaced by L, and iμ by i 332 Appendices current) The 10 measured inductance values are saved in the field L[0] L[9] After calculation of A [5,5] and b [5] the linear equation system (12.27) or (12.28) can be relatively simply solved by using the Gauss elimination method The first step of the method is the forward elimination, which can be summarized (cf [Sedgewick 1992]) as follows: The first variable in all equations, with exception of the first one, has to be eliminated by addition of suitable multiples of the first equation to each of the other equations Then the second variable in all equations, with exception of the first two, has to be eliminated by addition of suitable multiples of the second equation to each of the equations from the third up to the last one (now named as the N-th) Then the third variable in all equations, with the exception of the first three, has to be eliminated etc To eliminate the i-th variable in the j-th equation (for j between i+1 and N), the i-th equation must be multiplied with a ji aii and subtracted from the j-th equation The described procedure is too simple to be completely right: aii (now named as pivot element) can become zero, so that a division by zero could arise This can be avoided because any arbitrary row (from the (i+1)-th to Application of the method of the least squares regression 333 the N-th) can be swapped with the i-th row, so that aii in the outer loop is different from zero For swapping it is the best to use the row for which the value in the i-th column is the greatest with respect to the absolute amount The reason is, that in the calculation considerable errors can arise if the pivot value, which is used to multiply a row by a factor, is very small If aii is very small, a ji aii can become very big This process, called the partial pivoting, is realized in the example of the Ls identification by the following program section After the step of the forward elimination is completed, the field below the diagonal of the modified matrix A[5][5] contains only zeros The step of the backwards insertion can be executed now to calculate the coefficients a0, a1, a2, a3 and a4 With the calculated coefficients a0, a1, a2, a3 and a4, the magnetizing curve L(i) in form of a polynomial (cf equation (12.25)) is now available 334 Appendices 12.4 Definition and calculation of Lie derivation An unexcited system (input vector u = 0) is defined (cf [Wey 2001, appendix B], [Phuoc 2006, section 5.1.2]) as follows: dx = f ( x) (12.29) dt A scalar function v(x) is given The derivation of this scalar function along the freely moving state trajectory (along the vector field f (x) of the unexcited system (12.29): x (t ) = f t (x0 ) n ) (12.30) can be given as follows: L f v (x) = n i=1 v ( x) fi ( x1 , xi , xn ) (12.31) with: v (x) = v , x1 v , x2 , v xn (12.32) x Using (12.32), the Lie derivation Lfv(x) can also be formulated as a scalar product (a scalar function): v (x) v (12.33) Lf v = L f v ( x) = f f (x) x x The function Lfv(x) returns the quantitative change of v(x) along the trajectory (12.30) The figure 12.1 illustrates this fact Fig 12.1 Derivation of the scalar function v(x) along the state trajectory x(t) References 335 The dashed curves in the figure 12.1 represent the sets of points inside at which the function v(x) has the same values The dashed curve with the point x contains the set of points which fulfills v(x) = k1, and the curve with xT the set of points fulfilling v(xT) = k2 In this case, the speed of the quantitative change of v(x) along x(t), from point x to the point xT, will be: n L f v (x) = lim T = k2 k1 = lim T v lim xT T f T v (xT ) v (x) T (x) x T = T v ( x ) dx x = lim dt = v ( f T (x)) v (x) T v (x) x f (x) (12.34) The Lie derivation has the following properties: • The multiple Lie derivation of v(x), at first along the vector field f(x) and then along g(x), can be written as follows: L f v (x) v (x) Lg L f v (x) = Lg L f v (x) = g (x) = f (x) g (x) x x x (12.35) • Let w(x) be an additional scalar function, then the following relation is valid: Lwf v (x) = L f v (x) w because Lwf v (x) = v (x) x (w f ) = v (x) x f (x) w(x) (12.36) L f v(x) • Let k be an integer number, then the k-fold Lie derivation of v(x) along f(x) can be recursively calculated as follows: Lkf v (x) = Lkf 1v (x) x f (x) with L0f v (x) = v (x) (12.37) 12.5 References Rojiani KB (1996) Programming in C with numerical methods for engineers Prentice-Hall International, Inc Sedgewick R (1992) Algorithmen in C Addison-Wesley Phuoc ND, Minh PX, Trung HT (2006) Nonlinear control theory Publishing House of Science and Technique, Hanoi (in Vietnamese) Wey T (2001) Nichtlineare Regelungssysteme: Ein differentialalgebraischer Ansatz B.G Teubner Stuttgart - Leipzig - Wiesbaden Index A adaptation 251, 288 on-line 225, 231 adaptation fault 236 anti-reset windup 172 B bilinear 62, 77, 