Tài liệu High Performance Driver P3 doc

HIGH PERFORMANCE DRIVES ---------------------------------------------------------------------------------------------------------------------------------------  E Levi, 2001 41 3. ROTOR FLUX ORIENTED CONTROL OF INDUCTION MACHINES 3.1. FIELD ORIENTATION POSSIBILITIES IN INDUCTION MACHINES Decoupled flux and torque control of an induction machine can be realised by control of instantaneous phase current values, which in turn can be achieved by stator current space vector magnitude and position control with respect to the chosen flux space vector. As already discussed in the Section 2.3.3., a vector is composed of appropriate d-q axis components, d-axis component being the real part and q- axis component being the imaginary part of the vector (i.e. d-q axes are mutually perpendicular, as already emphasised). Stator current space vector can be orientated with respect to space vector of rotor flux, stator flux or air gap flux and machine can be fed from either voltage or current type source. Comparative analysis of different orientation possibilities reveals the main reasons for prevailing application of current-fed induction machines with rotor flux oriented control. If the decoupled flux and torque control are to be achieved in an induction machine by means of either stator flux or air gap flux oriented control, an appropriate decoupling circuit has to be inserted in the control part of the system, regardless of whether the machine is fed from a voltage or a current type source. A similar conclusion is valid for a voltage-fed induction machine with rotor flux oriented control. Orientation along air gap flux space vector has been a rarely discussed and applied orientation possibility until recently. Orientation along stator flux remained as well beyond the scope of interest for a long time. Recent advancement in development of microprocessors and DSPs enables however significantly easier implementation of decoupling circuits. As a consequence, an increase in amount of research devoted to these two orientation methods has occurred recently, with emphasis on orientation along stator flux. If the flux space vector is to be estimated from measured values of stator currents and stator voltage, then the stator flux can be estimated more accurately than the rotor flux, due to inevitable parameter variation effects in the machine. This is exactly the vector control scheme where orientation along stator flux is suggested to be a better solution than the rotor flux orientation. If the machine is of such a design that rotor slots are closed, orientation along stator flux becomes especially advantageous. The variation of rotor leakage inductance in machines with closed rotor slots can be very significant and can cause even static instability if rotor flux oriented control is applied. More specifically, pull-out torque, which is theoretically infinite in the absence of parameter variations, is a function of rotor leakage inductance and if it is reduced to such a level that maximum allowed torque exceeds its value, static instability will result. On the contrary, with orientation along stator flux pull-out torque always has finite value which is however not affected by rotor leakage inductance variations. Therefore stability problems can be avoided by appropriate design of the torque controller limit. Very much the same discussion and conclusions apply for the field-weakening region as well. If there are no parameter variations, orientation along stator flux with appropriate decoupling circuit can provide the same quality of dynamic response as the rotor flux oriented control. However, if there are parameter variations, stator flux oriented control becomes superior. All the previous remarks are restricted to the case when flux estimation is performed by means of measured stator voltages and currents. The most advanced achievement in field oriented control of induction machines is the universal vector controller (UFO), which enables realisation of orientations along stator, air gap and rotor flux. Different orientation types and appropriate decoupling circuits are available. Universal vector controller enables combined application of indirect and direct vector control methods with orientations along rotor, air gap and stator flux in different operating regimes. A possible combination suggested consists of indirect rotor flux oriented control in low speed region and direct stator flux oriented control in high speed and field-weakening region (the notions of indirect and direct vector control will be explained shortly). HIGH PERFORMANCE DRIVES ---------------------------------------------------------------------------------------------------------------------------------------  E Levi, 2001 42 Wider application of universal vector controllers with combined orientation schemes is however not likely to take place in the near future. Of far the greatest importance is still rotor flux oriented control of a current-fed induction machine, as it offers both theoretically ideal dynamic response and the simplest realisation. This vector control scheme has become during the last five years an industrial standard and there is at present a variety of commercially available drives that utilise the principle of indirect rotor flux oriented control of a current-fed induction machine. Notions of indirect and direct vector control were introduced in the preceding paragraph and they ask for further clarification. These terms are related to the way in which orientation is achieved but they are not related to any specific choice of flux along which orientation is performed. Different subdivisions of vector control techniques on direct and indirect methods are available. As will be shown in the next Section, vector control requires precise knowledge of the instantaneous angular position of the selected flux space vector. Conventional approach groups together as direct methods all the methods which utilise some measured data related to either terminal or magnetic variables (air gap flux, stator currents and voltages) for flux space vector position calculation. In the sense of this definition, indirect vector control schemes perform calculation of flux space vector position without any measurement of electromagnetic variables. Commanded axis currents enable calculation of angular slip frequency which is later on used together with measured rotor speed (position) to produce an estimate of flux position. Indirect and direct vector control defined in this way are often termed as feed-forward and feedback control, respectively. Notions of indirect and direct field oriented control will be used according to this subdivision. Summarising, indirect vector control methods here are those that utilise reference values of induction motor variables for calculation of rotor flux space vector position (plus measured speed or position), while direct methods perform the same calculation by using measured electro-magnetic variables, and, if necessary, measured rotor speed or position. Methods of direct and indirect rotor flux oriented control will be discussed in detail in Section 3.3. Only rotor flux oriented control is elaborated in what follows. Current-fed induction machine is treated first. This is followed by analysis of the voltage-fed induction machine. One important aspect of the discussion that follows is related to the power supply. As emphasised in Section 2.4., an ideal variable magnitude, variable frequency sinusoidal voltage source does not exist and an inverter has to be used instead. This is however just an unavoidable nuisance which has no consequences on the principals of the vector control. Vector control theory is developed under the assumption of ideal sinusoidal supply under all operating conditions. This means that the control is related to the fundamental harmonic of the supply in the inverter case. The fact that the inverter supply produces higher order voltage harmonics as well does not influence in any way the principles of vector control. The presence of higher order voltage harmonics in the supply simply means that the currents of the machine and the torque contain higher order harmonics as well. While these do adversely affect the behaviour of the machine (additional losses and heating, torque ripple, reduced efficiency), they do not affect the control. Hence the control system of any vector controlled drive essentially generates current (or voltage) references that are in any steady- state operation pure sine waves. The complete theory of vector control that follows in Section 3.2. assumes that the machine is supplied from a pure sinusoidal voltage source. 