Speed Regulation of a Small BLDC Motor using Genetic-Based Proportional Control

Transcription

1 World Aademy of Siene, Engineering and Tehnology 47 8 Speed Regulation of a Small BLDC Motor using Geneti-Based Proportional Control S. Poonsawat, and T. Kulworawanihpong Abstrat This paper presents the speed regulation sheme of a small brushless d motor (BLDC motor) with trapezoidal bak-emf onsideration. The proposed ontrol strategy uses the proportional ontroller in whih the proportional gain, k p, is appropriately adjusted by using geneti algorithms. As a result, the proportional ontrol an perform well in order to ompensate the BLDC motor with load disturbane. This onfirms that the proposed speed regulation sheme gives satisfatory results. Keywords BLDC motor, proportional ontroller, geneti algorithms. A I. INTRODUCTION brushless DC motor is one of a small-sale motor used in small eletri devies suh as CD players, hard disk drives, or even small eletri ars. Its rotor is mounted with permanent magnet. There is no need for extra field exitation. This motor is well-known and popular for position and speed ontrol drive appliations [1,]. The key advantage of this motor over other types in the same rating is higher ratio of produed torque per weight, faster response, aurate position ontrol, lower moment of inertia, less maintenane, et [3,4]. In this paper, mathematial modeling of a small BLDC motor and its eletrial drives are the main purpose as desribed in Setion II. Setion III presents power loss and effiieny alulation for the BLDC motor. Setion I presents speed regulation tehnique. Setion and I are simulation results and onlusions, respetively. II. PRINCIPLES OF THE BLDC MOTOR A. Mathematial Model of BLDC Motors Modeling of a BLDC motor an be developed in the similar manner as a three-phase synhronous mahine [1,]. Sine its rotor is mounted with a permanent magnet, some dynami harateristis are different. Flux linkage from the rotor is dependent upon the magnet. Therefore, saturation of magneti flux linkage is typial for this kind of motors. As any typial three-phase motors, one struture of the BLDC motor is fed by a three-phase voltage soure as shown in Fig. 1. The soure is Manusript reeived Marh 31, 8. This work was supported in part by the Hard Disk Drive Institute, NECTEC, Thailand, and Suranaree University of Tehnology, Thailand. S. Poonsawat and T. Kulworawanihpong are with the Shool of Eletrial Engineering, Suranaree University of Tehnology, Nakhon Rathasima Thailand (phone: ; fax: ; thanathai@gmail.om). not neessary to be sinusoidal. Square wave or other waveshape an be applied as long as the peak voltage is not exeeded the maximum voltage limit of the motor. Similarly, the model of the armature winding for the BLDC motor is expressed as follows. dia v = Ri + L + e a a a (1) dib v = Ri + L + e b b b () di v = Ri + L + e (3) Or in the ompat matrix form as follows. va R + pl ia ea v b R pl i b e = + + b v R + pl i e Where L = L = L = L = L [H] a b S M L S is the armature self-indutane M is the mutual indutane R = R = R = R : armature resistane [ Ω ] a b va, vb, v : terminal phase voltage [] ia, ib, i : motor input urrent [A] ea, eb, e : motor bak emf [] p in the matrix represents d U d I d Fig. 1 BLDC Motor Control System Due to the permanent magnet mounted on the rotor, its bak emf is trapezoidal as shown in Fig.. The expression of bak emf must be modified as expressed in () (7). a ( ) φ ( θ ) ω ( ) E i a i b i b a n (4) e t = K t () 3

