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The flux path in this model is similar to the field circuit and the torque path is similar to the armature circuit of a separately excited DC motor. As it is normally done in a DC motor, the flux level in the machine should be maintained at a maximum level depending on the AC voltage avail- able and the Operating flux density of the motor. It should be maximum up to the base speed, above which it may be weakened. The speed control can be imposed on i” and the flux control on i“. When reference values for i“ and in! are calculated, reference for the field oriented quantities um and use are established. CHAPTER III CONTROL SCHEME The control scheme consists of speed, torque and flux controllers. The flux controller, which operates on the diffirence between the rotor flux refer- ence and the actual rotor flux, sets the reference for i“, which is denoted by id". Similarly, the speed error (m,’-m,) serves as the input to the spud controller, which sets the reference for the torque. This reference value when . C divided by the magnitude of the actual flux vector gives 1,4 . To achieve a fast response drive system, the corresponding voltage has to be impressed on the motor. The reference voltage, a“. and uw‘, can be obtained from (11) and (13), which define the relation between the stator current and the terminal voltage in the field coordinates. Since these equations include motor-parameters, they will produce errors because of the variation of the motor-parameters due to the temperature and the accuracy with which they are measured. Hence, a closed-loop control sys- tem is required for i“ and isq , which is achieved by the flux controller and torque controller as shown in Figure 4. Once the manipulated values a“. and “sq. are determined, they are to be properly positioned in the stationary reference frame. Equation (14) defines the desired slip. Actual slip is calculated with the indirect rotor EMF method discussed in the next section. The output of the flux controller gives the incremental value Ap of p, which, when added to the actual flux position p, gives the desired position, p0 = p + Ap, of the flux vector. Knowing the desired position of the flux vector, the terminal voltage of the stationary reference can be calculated. This is then converted to the three phase quantities serving as the reference voltage for the inverter (see Chapter VI). 10 cur—flu):— e‘szn_ see. :VT emaoa mm m Statues Sketch 8398.300 .V mkfiuwk Lozocvcou Bowen 11 CHAPTER IV INDIRECT ROTOR EMF AND SLIP CALCULATION FOR SPEED SENSORLESS SYSTEM 4.1 Diffirential Equations in Stationary Reference Frame An improved method of computing the actual slip was developed [2], based on the difbrential equations describing the machine behavior in the sta- tionary reference frame. As state variables are used the flux components in the stator fixed rec- tangular coordinates ((1,8), while the currents in these coordinates are auxili- ary variables. Based on the corresponding circuits (Figure.5), the state equa- tions are: d . ‘d—t‘psa = um - Rs‘sa ’ (16) d . E¢ra = -erra - cur¢rB 9 (17) d . E¢rfl = -Rr’rfl + mr¢ra ’ (18) d . ' This = use - Rs'sa- (19) 12 1'2 Rs Ls-Lo _ L7”— L" R” 1m Um LO (“’7' 94-5 Rs Ls-Lo 1"" L0 Rr in? m—m—4— LO <15 0), 94-“ “—255 an Figure 5. Induction motor equivalent circuit in the rotor flux fixed frame. In matrix form Pqu’sa- [$4,141.1 1 qu’ra- Lo¢sa [,er 4» L02 Lr¢sfl - Lo¢rfl ’ LLs¢rB - Lo¢sB. i= k¢ =(L)-1¢ = (20) where i = (ism irw is B! irB)T 9 ¢ = (¢SG’ ¢ra9 ¢3Bo ¢rB)T 9 13 Taking derivatives ¢ra 9A: + ¢rfl ¢rB = 0 - Then, . ¢,B . a = — r , 28 ¢r ¢ra¢ B ( ) and ° ‘pra ' r = __ r . 