HIGHPERFORMANCE 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 HIGHPERFORMANCE DRIVES --------------------------------------------------------------------------------------------------------------------------------------- E Levi, 2001 57 applied through a rate-of-change limiter. Load torque equals zero throughout the transients. A steady- state, similar to the one of Fig. 3.13, is established. The subsequent simulation, illustrated in Fig. 3.14, involves so called reversing transient: speed reference is changed from −40 % torque limit T e ψ r 0 ψ r ω * ω 0time Fig. 3.13 - Dynamic behaviour of current-fed direct rotor flux oriented induction machine: Initial excitation and rapid acceleration in the constant flux region. 0 0.2 0.4 0.6 0.8 1 Rotor flux (Wb) -200 -100 0 100 200 300 Rotor angular speed (rad/s) 0 0.01 0.02 0.03 0.04 0.05 Time (s) ψ ψψ ψ ψ ψψ ψ ψ ψψ ψ e r * r r ω ωω ω ω ωω ω * = -40 -30 -20 -10 0 10 20 30 40 Torque (Nm) -5 0 5 10 15 20 25 30 35 Stator current components (A) 0 0.01 0.02 0.03 0.04 0.05 Time (s) e ds i e qs e e e T ,T =i i =i * qs * ds Fig. 3.14 - Reversing transient of the drive of Fig. 3.9. HIGHPERFORMANCE DRIVES --------------------------------------------------------------------------------------------------------------------------------------- E Levi, 2001 58 to +40% of the rated speed in a ramp-wise manner. Simulation results, summarised in Fig. 3.14, depict rotor flux reference, actual rotor flux and estimated rotor flux (superscript ‘e’ denotes estimated values), actual and estimated torque, actual and commanded speed, and actual and estimated stator current d-q axis components. Excellent dynamic behaviour is evident. The actual flux in the machine remains constant during the reversing transient, indicating that the variation of the stator q-axis current does not affect the rotor flux. Flux and torque control are therefore fully decoupled. The drive operates for a prolonged period of time in the torque limit (and hence in the current limit as well) during the reversing. Once when the new operating speed is established, torque falls to zero since the reversing is simulated under no-load conditions. As the last simulation example of the scheme of Fig. 3.9, step loading and unloading of the drive is investigated. The drive operates initially in steady-state with zero load torque and with constant value of speed and rotor flux. Step load torque, equal to the rated value, is then at first applied and then removed. Rotor flux and motor torque are shown in Fig. 3.15. Torque build-up is extremely quick. The torque initially overshoots the load torque in order to compensate the speed dip caused by the load torque application. Speed is returned to the previous steady-state value and the motor torque equals load torque until the load torque removal takes place. Torque of the motor quickly goes down and becomes negative for a very short period of time in order to compensate for the increase in the speed caused by the load torque removal. Once when the speed is returned to its previous value, torque becomes equal to zero. Rotor flux remains completely undisturbed during these transients, indicating once more fully decoupled rotor flux and torque control. Fig. 3.15 - Response of the drive of Fig. 3.9 to step loading and unloading. Indirect rotor flux oriented induction motor drive of Fig. 3.11 is investigated next. A series of experiments is performed on a commercially available drive (manufactured by Vickers company) and some of these are presented in what follows. However, before depicting the transient behaviour, let us at first illustrate the current waveform in steady-state operation. Fig. 3.16 shows recorded phase current waveform and its spectrum. Obviously, the current is not an ideal sine wave. However, all the higher harmonics are of very high frequency and are situated around multiples of the 10 kHz frequency, which is the switching frequency of the inverter. The CRPWM inverter thus enables approximate satisfaction of the condition (3.1), as already noted in the beginning of this Chapter. Acceleration transient is investigated next. The machine initially operates at 200 rpm. A speed command of 1500 rpm is then applied, under no-load conditions. The motor speed and the phase current are shown in Fig. 3.17. As can be seen, current quickly goes into the limit and stays in the limit until the motor speed approaches the set speed. The speed slightly overshoots the reference, leading to the braking action of the motor. Once when the acceleration transient is over, motor phase current becomes equal to the magnetising current of the machine (i.e., neglecting losses, stator q-axis current is zero and the phase current is equal to the stator d-axis current which is the magnetising current). HIGHPERFORMANCE DRIVES --------------------------------------------------------------------------------------------------------------------------------------- E Levi, 2001 59 -8 -6 -4 -2 0 2 4 6 8 Current (A) 0 0.005 0.01 0.015 Time (s) a. current waveform 0.001 0.01 0.1 1 10 Current (A) 0 2 4 6 8 10 12 Frequency (kHz) b. spectrumupto12.8kHz 0.001 0.01 0.1 1 10 Current (A) 0 100 200 300 400 500 600 700 800 Frequency (Hz) c. spectrum up to 0.8 kHz Fig. 3.16 - Current waveforms and spectra during no-load operation at 2100 rpm. Fig. 3.17 - Acceleration transient of an indirect rotor flux oriented induction machine. As the next example, deceleration transient is investigated. The previous steady-state is the one of Fig. 3.17 (1500 rpm) and speed command is now stepped down to 200 rpm. Stator phase current and speed are illustrated in Fig. 3.18. Note that the phase current goes in the limit again, indicating that the motor is developing the maximum permitted torque, but now of negative value (braking action). The speed quickly reduces and once when it becomes equal to the set value, stator current returns to its previous (magnetising current) value. Finally, Fig. 3.19 shows an experimental result obtained again with indirect rotor flux oriented current fed induction machine. Stator d-axis current command is set to 70 % of the rated value so that machine operates with 70 % of the rated flux (the reason for operation with reduced flux is beyond the scope of HIGHPERFORMANCE DRIVES --------------------------------------------------------------------------------------------------------------------------------------- E Levi, 2001 60 interest here). Speed command equals 600 rpm and is constant. Step loading and later on unloading is applied, with load torque equal to the machine’s rated torque. Speed response and commanded stator q- axis current are shown in Fig. 3.19. Fast build-up of the torque and consequently very fast recovery of the speed with relatively very small drop and subsequent increase during the transients can be observed in Fig. 3.19. Fig. 3.18 - Deceleration transient of an indirect rotor flux oriented induction machine. Fig. 3.19 - Experimentally recorded speed and commanded q-axis current response to step loading and unloading of an indirect rotor flux oriented current fed induction machine. 