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633 High Performance Induction Motor Control Via Feedback Linearization M. P. Kazmierkowski and D. L. Sobczuk Institute of Control and Industrial Electronics, Warsaw University of Technology, u l . Koszykowa 75, 00-662 WarszawaPoland Phone: +48/2/6280665; Fax: +48/2/6256633 E-mail: .pl; .pl Abstract - This paper presents a feedback linearization approach for high performance induction motor control. The principle of the method is discussed and compared with most popular in AC motor drive technology field oriented control technique. Some oscillograms illustrating the properties of the PWM inverter-fed induction motor with control via feedback linearization are presented. INTRODUCTION The induction motor thanks to its well known advantages as simply construction, reliability, raggedness and low cost has found very wide industrial applications. Furthermore, in contrast to the commutator dc motor, it can also be used in aggressive or volatile environments since there is no prob- lems with spark and corrosion. These advantages, however, are occupied by control problems when using induction motor in speed regulated industrial drives. Ths is due primarily three reasons: (a) - the induction motor is high order nonlinear dynamic system with internal coupling, (b) - some state variables, rotor currents and fluxes, are directly not measurable, (c) - rotor resistance (due to heating) and magnetising inductance (due to saturation) varies consider- ably with a sigluficant impact on the system dynamics. The most popular high performance induction motor control method known as Field Oriented Control (FOC) or Vector Control has been proposed by Hasse 4 and Blaschke 11. In this method the motor equation are (rewritten) transformed in a coordmate system that rotates with the rotor flux vector. These new coordinates are calledfield coordinates. In field coordinates - for the constant rotor flux amplitude - there is a linear relationshp between control variables and speed. Moreover, as in separately excited DC motor, the reference for the flux amplitude can be reduced in field weakening region in order to limit the stator voltage at high speed. Transformation of the induction motor equations in the field coordinates has a good physical basis because it corresponds to the decoupled torque production in separately excited DC motor. However, from the theoretical point of view other type of coordinates can be selected to achieve decoupling and linearization of the induction motor equations. Krzeniinski 7 has proposed a nonlinear controller based on multiscalar motor model. In this approach, similarly as in field oriented controller, it is assumed that the rotor flux IEEE Catalog Number: 95TH8081 amplitude is regulated to a constant value. Thus, the motor speed is only asymptotically decoupled from the rotor flux Bodson et al. 2,3 have developed a nonlinear control sys- tem based on iiiput-output linearization. In this system, the motor speed and rotor flux are decoupled exactly. The system, however, use the transformation in field coordinates. Marino et al. 8,9 have proposed a nonlinear transformation of the motor state variables, so that in the new coordinates, the speed and rotor flux amplitude are decoupled by feedback. Similar transformation have been used by Sabanovic et al. E for decoupled rotor flux and speed sliQng mode controller. In the paper the feedback linearization control of induction motor is presented. In contrast to the works 2,3 the block diagrams and relationships to field oriented control are dis- cussed. Also, figures illustrated the properties of the control system when the motor is fed by PWM inverter are shown. MATHEMATICAL MODEL OF THE INDUCTION MOTOR Mathematical description of the induction motor is based on complex space vectors, which are defined in a coordinate system rotating with angular speed oK. In per unit and real- time representation the following vectorial equations describe behaviour of the motor 6: The electromagnetic torque m can be expressed as In the case of the squirrel-cage induction motor, the rotor voltage-vector vanishes from Eq. 2, having zero value. If a 633 current controlled PWM inverter is used, the stator voltage Eq. 1 can be neglected because it does not affect the control dynamics of the drive. FIELD ORIENTED CONTROL (FOC) In the case of field oriented control, it is very convenient to select the angular speed of the coordmate system oK equal a , . Under these assumptions, substituting the rotor current vector from the rotor voltage Eq. 2 by Eq. 4 ve obtain a differential equation for the rotor flux vector: Equations Eq. 10, Eq. 12 and Eq. 5 form the block diagram of the induction motor in the field oriented coordmates x y (Fig. 1). Diagram of control