29 ¢ 8 ¢rfl¢ a ( ) Substituting (29) in (25) gives -q’r 2 ' ¢ °‘ ¢,.,- 4». m m— 'p 2 2 ¢ra + ¢r8 e. = __ . 30 mp ( ) Substituting (28) in (25) gives 2 ¢rfl ¢ra +11% ¢r8 (I): ’0 ¢ra2+ (prflz = fl . (31) 4M: From (30) and (31) and with slowly changing flux 4, z .L¢' (32) rd 0) r8 v and ¢rB :: -—ml¢ra - (33) 14 and _ 1 . Lr -Lo Lr _Lo L 1= dt . (L,L,+L,,2) “8 {[-L0 L3 ] {-Lo L. 4.2 Slip Calculation from Flux The torque expressed in rotor quantities is Td = "p ara¢r8 _ irB¢ra> (21) where np is the number Of pole pairs and to, = m— a), . (22) Substituting (17) , (19) ,and (22) in (21) gives Td = -;L [(0), - (0) (¢ra2+ ¢r82) + (¢rB ¢ra— ¢ra¢rfi)] and Rer _ ¢r8¢ra- (pita-(”8 rem»: + (p.52) on} + 4432) ' e-w= Calculating o) from Figure 2 gives p = tan’l-qiE , ¢ra 15 and 33 = m: 4». M- m 4». w h£+mfi ' Finally 1 Td = up 0.), (R X¢ra2+ ¢r82) 9 T4 Rr up (¢ra2 + ¢rflz) ’ (03: iii-B ¢ra " ¢ia $1 (I)2 2 2 . ra. + ¢r8 The torque also can be expressed by stator currents and rotor flux as L0 . . Td = "p T(¢ra ‘sB _' ¢rfl ‘30) - 7 Substituting (26) in (24) yields Lo ¢ra isfl - ¢rflisa 3 = —R' 2 2 ' L7 (¢ra + ¢rfl ) 4.3 Rotor EMF Method to Calculate Slip Using Stator Components (23) (24) (25) (26) (27) Equation (27) is not adequate to calculate slip because it contains rotor components which are not measurable. Instead we derive the slip equation using stator current components which are measurable. A useful expression is obtained if |¢,2I = ¢,a2+ 9,132 can be made constant, or, only changing slowly as ¢m2 + ¢,32= constant . 16 4);“ , 9:113 can be interpreted as the components of the rotor EMF in the (a , B) frame. Substituting (32),(33) in (27) yields flew (”fin O) = R Lo 0) S r L’ ¢ra2+ ¢rflz of L0 m(¢rBisB+ ¢raisa) — R, L . 2 . 2 . ' (bra + ¢rB So, L e i + e i msszr 0 r8 :8 rd sa. (34) Lr em 4' e33 where (but: era and ¢r8 = erfl ° The rotor EMF e"IL , e,5 can be Obtained from measured stator quantities and Figure 6 shows an implementation of the following equations: L . 1 di era = L—r(u3a - R3 Isa - k—ll' d:a) 9 (35) 0 L . 1 di e,B= 7:04,, - R, 1,5 km d’t" . (36) where 17 5533: SE do 83339.8 .9 23. .1 La 18 Equations (34), (35) and (36) are the key equations for calculating the actual slip from measured stator quantities instead as the diffirence between (uando),. By introducing the rotor EMF method, it is possible to eliminate the speed sensor which is used to measure the rotor speed to, . 4.4 Analysis of Transient Rotor Flux Calculation In the previous section, we assumed that l¢,2| = constant, which is suitable for the steady state. Now let us consider rotor flux in transient state. From (7) we get t...(:)= Lm+ (Renews (7) = L(t)+ ffttmit (37) where (l+o,)Lo = L, . We can define in) = L,r.(r)ei‘+L.L(:) L, . . = Lotta )+L—1.