3.2. ROTOR FLUX ORIENTED CONTROL 3.2.1. Current fed induction machine As already noted, if the machine is regarded as current-fed the stator currents can be treated as known and, under ideal conditions, equal to their reference values. In other words, ii ii ii aa b bcc *** === (3.1) where the asterisk denotes reference (required) phase current values generated by the control system, while current values without an asterisk are those in the machine. Equation (3.1) applies to any instant HIGH PERFORMANCE DRIVES ---------------------------------------------------------------------------------------------------------------------------------------  E Levi, 2001 43 in time and cannot be ever fully satisfied in reality. However, by using current control of a VSI, one approaches this ideal situation. The methods of current control of a VSI are beyond the scope of interest at present and are discussed in detail in Chapter 5. From the preceding discussion it follows that the output of the control system are phase current references. Recall that the same was said in Chapter 1 in conjunction with armature current of a high performance separately excited DC motor drive. Since the stator currents are known, there is no need to consider stator voltage equations for realisation of the control scheme. In order words, stator voltage equation of the model (2.36) can be omitted from consideration. This is obviously a simpler case than when stator voltage equation has to be considered as well, as the case is with rotor flux oriented control of a voltage-fed induction machine. The model (2.36) is given in the arbitrary reference frame and is repeated here once more for convenience: vRi d dt jRi d dt j s s s s a s r r r a r =+ + =+ + − ψ ωψ ψ ωωψ 0() ψ ψ s s s m r r r r m s Li L i Li L i=+ =+ } TP L L i e m r s r = ì í î 3 2 Im ψ (2.36) () () vv jv ve ii ji ie s ds qs s j s ds qs s j ss ss =+ = =+ = −− βθ εθ The clue to vector control consists in selection of the speed of the common reference frame. For rotor flux oriented control the speed of the reference frame is selected as equal to the speed of rotation of the rotor flux space vector at all times. The real axis of the reference frame, d-axis, is at all times aligned with the rotor flux space vector, while the q-axis is perpendicular to it. As the common reference frame is fixed to the rotor flux space vector and moreover, as the d-axis (real axis) of the common reference frame coincides with rotor flux space vector, then θ φ θ φ θ ωω ω φ sr rr ar r r d dt ==− == (3.2) Rotor flux space vector, Eq. (2.34), becomes a pure real variable in this special frame of reference, ψ ψ ψ ψ r dr qr r j=+ = (3.3a) i.e., it follows that ψ ψ ψ ψ dr r qr qr dt== =d00 . (3.3b) Rotor flux and stator current space vectors in the common reference frame fixed to the rotor flux space vector are illustrated in Fig. 3.1. Taking into account Eq. (3.3), torque equation of (2.36) takes, using the stator current space vector equation of (2.36), the following form: {} TP L L iP L L i e m r r s m r rqs == 3 2 3 2 ψψ Im (3.4) Torque equation (3.4) is of the same form as the torque equation met in DC machine theory (see equation (1.4)) and it shows that, if the magnitude of the rotor flux is kept on constant value, torque can be independently controlled by stator q-axis current. In order to accommodate rotor voltage equation (2.36) to the chosen reference frame, rotor current space vector has to be expressed from the rotor flux linkage equation of (2.36) (note that Eq. (3.3) is HIGH PERFORMANCE DRIVES ---------------------------------------------------------------------------------------------------------------------------------------  E Levi, 2001 44 accounted for and hence the rotor flux space vector is a real quantity; it is therefore not underlined any more): q-axis (Im) ω r i s ε s −φ r ψ r d-axis φ r (Re) ε s θω A-axis a-axis Fig. 3.1 - Illustration of space vectors in common reference frame fixed to the rotor flux space vector. () iLiL r rm s r =− ψ / (3.5) Substituting (3.5) into rotor voltage equation of (2.36) and taking into account (3.2), the following rotor voltage equation results: 0 11 =++−− T d dt j T Li r r r rr r m s ψ ψ ωωψ () (3.6) where T r = L r /R r is the rotor time constant. Separation of (3.6) into real and imaginary part yields ψ ψ rr r mds T d dt Li+= (3.7) () ω ω ψ rrrmqs TLi−= (3.8) Equation (3.7) reveals that, in this special common reference frame fixed to the rotor flux space vector, magnitude of rotor flux can be controlled by stator d-axis current and that magnitude of rotor flux is constant if the stator d-axis current is constant. According to Eq. (3.8), angular slip frequency ω ω ω sl r =− is linearly dependent on stator q-axis current if the magnitude of rotor flux is constant. Consequently, developed torque is proportional to slip frequency. If the stator d-axis current is held constant, torque can be instantaneously altered if it is possible to change stator q-axis current instantaneously. Note that there is no pull-out torque (i.e., pull-out torque is theoretically infinite) and any desired value of the torque can be obtained by applying appropriate value of the stator