SCALAR CONTROL SYSTEMS WITH PERMANENT MAGNET SYNCHRONOUS MOTORS WITH SINUSOIDAL CURRENT CONTROL. CALCULATION OF SPEED CONTROLLER PARAMETERS

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1 Automatique et ordinateurs SCALAR CONTROL SYSTEMS WITH PERMANENT MAGNET SYNCHRONOUS MOTORS WITH SINUSOIDAL CURRENT CONTROL. CALCULATION OF SPEED CONTROLLER PARAMETERS NICOLAVALERIU OLARESCU, SORIN MUSUROI, CIPRIAN SORANDARU, FLAVIUMIHAI FRIGURAILIASA ey words: Permanent magnet synhronous motor, Salar ontrol, Speed ontroller, Calulation o parameters. The present paper presents the alulation o the speed ontroller parameters o the salar ontrol systems with permanent magnet synhronous motors with sinusoidal urrent ontrol used in low power appliations. The shat position sensor used to determine the rotor position, respetively its rotation speed is made o two Hall sensors. In the onsidered numeri example, we presented the Bode diagrams o the speed loop with and without onsidering the visous rition oeiient, in the ase the lowpass ilter does or does not interere in the negative reation. 1. INTRODUCTION Permanent magnet synhronous motors (PMSM) are used in the regulated systems with variable speed, by their assoiation with requeny onverters. Generally, in the ase o PMSM, in order to determine the rotor position, a shat position sensor is used, while to determine the rotor speed, a speed sensor is used, or it an be alulated using the position data, thus the speed sensor an be removed. The requeny onverter used with suh ontrol system is the traditional one, exept or the ontrol module o the phase power devies whih must be orrelated with the rotor position. The present paper will approah the permanent magnet synhronous motors with sinusoidal urrent ontrol (BLAC). These mahines have, rom a onstrutive point o view, the stator windings sinusoidal distributed or onentrated, being haraterized by a rational number o slots per pole and phase. The permanent magnets an be plaed on the surae o the rotor (it results BLAC with ull pole, to whih L d = L q ) or in its interior (BLAC with prominent poles are obtained, L d L q ). The ontrol systems with BLAC an be Politehnia University o Timisoara, Faulty o Eletrial Engineering, Bd. Vasile Parvan, RO3003, Timisoara, Romania; vali_olaresu@yahoo.om Rev. Roum. Si. Tehn. Életrotehn. et Énerg., 57, 1, p , Buarest, 01

2 Salar ontrol systems with PMSM with urrent ontrol 71 lassiied as ollowing: ontrol systems in open loop, salar ontrol systems, ield oriented vetor ontrol systems and diret torque ontrol systems. The present study ouses on alulating the speed ontroller parameters o the salar ontrol systems used in low power appliations.. THE SCALAR CONTROL SYSTEM TOPOLOGY WITH BLAC The topology o a salar ontrol system (SCS) used in low power appliations is presented in Fig. 1. In the ase o the SCS, a shat position sensor is used (two Hall sensors in our ase) to determine the rotor position, respetively its rotation speed, whih is generally inorporated in the BLAC. For the SCS with BLAC, the requeny onverter has no sensors to measure the line urrent o the BLAC. The requeny onverter supplied rom a singlephased network presents the ollowing modules: the retiier represents a singlephased unontrolled bridge retiier o a apaitive harge; the inrush urrent limitation module by whih the inrush urrent is limited, the one ourring when the requeny onverter is onneted to the network i the apaitor is not harged yet. This urrent will be limited by a resistor that will be shortiruited using a relay when the apaitor is harged. the ilter the indutive type is used to limit the harmonis o the inrush urrent in the requeny onverter, determining a passive orretion o the power ator; the power module made o 6 IGBT s with 6 run diode in antiparallel, the driver used or the ontrol o the IGBT s in bootstrap typology, respetively a NTC temperature sensor through whih the temperature o the module an be measured. The generating o the waveorms is done by µcdspdsc. This physial module an be integrated (MIP), or disreet. the eletroni iruit module in whih there are inluded the generating eletroni iruit o the low d voltage (15 V, 5 V or 3.3 V), the eletroni iruit o the serial ommuniation as well as dierent eletroni iruits neessary to measure dierent values (voltages, urrents and temperature). The 15 V d voltage is obtained by a step down onverter whih has the as input value the voltage rom the intermediary iruit o d voltage link. The 5 V or 3.3 V d voltage is obtained rom the 15 V one through a linear retiier. In order to measure the supply voltage, respetively the d link voltage a two resistor voltage divider is used. The mains urrent is measured using the R reshunt and an operational ampliier, while the urrent o the intermediate d link voltage iruit is measured using the shunt R dshunt and an operational ampliier. Using the shunt R dshunt and a omparator the maximum value o the urrent I p an be determined, or whih the power module must be