(t)e") . (38) Substituting (37) in (38) gives SLO) = LAW“). (39) 19 and Substituting (39) in (13) and (14) gives ‘14) . Tr'j' + (pr = Lo‘sd ’ and (14) (39) (40) From equations (39) and (40) we can calculate rotor flux 4), and slip fre- quency to, during the transient The estimation flux diagram for the rotor flux and slip frequency is shown in Figure 7. Jae—l: e E is . q I Lo Tlr Figure 7. A diagram for the estimation of the rotor flux and slip frequency. CHAPTER V COMPENSATION OF FLUX DEVIATION 5.1 Voltage Control Method If the flux 4)", is kept constant and the flux it)", kept zero, the torque is generated without delay in proportion to the current component i“, . How- ever, there will be a flux deviation due to the various disturbances. Therefore, it is important to find a control measure which makes 6,4 zero. This method is denoted. as 4),, zero control (see 5.2). The voltage equation of a induction machine [4] is given by the following formula in rectangular d-q coordinates with stator angular frequency to . From equation (3) we get u,eiw = R,i,ej“+L,7dt-(i,ej“)+Lo-%(i,ej°°‘) . di . di . . =R,i_,e’“+L,7’-+jtuL,i,e1“+L07'el”+jcuLoi,eJ‘”. Dividing by ej‘" . dis . 41} u, = R313 + L, 7 + jWLSls + L0 7 + ijoi, . (42) From equation (4) we get 0 = mam+L.§-a.eW>+L.-§,—a.eim> . jto,t dir jm,t . . joit dis jtu,t . . jm,t = R,z,e +L’7e +1 (u,t,L,e +L°78 +jm,Loz,e . Dividing by ej m" and arranging yields 20 21 dis . . . dir . . 0= Lo 7+ 1 wsLot, + R, t, + L, 7+ 1 (03L,t,. (43) Where i, = i“ + jig, i, = i", + 17,, , u..(t)=u,e"°°‘. 1,(:)=i,ei°°', j,(:)=i,e"°°", e540,: , and 03380-0») is a slip angular frequency . From (43) we get dis dir . . . . . Lo d! + Lr dt = "Rr tr _ J msLo's _' J (0er5 R R = T'Loi, _ (L—' + jog) (Lot, + L,i,). (44) r r Since ¢r = Lois + Lrir 9 (45) equation (44) can be expressed as d . R . E¢,=L—'L,t,-(L'+)m,)¢,. (46) r r Substituting (45) and (46) in (42) and arranging yields 2 di 2 L 2 (L— ") ’ =[-(R +-—°—R )-J'c0(L- ° )1: S L, d: S er f S L, S 22 0 Lo . + [14,2 Rr-t1(m_ms)]¢r + “S ' SO, a dis ' . ’ . Rf, . L0 L dt=[-(R,+R,)-joi.]t,+(Lo -j(0,Lr)¢,+u, (47) where . L 2 R, = [:2 R, , (0, = (D- 0): t and , L,2 L = L - S L7 The reference for the inverter output voltage is calculated from (48), assuming that the stator voltage components u“ and an are controlled pro- portionally to the respective reference u“. and u”. by the PWM inverter. 14,4 = a; = R, 1,, - mL' in + AV (48) But 14,, can be calculated as the real part of (47) I . I. r L0 I did usd = (R3+Rr )lsd'ai‘ 'sq- L ¢rd-mr-E—¢rq+l‘ —dt- - (49) o r Since the output of the flux controller, i“, is kept constant with some gain, , di L $31 in (49) is relatively small compared to other terms and can be neglected while on, = Lei“. Equation (49) then becomes I L .. u,d=R,i,d-cuLi,q-ta,Z°—¢,q. (50) f 23 Comparing (48) with (50) gives La AV = -0), z— ¢rq . (51) Equation (51) means that 4),, can be estimated from AV to which it is pro- portional. Furthermore, since 9n, must be zero, AV nwd only to be zero. 5.2 9n; Zero Control Taking (46) into consideration, the sign of 9n; varies as shown by equa- tion of (51) and (52). From (46) we get Lo isq-¢rq-Tr 0), ¢rd=09 ¢rq = Lo isq - Tr (I), ¢rd ' where T, = g. 80, 9n; 2 0 when {is 131:", , (52) 4),, < 0 when 23> r3)", . (53) Therefore, it is possible to keep the flux 4% zero by controlling the value to of 7%, through AV. In practice, this could be done controlling i,q sq ‘. (equivalent to u”) or to, (equivalent to to). 