82, 88, 287 blanking time 29 C component direct field-forming 3, 205 field synchronous quadrature torque-forming 7, 207 controller approarch 157 current 306 flux 283 on-off 145 predictive 149 state space 167 three-point 147 two-point 146 coordinate field 2, 130, 147, 158 grid voltage orientated stator-fixed 2, 74, 79, 130, 149, 162, 185 copper loss 258 current displacement 191 current-voltage characteristics 213 D DC link voltage 17, 143, 168, 301 dead-beat 139, 155, 159, 163, 307, 313 decoupling 12, 128, 143, 155, 159, 168, 305 direct 13, 293, 299, 315 input-output 294 decoupling network 7, 83 difference order 98 discretization 65, 84, 137, 328 doubly-fed induction machine 5, 90 E eddy-current 191 efficiency 258 optimal 281 eigenvalue 67 encoder 111, 143 equivalent circuit 185, 190 inverse 188, 193 189 T, transformer- 186 Euler 66, 329 excitation direct current 221 single-phase sinusoidal 215 excitation frequency 212, 217 F fault model 239 field displacement 194 field weakening 8, 107, 262 338 Index lower 269 upper 265 finite adjustment time 156, 165 flux model 7, 81 forbidden zone 28 frequency response 215 friction loss 192 front-end converter 301, 309 G Gauss elimination 332 H hysteresis loss 191 I identification off-line 212, 287 incremental encoder 111 inductance leakage 211, 254 main 199, 211, 219 mutual 3, 70 rotor 3, 70 rotor leakage 197 single-phase main 220 stator 70, 211 three-phase main 220 transient leakage 213 induction motor iron loss 191, 249 iron loss resistance 192 K Kalman filter 116, 231 L leakage factor 73, 208, 277, 284 least squares regression 330 Lie derivation 100, 334 limitation 172, 261, 275 output 144 linearization exact 13, 98, 288, 316 Ljapunov 119 losses copper 258 friction 192 hysteresis 191 iron 191, 249 Luenberger observer 118 M magnitude-optimal 283 matrix feedback 167 input 65, 92, 94 output 65 pre-filter 167 system 74, 92 transition 65, 79, 89, 94 measurement current 107 instantaneous value 108, 159 integrating 108, 110, 163 speed 107 model fault 239 power balance 243 stator voltage 239 voltage vector fault 241, 248 modulation stochastic 53 synchronous 51 vector 17 modulation with two legs 49 N name plate 204, 210 natural field orientation 116, 131 nonlinear control 14, 144, 295 nonlinear coupling 77 nonlinear parameter 97, 288 nonlinear process model 287 Index nonlinear properties 97 nonlinear structure 97 normalization 325 O observer 118, 131, 231 nonlinear 233 rotor resistance 237 observer approarch 236 operation generator 177 motor 177 oversynchronous 302 P permanentmagnet-excited synchronous motor 3, 85 pivot element 332 pole flux 85, 117 power active 6, 304 reactive 6, 304 power factor 6, 205, 208 prediction 168 priority 175, 179 process model 81, 89, 95, 97, 157, 310 protection time 29 pulse frequency 22, 145 pulse period 19 Q quadrant 18 R randomizing sequence 20 zero vector 57 reactor 311 resolution 112 voltage 27 resolver 113, 143 339 reverse correction 173, 180 ride-through 13, 315, 321 rotor flux 3, 70, 85, 90, 128 rotor flux control 282 S sampling period 22, 65, 110, 135, 155 saturation 132, 191, 198, 203 scaling 325 scaling factor 327 sector 18 sensorless 107, 116, 125, 287 sensitivity parameter 244 sensitivity function 217 sequence 20, 109 sequence randomizing 54 set point 159, 257 current 134 flux 270 torque optimal 261 Shannon 138 speed range basic 10, 192, 261 splitting strategy 175 standstill 211, 215 state space controller 167 state space model 305 continuous 70, 72, 75, 87 discrete 78, 82, 88, 92 stator flux 11, 70, 90 stator resistance 208 subsynchronous 302 Sylvester-Lagrange 67, 329 synchronization 28, 115 system bilinear 62 linear 62 sampling 64 system matrix 74 switching pattern 21, 55 340 Index switching state 18 switching time 23 T time constant rotor 3, 73, 207, 226 stator 73, 208 torque 70, 86 motor generator torque of inertia 70 torque optimal 257, 281 transfer ratio 186 transformation 84, 129 coordinate 99, 291, 297, 318 transition matrix 65, 79, 89, 94 V vector boundary 19 orientation 107 standard voltage 18, 150 zero 18, 108 vector modulation 7, 18 voltage resolution rotor 70, 90 stator 70, 90 W windup 144 anti-reset 172 Z zero-crossing 113, 215 zero-order hold 65, 135 zero vector 18, 70, 108 zero vector randomizing 57 ... Quang and Jörg-Andreas Dittrich Vector Control of Three- Phase AC Machines Nguyen Phung Quang · Jörg-Andreas Dittrich Vector Control of Three- Phase AC Machines System Development in the Practice With... Table of Contents A Basic Problems Principles of vector orientation and vector orientated control structures for systems using three- phase AC machines Formation of the space vectors and its vector. .. Zürich Switzerland andreas_d@swissonline.ch ISBN: 97 8-3 -5 4 0-7 902 8-0 e-ISBN: 97 8-3 -5 4 0-7 902 9-7 Power Systems ISSN: 161 2-1 287 Library of Congress Control Number: 2008925606 This work is subject to