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 HIGH PERFORMANCE DRIVES ---------------------------------------------------------------------------------------------------------------------------------------  E Levi, 2001 51 position of the rotor flux space vector can then be obtained from flux linkage equations (2.21), taking into account that ψ ψ αα ββ ααα βββ mmm mmm msr msr Li Li iii iii == =+ =+ (3.20) and that ψ ψ ψ ψψ ψ αγαα γαα α βγββ γββ β rrrmrms m rrrmrms m Li L i i Li L i i =+= −+ =+= −+ () () (3.21) according to the following equations ψ σ ψ ψ σ ψ ψψψ ϕψψ ϕψψ α α γα β β γβ αβ α β rrmrs rrmrs rrr rrr rrr Li Li=+ − =+ − =+ = = () () cos sin 11 22 (3.22) where σ γ rrm LL= /. Torque (2.19) can be given as a function of the measured variables, () TPi i emsms =− 3 2 ψψ αβ βα (3.23) Calculation of magnitude and position of the rotor flux space vector using the measured stator currents and air gap flux is illustrated in Fig. 3.7 (output related to the rotor flux position is given as ‘cos’ and ‘sin’ function of this angle; note that that is exactly the form required by the co-ordinate transformation block exp( ) j r φ ). This estimation procedure will be called ψ m s i− estimation further on. ψ α m ψ α r 1+ σ r −ψ r ψ β m ψ β r 1+ σ r − cos φ r i a 3L γ r i b sin φ r i c 2L γ r Fig. 3.7 - Calculation of rotor flux space vector from sensed stator currents and main flux. An important feature of the ψ ms i− estimator is that acquisition of rotor speed or position signal is not needed for the estimation procedure. This enables realisation of a cheaper drive with still very good quality of dynamic response. Due to the numerous shortcomings experienced in application of Hall sensors for flux measurement this method of rotor flux position estimation is of historical rather than practical value. The second rotor flux space vector estimation method asks for measurement of stator voltages and stator currents. If the stator phase currents and voltages are sensed and transformed into two-phase stationary reference frame, magnitude and position of the rotor flux space vector can be calculated either by means of analogue or digital circuitry. Model of an induction machine in stationary reference frame, (2.24)-(2.26), together with flux linkage equations (2.21), suggests the following procedure: HIGH PERFORMANCE DRIVES ---------------------------------------------------------------------------------------------------------------------------------------  E Levi, 2001 52 iiii iii sabc sbc α β =−− =− − + (/)( . . ) (/)( / /) 2 3 05 05 23 32 32 (3.19) vvvv vvv sabc sbc α β =−− =− − + (/)( . . ) (/)( / /) 2 3 05 05 23 32 32 (3.24) () () ψ ψ ααα βββ ssss ssss vRidt vRidt =− =− ò ò (3.25) ψσψσ ψσψσ ααα βββ rrs sr m s rrs sr m s LL L i LL L i =+ − =+ − () () 1 1 (3.26) ψψψ ϕψψ ϕψψ αβ α β rrr rr rr =+ = = 22 cos sin rr // Equation for torque calculation (3.23) remains unchanged, () TPi i emsms =− 3 2 ψψ αβ βα (3.23) Block diagram of the estimator which relies on sensed stator currents and voltages, vi ss − estimator, is showninFig.3.8. The most pronounced shortcoming of the v i s s − method is that integration is involved, according to (3.25). This restricts practical implementations of the estimator to drives which are not aimed for operation at zero speed. Typical limit is frequency value of 3 Hz. Comparative analysis of v i s s − and ψ ms i− estimators reveals that their behaviour is practically identical at frequencies above 10 Hz, while at lower frequencies ψ m s i− estimator is advantageous and can be used down to the frequency of 0.5 Hz. Typical applications of vi ss − estimatorarecorrelatedtodriveswherespeedandpositionsensors are to be avoided. It should be noted that v i s s − estimator does not necessarily ask for voltage measurement, because the output voltages of a transistor inverter can be reconstructed from the driving signals of power semiconductor switches and measured DC link voltage. 3v α s ψ α r ψ r v a ò 1+σ r v b v β s −−ψ β r ò 1+σ r v c 2 −− R s R s cos φ r i α s i a 3 σ L s L r /L m i b i c 2 σL s L r /L m sinφ r i β s Fig. 3.8 - Rotor flux space vector estimation process involving measured stator currents and voltages. HIGH PERFORMANCE DRIVES ---------------------------------------------------------------------------------------------------------------------------------------  E Levi, 2001 53 The most frequently utilised method of rotor flux space vector estimation asks for measurement of stator currents and rotor speed or position (it will be denoted as i s − ω estimator). The main reasons for such a widespread application of this scheme are that there is no need for special construction or modification of the machine, integration is avoided and estimation is operational at zero speed. It is customarily used in the vector control system illustrated in Fig. 3.6, where current control is performed in rotational reference frame and the machine is either treated as being voltage-fed, or it is assumed to be current-fed and decoupling circuit is omitted. Current control in rotating co-ordinates