Steady State Stability Analysis of a CSI-fed synchronous motor drive using Digital modeling

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1 ISSN (Online) : ISSN (Print) Steady State Stability Analysis o a CSI-ed synchronous motor drive using Digital modeling 1 *Shazia Hasan, A.B.Chattopadhyay, 3 Mohammed Abdul Jabbar, 4 Sunil Thomas 1 Electrical and Electronics Engineering Dept, BITS Pilani Dubai, Dubai Electrical and Electronics Engineering Dept, BITS Pilani Dubai, Dubai chattopadyay@dubai.bits-pilani.ac.in 3 Azbil Corporations, Dubai ajabbar.mohd@gmail.com 4 Electrical and Electronics Engineering Dept, BITS Pilani Dubai, Dubai e -mail: sunilthomas@dubai.bits-pilani.ac.in. *corresponding author, e -mail: dr.shaziahasan@gmail.com Abstract This paper develops a digital model o a current source Inverter ed three phase synchronous motor drive system rom the view point o steady state stability aspect. The motivation lies in the act that to control any electrical drive system digital controller is needed. To develop the sotware and hardware o such controller, a suitable digital model o the original drive system becomes necessary. Approach to develop the model in s-domain has been outlined and then z-transorm has been applied. Dierent aspects o the model like the stability assessment using pole-zero mapping, Jury s test, range o coeicients o characteristic equation or stability etc., have been computed leading to various graphical plots. Furthermore perturbation o machine design parameters have been modeled rom the view point o stability assessment with necessary computational results. Keywords : CSI ed Synchronous Motor, Z transorm, Jury s test, Impulse response, stability analysis. 1. Introduction Permanent magnet Synchronous Motor (PMSM) drive systems are becoming more popular due to their advantages such as utility o sel control, good eiciency and operation near to unity power actor, small inertia etc. Due to these advantages synchronous motors are also serious competitors to both dc motor drives and induction motor drives [XI].When compared to an AC induction motor, PMSM has superior advantage to achieve higher eiciency as it produces the rotor magnetic lux with permanent magnets. For this reason PMSMs are used in high end appliances and equipments that require high eiciency and reliability [XII]. PMSMs are gaining increasing popularity and demand in various areas such as 68

2 automobiles, robotics and aerospace engineering [X]. In addition, Z Source Based Permanent Magnet Synchronous Motor Drive System [IX] is also gaining importance. Design o controller or PMSM has been reported in [XIII]. PMSM has many attractive characteristics such as high-power density, high-torque-toinertia ratio, wide speed operation range, and ree rom maintenance, it is suitable or many industrial applications[vi,vii]. So ar the literature review is concerned, digital implementation o an adaptive speed regulator or a PMSM has been reported in [VIII]. However this work does not ocus on the stability analysis rom the digital domain point o view To evaluate the ield perormance accurately we need accurate digital simulation tools. Hence design o digital controllers merges to be the main area o interest or such drive systems. One disadvantage with PMSM is that or accurate analysis o the perormance o the drive systems, circuit theory loses its advantage whereas in such problem, electromagnetic ield theory becomes quite successul to analyze and then to come to a concrete conclusion. That s why analysis and control o a normal synchronous motor drive system (without PMSM) becomes advantageous. A normal synchronous motor drive system does not have variety o applications due to the eature o constant speed, but it has some critical applications in steel industries [III] in the orm o drives or big blowers and rolling mills. Without using power electronics devices, classical control o synchronous motors have not much practical importance. However, voltage source inverter (VSI) or current source inverter (CSI) can be used as a speciic driver or the synchronous motor depending on the necessity. Based on these acts, the authors o this paper have been motivated to take up the digital modeling o a classical analysis perormed in the complex requency domain or a current source inverter ed Synchronous motor drive system [I,II]. Such problem carries utmost practical importance because: (a) Due to the reduction o a problem o two variables (armature current and ield angle) to a problem o single variable(only, ield angle). (b) Non-uniormity in air-gap within the synchronous motor can be modeled with suicient accuracy. Thereore the concept in point (a) as stated above clearly indicates that inverter ed synchronous motor drive system leads to a more simpler mathematical model. Controlling o electrical machines by means o power electronics is one o the most important concerns o many engineers, whether it concerns the designing o complex control algorithms or turning on or o the control parameters. Parameter variations, error measurements and distortions need to be taken into account while designing phase. A model o control loop in discrete time domain or Z-domain needs to be obtained as it involves power electronics and as we are going to use digital controllers. Many studies have been carried out or this purpose in the past [IV,V]. Steady State Stability analysis o a CSI ed Synchronous Motor drive system with 69