system applied to induction motor (direct field orientation) is.presented in Fig 2. In many cases as flux, speed, i , , i , controllers, simple PI regulators are used. FEEDBACK LTNEARTZATION CONTROL (FLC) Using p.u. time we can write the induction motor equations in the following form 6,11: (7)- where T , is the rotor time constant expressed as x =f(x + %.ga + usp g p (13) X Tr=ITN rr For the field oriented coordinates x-y we have Wm= Yr y = o ry where and m. I Note that am, yra, yq are not dependent on control signals U , , U$. In this case it is easily to choose two variables dependent on x only. For example we can define 5,10. Fig 1 Block diagram of induction motor in x-y field coordinates Eq. 10 describes the influence of the flux stator current according to Eq 5, can be expressed as follows: components i, on the rotor flux. The motor torque, XI = Vra2 + vr3 = V: (18) &(x) = 0, (1 9) 635 d P-P 1 sa 1 SG - 1rp Vector 1 Sigoals . +r ffiatioc &a Estimaiion - Transfor- vr* * vrB * % Fig. 2. Control of induction motor via field orientation Let +,(x), I$(X) are the output variables. The aim of control is to obtain: 0 constant flux amplitude, 0 reference angular speed. P a r t of the new state variables we can choose according to Eq. IS, Eq. 19. So the full definition of new coordinates are given by 8,9: In the further part of this section w e will consider the system consists of the first fourth equations. Note, that the fifth equation is as follows: We can rewite the remain system Eq. 22 in the following form: D is given by: After simple calculations one can obtain: 6 where D-I we can calculate using the following formula It is easily to show that if+l f 0 then det(D) # 0. In this case we can define linearizing feedback as: The resulting system is described by the equations: 2, =z2 z,= , z3 = z4 2 , = v 2 Block diagram of induction motor with new control signals is presented in Fig 3. Control signals v1 ,v2 one can calculate using linear feedback: where coefficients k, k, k, k, are chosen to determinate closed loop system dynamic. Control algorithm consists of two steps: 0 calculations v1 ,v2 according to Eq. 32, Eq 33, 0 calculations U , , uSp according to Eq. 29. Diagram of control system applied to induction iiiotor (feedback linearization) is presented in Fig. 4. Fig. 4. Control of induction motor via feedback linearization 637 -0461 1 -09El RESULTS -04FI , -09.! The simulated oscillogranis obtained for FLC and FOC sys- tems with linear speed and rotor flux controllers (motor, in- verter and controllers data are given in Appendix) are shown in Fig. 5. These oscilllograms, show the speed reversal over the constant flux amplitude and field weakening ranges when motor is fed from VSI inverter with sinusoidal PWM. As can be seen from Fig. 5B the field oriented control does not guarantee full decoupling betwen speed and flux of the motor. With linear speed controller the FOC systein imple- ments torque current limitation, whereas the FLC system limits the motor torque (see Fig. 5A). Therefore, in the FOC system the torque is reduced in field weakening region and the speed transient is slower in FLC system. A-FLC B - FOC lineat r O C 6 1 C 2 0.3 6 4 05,-0.0 012 0.24 0.36 048 0 8 Cl ; 0 10 05 05 00 00 -0 5 -0 5 - 1 c -! 0 d)oC 0 1 0 2 03 0 4 0 5 d)OO 012 024 036 018 06 j -0961 I c 9f 1 t)C7 L 1 ( 7 0 3 0 4 5 e ) O 0 Ci 024 036 048 O t O 01 D: 03 0 4 05 0 0 O l i 024 036 0 4 B 06 Flg 5 Conk01 of mduclon motor via feedback linearization and field oriented control (Speed reversal mcludmg field weakening range). a) actual and reference speed (comer con,) b) torque m. c) flux component and amplrtude ( y,. yr), d) flux current isX. e) torque current i s , , . f ) current component lso To guarantee full decoupling in FOC system working with field weakening region a PI speed controller with nonlinear part (controller output signal should be divided over the rotor flux amplitude mrdyr) has to be applied. This division compensates for the internal multiplication (m = vr i ) sv needed for motor torque production in field oriented coordi- nates (Fig. 1 .). With such a nonlinear speed controller a very similar behaviour to FLC can be achieved (see Fig. 6.). In Fig. 7. the response to speed reference change for constant flux amplitude is presented. Note, that in contrast to the FOC system (where the control variables are U , , u ) , the control variables in FLC system (vl, VJ are exactly decoupled Fig. 6. Control of mdudion motor ia feedback linearization and field oriented control with nonlniear term (Speed reversal including field weak&y range): a) adual and reference speed (omrep om) b) torque m, c) flux component and amplitude (iy,. t y , ) . d) flux current isx. e) torque current is,., 9 cunent component i SP A - FOC 0) ;y 04 t,CO 003 ODE 009 12 GI5 -, 1 2 0 6 d)G0 003 006 005 G I ? 