q-axis current, while maintaining the constant rotor flux. Illustration of flux and torque production in a current fed rotor flux oriented induction machine is given in Fig. 3.2. From Fig. 3.2 it directly follows that rotor flux amplitude is controllable by stator d-axis current component only (as excitation flux can be controlled in a DC machine by excitation winding current) while for constant flux operation torque depends on stator q-axis current component only (which corresponds to armature current in a DC machine). The principle difference is however that in an induction machine both d- and q-axis stator current components are actually parts of the same stator current vector, while in a DC machine excitation and armature current are completely independent. If the machine is current-fed, stator voltage equations can be omitted from consideration and the equations derived so far constitute the complete model of a rotor flux oriented induction machine. Equations (3.4), (3.7) and (3.8) show that decoupled torque and flux control can be obtained with a HIGH PERFORMANCE DRIVES ---------------------------------------------------------------------------------------------------------------------------------------  E Levi, 2001 45 current-fed machine in rotor flux oriented reference frame. Difficulties experienced in practical analogue realisations, before the microprocessor era, are a consequence of the need to perform co- ordinate transformation from rotational to stationary reference frame. The control system has to operate in rotor flux oriented reference frame if decoupled control is to be achieved, while actual machine operates in stationary phase domain. A schematic representation of a current-fed induction machine with rotor flux oriented control is given in Fig. 3.3. Block CRPWM denotes current regulated PWM inverter, speed drive is shown, and provision for operation in field-weakening region is included. Closed loop current control is not explicitly shown in Fig. 3.3: however, it is contained within the block CRPWM, that ensures that Eq. (3.1) is (approximately) satisfied. Closed loop speed control is of course present. Hence the situation is the same as in a separately excited DC machine: the inner current control loop is contained within the outer speed control loop and measurement of currents and the speed is required. Note that all the variables associated with the control system have an asterisk, while those without an asterisk are the variables within the machine. The configuration of the drive displayed in Fig. 3.3 corresponds to category of direct orientation schemes (superscript ‘e’ in the transformation block indicates that an estimate of the rotor flux space vector position is used for co-ordinate transformation). Structures of rotor flux estimators are dealt with in Section 3.3. i a i αs i ds 1 ψ r L m i b 3 −jφ r 1+sT r e i c 2i βs i qs L m ω sl T r φ r 3P L m T e P ω 2L r Js T L ω r 1 s Fig. 3.2 - Block diagram of an ideal current fed rotor flux oriented induction machine (note that if stator d-axis current is constant rotor flux is constant and torque depends on stator q-axis current only). Friction torque is neglected and s represents Laplace operator. signals ω * ψ r - - - Field contr. contr. T e * i qs * i ds e φ r e ω j 2 3 IM ω i i i i i i a b cc b a ψ r T e φ r i i i ω a b c rotor flux C R P W M ** * * * Calculation of vector ψ r from motor Torque Flux weak. Speed contr. i i * * α β s s Ideal CRPWM: i a *= i a i b *= i b i c *= i c Fig. 3.3 - Control of a current-fed direct rotor flux oriented induction machine: the scheme for operation in both the base speed region and the field weakening regions is shown. HIGH PERFORMANCE DRIVES ---------------------------------------------------------------------------------------------------------------------------------------  E Levi, 2001 46 Control system in Fig. 3.3 contains two parallel channels, one for flux control and the other one for torque control. It is just one of possible versions of the direct rotor flux oriented control system. For example, if the drive is required to operate in the base speed region only, then the rotor flux reference is an independent constant input. If rotor flux reference is constant, then it follows from (3.7) that ψ rmds Li = and flux controller is often omitted. Stator d-axis current reference is in such a case obtained as iL ds r m ** = ψ . Next, under constant rotor flux reference operation it follows from (3.4) that stator q- axis current is at all times directly proportional to the torque. Hence the torque controller may be omitted and the speed controller