3 Damper Windings in continuous time domain was done in [I,II]. The authors o [I,II]. expressed the transer unction o a Current Source ed synchronous motor drive system in a convenient manner ater developing the voltage-balance equations and torque-balance equations in time domain. The transer unction obtained in this study was the ratio o Laplace transorm o change in angle (β) between ield and armature axis to the Laplace transorm o change in load torque (T L ). The detailed mathematical analysis was presented [I,II]. The base paper used a block diagram o the CSI (current source inverter) ed Synchronous motor drive system as shown in Fig.1 [I]. Fig.1: Drive coniguration or open loop current ed synchronous motor In continuation with the investigation in [I]. the proposed analysis in the present paper targets to convert the transer unction rom continuous domain to discrete domain. Z transorm is used to analyze the transer unction in discrete domain. The main objective o this paper is to conduct a steady state stability analysis o CSI ed synchronous motor in discrete time domain and to calculate the range o stability. A detailed mathematical analysis in this direction is presented in this paper. The main advantage o having a transer unction in digital domain is that it becomes easier to design a digital controller. Since digital control oers more lexibility, it can perorm really complicated control activities keeping scope o modiication as per necessity compared to analog control which is slow to develop and diicult to make accurate designs. Furthermore such digital modeling o Synchronous motor can be used as a design tool to investigate the range o certain machine design parameters rom the view point o steady state stability phenomenon. Basically the acts can be looked upon on the ollowing lines: I there occurs some small perturbations o load torque the corresponding change in the ield angle is to be noticed. As, conceptually the ield angle is very near to our traditional concept o torque angle (or load angle), such problem leads to the Steady State stability problem o a Synchronous motor drive system. Field angle is the space angle between armature m.m. position (axis) and the direct axis(d-axis). 70

4 Based on such concept, the result and discussion section shows the complete calculation methods or range o parameter like R kd, R, R kq and L md on the basis o maintaining steady state stability perormance o the said drive system. Such analysis demands or Z-transorm applications along with application o Euler s equation involving partial derivative concepts. These detailed treatments along with appropriate numerical results have been presented in the concerned section. The next section has been presented, based on the above said philosophy. In the Problem Formulation section, or a better clarity o understanding the proposed research work has been decoupled into two parts. The irst part gives a brie outline o the modeling in s-domain with the expression or transer unction. The second part deals with the digital modeling o the said system. The digital modeling initiates with the mathematical process o the derivation or obtaining transer unction in Z-domain. The transer unction obtained in complex requency domain (s- domain) in [I,II]. did take care o air-gap saliency o the motor and that s why the machine equations were developed in d-q rame (using axis model, not phase model o the machine). Ater this, conceptually the subsequent analytical parts like computation o the range o stability, eect o unit impulse input on stability aspect etc. have been taken up in subsequent sections.. Problem Formulation: Primitive machine model o the Synchronous motor pertaining to the model shown in Fig.1 has been developed and the detailed mathematical analysis is presented in [I]..1 Problem Formulation Part-1: Development o Transer Function The torque dynamic equation o a synchronous motor can be written as, d T T J dt e (.1.1) L where, T e is electromagnetic torque in N-m T L is Load Torque in N-m is Motor speed in mechanical rad./sec. J ispolar moment o inertia o motor and load (combined) kg-m e () T s is perturbed quantity (transormed) o electromagnetic torque and the expression or it is derived rom [I] is given as 71