0 1 5 ) G O 003 GO6 00s 012 015 0 6 00 -0 6 -0 E - 1 i -1 2 e)00 003 GO& 009 C l I Ol5e)GC 003 GO6 005 G12 0 0 0 i I 0 0 003 006 003 012 D!: C C 063 031 005 01; 0 : Fig 7. Control of mdudion motor via feedback lmeariiatton and field onitcd conk01 (the response to speed rcfercnce dmigc - mstmt f l u s raigc) a) adual and refaence speed (o um). b) refercnce control agial A - u s , . B - vl. c) reference mtrof%&l A - u s ) . . B - v2. d) flux currail ish e) torque currat I 6) 638 CONCLUSIONS Kurzchlusslauferniotoren, Reglungstechnik, 20: pp. 60- 66. 1972. In this work a high performance Feedback Linearization Control (FLC) system for PWM inverter-fed induction motor drives is presented. The block diagrams and relationshps to conventional Field Oriented Control CFOC) are discussed. SI A Isidori: Nonlinear Control Systems, Comniunica- tions and Control Engineering. Springer Verlag, Berlin, second ehtion, 1989. .I The main features and advantages of the presented control 6 M. P. Kaimierkomki and H. Tunia: Automatic Control systems can be summarised as follows: of Converter-Fed Drives, ELSEVIER Amsterdam- with control variables v, 17, the FLC guarantee the exactly London-New York-Tokyo, 1994. decoupling of the motor speed and rotor flux control in both dynamic and steady states. Therefore, lugh performance drive system working in both constant and field weakening range can be implemented using a linear speed and flux 7 Z. Krzeminski: Multi-scalar models of an induction motor for control system synthesis, Scieiiia Electrica, 33(3): pp. 9-22, 1987. controllers. Adaptive partial with control variables is, is, the FOC cannot guarantee feedback linearization of induction motors, In Pro- the exactly decoupling of the motor speed and rotor flux ceedings o f the 29th Conference on Decision aid control in dynamic states. Therefore, high performance drive Control, Honolulu, Hawaii, pp. 3313-3318, Dec. 1990. 8 R. Marino, S. Peresada, and P. Valigi: 9 R. Marino and P. Valigi: Nonlinear control of induction motors: a simulation study, In European Coiitrol Corifereme. Greiioble, France, pp. 1057-1062, 1991, system working in both constant and field wakening range requires a speed controller with nonlinear (division over the rotor flux amplitude) part. FLC is implemented in a state feedback fashion and needs more complex signal processing (full information about motor state variables and load torque is required). Also, the lo H. Nijmeijer and A van der Schaft: Nonlinear dy- namical control systems, Springer Verlag, 1990. ll D. L. Sobczuk: If Nonlinear control for induction nie transformation and new control variables vl, v2 used in FLC have no so dlrect physical meaning as in, is, (flux and torque current, respectively) in the case of FOC system. tor, In Proceedings PEIIK 94, pp. 684-689, 1994. FOC can be implemented in classical cascade control (121 A Sabanovic and D. B. Izosiniov: Applications of structure and, therefore, an overload protection can easy be sliding modes to induction motor control, IEEE achieved using reference currents limiters on the outputs of Transaction on Industry Applications, 17( 1): pp. 41-49, 1981. It can be expected, however, that - because in FLC vari- ables (am, v:) and its derivative (Om, W,) are used as new coordinates - this approach will .be well suited for sliding mode speed and position controllers. Therefore, FLC create an interesting alternative to FOC for applications where high performance induction motor drive system are required. the flux and speed controllers, respectively. APPENDIX Motor Data: Controllers Data: REFERENCES rr = 0.0464 p.u. rs = 0.0314 p.u. Feedback linearization: k, = 2.25 p.u. 11 F. B1aschke:Da.s Verfahren der Feldorientirung zur xr = 2.237 PU. xs = 2.194 P.U. XM = 2.133 P.U. TM= 0.2 s k, = 3.0 p.u. k21 = 6.25 p.u. kZz = 5.0 p.u. Reglung der Asynchronmaschne, Siemens Forschungs- . und Entwicklungsber, l(1): pp. 184-193, 1972 2 M. Bodson, J. Chiasson, and R. Novotnak: High per- - _ _ formance induction motor control via input-output line- arization, IEEE Control Systems, pp. 25-33, August 1994. Data: Field Orientation: Flux PI Controller: d = 4.0 P.U. * KR= 10; TR=318 ms Tv-50ps Speed PI controller: 3 .J. Chiasson, A Chaudhari, and M. Bodson: Nonlinear controllers for the induction motor, In IFAC Nonlinear Control System Design Symposium. Bordoeaux France, TR=318 ms pp. 150-155, 1992. Current controllers: K, = 150; 4 K. Hasse: Drehzahlgelverfahren fur schnelle Um- kehrantriebe mit stromrichtergespeisten Asynchron - A High Performance PWM Current Source Inverter Fed Induction Motor Drive with a Novel Motor Current Control Method Mika Sal0 and Heikki Tuusa Department of Electrical Engineering, Power Electronics Tampere University of Technology P.O.Box 