output becomes equal to the stator q-axis current reference. Hence one obtains the scheme of Fig. 3.4 instead of the one in Fig. 3.3. From the rotor flux position estimator φ r e ω * i qs * i β s * i a * i a PI 2 C _ j φ r e R e i b * P i b IM ψ r *1 i ds * i α s * 3 i c * W i c L m M ω Fig. 3.4 - An alternative form of the direct rotor flux oriented control of a current-fed induction machine (operation in base speed region only). Control system of Fig. 3.4 is undoubtedly considerably simpler than the one of Fig. 3.3, since there is only speed controller rather than speed, torque and rotor flux controllers. Current controllers are once more hidden within the block CRPWM that ensures satisfaction of (3.1). Fig. 3.4 is in essence fully equivalent to the control system of a DC motor shown in Fig. 1.2. Speed controller output is the stator q-axis current reference; the same applies to Fig. 1.2 where the output of the speed controller is the armature current reference (scaling between torque reference and armature current reference in Fig. 1.2 can be included in the speed controller parameters in the same way as it was done in Fig. 3.4). The second branch of the control system in Fig. 3.4 sets the constant stator d-axis current reference that will provide operation with constant rotor flux; it corresponds to the excitation winding current setting in a DC machine. The important difference between Figs. 1.2 and 3.4 is that in the case of the DC machine there is no need for any kind of co-ordinate transformation. The commutator ensures that torque and flux control are inherently decoupled. In an induction machine decoupled control of rotor flux and torque results in the rotor flux oriented reference frame. Hence the control system operates in that reference frame. As the machine has to be supplied with physical phase currents, it is necessary to perform co-ordinate transformation in order to relate the fictitious d-q axis currents with physical phase currents. Thus the co-ordinate transformation in an induction motor control system performs the role of the commutator in a DC machine control. The co-ordinate transformation requires knowledge of the rotor flux space vector instantaneous position. The methods of calculating the position of the rotor flux space vector will be elaborated in Section 3.3. It should be noted that the co-ordinate transformation in Figs. 3.3 and 3.4 is shown as a two-step transformation (blocks () exp j r e φ and ‘2/3’). These two blocks describe the transformation from d-q axis reference frame to the phase reference frame, described as a single transformation step in (2.23). To summarise the discussion so far. Model of a rotor flux oriented current-fed induction machine fully resembles the model of a separately excited DC machine. However, since the stator current d-q axis components are fictitious currents, it is necessary to convert these current references (created by the control system) into the actual phase current references using a co-ordinate transformation. The HIGH PERFORMANCE DRIVES ---------------------------------------------------------------------------------------------------------------------------------------  E Levi, 2001 47 equations that constitute the model of a current-fed rotor flux oriented induction machine are (3.4), (3.7) and (3.8): ψ ψ rr r mds T d dt Li += () ω ω ψ rrrmqs TLi−= (3.9) TP L L i e m r rqs = 3 2 ψ The control system creates stator d-q axis current references on the basis of these equations. The co- ordinate transformation is governed by (2.23), ii i ii i ii i ads rqs r bds r qs r cds r qs r ** * ** * ** * cos sin cos( /) sin( /) cos( /) sin( /) =− =−−− =−−− φφ φπ φπ φπ φπ 23 23 43 43 (3.10) and it requires knowledge of the instantaneous rotor flux spatial position angle. Its calculation is discussed in Section 3.3, since it does not depend on whether the machine is current-fed or voltage-fed. 3.2.2. Voltage fed induction machine If the machine can not be treated as being current-fed, voltage-fed case results and stator voltage equation of (2.36) has to be considered as well. It is important to note that the equations (3.9), that describe rotor flux oriented current-fed induction machine, remain to be valid and constitute now one part of the model, rather than the complete model. The final goal remains orientation of stator current space vector along rotor flux space vector, however the appropriate stator voltage space vector which enables achievement of the goal has to be determined. In other words, stator voltage space vector has to have such a magnitude and has to be put in such a position that