5 3 ( ) x1s xs x3s x 4 T ( ) e s s 3 l1s ls l3s l4 (.1.) The small change in speed equal to can be related to small change in ield angle, as given by, d( ) (.1.3) dt The negative sign in equation physically indicates a drop in speed () due to increase in ield angle (). Based on equation (.1.3), the ollowing expression can be written, d( ) d d d J J ( ) J ( ) dt dt dt dt The small-perturbation model o equation (.1.4) can be written as, d( ) Te TL J dt Combining equations (.1.4) and (.1.5), it yields (.1.4) (.1.5) d Te TL J ( ) (.1.6) dt The transormed version o eqn. (.1.6), with initial condition relaxed, comes out to be T ( s) T ( s) Js ( s) (.1.7) e L Substituting the expression or Te(s) rom eqn. (.1.) in eqn. (.1.7), it yields 3 x1s xs x3s x 4 3 l1s ls l3s l 4 ( s) Js ( s) TL ( s) (.1.8) Equation (.1.8) gives ater manipulation, a Transer Function, T(s) expressed as, where, Ts = () s l s l s l s l T () L s K s K s K s K s K s K K K Jl 1 1 Jl (.1.9) 7

6 K Jl x K Jl x K K 4 4 x 5 3 x 6 4 Once the transer unction has been developed in s-domain, the z-transer unction has been obtained using MATLAB as explained in the next section.. Problem Formulation Part- : Transer Function Development in Discrete domain The transer unction in the continuous time domain was given as [I]: Where T(s)= = K s 1 5 K l s s l K s 3 s 3 l 3 s K 4 s l 4 K s K 5 6 (..1) I. The algebraic expressions o l 1, l, l 3, l 4 and K 1, K, K 3, K 4, K 5, K 6 are available in [I] II. III. is the transormed ield angle (small perturbation). is the transormed load torque (small perturbation). Here the terminology transormed means that the mathematical tool Laplace Transorm has been applied. The polynomial coeicients (K 1 to K 6 ) [I] were calculated using the machine data present in the appendix. The values o K 1 to K 6 are as ollows. K 1 =.871 K = K 3 = K 4 = K 5 = K 6 =.419 x 10-6 The numerator polynomial coeicients were calculated using the equations and the machine data [I]. The values o l 1 to l 4 are as ollows: l 1 = 0.857, l = , l 3 = 1.615*10-3, l 4 = 1.755*10-6. The above equation in continuous orm is converted to discrete orm (i.e. converted 73

7 to z domain). The discrete domain conversion was done using MATLAB sotware with sampling time as 0.1 seconds. Z transorm was taken ater substituting the coeicient values. The transer unction obtained is represented as T(z)...1 Stability assessment using POLE-ZERO mapping (..) The stability analysis o the discrete time equation is done by plotting a pole zero map using MATLAB sotware. Fig: pole zero map o the transer unction Fig: shows the pole zero map obtained or the z-transer unction. Since all the poles lie inside the unit circle, the steady state stability is assured or the system in discrete domain. Even though pole-zero locations give a basic inormation about the system stability, investigations on the coeicients o the characteristic equation o the same system gives a more clear picture on the stability assessment. Such investigation is known as Jury's test and it is taken up in the next section or a through result and discussion... Jury's Test Jury's stability criterion is a method to analyze the stability o discrete time system using the coeicients o the characteristic equation derived rom the transer unction. Using Jury's test we can do stability analysis o a discrete time system without having to calculate the poles o the system. 74