3 7 NiolaValeriu Olaresu et al. 3 proteted; the same proedure is used to determine the maximum peak urrent I max that an pass through the BLAC phases. L N PE Frequeny Converter Filter Retiier DC Voltage Ciruit Power Modul V1 D1 V3 D3 V5 D5 GV1 GV3 GV5 Inrush urrent limitation modul C NTC V6 D6 V4 D4 V D a b i ib ia GV6 GV4 GV Rreshunt Rdshunt n Srxd Eletroni Ciruit Modul GV1 GV GV3 GV4 GV5 GV6 BLAC Stxd Sm Vdmax Rxd Txd vre ire Vd Tem Id Imax Vd Ip U W V V W U SCI ADC Modul I/O PWM Modul Hall1 PWM µ C DSP DSC TICM1, Hall Control Algorithm o BLAC Hall Sensors Fig. 1 Blok diagram o SCS with BLAC o redued power. In order to detet the maximum admitted value o the d link voltage a hardware solution is used, provided by a omparator. Thus, the power module, respetively the apaitor rom d link iruit is proteted, against the overvoltage that may be aused by the indued voltage o the BLAC. µcdspdsc is the module around whih the requeny onverter is built. This module represents in at one o the three variants µc (miroontroller), DSP (digital signal proessor), DSC (digital signal ontroller). Generally, in the ase o SCS a µc is used due to the lower ost. In this module, the ontrol algorithm is implemented, respetively the diverse requirements imposed in the ase o the requeny onverters used in low power appliations. 3. SPEED CONTROLLER DESIGN For the SCS see Fig. 1, the hosen ontroller is o PI type, with the limitation o the output and antiwindup whih has as input the speed dierene between the reerene speed and the alulated speed and as output the value o the reerene

4 4 Salar ontrol systems with PMSM with urrent ontrol 73 torque or the reerene value o the stator urrent. Other antiwindup variants are presented in [5]. Fig. presents two PI ontrollers: in Fig. a, only with the limitation o the output, respetively in Fig. b, with the limitation o the output and antiwindup. The transer untion o the PI ontroller an be written in Laplae as: y pu k k ( ) = i k = = i p PI ( s) k p 1 s. * (1) x x s s k inpu inpu i Coeiient k aw (Fig. a) will be hosen: k aw = k p. To determine the parameters o the speed ontroller we start rom the motion equation in perunit. Applying the Laplae transorm, it results: 1 rpu = rmpu = ( telmpu t spu ), r H s () where rpu is the angular speed in perunit, H the onstant o inertia, t elmpu and t spu the eletromagneti torque developed by the BLAC respetively the load torque in perunit. Thereore the transer untion in Laplae, o the movement equation, an be deined as: M () s 1 = H s r. (3) xinpu* xinpu PI yiepu xinpu* ypu yiepu k p xinpu k i s xinpu* xinpu PI (Antiwindup) yiepu xinpu* ypu yiepu k p xinpu 1 k aw a) b) Fig. Types o PI ontrollers: a) PI ontroller with the limitation o the output; b) PI ontroller with the limitation o the output and antiwindup. The mathemati model () or (3) is used to determine the parameters o the speed ontroller, whih is represented in Fig. 3a as a ontinuous model (applying k i s