resulting stator current space vector attains exactly the magnitude and position with respect to rotor flux space vector that are needed to realise decoupled flux and torque control; i.e., stator current space vector is controlled indirectly by direct control of stator voltage space vector. Current control remains to be present inside the drive control structure. However, in contrast to the case of the current-fed induction machine, where current control was performed using the phase currents and it was therefore possible to regard CRPWM as an ideal current source, current control is now performed using d-q axis stator current components. The ultimate outputs of the control system are now stator phase voltage references, rather than stator current references. Since current control is performed in the rotating reference frame, it is necessary to transform measured stator phase currents into d-q axis components. Two co-ordinate transformations are therefore now required: one transforms stator d-q axis voltage references into phase voltage references, while the other one (inverse one) transforms measured phase currents into their d-q axis components. The overall complexity of the control system is therefore significantly higher than the case is with a current-fed induction machine. The reason why the concept of voltage-fed machine, with current control in rotating reference frame, is frequently applied is that it is extremely well suited for pure digital realisation (in contrast to the case of the current-fed machine, that is usually built using combined digital-analogue realisation; current control is usually analogue, while the remainder of the control system is digital). Stator flux space vector can be given from flux linkage equations of (2.36), under the conditions of rotor flux orientation (3.2)-(3.3), as ψψσ s m r rs s L L Li =+ (3.11) HIGH PERFORMANCE DRIVES ---------------------------------------------------------------------------------------------------------------------------------------  E Levi, 2001 48 where σ =−1 2 LLL m s r /( ) is the total leakage coefficient. Substitution of (3.11) into stator voltage equation of (2.36) and resolution into d-q components gives stator voltage d-q axis equations in the reference frame fixed to the rotor flux: vRi L di dt L L d dt Li vRi L di dt L L Li ds s ds s ds m r r rsqs qs s qs s qs r m r rrsds =+ + − =+ + + σ ψ ωσ σωψωσ (3.12) If the stator transient time constant TTLR s s s s ' /== σσ is introduced, Eqs. (3.11) become iT di dt R v R d dt Ti iT di dt R v R Ti ds s ds s ds sr r rsqs qs s qs s qs r s r rrsds +=− + + +=− + − '' '' 111 1 111 1 σ ψ ω ω σ ψω (3.13) Equations (3.13) show that d- and q-axis components of stator voltage and stator current are not decoupled. In order words, each of the two voltage components is a function of both stator current components. If the decoupled control of stator d- and q-axis currents is to be achieved, it is necessary to introduce appropriate decoupling circuit in the control system. If the output variables of current controllers are defined as vRiT di dt vRiT di dt ds s ds s ds qs s qs s qs '' '' =+ æ è ç ö ø ÷ =+ æ è ç ö ø ÷ (3.14) required reference values of axis voltages vv ds qs ** and are obtained as vve vve ds ds d qs qs q *' *' ==++ (3.15) where auxiliary variables ee dq and are calculated as eR R d dt Ti eR R Ti ds sr r rsqs qsr sr rrsds = + − æ è ç ç ö ø ÷ ÷ = + + æ è ç ç ö ø ÷ ÷ 11 1 11 1 σ ψ ω ω σ ψω ' ' (3.16) Equations (3.16) describe decoupling circuit which has to be introduced into the control system of a voltage-fed rotor flux oriented induction machine, if decoupled flux and torque control is to be achieved. Decoupling circuit is shown in Fig. 3.5. Block diagram of a voltage-fed rotor flux oriented induction machine is given in Fig. 3.6. Application of voltage-type source enables almost the same quality of the dynamic response as is obtainable with current source, provided that decoupling circuit is included in the control system. The quantities required in the decoupling circuit of Fig. 3.5 (rotor flux magnitude and rotor flux angular velocity) have to be estimated together with the rotor flux position, using either principles of indirect or direct rotor flux oriented control. The signals needed in the decoupling circuit are either measured or commanded quantities, in accordance with the applied orientation method. Term d ψ r /dt is usually neglected in Eq. (3.16), as for operation in the constant flux region it should anyway be equal to zero. This simplifies Eqs. (3.16) since they become eLi e L L drsqs qr s m r =− = ω σ ωψ (3.17) HIGH PERFORMANCE