3.5. PARAMETER VARIATION EFFECTS IN ROTOR FLUX ORIENTED INDUCTION MACHINES As is obvious from the considerations of rotor flux oriented control principles, all the methods of rotor flux position estimation, as well as rotor flux amplitude and torque estimation, heavily rely on mathematical model of the induction machine. The model assumes that all the parameters are constant and accurately known. Discrepancy between parameter values utilised in an estimator and actual parameter values in the machine leads to incorrect rotor flux space vector estimation and erroneous orientation of stator current space vector results. HIGHPERFORMANCE DRIVES --------------------------------------------------------------------------------------------------------------------------------------- E Levi, 2001 61 Fig. 3.20 illustrates the consequence of the incorrect parameter values. Rotor flux position is erroneously estimated and hence the true rotor flux space vector position differs from the one assumed by the control system. There is an error in the orientation angle, meaning that the decoupled rotor flux and torque control is lost. In other words, the actual rotor flux is not aligned with the commanded d- axis, so that there is a projection on both the d- and q-axis of the reference frame dictated by the controller. q-axis q*-axis ω r d-axis ψ r ∆ φ r ψ r * d*-axis φ r φ r e a-axis Fig. 3.20 - Error in rotor flux orientation angle leads to the loss of decoupled flux and torque control since the actual rotor flux axis and the axis assumed by the control system do not coincide. Rotor flux estimation by the aid of ψ m s i− method depends on rotor leakage inductance and on magnetising inductance. Torque estimation is independent of parameter variations, as only measured values enter the equation. If the vi ss − estimator is utilised, rotor flux calculations are affected by variation in stator resistance, stator and rotor leakage inductance and magnetising inductance. Torque estimation is dependent on stator resistance only. Rotor flux space vector estimation in i s − ω scheme is influenced by rotor resistance, rotor leakage inductance and magnetising inductance variation. Torque computation, being a function of estimated rotor flux magnitude, is dependent on the same parameters. Indirect vector controller relies on the same model of an induction machine as the i s − ω estimator; therefore the same parameters exhibit influence on the drive behaviour. As is obvious from this short discussion, different parameters will affect operation of the drive in a different manner, depending on the selected flux orientation scheme. The most frequently applied method of rotor flux oriented control is indirect scheme of a current fed machine, which relies on accurate knowledge of magnetising inductance, rotor leakage inductance and rotor resistance. As the rotor leakage inductance enters equations as a term in summation, the other term being magnetising inductance, it is customary to assume that rotor leakage inductance is really a constant parameter. The ratio of rotor leakage inductance to magnetising inductance under rated conditions is from 0.01 for large machines up to 0.1 for small machines. This clearly indicates that magnetising inductance has the dominant influence. Even a 100% change of a rotor leakage inductance in a small machine would cause just a 10% change in the sum, i.e. in rotor inductance. If the main flux saturation is neglected, this being a conventional approach, magnetising inductance is assumed to be constant parameter as well and the only parameter which remains dependent on the operating regime is rotor resistance. Rotor resistance, which is present in the rotor time constant, changes with operating temperature. Discrepancy between actual rotor resistance value and the value employed in indirect vector controller affects torque response in such a way that it becomes oscillatory instead of being instantaneous and an error in actual torque and flux values takes place in steady-state. Static torque HIGHPERFORMANCE DRIVES --------------------------------------------------------------------------------------------------------------------------------------- E Levi, 2001 62 curve expressed as a function of the stator q-axis current component becomes non-linear and pull-out torque attains finite value. Detailed analysis and discussion of parameter variation effects and means for their compensation is beyond the scope of interest here. Only one illustration is provided, obtained using the experimental system of Fig. 3.11. The machine is initially brought to operate at certain constant speed. For the sake of this experiment the speed control loop is now opened. Stator q-axis current command is taken as an independent input and is applied as an alternating square wave, of rated stator q-axis current value. The parameter that influences the speed response is the rotor time constant. If the value is correctly set, then the motor torque exactly follows the reference and the motor speed response is of triangular wave-form. If the value of the rotor time constant is not correct, the motor torque will not be of a square wave-form, so that the speed response will deviate from the triangular one. Experimentally recorded speed response and the stator q-axis current reference are illustrated in Fig. 3.21 for the correct and an incorrect value of the rotor time constant. This is simultaneously a relatively simple method of rotor time constant identification in indirect vector controlled induction machine. When the rotor time constant in the control system is adjusted to the correct value, speed response is a triangular alternating function. a) TT r r n * = b) TT r r n * .= 17 Fig. 3.21 - Speed response to the alternating square wave q-axis current command for the rated d- axis current and open speed loop. . ω ωω ω ω ωω ω * = -40 -30 -20 -10 0 10 20 30 40 Torque (Nm) -5 0 5 10 15 20 25 30 35 Stator current components (A) 0 0.01 0. 02 0.03 0.04 0.05 Time (s) e. E Levi, 20 01 59 -8 -6 -4 -2 0 2 4 6 8 Current (A) 0 0.005 0.01 0.015 Time (s) a. current waveform 0.001 0.01 0.1 1 10 Current (A) 0 2 4 6 8 10 12 Frequency