5 74 NiolaValeriu Olaresu et al. 5 the Laplae transorm), respetively in Fig. 3b, as a disreet model (applying the Z transorm). The passing rom the plan s to the plan Z (disreet time) an be done through the ollowing relations: a) 1 z s = t ; b) 1 z 1 z s = ; ) s =. (4) t z t 1 z In Fig. 3, it an be notied that the negative reation o the speed is iltered by using a lowpass ilter (LPF) or a PT1 ilter. The ilter transer untion in plan s is: F () s =, (5) s where: =π, representing lowpass requeny. Using the relation (4a) the transer untion o the ilter in plan Z, an be written: F( z) = t t 1 z. (6) I zeroorder hold eet is applied to obtain the transer untion o the ilter in plan Z, the ollowing relation is obtained: t F ( ) ( ) ( s) ( 1 e ) F z = 1 z Z = t. (7) s 1 z e tspu tspu rpu* PI (Antiwindup) tpu* 1 r H s rpu rpu*(z) PI(z) (Antiwindup) tpu*(z) M(z) rpu(z) Filter(s) Filter(z) a) b) Fig. 3 The model used to determine the speed ontroller parameters: in ontinuous, using Laplae transorm: in disreet, using Z transorm. In order to determine the PI speed ontroller parameters, we onsider the transer untion in open loop FRV BD (s) in plan s, without onsidering the lowpass ilter (LPF) in negative reation: ( s) PI( s) M ( s) FRV BD = (8) respetively, the transer untion in open loop FRV BDFT (s) in plan s, onsidering the lowpass ilter in negative reation:

6 6 Salar ontrol systems with PMSM with urrent ontrol 75 ( s) PI ( s) M ( s) F(s) FRV BDFT =. (9) Thereore, the transer untions (with speed iltering, respetively without speed iltering) in lose loop FRV BÎ (s), respetively FRV BÎFT (s), in plan s o the model with negative reation, an be written: FRVBD ( s) FRV ( s) a) FRV () s = BÎ ; b) FRV () s 1 FRV ( s) = BD BÎFT 1 FRV ( s) F s. (10) BD To determine the ontroller parameters we onsidered the ollowing requirements: a zero speed error in stationary regime, a rapid transitory answer and a redued value o the overregulation. In order to determine the PI ontroller parameters k p, k i, the Bode diagram o the transer untion in open loop is used. The required onditions are: a) FRV ( j ) = 1; b) ( j ) = 1 BD BDFT BD () FRV ; (11) a) FRV ( j ) = 80 RF ; b) FRV ( ) = 80 RF BD j, (1) BDFT where: RF represents the phase reserve and it generally has values between [45º, 60º]; = π, where is the breaking requeny (the requeny to whih the amplitude o the transer untion in open loop is 1 and the dephasing introdued by the transer untion in the loop is lower than 180, and the angle is negative). Further on, to simpliy the alulations it is noted: respetively (1a) it results: τ = k k p i. Thus, rom (11a), r tg( 80 RF) τ =, (13) r tg( 80 RF) r H ki =, (14) r r τ τ a) k k τ ; b) kih = t ki, (15) p = i

7 76 NiolaValeriu Olaresu et al. 7 where: t represents the sampling period o the speed loop. I the lowpass ilter in negative reation is onsidered, then rom (11b), respetively (1b) it results: τ FT tg( 80 RF) = r tg ( 80 RF ) 3 ( 80 RF ) tg( 80 RF ) r tg r r, (16) k ift = τ FT r τ FT H r r τ ( ) FT τ FT r r (17), a) k pft = k ift τ ; b) kihft = tkift. (18) Further on, a numeri example is presented in whih: I b = 8.81 A, V b = 3488 V, n = 14, rmb = rad/s, T b = Nm, = 0.36 kgm, H = kgm rad/vas, r = Nm/rad/s, t = s, RF = 60º and = 36 Hz. The results proessing was done by a program in Matlab 7.1 R14. Thus, i in the lose loop o the speed ontroller the lowpass ilter is not onsidered, by help o (11a), (1a), respetively (13), (14) we obtain the ollowing values or the speed ontroller o a PI type: k p( 0 ) = ; k p( = 0 ) = ; k ih( 0 ) = ; k ih( = 0 ) = r r In Fig. 4 there are Bode diagrams (the response in requeny) o the speed loop with and without onsidering the visous rition oeiient r, in ase the lowpass ilter does not interere in the negative speed regulation loop. It an be notied that to simpliy the alulations, in determining the parameters o the speed ontroller o PI type, the visous rition oeiient r may be negleted. In Fig. 5 there are Bode diagrams o the speed loop with and without onsidering the visous rition oeiient r, in ase the lowpass ilter intereres in the negative reation. In this situation, rom (11b), (1b), respetively (16) and (17) it results: k ( 0 ) = 7. pft 9890 ; k pft ( =0 ) = ; r k ihft ( 0 ) = ; k ihft ( =0 ) = r r r r r

