THE COMPLEX CONTROLLER FOR THREE-PHASE …

1 THE COMPLEX CONTROLLER FOR THREE-PHASE INDUCTIONMOTOR direct torque CONTROLA lfeu J. Sguarezi Filho Ruppert Filho Universidade Estadual de Campinas, FEEC-DSCEAv. Albert Einstein, 400 Cidade Universit aria Zeferino Vaz, Campinas, SPABSTRACTThis paper proposes a design and tuning method for acomplex gain CONTROLLER , based on the THREE-PHASE induc-tion motor mathematical model COMPLEX transfer func-tion to be used in the direct torque control at low speedwhich is a problem so far. The design and tuning ofthe COMPLEX gain is done by using the closed loop sys-tem frequency-response function. Experimental resultsare presented to validate the CONTROLLER and operation atlow speed is also : COMPLEX gain CONTROLLER , COMPLEX transferfunction, induction motor , direct torque trabalho prop oe um m etodo de projeto para umcontrolador de ganho complexo baseado na fun c ao detransfer encia complexa do motor de indu c ao trif asicopara uso no controle direto de torque em que a opera- c ao do motor em baixas velocidades e um problema.

2 Oprojeto e a sintonia do ganho complexo e realizado como emprego da resposta em frequ encia do sistema em ma-lha fechada. Resultados experimentais s ao apresentadospara a valida c ao da proposta do : Controlador de ganho complexo,Artigo submetido em 22/11/2008 (Id.: 00924)Revisado em 26/02/2009 Aceito sob recomenda c ao do Editor Associado Prof. Darizon Alves de An-dradeFun c ao de Transfer encia Complexa, motor de Indu c ao,Controle direto de INTRODUCTIONThe dynamics of the THREE-PHASE induction motor istraditionally described by differential equations. Thespace-vector concept (Kov acs e R acz, 1984) is used inthe mathematical representation of the motor state vari-ables such as voltage, current, and concept of COMPLEX transfer function derives fromthe application of the Laplace transform to differen-tial equations where the COMPLEX coefficients are in ac-cordance with the spiral vector theory by Yamamura(1992).

3 Holtz (1995) proposed a THREE-PHASE induction motormathematical model using the COMPLEX transfer functionand presented the induction motor root locus. Otherprocedures for modeling and simulating the three -phaseinduction motor dynamics using the COMPLEX transferfunction concept are also presented by Cad (2000).Briz et al. (2000) applied the COMPLEX transfer func-tion concept to design current regulators for RL loadsand induction motors. The regulator design was imple-mented through the frequency-response function of theclosed-loop COMPLEX transfer function of the controlledinduction machine system in the synchronous referenceframe.

4 Experimental results showed good performancealthough thedqstator currents had presented cross-256 Revista Controle & Automa c , Maio e Junho 2009coupling in the induction machine transients and low-speed tests had not been solve the cross-coupling between induction machinedqtransient stator currents Holtz et al. (2004) de-signed a stator-current CONTROLLER using COMPLEX nota-tion. From this, the current CONTROLLER structure em-ploying single- COMPLEX zeros is synthesized. Experimen-tal results demonstrate that high dynamic performanceand zero cross-coupling are achieved even at very lowswitching frequency although the speed control had notbe alternative for induction motor drives is the directtorque control (DTC), which consists of the direct con-trol of the stator 1and the electromagnetic controllers generate a stator-voltage vector thatallows quick torque response with the smallest varia-tion of the stator flux.

5 The principles of the DTC usinghistereses controllers and variable switching frequencywere presented by Takahashi e Noguchi (1986) and De-penbrock (1988).Xue et al. (1990) proposed a DTC strategy using PI con-trollers and space-vector modulation to generate a sta-tor voltage based on torque and stator-flux errors. Thisstrategy has presented good torque response althoughlow-speed tests had not been shown so literature shows the application of the control the-ory to some DTC strategies, as that one presentedby Buja e Kazmierkowski (2004) and other presentedby Stojic e Vukosavic (2005). Each strategy aims to thetorque and to the rotor or stator flux control althoughthe COMPLEX transfer function of induction motor is paper proposes a design and tuning method fora COMPLEX gain CONTROLLER which is designed by usingthe induction motor COMPLEX transfer function to controlthe motor .