normally leads to fully digital realisation of the vector control, in contrast to the scheme shown in Figs. 3.3-3.4 where current controllers operate in stationary frame of reference and realisation is usually combined analogue-digital. Estimation of rotor flux space vector by means of measured stator currents and rotor speed (position) utilises model (3.9) of an induction machine in the rotor flux oriented reference frame. Consequently, values of measured stator currents have to be transformed from stationary to rotational, rotor flux oriented, reference frame using 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 (3.27) Equations (3.9) 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 canberewrittenas () iT d dt L Li T ds r r r m sl m qs r r =+ æ è ç ö ø ÷ = ψ ψ ωψ / (3.28) Spatial position of the rotor flux space vector is determined with () ϕωω rsl dt=+ ò (3.29) while torque estimation can be done by the aid of (3.9) TP L L i e m r rqs = 3 2 ψ (3.9) The most frequent outlook of the estimator is given in upper part of the Fig. 3.9, where it is shown as an integral part of the drive of Fig. 3.3. Superscript ‘e’ denotes all the quantities that are the estimates of actual variables. There are some other realisations, with minor modifications which do not affect the fundamental operating principles of the scheme. Estimation of rotor flux space vector, out of the measured stator currents and rotor speed, can be performed in stationary reference frame as well. In this case there is no need for transformation of measured currents from stationary to rotational reference frame. However, the equations that have to be used are much more complicated and this method is rarely applied in practice. Furthermore, if current controllers operate in rotational reference frame, Fig. 3.6, measured stator currents have to be transformed from stationary to rotational reference frame anyway and it is far simpler to apply estimator structure depicted in Fig. 3.9. All the three estimators discussed so far are used for direct field orientation, since measured electromagnetic variables are involved in the process of rotor flux position calculation. HIGH PERFORMANCE DRIVES ---------------------------------------------------------------------------------------------------------------------------------------  E Levi, 2001 54 ωω T r L m 1/(1+ sT r ) i α s i a ω sl e ψ r e i ds e 2 − j φ r e i b i qs e e i β s i c L m 3 ω r L m / L r 3 P /2 1/ s φ r e T e e _ φ r e ω * T e * i qs * i β s * i a * i a PI PI 2 C _ j φ r e R _ ψ r e e i b * P i b IM Field ψ r * i ds * i α s * 3 i c * W i c weak M ω PI Fig. 3.9 - Rotor flux space vector estimation by means of measured stator currents and rotor speed (position). 3.3.2. Indirect (feed-forward) estimation of rotor flux position Vector control can be achieved by indirect orientation as well, where position of rotor flux space vector is estimated again on the grounds of induction machine model in rotor flux oriented reference frame. In order to achieve orientation it is theoretically necessary to measure only rotor speed or position. Indirect vector controlled induction machine is conventionally fed from current-regulated PWM inverter and current control is performed in the stationary reference frame. Indirect orientation principle follows directly from (3.9) and is described with i P TL L i L T d dt qs e r r m ds m rr r * * * ** * = =+ æ è ç ç ö ø ÷ ÷ 2 3 1 ψ ψ ψ (3.30) ω ψ ϕωω sl m r qs r rsl L T i dt * * * * () = =+ ò (3.31) Indirect method of calculating the rotor flux position is illustrated in Fig. 3.10, where asterisk denotes once more reference quantities. Prevailing applications are for drives which require operation in the base speed range only. Consequently, only operation in the constant flux region is needed and the scheme can be further simplified. Such a drive is shown in Fig. 3.11, where due to ψψ rdsrm const i L *** ., /== is a constant as well. Torque command (or stator q-axis current command) is obtained as output from the speed controller. Indirect vector control has gained enormous popularity in practical realisations as the overall control system complexity is significantly reduced compared to direct orientation methods. HIGH PERFORMANCE DRIVES ---------------------------------------------------------------------------------------------------------------------------------------  E Levi, 2001 55 T r ψ r e T * * s 2 3P 1 L m L L m r L r sl qs ds i i ω * * * Fig. 3.10 - Principle of indirect vector control of a current-fed induction machine. ω * - 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 φ r C R P W M * * * * Speed contr. i i * * α β s s K 1 K 2 K P L Li K Ti r m ds r ds 1 2 2 2 3 1 1 == ** Fig. 3.11 - Current-fed induction machine with indirect vector control. Indirect field oriented control may be applied in conjunction with voltage-fed induction machine as well. One possible implementation is shown in Fig. 3.12, where although measured d-q axis currents are available in rotational reference frame, commanded values are used for achieving both