8 Detailed mathematical analysis o jury test on the system is given below. The characteristic equation is: ( z) z 4.98z 9.931z 9.899z 4.935z (..3) The necessary and suicient conditions to satisy or the system to be considered stable are: Rule 1 I z is 1, the system output must be positive: ( 1) Rule 0 I z is -1, then the ollowing relationship must hold: n ( 1) ( 1) 0 Where n is highest power o the characteristic equation. Rule 3 The absolute value o the constant term (a 0 ) must be less than the value o the highest coeicient (a n ): a 0 < a n, where the polynomial is given as 1 3 ( z) an an 1 z anz an3z... a0 I Rule 1 Rule and Rule 3 are satisied, construct the Jury Array. Rule 4 Once the Jury Array has been ormed, all the ollowing relationships must be satisied until the end o the array: b 0 > b n-1 c 0 > c n- d 0 > d n-3 And so on until the last row o the array. I all these conditions are satisied, the system is stable. Jury array or the system is shown in Table.1. z n 75

9 Table.1: Jury's array pertaining to the stability assessment in discrete time domain Z 0 Z 1 Z Z 3 Z 4 Z x x x 1.594x x10-6 x x x x x 1.3x1 x x -6 x x x x10-8 x x x The results in Table.1, hereby assures the stability o the system. It is well known that i the coeicients o a characteristic equation are perturbed, the system may move to the unstable zone. This will give a rough indication o the degree o robustness o the system, Such treatment is taken up in the next section..3 Range o coeicients or stability: The stability ranges or coeicients were ound by changing the value o one coeicient while keeping the other coeicients constant. The stability analysis was done by plotting the pole zero maps using MATLAB or the transer unction. I the poles o the equation lie inside the unit circle then the system was concluded to be stable and i the poles lie outside the unit circle then the system was concluded to be unstable. Table represents the range o co eicient. Table : Range or the coeicients Range Original Values(stable) K 1.4 (.871) K 15.9 (0.3585) K (0.5577) K (0.0776) The original value o the coeicients is represented in the brackets. For example when K 1 is given value more than.4 the system becomes unstable. Similar analysis was done or rest o the coeicients as well. 76

10 Fig 3: pole zero map or stable system Fig 4: pole zero map or unstable system Fig 3 represents the pole zero map o the stable system when there was no change in the coeicient values. Fig 4 represents the pole zero map o unstable system. In this transer unction the K1 value was taken as.5 which clearly exceeds the maximum range which is.4. Thereore the system becomes unstable. Once the range o coeicients o the 77

11 characteristic equations are known rom the view point o stability, it becomes necessary to ind the impulse response o the system as an important part o the whole digital modeling. The analysis involving the impulse response is presented in the next section. 3. Impulse response: It is well known that an Impulse response o a system can be looked upon as Inverse Laplace Transorm o the Transer Function. Hence it is a very important analysis o a control system. For example i we consider an AVR control system that controls the output voltage o the generator, when designing a controller or this system, one has to consider worst case scenarios or conditions. For example i we consider the worst case scenario can be a lightning or a very short period o time (in microseconds ). This causes a very big voltage to be impressed at the terminals o the generator. In this case the control system must be able to secure the generator rom the worst possible scenario. In such case stability analysis using impulse response method becomes a strong tool or analysis. In the ollowing section the stability analysis o the system with input as unit impulse is perormed and subsequently a detailed mathematical analysis is represented 3.1 Unit impulse input in s-domain: With reerence to Fig. 1, there may occur so many types o variations in the magnitude o change in load torque. Due to abnormal behaviour o the mechanical coupling system or any other reason there may appear a sudden change in load torque. In other words change in load torque may be considered as a unit impulse unction and our objective in this subsection is to investigate the stability assessment o the particular system with input as δ(t) through the method o Pole-Zero mapping. Where L and L -1 are Laplace and inverse Laplace transorm operator respectively. Hence (3.1.1) (3.1.) Δß(t)= +B +C +D +E (3.1.3) Where p1,p,p3,p4,p5 are the roots o the characteristic equation and A,B,C,D,E are the coeicient values. p1= i 78