6 The CONTROLLER is adjusted by the frequency-response function of the closed loop system. Experimen-tal results are presented for validation of the proposedcontroller including low speed THE COMPLEX MATHEMATICALMODEL OF THE three -PHASEINDUCTION MACHINEThe THREE-PHASE induction machine mathematical modelis written with the variables referred to thedqsyn-chronous reference frame and the COMPLEX state vari-ables are the stator current~i1dqand the stator flux~ 1dq.~v1dq=R1~i1dq+ ~ 1dq+j 1~ 1dq(1)0 =R2~i2dq+ ~ 2dq+j( 1 P mec)~ 2dq(2)The relationship between currents and fluxes are givenby"~ 1dq~ 2dq#= L1 LmLmL2 ~i1dq~i2dq (3)The electromagnetic torque and mechanical speed aregiven byTe=32P Im[~i1dqconj(~ 1dq)](4)andJ mecdt=Te TL(5)

7 The subscripts1,2andmrepresent the stator, rotorand magnetization parameters respectively, 1is thesynchronous speed, mecis the machine speed,R1andR2are the estator and rotor windings per phase elec-trical resistance,L1,L2andLmare the proper andmutual inductances of the stator and rotor windings,~vis the voltage vector ,Pis the machine number of pairof poles,Jis the load and rotor inertia moment andTLis the load combining equations (1), (2) and (3), after some al-gebraic manipulations, one can write the COMPLEX spacestate equations as: ~ 1dq ~i1dq = a1a2a3a4 ~ 1dq~i1dq + ~v1dq~v1dq L1 (6)wherea1= j 1(7)a2= R1(8)a3= R2 L1l2 jP mec L1 (9)a4= R1 L1+R2 L2+j( 1 P mec) (10)Revista Controle & Automa c , Maio e Junho 2009 257 Figure 1: IM block = 1 L2mL1L2is the leakage order to obtain the induction motor COMPLEX trans-fer function the Laplace transform is applied to (6) inaccordance with the COMPLEX transfer function concept,it is assumed that the mechanical time constant of themotor is much larger than the transient electromag-netic time constants and the saturation effects is ne-glected.

8 Thus, mec= constant is a valid approxima-tion (Yamamura, 1992; Holtz et al., 2004). The induc-tion motor block diagram originated by use of (4), (5)and (6) is shown in Figure designing the DTC control system,~v1dqis consid-ered as the input and the~i1dqis considered as the out-put. Therefore the induction machine COMPLEX transferfunction is given byH(s) =I1dqV1dq= s+j 1 L1 +a3(s+j 1) (s+a4) +R1a3(11)whereI1dq=Ln~i1dqoandV1dq=L{~v1 dq}.3 direct torque CONTROLThe direct torque control strategy allows a quick torqueresponse and consists of the direct control of the sta-tor flux 1and the torqueTe. The flux and torquecontrollers generate a stator-voltage vector that allowsquick torque response with the smallest variation of thestator work is based in Xue s strategy as shown in (Xueet al.)

9 , 1990) which uses conventional PI controllers andspace-vector modulation to generate a stator voltagebased on torque and stator-flux this present work, by using stator-field orientation,the torque and stator flux must become parts of a com-plex number, where the magnitude of the stator flux 1is the real component and the torqueTeis the imagi-nary component. Hence, the reference signals and theerror become a COMPLEX number and the proposed con-troller is a COMPLEX gain (a+jb). This gain has thefunction to generate a voltage reference vector using thestator flux- torque vector error ( +j T). This way thestator-voltage vector in this control strategy is given by~v1dq= ( +j T) (a+jb)(12)which meansv1d= ( Tb+ a)(13)v1q= ( Ta+ b)(14)whereais the real part of the COMPLEX gain,bis theimaginary part of the COMPLEX gain, is the flux errorand Tis the torque block diagram of the mathematical model of thecontrol system proposed with the COMPLEX gain con-troller is presented in Figure can be observed in equation (12)

10 That the complexgain changes the amplitude and phase of the vector volt-age due to the cross-coupling between the COMPLEX gainand the error reference stator-voltage vector~v1dqis transformedby using stator-flux position sto obtain the stator volt-age in the stationary reference frame , as shown in thenext Revista Controle & Automa c , Maio e Junho 2009 Figure 2: DTC strategy with COMPLEX ESTIMATION BLOCKThe estimation of the stator flux is calculated by usingthe stator currents and voltages, given by~ 1 =Z(f em )dt=Z(~v1 R1~i1 )dt(15)where the subscript is used to designate the statorstationary reference frame which is being stator-flux angle is estimated by using the trigono-metric transfer function s=tg 1 1 1 (16)In order to achieve the stator-flux estimation for awide speed range in drives using (15) an integrationmethod (Xu et al.)