692, FIN-33 101 Tampere, Finland Abstract - This paper presents a high performance vector control- led PWM current source inverter (PWM-CSI) fed induction mo- tor drive where only the measured rotor angular speed and the de-link current are needed for motor control. Novel methods for compensating the capacitive currents of the motor filter and damping the motor current oscillations in the transient conditions are presented. The validity of the proposed methods are verified by simulation. I. INTRODUCTION The rapid development of power and micro electronics in re- cent years allow the use of induction machine also in high per- formance motor drives. At low- and medium power level the variable speed induction motor drives are usually realized us- ing a PWM voltage source inverters (PWM-VSI). However, the switched voltages yield high dddt-voltage slopes over the stator windings, which stresses the insulations and causes bear- ing current ploblems. A possible solution for this ploblem is the use of PWM current source inverter (PWM-CSI) (Fig. 1). Both the voltages and the currents of the machine are nearly sinusoi- dal and therefore the voltage stresses in the machine windings are low. In the PWM current source inverters a C filter has to be in- serted on the load side to reduce the current harmonics. Due to the capacitive currents of the filter the motor current references are not realized accurately, which can be the cause for unsatis- factory performance and instability problems. A few meth- odsl,2, which are based on the measurement of the load capacitor voltages, have been reported to solve the problem. However, with the compined steady state equations of the load filter and motor the capacitive currents can be compensated without any measurements. Line bridge On the other hand, the C filter and the machine inductances form a resonance circuit which is stimulated especially when the motor current references are changed. Some methods3,4, based on the measurement of the motor voltages and/or cur- rents have been proposed to damp the motor current oscilla- tions in the transient conditions. However, in PWM-CSI drives motor current measurements are not needed for protection since the overcurrent can be detected with the dc-link current sensor. So, it is preferred to use control methods where motor current measurements are not needed because in that case the motor current sensors can be totally eliminated. In the present work the control system of the PWM-CSI fed drive is under investigation. The line side converter has been studied earlier5,6 when also the prototypes of 5 kW and 100 kW have been built. The final goal is to develop a high per- formance motor drive with minimum hardware requirements. The proposed vector control system is realized in the rotor flux oriented reference frame. The capacitive currents of the load filter are compensated without any measurements using the combined steady state equations of the load filter and the mo- tor. Also, a new method for damping the motor current oscilla- tions in the transient conditions is presented. The method is based on the combined dynamic equations of the load filter and the motor and does not need any measurements. However, the speed sensor is included to get the drive also to work well near zero speed. 1 1 . VECTOR CONTROL OF THE PWM CURRENT SOURCE INVERTER FED INDUCTION MOTOR DRIVE Fig. 1 shows the main circuit of the PWM current source in- verter fed induction motor drive. Llif and Clif are the induct- ance and capacitance of the line filter and usup the supply Lrlr Load bridee Fig. 1. The main circuit of the PWM current source inverter fed induction motor drive 0-7803-5421-4/99/$10.00 0 1999 IEEE 506 voltage. Clof is the load filter capacitance. The line and load bridges are identical. Both bridges consist of six controllable switches such as IGB transistors (IGBTs). Antiparallel diodes of the IGBTs in the commercial power modules are also shown in the figure. Because of these diodes and very low reverse voltage blocking capability of the IGBTs, additional diodes have to be connected in series with the transistors. A smoothing inductor ( Ldc) is connected between the bridges. In the PWM-CSI drives the line converter is used to control the dc-link current. The function of the line converter is syn- chronized with the supply voltages. By changing the modula- tion index in the line bridge the dc-link voltage, i.e. the dc-link current, can be controlled. In the line-voltage-oriented refer- ence frame the active and reactive power of the line converter can be simply controlled with the real and and imaginary axis components of the supply current vector5,6. The line filter takes reactive power which can be compensated by the control system5,6. The stator currents are generated by the load