24 The method in detail (see Figure 8) is as follows: L (,3- - o ) is zero, no Au“, will be fed back to u, ' and if ‘34 Tr¢rd q L (70),— - o ) is not zero, Au”, the diffirence between 23— and a , will I” T,¢,d ‘34 Tr¢rd be fed back to u” . ___fi‘*’s T l isq inverter ‘4' LP 0 Rr ' usq LO Figure 8. Compensation for flux deviation. CHAPTER VI FLUX POSITION CONTROLLER AND COORDINATE TRANSFORMATION 6.1 Flux Position Controller Since we have analyzed the control scheme in d-q coordinates, we have to control the rotor flux position. The position of the voltage components in the stationary reference frame to establish the demanded field components is quite import ant . The acceleration or deceleration of the flux vector depends on the slip reference (demanded slip) and actual slip calculated from (34). The slip refer- ence is calculated from (14), repeated here for convenience. to, = -i‘!— (14) The slip error quantity (0,. — a), is the error in the angular speed at any instant required for the flux vector to attain the desired position. The integral of the error quantity gives the required displacement of the flux vector. This is done by the flux position controller, which is basically a PI con- troller with small gain. The output of the controller gives Ap, the increment of p. The desired position of the flux vector p, is then given by p0 = p + Ap and a diagram is shown in Figure 9. 25 26' flux position tmtroller _°’_s;‘§_ l/l AP ’3? >100 Figure 9. A diagram of flux position controller. 6.2 Coordinate Transformation From the d and q-axis components of the stator voltage and using the following inverse coordinate transformation, the voltage components are obtained as Vsa _ OOSOO -mpo Vsd. _ . . . (54) V38 smpo cospo V” The three phase quantities are obtained from V,“ , VsB by 2-axis to 3-axis transformation as V“ [.11 v V32 = ‘2— V33 —1 - _2 leo '<<:' ‘°‘ . 55 ,3] . . 4N ml NI 27 These three phase quantites save as the reference voltage to the inverter. The inverter was modeled as a small delay element, which can be ignored con- sidering the whole dynamic response of the system. Hence, the reference vol- tage saves as the input to the motor. For the 2-axis model of the motor, the three phase quantities are con- verted to two phase quantities through the following transformation as 0 VS 1 1.91 V: 2 - (56) 3—3; —'3-: V33 CHAPTER VII PI Controller The controllas used in this scheme are the proportional and integral(PI) type. The equation of a continuous PI controller is written as =_Y_$_i= i H(s) E(S) 190+“) 1- =R, (1+ :3 1). where t is the time constant, and K c is the gain. (57) Using the simple rectangular approximation for s“ with T, as a sampling time, we get 28 (58) 29 where s _ 1- z‘1 T: Then, K _. .: :1: - where. = 1' 1:+ T, ° So, KC Y(z)(z— 1): 7(2- 0015(2). Taking invase z transform yields K y(n)= y(n-l)+ 7" e(n)- K, e(n-l). (59) (60) (61) CHAPTER VIII GATE PULSE GENERATION 8.1 Pulse Width Modulation for 3-Phase Inverter The genaation of the gate pulse signals can be divided into two parts. One is to determine the firing time of each transistor and the otha is to gen— erate the gate pulse at some determined time. The PWM signal is generated as a three-phase square signal as shown in Figure 10, by comparing a three-phase sine wave signal (modulation signal) to a triangular wave signal (carria signal). Figure 10. The principle of PWM signal genaation. 