decoupling and rotor flux oriented control. Note that Fig. 3.12 is essentially the same as Fig. 3.6. The only difference is that the calculation of rotor flux position and the decoupling terms are now explicitly shown. As the rotor flux space vector can not be measured, all estimation methods inevitably make use of an appropriate induction machine model to carry out necessary calculations and provide an estimate of rotor flux space vector. The crucial information is the one regarding the position of the rotor flux space vector, because stator current vector will be placed at exactly required position only if the rotor flux position is accurately known. The other estimator outputs, e.g. estimates of torque and rotor flux magnitude, may or may not be required, depending on the structure of the control system. Induction machine models, applied in the estimation process, are constant parameter models. Actual machine parameters are subject to variation and will predominantly vary due to operating temperature variation (resistances) and due to saturation of flux paths (magnetising inductance and leakage inductances). Consequently, accuracy of all the estimation schemes depends on parameter variations in the machine. Different estimators are influenced by different parameters in different manner. The most frequently applied method of rotor flux oriented control, indirect feed-forward method, is not affected by variation of the stator parameters since these are not involved in the control system. However, its operation is strongly affected by variation in the rotor resistance, that will inevitably vary as operating temperature in the machine changes. In addition, especially in the field-weakening region, this scheme is affected by variation in the magnetising inductance as well. HIGH PERFORMANCE DRIVES ---------------------------------------------------------------------------------------------------------------------------------------  E Levi, 2001 56 v qs * v ds e φ r j ω φ r i i i a b c * v v * * α β s s v qs ' v ds ' - i ds * qs i * ω * - ω P+I - P+I P+I i ds * i qs * ω r * L s L σ s qs i ds i - + e -j φ r 2 3 VOLTAGE SOURCE PWM INVERTER T r i ds * 1 ω sl * ω r * + + INDUCTION MACHINE Fig. 3.12 - Indirect (feed-forward) voltage fed rotor flux oriented induction machine. 3.4. PERFORMANCE OF A ROTOR FLUX ORIENTED INDUCTION MACHINE Some characteristic simulation and experimental results are given in this Section, that illustrate behaviour of rotor flux oriented induction machines. Current-fed indirect rotor flux oriented machine is discussed at all times. Both the scheme of Fig. 3.11 with indirect vector control and the scheme of Fig. 3.9 with direct vector control are under consideration. Consider at first the scheme of Fig. 3.9. Simulation results are presented in what follows. Rated rotor flux reference is applied at time instant zero. Speed reference and the load torque are equal to zero. Once when the rotor flux is established, a speed reference is applied. The simulation results are given in Fig. 3.13. As can be seen from Fig. 3.13, the initial excitation of the machine follows exponential law. When the flux settles and speed command is applied, actual speed follows reference speed with a very small delay, caused by inertia of the drive. Torque rises almost instantaneously and reaches the maximum value determined by the imposed limit and hence the stator current is in the limit as well. Note that rotor flux remains constant during the torque variation, indicating that flux and torque control are fully decoupled (change of stator q-axis current does not cause any variation in the rotor flux). When the reference speed reaches steady-state value the actual speed overshoots (due to action of the speed controller, which is PI) and hence torque rapidly goes out of the limit, reduces and becomes negative, which means that electric braking operation takes place. Once when the speed settles at the value equal to the reference value, torque falls to zero, because the case shown in Fig. 3.13 is acceleration with zero load torque. Comparison of Fig. 3.13 with Fig. 1.4 shows that the responses are identical: hence the rotor flux orientation control successfully converts an induction machine into its DC machine equivalent. Direct rotor flux oriented current fed induction machine of Fig. 3.9 is simulated again. CRPWM inverter is taken as ideal so that commanded stator phase currents are directly impressed into the motor stator phase windings. Once more, the machine is at first excited with rated rotor flux command. Once when the rotor flux is established in the machine, speed command equal to −40 % of the rated speed is . HIGH PERFORMANCE DRIVES ---------------------------------------------------------------------------------------------------------------------------------------. used in the air-gap of the machine, displaced by 90 degrees). Magnitude and HIGH PERFORMANCE DRIVES ---------------------------------------------------------------------------------------------------------------------------------------