converter. The load filter takes capacitive currents which are proportional to the square of stator frequency in the constant torque region and linearly proportional to the stator frequency in the field weak- ening region. A. Rotorflux based vector control system In the vector control strategies the AC motors are controlled like dc motors which have independent channels for flux and torque control. Fig.2(a) shows the vector control system which is realized in the rotor-flux-oriented reference frame and is based on indirect vector control scheme7. It should be noted that the control system of the line converter is not shown in the figure. Detailed description of the line converter control can be found in 5,6. The electromagnetic torque of the induction motor in the ro- tor-flux oriented reference frame can be written as 3 L m - t e = 2 -p+$ Lr SY where p is the number of pole pairs, L, magnetizing induct- ance, Lr rotor self inductance, limd rotor magnetizing current and i the imaginary axis component of the stator current vector in the rotor flux based coordinate system. Below nomi- nal rotor speed is kept constant and the electromagnetic torque is controlled with is, of which reference value is the output of the speed controller. Above nominal rotor speed the reference value of the magnetizing current is inversely propor- tional to the stator frequency. lirnr1 can be controlled with the real axis component of the stator current vector is, expressed in the rotor-flux-oriented reference frame, as follows: sY d - T -limd + = is, dt ref dc I Machine Load Converter Soeed Controller + . I+ Trit a) .ref + invy ref SY campy Phase-error compensation eompx .ref + .ref LgX invx b) + invy Referenc filter Oscillations damping Referen filter + .ref ref 8x 1inv.x e) Fi 2 a) Vector control of the PWM-CSI fed drive in the rotor-flux-oriented reference frame. b) Compensation of the motor current phase-error. c) Damp- ing of the motor current oscillations. In the indirect vector control system the rotor flux angle is calculated as a sum of the measured rotor angle and the refer- ence value of the slip angle in the following way: :ref e , , = er+ - sy dt T :I (3) If the angular rotor velocity w, instead of 8, is measured, as is the case in Fig. 2(a), (3) can be written as (4) Rotor flux angle is needed to transform the inverter current ref- erence vector ; ; : $ to the stationary coordinates &$. Su- perscript mr refers to the rotor flux based reference frame. In the proposed vector control system only the rotor angular speed and the dc-link current measurements are needed for mo- tor control. The measured dc-link current is needed for the modulator realization 8,9 in both converters and for dc-link current control in the line converter. The dc-link current refer- ence value is generated in the load converter as follows: 507 where the constant c 2 1 i.e. the magnitude of the dc-link cur- rent should be equal or greater than the length of the inverter current reference vector in order to keep the modulation in the linear region. B. Compensating the motor current phase-error The problem in Fig. 2(a) control system is that the stator cur- rent reference vector is not realized accurately because of the capacitive currents of the load filter. With the combined steady state equations of the load filter and motor the capacitive cur- rents can be compensated without any measurements. Next, the equations needed for compensation control are derived. The stator voltage equation of the induction motor in the sta- tionary reference frame can be expressed as where o is the resultant leakage constant. The load filter ca- pacitor voltage can be written as (7) and the load capacitor current as ilofc = iinv- is. (8) By substituting (8) into (7) and the resulting equation into (6) (Es= Elofc) following expression is obtained: d i d imr ( i n V - , ) d t =RSis+oL -+(I-o)L - (9) Sdt Sdt When (9) is expressed using the quantities of the rotor flux-ori- ented reference frame we have -mr By solving (10) for iinv the following equation is obtained: -mr di, -mr m r z a m r is where w mr = &,/dt. According to (1 1) the effect of load fil- ter in steady state can be compensated as follows: When (12) is expressed in terms of direct and quadrature axis components we have and 2 .ref = R c 0 jref .ref compy s lof mr sx -(3Lsclofwmrsy (14) L where the reference values of the stator current components and the rotor magnetizing current are used. In the constant flux region I imrl = is, and (13) can be written as .ref ref 2 .ref compx = -RsClofWmrisy -Lsclofw mrsx (15) The proposed compensation method is shown in the block diagram form in Fig. 2(b) which replaces the area surrounded by the broken line in Fig. 2(a). C. Damping the motor current oscillations The load filter capacitance