30 31 8.2 Genaation of Pulses from Frequency, Magnitude and Phase Angle The modulation signal corresponds to the reference values of the voltage frequency f , magnitude u,, and phase angle Ap. The width of the square signal changes at each crossing point. This relation is u, = \] (u,,")2+ (u,,,,‘)2 and 1 “84 “3d Ap = tan’ When a microprocessor is used for the genaation of the firing pulse, it reads the values of p,u, and Ap from memory at the controlla ['7]. The instantane- ous form of the modulation signal Va is expressed as V, = u, sin(p+ Ap). The powa converta performs a switching Opaation in response to the gate pulse signals. Consider the three-phase bridge invata in shown Figure 11. It has three parallel branches, each corresponding to one output phase and consisting of two fully controlled semiconductor power switches and two freewheeling diodes. At any instant time, one of the switches in each branch must be ON while the otha must be OFF [8]. By this condition, we can define logic states as follows: 32‘ if Si is OFF and Si ’ is ON, the logic state of this branch is 0 for i=A ,B ,C . The logic state of the whole inverter can be designated by a three bit binary number abc . For example, logic state 6 corresponds to (abc)6 = 110, i.e., thyristor SA , SB and SC’ are in ON and SA ’, SB', and SC are in OFF. The line to neutral and line to line voltages of the inverter can be expressed in terms of the logic variables and supply voltage 5,, as Ed VXN = 7(2x—y-z) (62) and Wu = 5.: (1'1) (63) where X andY denote any two of A, B and C phases and x,y and z is a corresponding sequence of a,b and c logic variables. For example, a = 1, b =1andc = Ointhecase(abc)6. 33 “T is sea 2: self 2: + + +' Van Vb Vcn : Vab - + Vbc " T Vca | Figure 11. Three phase bridge inverter. 8.3 Harmonics Due to PWM The PWM signals a(ox ), b(ar) and c(cot) of the switching function as well as the corresponding sequence of logic states of the inverta are shown in Figure 12. The 21: output voltage paiod is divided into N equal switching intavals Act. The central angle of nth switching intaval is designated by (1,, . The switching angles are denoted by Oil, for switch ON angles and 002,, for switch OFF angles. Duration of individual logic states are designated by A, . 34 a, q, '5 q" El : I ll—l— l t l I (A) I (A) 4 I I I “it I "81 “i i | i a)” “31" b(wt) 5 E— : . . Wt“ : )i i h, I chat) | | : ‘ low: I state I I I l I l l I II A’ ll ll ll 1 Figure 12. Switching function of a PWM invata. The ratio of a switching function pulse width to the switching intaval is refared to as switching ratio given by on-“ (X) t.“’= a” 1” (64) (X) = 7 (or) A“ for a,—-A7a-Stlr5an+A29-. Since a close relationship between the harmonic spectrum of 70!) and that of invater output voltage can be expected, a strong adjustable funda- mental of 10‘ >(ar) is desirable 35 Obviously, the switching ratio am only assume values between 0 and 1. Hence, the simplest switching function is as follows and is shown to Figure 13. Figure 13. PWM configuration. Hence, Aa . x = (V u SING/z") . 4V”, ”" " Therefore, at), - a1, = Aa- A“ (v,,, u, sinaln)—x 4V,” At: A . 2 + 2V,” u,slna,, where u! m -- V,” and u, Sinai, + u, 8111(1)" . 2 = u, smut,I . Finally, 7,,(A)=-;-(l+rn sinan), l . 2 Yn(B)='E(]-+m smug-3%)) . and Yn(c)=%(1+m sin(a.--§u)). (65) (66) (67) (68) (69) I (70) 37' where m denotes magnitude control ratio (0 < m < l) . In practice, it is sufficient to only modulate the switching function a(tut) according to (68) and shift the resulting pattern of switching angles by -§-1t and gut to obtain b(ox) and C((nt ), whose switching ratios satisfy (69) and (70) respectively. Using (64), the switch-ON and switch-OFF angles can be calculated as Act a1..“’=a.-v.°”2 (71) and %“’=a.+v.°"329- (72) whae {Wig}, a.