2 World Aademy of Siene, Engineering and Tehnology 47 8 π eb t KE t ( ) = φ θ ω ( ) π e t KE t ( ) = φ θ + ω ( ) Where K E is the bak emf onstant and ω is the mehanial speed of the rotor. (6) (7) Power supply for BLDC motor drives an be in various forms. Sinusoidal supply is typial as the standard power supply of an eletri utility, while square waves and PWM are widely used in small power appliations. In this paper, only PWM shape is used for study. It is slightly ompliated. To generate the PWM shape, it requires operation of highfrequeny swithing devies of eletroni inverters, e.g. transistor, MOSFET, et. The example of natural sampling or sinusoidal PWM wave shape is shown in Fig. 3. Fig. BLDC Motor bak emf and the motor phase urrents The permanent magnet also influenes produed torques due to the trapezoidal flux linkage. Given that K T is the torque onstant. The produed torques an be expressed as desribed in (8). ( ) / T = e i + e i + e i ω (8) E a a b b Substitute () (7) into (8), the resultant torque, T E, an be obtained by the following expressions. ( ) φ ( θ ) ( ) T t = K i t (9) a T a Tb t K π T ib t ( ) = φ θ ( ) T t K π T i t ( ) = φ θ + ( ) ( t) ( t) ( t) ( t ) E a b (1) (11) T = T + T + T (1) With the Newton s seond law of motion [], the angular motion of the rotor an be written as follows. ( t ) dω T ( t E ) TL ( t ) = J + B ω ( t ) (13) T Where : load torque [N m] L J : rotor inertia [kgm ] B : damping onstant Fig. 3 Natural sampling PWM wave form III. POWER LOSS AND EFFICIENCY CALCULATION In the BLDC motor, the power losses onsist of ore losses in the magneti ore, opper losses in the winding and mehanial losses [6]. A. Copper Losses The opper losses are I R loses. All three-phase windings must be taken into aount. Thus, the total armature opper loss ( P u ) is equated as follows. ( ) P = i + i + i R (14) u a b B. Core Losses and Mehanial Losses Mehanial losses are aused by frition (mostly in the bearings) and the dynami drag to oppose the motion of the rotor or so-alled the frition and windage losses (P fw ). Core losses (P ore ) are the open iruit losses due to hysteresis property and indued eddy urrent in the ore, whih exist as long as the exitation winding energized. To be onvenient and very useful for effiieny omputation, the ore losses, frition losses and windage losses are summed up to give rise to the no-load rotational loses (P rot ). Prot = Pore + Pfw (1) C. Effiieny The effiieny of a motor is an important performane harateristi. Although its formula is as very simple as (16), to ompute the effiieny is problem-dependent due to the assumption of power onversion in the mahine. In this paper, input and output powers are defined aordingly. B. Supply Soure 33

3 World Aademy of Siene, Engineering and Tehnology 47 8 Pout η = 1 (16) P in Pin = vaia + vbib + vi (17) 1 = + ω (18) Pout Jω TL Geneti Algorithms (GADS TOOLBOX in MATLAB ) [9] is employed to generate a set of initial random parameters. With the searhing proess, the parameters are adjusted to give response best fitting lose to the desired response in the ab referene signals. To perform the searhing properly, its objetive funtion is the key. In this paper, the objetive funtion is defined by using the power loss funtion in (14-1). I. SPEED REGULATION TECHNIQUE To regulate the motor speed at the desired level, a speed regulation sheme is proposed. Although disturbanes an be aused by several events, e.g. supply hange, sudden load hange, et, in this paper the speed regulation under mehanial load disturbanes is only our partiular study. To regulate the speed level, the proportional ontrol sheme is employed and an be summarized as shown in the blok diagram of Fig. 4. ω ref ω k p ω ω δ ω sen Fig. 4 Closed-loop speed regulation sheme To demonstrate the proposed speed regulation sheme, the proportional ontrol (P-ontroller) is introdued. As obviously found in other ontroller design problems, the proportional gain (k p ) is the key to aomplish this regulation. From Fig. 4, the Proportional ontroller output is assigned by (19). To find an appropriate value of k p, some effiient searh algorithm, suh as geneti algorithms, is employed. Where ω = k ω (19) p ω is proportional ontroller output k is gain of proportional ontroller p ω is the speed error A. Geneti Algorithms: GAs There exist many different approahes to tune ontroller parameters. The GAs is well-known [6] there exist a hundred of works employing the GAs tehnique to design the ontroller in various forms. The GAs is a stohasti searh tehnique that leads a set of population in solution spae evolved using the priniples of geneti evolution and natural seletion, alled geneti operators e.g. rossover, mutation, et. With suessive updating new generation, a set of updated solutions gradually onverges to the real solution. Beause the GAs is very popular and widely used in most researh areas [7-8] where an intelligent searh tehnique is applied, it an be summarized briefly as shown in the flowhart of Fig.. In this paper, the GAs is seleted to build up an algorithm to tune k p parameters. The proedure to perform the proposed parameter tuning is desribed as follows. First, time-domain results of the motor speed obtained by simulating the BLDC motor system in MATLAB [9] are olleted. Seond, the Fig. Flow hart of the geneti algorithm. SIMULATION RESULTS In this setion, demonstration by simulation is illustrated. A small brushless DC motor is assumed to be operated with an adjustable voltage soure inverter of the PWM type. The peak voltage magnitude is fixed at 1. For parameters of the test motor and GADS s MATLAB TOOLBOX are given below. TABLE I PARAMETERS OF THE TEST BLDC MOTOR Parameter Symbol Data Armature Indutane L 1.13 mh Armature Resistane R 3.1 Ω Rotor Inertia J kg.m Damping Constant B N.m.s/rad Bak EMF Constant K e se Torque Constant K t N.m/A 34