and the machine inductances form a resonance circuit which is stimulated especially when the motor current references are changed. One solution to over- come this problem is to use combined dynamic equations of the load filter and the motor. By taking into consideration the dynamic terms of the stator current vector in (1 1) we have -mr 7-mr The dynamic terms of rotor magnetizing current are not in- cluded in (16) because limrl changes much more slowly than .mr L and also because the rotor magnetizing current is normally kept constant. When (16) is expressed in terms of direct and quadrature axis components we have and 508 However, because in practice real stator currents cannot follow step responses of the supply current references, modified (fil- tered) current references( z;lf and isryef) are used in (17) and (18). The proposed damping method is shown in block diagram form in Fig. 2(c), which replaces the area surrounded by the broken line in Fig. 2(a). An example of filtering the stator cur- rent references in discrete case is shown in Fig. 3 where a change in stator current reference value is obtained at time tk. The realization of the reference value is begun at time tk+l be- cause of the one time interval calculation delay. After that the orginal reference value is realized during four time intervals. For microconpoller implementation (17) and (18) have to be discretized when we have At - .k + 1 ldampx= RsClof -ref At Ais, k + 1 + oLsCl0f L At k + l (19) and k+l -ref k + 1 -ref k = ( A * - % )/At (22) In discrete realization the average values of modified stator current references during a time interval should be used in the summing point shown in Fig. 2(c). These can be expressed as :ref,k+1 :ref,k+l -ref,k+2 lsxy,av = (zsxy + isxy )/2 (23) Finally, the modified current reference, which is filtered ac- cording to Fig. 3, can be written as SXY ref,k+l - i -ref,k+2 - :ref,k+l isxy - zsxy + 0.25 ( isxy ref, k . ref, k - 1 ref, k - 1 - irefk-2) (24) ) + 0.3 ( isxy SXY + 0.45 (isxy - zsxy and r i k + 1 1 1 1 . SIMULATION RESULTS L J where (both components combined in one expression) -ref,k+2 :ref,k+l = ( isxy - xy /At % . ref tk tk+l tk+2 tk+3 tk+4 tk+5 Fig. 3. Example of filtering the stator current references. The simulation is based on the parameters shown in Table I. However, due to the skin effect the stator resistance in the res- onance frequency of the load filter (360 Hz) is considerably larger than that shown in the table. Therefore, three times the value given in the table has been used in the simulation model. The model has been built in discrete form to have close analogy with the future microcontroller implementation. The model has been built using per unit values. The base values are: current io= A, voltage uo= ./zUs, angular speed wo= 21150/p, flux wo= U,/ (21150) and torque to= (3/2) uoio/wo. Fig. 4 shows simulation results of the proposed damping method where the y-axis component of the stator current refer- ence vector is suddenly changed. The stator current references are filtered as shown in Fig. 3. The discrete time interval At in (19)-(24) is 200 ps. Fig. 4(a) shows the phase-A stator current when the damping method is not used. Fig 4(b) shows the sim- ulation result when the damping method is used. It can be seen that with the proposed damping method the current oscillations can be considerably reduced. Fig. 5 shows the simulation results of the entire vector con- trol system. The reference values are shown with the broken line and realized values with the solid line. The proposed con- trol methods of compensating the reactive power drawn by the load filter and damping the stator current oscillations are used. The magnetization of the motor is beginned at 10 ms. The ref- erence value of the rotor flux is rate limited in order to keep iLE* at an acceptable level. The final value of the rotor flux in the constant flux region is set to 0.9 p.u. Then, at 100 ms the 509 TABLE I SIMULATION PARAMETERS Nominal Stator phase Voltage us 230 v Nominal stator current 1 , 16 A NOmindl Shaft power PN 1.5 kW Magnetizing inductance L, 80 m H Stator leakage inductance L,I 4.5 mH Rotor leakage inductance L,I 4.5 mH Stator resistance 50 Hz Rs 0.6 Q Rotor resistance 50 Hz R, 0.7 Q Inertia moment J 0.1 kgm2 Friction constant B 0.01 Nms Nominal speed nN 1440 r/min Load filter capacitance CI, Line filter capacitance Clif Line filter resistance50 Hz Rlif Number of po:e pairs p 2 22.5 pF 22.5 pF 1.2 mH 0.1 Q Dc-link inductance Ldc 20 mH Line filter inductance Llif I 0 0.05 0.075 0 025 f , r n e ( s ) I I 0 0.05 0 075 07 ,me(=) 0 025 Fig, 4. Simulated waveforms of phase-A stator current at sudden change in ig&. a) Without oscillation damping control. b) With oscillation damping cont
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