12 p= i p3= p4= p5= A= i B= i C= 0 D= E= 0 Since all the poles o the characteristic equation have negative real parts assured stability is here by assured. In this context plot o change in amplitude versus time has been presented in Fig. 5. This plot clearly indicates a decaying sinusoid having both upper envelope and lower envelope. It is also a point o interest that such plot is treated as a symmetrical wave orm leading to the major physical conclusion, i.e., the DC component is absent. Such concept appears to be automatically correct because o the ordinate o Fig.5 is a change in variable (not absolute value). Fig.5: Impulse response o the transer unction in continuous time domain 3. Unit impulse input in Z-domain: It may be noted that the concept put in section.1 was valid in continuous time domain and that is why Laplace Transorm operator was used to investigate the satiability assessment phenomena. However, as the present paper deals with the 79

13 digital modelling o the system, at this stage it becomes necessary to convert the above said philosophy in section. to discrete time domain. As a natural responsibility the use o 'Z' transorm must be involved in such case to investigate the stability phenomenon in discrete time domain. Based on such philosophy the related mathematical equations are presented as ollows: Since the Z[ δ(n)] =1, Hence Δß (z)=t(z) * 1 (3..1) Δß(n) = T(z)] (3..) Δß(n) =A δ(n)+ B + C + D + E + F (3..3) Here l 1,l,l 3,l 4,l 5 are the roots o the characteristic equation and A,B,C,D,E,F are the coeicients. l1= i l= i l3= i l4= i l5= A= B= i C= i D= i E= i F= The explanation behind the presentation o Fig.5 can be reproduced in similar lines or simulations in discrete time domain. Such approach leads to the development o Fig.6; where impulse response o transer unction is presented in discrete domain. 80

14 Fig 6: Impulse response o transer unction in time discrete domain It is natural to put a question on the act that Δß(t) or Δß(n) are considered to be variables o importance, and why not ß(t) or ß(n). The answer lies in the act that the very purpose o this research is to investigate the steady state stability o synchronous motor drive system rom various viewpoints. Moreover the concept o the steady state stability is based on the theory o small perturbations. At this stage it is elt that even though necessary analysis has been presented involving the coeicients o the characteristic equation, still some more study rom the view point o design aspects o the motor can be done. This study will be based on perturbing the real machine design parameters like R (Field winding resistance), R kd (Direct axis damper windings) etc. Due to these changes, the coeicients like K1, K etc (..1) will be perturbed and hence steady state stability o the system may get aected. Such analysis with the relevant results are presented in the next section. 3.3: Analysis o parameter perturbation Based on the stability range or the coeicients given in the table 1, it reveals that changing the parameter aects the design rom the view point o steady state stability assessment through digital modeling. To ind out which machine parameter aects the stability o the system most a method is proposed with a detailed explanation in an example below. I consider coeicient K 6 as K 6 =x 4 as in (.1.) x 4 =n 4 i s cosδβ m 4 i s sinδβ (3.3.1) 81