=(n--)Aa. and n = 1,2,3, ........... ,N-1,N. We can then express a (cut) , b(tnt) and C(01) by using Fourier Saies as a(cut)= a0 + Zak cosktnt + bk sin/cox , k=1 2: _ 1 (A) de— ELY» 40‘ l m 1 = _ + — . _- _ , , . 41: L d a 4n gsmad 0t 2 (73) l 21: um = — 9‘) skatda N-noak 1! I)?" CO 12' m u = — + _ . = a 4 2“lgcoskada 2n£smacoskada O (7) Ian lim = — (A) . k d Nae-bk “£7" 8m (1 a 12:: 21: = —jsinkada+ -m-Isinasinkad0t, 2n 0 21: o and £5. 2 ifk=l b“ = 0 otherwise ‘ (75) Hence, the higha number of pulses N attenuates the Iowa-order harmon- ics of the invata switching functions while the fundamental approaches the value of half of the control ratio rn . Also, by selecting N as a multiple of 3, the triplen frequency-related currents can be avoided in the line. If, as assumed 2 4 a (08 )=b(0¥- 3'1!)=C(01-3-1t) then, the pa unit harmonic spectrum of line to neutral inverter voltage differs from that of the switching function. The magnitude of the remaining voltage harmonic is equal to those of the switching function, times the supply voltage 5,. The line to line voltages have the same harmonics as the line to neutral voltage only )I 3 times greater in magnitude. 39 Hence, in the described PWM scheme, the r.m.s. values of fundamentals line to neutral and line to line voltage are ‘1 2 VXN(nns) = m —'4 Ed (76) and )l 6 VXY(rrns)= m TEd - (77) Equations (76) and (77) show that the fundamental of the invater out- put voltage can be linearly controlled by adjusting the control ratio rn . The length of A ,- changes with the magnitude control ratio m but the sequence of A ,- (10gic state intaval) remains unvaried. 8.4 Hardware Implementation The basic hardware configuration of the PWM signal genaation circuit is shown Figure 14. 40 ””M I CRDPRDCESSDR I: i I < SYSTEM BUS "' W Eugfi‘ Rm ‘ ”—ng 8 RUM , 1,0 , CLOCK PULSE IIIUU'U'L Figure 14. Hardware configuration of PWM generation. We define time intervals of logic states as A- -= —1— (78) 1 21¢ ° Output bit signals a , b and c represent the logic variables of the inverter. A programmable counter is used to generate a pulse of width 1:1" To produce switching functions a(o)t) , b(cor) and c(tut) appropriate three bit numbers (abc) are loaded in the I/O device. Counter—l generates a pulse of width 1:,- while counter-Z is being programmed for the next pulse. Counter-3, activated by the clock pulse generator, serves as fre- quency divider to produce the clock signal for counter-l and counter-2. 41 When the interval tj has elapsed, the microprocessor is interrupted by the counter. The interrupt service routine sends new data to the proper counter and output device. The microprocessor reads input f, m and p from common memory. If f and/or m are found to have changed, a next look-up table of logic states (abc)2 and time interval 1:,- is computed. Firstly, the number N of the pulses per period is deter- mined according to an assumed N versus f relation. Secondly, time intervals cal- culated from (78) are assigned to successive logic states in an appropriate sequence. Until the next look-up table is computed, the present table is used in interrupt service routine. Then, the next table replaces the present one. The phase control is activated by a change of the reference phase angle p and causes the program to branch to the pertinent location in the look-up table. CHAPTER VIIII RESULTS AND CONCLUSIONS 9.1 Simulation Results A number of numerical experiments were conducted to evaluate the per- formance of the controlla. The program was written in Fortran and run on a VAX 8600 computa. Also motor parametas (Table l), the flow chart of the controlla (Figure 15) and a flow chart of the main program (Figure 16) are shown. Tablel Motor Parameters P, 5 HP f ,, 60 Hz Vn 220 V N, 1800 RPM L, 0.0704 H L, 0.0718 H L, 0.0675 0, 0.0421 6, 0.0631 0 0.0973 R, 0.4440 R, 0.2749 42 43 cacaoga c_o: wry mo wccru ao.m .m_ mc:o_m < .