DRIVES ---------------------------------------------------------------------------------------------------------------------------------------  E Levi, 2001 49 σL s - v ds ' σ 1 r 1+ s v qs ' v ds * v qs * i qs ψ r ω r i ds Voltage-fed rotor flux oriented induction machine σ 1 r 1+ σL s Fig. 3.5 - Decoupling circuit of a voltage-fed induction machine with rotor flux oriented control (note that the required rotor flux amplitude and angular velocity values are obtained either from measurements or using reference values, as discussed in Section 3.3). ψ r *i ds * ∆ i ds v ds *v a * 1/L m PI 2 − e d v b * Decoupling jφ r circuit, Fig. 3.5 e e q ω *i qs * ∆ i qs v qs *v c * Speed c. PI 3 −− φ r estimation i a i ds 2 − j φ r i b i qs e 3i c ω ω Fig. 3.6 - Voltage-fed induction machine with rotor flux oriented control aimed at operation in the base speed region only. It is possible to omit decoupling circuit from the control system, without significant influence on dynamic response, if the sampling frequency and inverter switching frequency are high enough, usually above at least 1 kHz. If the inverter switching frequency is under 1 kHz, decoupling circuit should be included. However, application of decoupling circuit enables improvement of dynamics even at inverter switching frequency above 1 kHz. At higher switching frequencies current controllers are capable of suppressing interaction between d- and q-axis and decoupling circuit is in most cases omitted. A good feature of a voltage-fed rotor flux oriented induction machine is reduced sensitivity to rotor parameter variation, compared to a current-fed machine. To summarise the discussion of the rotor flux oriented voltage-fed induction machine. Stator voltage d-q axis references are built as sums of the outputs of the current controllers (that operate in the rotating reference frame) and the so-called decoupling voltages (that take rather simple form if rotor flux is HIGH PERFORMANCE DRIVES ---------------------------------------------------------------------------------------------------------------------------------------  E Levi, 2001 50 regarded as constant). Equations derived for the current-fed machine remain to be valid. Two co- ordinate transformations are required: one transforms stator voltage d-q axis references into phase voltage references, while the other one transforms measured phase currents into d-q axis components. Hence the equations that are relevant for a voltage-fed rotor flux oriented induction machine are the following: vve vve ds ds d qs qs q *' *' ==++ eLi e L L drsqs qr s m r =− = ωσ ω ψ ψ ψ rr r mds T d dt Li += () ω ω ψ rrrmqs TLi−= TP L L i e m r rqs = 3 2 ψ (3.18) vv v vv v vv v ads rqs r bds r qs r cds r qs r ** * ** * ** * cos sin cos( / ) sin( / ) cos( / ) sin( / ) =− =−−− =−−− φφ φπ φπ φπ φπ 23 23 43 43 iii i iii i ds a r b r c r qs a r b r c r =+−+− =− + − + − ( / )( cos cos( / ) cos( / )) ( / )( sin sin( / ) sin( / )) 23 23 43 23 23 43 φ φ π φ π φφπ φπ It should be noted that the control system in both the voltage fed and current fed drive processes DC quantities (see discussion in Sub-section 2.4.1.) and that it fully corresponds to the one met in DC machines. The principal difference is the need for co-ordinate transformation in the case of an induction machine. 3.3. ESTIMATION OF MAGNITUDE AND POSITION OF ROTOR FLUX SPACE VECTOR 3.3.1. Methods for direct rotor flux oriented control If the decoupled flux and torque control is to be achieved, it is absolutely necessary to know instantaneous position of the rotor flux space vector, because the stator current space vector has to be orientated with respect to the rotor flux space vector. Estimation method suggested at the earliest stage of vector control development applies convenient induction machine model and measured values of stator currents and air gap flux (main, magnetising flux). Hall sensors are utilised for assessment of main flux components. Stator phase currents are measured and converted into alfa-beta components using (2.27), iiii iii sabc sbc α β =−− =− − + (/)( . . ) (/)( / /) 2 3 05 05 23 32 32 (3.19) while Hall sensors directly provide main flux components in the stationary reference frame, ψ ψ αβ mm , (two fixed sensors are used in the air-gap of the machine, displaced by 90 degrees). Magnitude and . oriented control in high speed and field-weakening region (the notions of indirect and direct vector control will be explained shortly). HIGH PERFORMANCE DRIVES. HIGH PERFORMANCE DRIVES ---------------------------------------------------------------------------------------------------------------------------------------