4 World Aademy of Siene, Engineering and Tehnology 47 8 TABLE II PARAMETERS OF GADS MATLAB TOOLBOX Total Generation Population Size 1 Stall Generation Limit Mutation rate. Crossover rate.,1 ab,36 Population Range k [ ], δ [ ], m [ 1,1] p The test ase employs a PWM voltage soure inverter to energize the BLDC motor as ommonly found for small BLDC motors in real-world appliations. To verify its responses, voltage and urrent waveforms together with mehanial rotor speed in rpm are haraterized. Due to the test of mehanial load disturbane, ontroller parameter tuning an be performed by using the GADS MATLAB TOOLBOX to find an optimal k p parameter. Fig. 6 shows solution onvergene for the k p tuning proess. To avoid too many voltage and urrent waveform presentations, only the results of the unompensated system are seleted as shown in Figs 7 1 for the motor s input voltages, the motor s urrent, the motor s bak emfs and the mehanial speed of the rotor, respetively. Similarly, Figs give responses for the ase of load disturbane event Fig. 8 Current of the BLDC motor ia ib i ea eb e Fig. 9 Trapezoidal motor bak emf 8 +%error Fig. 6 Convergene of GA solution of the k p parameter a 7 6 -%error with load without load Wm Speed (rpm) 4 3 b Fig. 1 Speed of the BLDC motor with and without load Fig. 7 PWM voltage supply soure 3

5 World Aademy of Siene, Engineering and Tehnology 47 8 a 8 +%error %error Wm b Speed (rpm) Fig. 11 PWM voltage supply soure Fig. 14 Speed of the BLDC motor with load Fig. 1 Input urrent of the BLDC motor ia ib i ea The results of the unompensated system as shown in Figs. 7 1 show that the rotor speed drops onsiderably to just over 6 rpm when the mehanial load torque disturbane is applied. To resume the rotor speed to its nominal value, the proportional gain needs to be tuned. With appropriately tuned proportional gain, the rotor speed an be regulated to operate within ±% speed error band. I. CONCLUSION In this paper, speed regulation sheme of a small brushless DC motor (BLDC motor) with trapezoidal bak-emf onsideration has been presented. The proposed ontrol strategy uses the proportional ontroller in whih the proportional gain, k p, is appropriately adjusted by using geneti algorithms. As a result, with the load torque disturbane applied to the BLDC motor operation, the rotor speed an be regulated to operate within ±% speed error band. To extend this study, thermal effet of the winding resistanes resulting from the motor urrent should be taken into aount as well as the use of different motor load models Fig. 13 Trapezoidal motor bak emf eb e REFERENCES [1] Y. S. Jeon, H. S. Mok, G. H. Choe, D. K. Kim, and J. S. Ryu, A New Simulation Model of BLDC Motor With Real Bak EMF Waveform, Computers in Power Eletronis,. COMPEL. The 7th Workshop on, pp. 17-, July. [] Lei Hao and Hamid A. Toliyat, BLDC Motor Full Speed Range Operation Inluding the Flux Weakening Region, Industry Appliations Conferene, 3. 38th IAS Annual Meeting. Conferene Reord of the, vol. 1, pp , Otober 3. [3] Joon-Ho Lee, Tae-Sung Kim, and Dong-seok Hyun, A study for Improved of Speed Response Charateristi in Four-Swith Three-Phase BLDC Motor, Industrial Eletronis Soiety, 4. IECON 4. 3th Annual Conferene of IEEE, vol., pp , November 4. [4] Quan Jiang, Chao Bi, and Abdullah, An Effetive Approah to Predit Performanes of High Speed BLDC Motor in Hard Disk Drives, Industrial Eletronis Soiety, 3. IECON '3. The 9th Annual Conferene of the IEEE, vol.3, pp. 1-1, November 3. [] Hamid A. Toliyat and Steven Campbell, DSP-Based Eletromehanial Motion Control, CRC Press LLC, Ameria, 4. [6] Seok-Myeong Jang, Han-Wook Cho, and Sang-Kyu Choi, Design and Analysis of a High-Speed Brushless DC Motor for Centrifugal 36

6 World Aademy of Siene, Engineering and Tehnology 47 8 Comprresser, Magnetis, IEEE Transations on, ol. 43, pp. 73-7, June 7. [7] S.O. Orero and M.R. Irving, Eonomi dispath of generators with prohibited operating zones: a geneti algorithm approah, Generation Transmission and Distribution, IEE Proeedings, olume 143, Nov. 1996, pp [8] D.E. Goldberg, and D. Edward, Geneti Algorithms in Searh, Optimization and Mahine Learning, Wiley, [9] The MathWorks In., Geneti Algorithms and Diret Searh TOOLBOX, CD-ROM Manual, 4. 37