- < D "Hr nu» 44(0 liq iii ‘ l_. mead TS _ w a _4nmeznu_ _ ewe 44 9 766051 4 2 90529‘ 1 09952‘ - 816238‘ .2 67700 I I I I I I I I I fl 1 00100 1 39870 1 79690 2 19910 2 59180 2 98950 3 38720 3 78990 9 18260 9 58030 9 97800 1. me 1:19.40 Speed reversal of flux compensation from 377 to ~37? rad/sec 60 9.2 CONCLUSIONS A field-oriented control (vector control) scheme for an induction machine without speed sensor has been studied along with a PWM method for a three phase inverter. The proposed drive system allows a high performance speed control that uses only current sensors, thereby eliminating the speed sensor. In this scheme, deviation of the rotor flux influences the stability and accuracy of the control system. For this reason, compensation for flux devia- tion was developed. Its effictiveness was verified by digital simulations. The simulation results verified that the speed and torque control is performed with good dynamic and robust state REFERENCES 1. W.Leonhard and C.Norby, "Field-oriented control of a standard AC motor using microprocessor", [BEE Trans.on IAS.,vol.IA-16, No.2, 1980, pp.186-l92. 2. R.Joetten and G.Maeder, "Control methods for good dynamic performance induction motor drive based on current and voltage as measured quantities", IEEE Trans.IA-19, No.3, 1983, pp356-362. 3. S.Sathikumar and J.Vithayathil, "Digital simulation of field-oriented con- trol of induction machine", IEEE Trans. on Industrial Electronics,vol.IE- 31,No.2,1984. 4. T.Okuyama,N.Fujimoto,T.Matsui and Y.Kubota, "A high performance speed control scheme of induction motor without speed and voltage sensors",Conference Record, IEEE-IAS. Annual meeting, 1986, pp.106-111. 5. W.Leonard, "Control of electrical drive", Springer-Verlag, 1985. 6. K.Lizuka,H.Uzuhashi,M.Kano,T.Endo and K.Mohri, "Microcomputer con- trol of sensorless brushless motor", IEEE Trans.on IAS, vol.IA-21, No.4, 1985. 7. K.Kubo,M.Watanabe,T.Ohmae and K.Kamiyama, "A fully digitalized speed regulator using multi-microprocessor system for induction motor drive", IEEE Trans. IA-21, NO.4, 1985. 8. S.Legowski and A.M.Trzynadlowski, "Incremental method of pulse width modulation for three phase inverters", Conf. Rec. 1986 Annual meeting. IEEE IAS. ”593-600, 1986. 9. B.K.Bose and H.A.Sutherland, "A high performance pulse width modulator for an inverter-fed drive system using a microcomputer", IEEE, Trans.IA-19, No.2, 1983. 61 62 10. Y.H.Kim and M.Ehsani, "An algebraic algorithm for microcomputer based(direct) inverter pulse width modulation", Conf. Rec. Annual meeting, IEEE, IAS. pp.586-592, 1986. ll. I.Boldin and S.A.Nasar, "Electrical machine dynamics", Macmillan pub- lishing company, 1986. 12. R.A.Pearman, "Power electronics", Reston publishing company, Inc. 1980. 13. B.H.Bose, "Power electronics and AC drives", Prentice Hall, 1986. 14. Benjamin.C.Kuo, "Automatic control systems", 5th edition, Prentice Hall, 198?. "Illllllllilfllllllfll‘lllfllfi