15 where, n 4 = c b 3 (3.3.) and c 1 = [(L d - L q )i q0 L mq i kq0 ] (3.3.3) c = [(L d - L q )i d0 + L md i kd0 + L md i 0 ] (3.3.4) = R kq (3.3.5) b 3 = R kd R (3.3.6) Let, i s cosδβ= P1 and i s sinδβ = P Thereore, (3..1) can be rewritten as x 4 = c b 3 P1 - c 1 b 3 P (3.3.7) K 6 = x 4 = R kd R R kq {[(L d - L q )i d0 + L md i kd0 + L md i 0 ] P1 - [(L d - L q )i q0 L mq i kq0 ] P } (3.3.8) Substituting the constant values given in appendix we get, K 6 = R kd R R kq {[(L d - L q )i d0 +L md i 0 ] P1 -[(L d - L q )i q0 ] P } (3.3.9) At this stage, it will be a matter o justice i the physical signiicances o variables with suice 0 (i.e. β o, i 0, i d0, i q0, i kq0, i kd0 ) are stated clearly. Basically, the theory o calculus states that Taylor s series expression o any unction becomes mathematically deined about an operating point (or quiescent point). i d0, i q0 etc are the quiescent values. Practically also, one cannot perturb a system until and unless the equilibrium state is known. We can see that K 6 is a unction o (R kd, R, R kq, L md ). Now rom the table 1 we know that the system is stable when K the system is stable. The system becomes unstable at K 6 = Thereore, dk 6 = or dk 6 = 3*10^-5. (3.3.10) Now using the Euler s dierential ormula we can write it as, dk drkd drkq dr dl (3.3.11) 6 md R R R L Kd Kq Case: 1 When we consider R Kd is changing, dk 6 drkd 0 (3.3.1) R Kd Since only R Kd is changing and the other coeicients are constant, the rest o the equation becomes zero. Now equating (3.3.10) and (3.3.1) we get 5 3*10 = drkd 0 R Kd md 8

16 { RKd R RKqP1[( Lq ) id 0 Lmd i 0 ]} { RKd R RKqP( Lq ) iq0} RKd RKd [ R RKqP1[( Lq ) id 0 Lmdi 0 R RKqP( Lq ) iq0] RKd (3.3.13) By substituting the values or the parameters given in the appendix we get R Kd = 0.37 (3.3.14) Case : When we consider R is changing, dk 6 dr 0 (3.3.15) R Since only R is changing and the other coeicients are constant, the rest o the equation becomes zero. Now equating (3.3.10) and (3.3.15) we get 5 3*10 = dr 0 R { RKdR RKqP1[( Lq ) id 0 Lmdi 0]} { RKdR RKqP( Lq ) iq0} R R ] [ RKdRKqP1[( Lq ) id 0 Lmdi 0]} { RKdRKqP( Lq ) iq0 R (3.3.16) By substituting the values or the parameters given in the appendix we get. R = 1.86*10 (3.3.17) Case 3: Now considering R Kq is changing, dk 6 drkq 0 (3.3.18) R Kq Since only R Kq is changing and the other coeicients are constant, the rest o the equation becomes zero. Now equating (3.3.10) and (3.3.18) we et 5 3*10 = drkq 0 R R Kq Kq { RKdR RKqP1[( Lq ) id 0 Lmdi 0]} { RKdR RKqP( Lq ) iq0} R ] Kq [ RKdR P1[( Lq ) id 0 Lmdi 0]} { RKdR P( Lq ) iq0 RKq (3.3.19) By substituting the values or the parameters given in the appendix we get. R Kq = (3.3.0) Case 4: Similarly considering L md is changing, 83

17 dk 6 dlmd 0 (3.3.1) L md Since only R Kq is changing and the other coeicients are constant, the rest o the equation becomes zero. Now equating (3.3.10) and (3.3.1) 5 3*10 = dlmd 0 L L md md { RKdR RKqP1[( Lq ) id 0 Lmdi 0]} { RKdR RKqP( Lq ) iq0} L =[ Kd Kq md md md [ R R R P1L i 0 ] L ] (3.3.) By substituting the values or the parameters given in the appendix we get. L md = (3.3.3) The physical interpretation o such change can be explained as ollows, with reerence to (3.3.3), the speciied value o the change in L md basically indicates a change in direct-axis air-gap. I the said value o perturbation in L md is not acceptable, then the direct axis air-gap will have to be changed and it will in turn aect the input power actor o the drive system. I, the particular value o change in input power actor is not allowed in practice, then ield winding current o the motor will have to be adjusted. 4. Conclusion This research paper concludes the ollowing salient points:- Assessment o steady state stability has been perormed in discrete time domain, by the method o "pole-zero" mapping and " Jury's Test". The result o both the methods do not conlict. As changing the coeicients o characteristic equation shows a direct impact on the stability o the system, a new view point based on machine design aspect is developed, which is written under the next point. The machine design parameters ( R, R Kd, L md etc.) can be perturbed. This aspects has been studied using necessary mathematical ormulations based on " Partial Dierentiation". It was observed that a minimum amount o perturbation on ield winding resistance aects the stability most. This is due to the act that a change in ield winding resistance changes the ield current. As a result the lux linkage changes and ultimately the electromagnetic torque is aected. I this eect happens in a negative direction, stability is aected and the motor may inally cease to take the load. 84