8 8 Salar ontrol systems with PMSM with urrent ontrol 77 Fig. 4 The Bode diagram o the speed regulation open loop, respetively losed loop, when the visous rition oeiient is r = Nm/rad/s, or r = 0. Fig. 5 Bode diagram o speed regulation open loop, respetively losed loop, when the visous rition oeiient is r = Nm/rad/s, or r = 0, in the negative reation a lowpass ilter is introdued to ilter the speed. In this ase also it an be stated that the visous rition oeiient has a redued inluene in determining o the PI ontroller parameters. The lowpass

9 78 NiolaValeriu Olaresu et al. 9 ilter used or speed iltration has a large inluene in determining o the PI ontroller parameters. Furthermore, in Fig. 6 Bode diagram is represented or the lowpass ilter, respetively or the speed ontroller. Fig. 6 Bode diagram o the lowpass ilter (in ase = 16 Hz) used to ilter the speed, respetively the Bode diagram o the speed ontroller o PI type. Fig. 7 The variation o the parameters o the speed ontroller o PI type aording to the moment o inertia.

10 10 Salar ontrol systems with PMSM with urrent ontrol 79 In ase o the appliations or whih the value o the moment o inertia is not onstant (Fig. 7) the values o the speed ontroller parameters are represented aording to the value o the moment o inertia. As it is expeted, the value o the parameters o the speed ontroller depends on the value o the moment o inertia, more exatly the parameters k p, k i varied linearly aording to the value o the moment o inertia. I a ompetitive adjustment o the BLAC speed is wanted, in the ase o the appliations or whih the moment o inertia is not onstant, the moment o inertia must be determined online in the adjustment system and the parameters o the speed ontroller must be modiied aording to the value o the moment o inertia. 4. CONCLUSIONS The design o the speed ontroller or the salar ontrol systems with BLAC is onsidered, obtaining a zero error in stationary regime, a rapid transitory answer and a redued value o the overregulation. To determine the parameters k p, k i o the PI ontroller, the Bode diagram o the transer untion in open loop was used. In ase that the lowpass ilter does not interere in the negative loop o the speed ontroller, the visous rition oeiient r an be negleted in determining the parameters o the PI speed ontroller. Also in the situation in whih the lowpass ilter intereres in the negative reation, the visous rition oeiient has a redued inluene in determining the parameters o the PI speed ontroller. The lowpass ilter used to ilter the speed has a large inluene in determining the parameters o the PI speed ontroller. The values o the parameters o the speed ontroller also depend on the value o the moment o inertia, k p, k i, linearly varying aording to the value o the moment o inertia. Reeived on 8 Marh 011 REFERENCES 1. D. ouve,.p. Rognon, D. Roye, Eetive Current and Speed Controllers or Permanent Magnet Mahines:A Survey, Proeedings o Fith Annual Applied Power Eletronis Conerene and Exposition, Marh 11 16, 1990, pp H. Panagopoulos,.. Astrom, T. Hagglund, Design o PID Controllers Based on Constrained Optimization, Proeedings o the Amerian Control Conerene, San Diego, une 1999, Volume 6, pp H. Panagopoulos,.. Astrom, T. Hagglund, A numerial Method or Design o PI Controllers, Proeedings o the 1997 IEEE International Conerene on Control Appliations, Hartord, CT Otober 5 7, 1997, pp D. Vrani, D. urii, S. Strmnik, R. Hanus, ClosedLoop Tuning o the PID Controller by Using MOMI Method, Proeedings o the Amerian Control Conerene, San Diego, une 1999, Volume 5, pp Y. Peng, D. Vrani, R. Hanus, AntiWindup, Bumpless, and Conditioned Transer Tehnique or PID Controllers, IEEE Control Systems Magazine, 16, 4, pp , 1996.