18 Reerences I. A.B. Chattopadhyay, Sunil Thomas and Ruchira Chatterjee, Analysis o Steady State Stability o a CSI Fed Synchronous Motor Drive System with Damper Windings included, Trends in Applied Sciences Research, 011. II. III. A.B. Chattopadhyay and Sunil Thomas, 014. Modeling and Simulation o Current Source Inverter Fed Synchronous Motor in Complex Frequency Domain Taking the Transition Zone From Induction Motor to Synchronous Motor Mode into Account. Research Journal o Applied Sciences, Engineering and Technology, 7(6): DOI: /rjaset A.K Paul, I Banerjee, B K Santra and N Neogi Application o AC motors and drives in Steel Industries Fiteenth National Power Systems Conerence (NPSC), IIT Bombay, December 008. IV. D.M. Van de Sype, Kleinsignaalmodellering van digitaal gestuurde schakelende energie-omzetters PhD dissertation, Ghent University, 004 V. D.M. Van de Sype, K. De Gussem, F. De Belie, A. Van den Bossche and J. Melkebeek, Small-signal z-domain analysis o digitally controlled converters IEEE Transactions on Power Electronics, Vol. 1, Nr., pp , VI. VII. F. J. Lin, Y. T. Liu and W. A. Yu, "Power Perturbation Based MTPA With an Online Tuning Speed Controller or an IPMSM Drive System," in IEEE Transactions on Industrial Electronics, vol. 65, no. 5, pp , May 018.doi: /TIE F. J. Lin, Y. C. Hung, J. M. Chen, C. M. Yeh, "Sensorless IPMSM drive system using saliency back-emf-based intelligent torque observer with MTPA control", IEEE Trans. Ind. Inormat., vol. 10, no., pp , May,014. VIII. H. H. Choi, N. T. T. Vu, J. W. Jung, "Digital implementation o an adaptive speed regulator or a PMSM", IEEE Trans. Power Electron., vol. 6, no. 1, pp. 3-8, Jan IX. H. Mahmoudi, M. Aleenejad and R. Ahmadi, "Modulated Model Predictive Control or a Z Source Based Permanent Magnet Synchronous Motor Drive System," in IEEE Transactions on Industrial Electronics, vol. PP, no. 99,pp.1-1.doi: /TIE X. L. Samaramayake and Y.K. Chin, "Speed synchronized control opermanent magnet synchronous motors with ieldweakening,"international Conerence on Power and Energy Systems, EuroPES003, vol. 3, pp , Sep

19 XI. Pillay, P and Krishnan. R Application Characteristics o Permanent Magnet Synchronous and Btushless dc Motors or Servo Wives, IEEE Trans on Ind Applications, Vol 7, pp , 1991 XII. Shiyoung Lee, Byeong-Mun Song, Tae-Hyun Won Evaluation o Sotware Conigurable Digital Controller or the Permanent Magnet Synchronous Motor using Filed oriented Control,4nd South Eastern Symposium on System Theory University o Texas at Tyler Tyler, TX, USA, March 7-9, 010. XIII. Yang Zheng, Z. Wang and Jianzhong Zhang, "Research o harmonics and circulating current suppression in paralleled inverters ed permanent magnet synchronous motor drive system," 013 International Conerence on Electrical Machines and Systems (ICEMS), Busan, 013, pp