A Novel Solution for Variable Speed Operations of Switched Reluctance Motors Abhinav A. Kalamdani Email: kalamdani@ieee.org
Contents Introduction Mathematical Model SR Motor Drive - Simplified C-Dump Converter - Commutation And Direction Control Position Control Results Conclusions
Introduction SRM is an old member of the motor family, due to electronics and complex control requirements, it did not come into light. SRM has a simple and robust structure with no windings on the rotor of the machine. It has high torque-to-inertia ratio and high starting torque without the problem of in-rush currents Each phase winding of the SRM is independent of the other phase windings and this makes the machine highly reliable.
Introduction (contd.) When the stator coils are excited the tendency of the nearest rotor pole is to align itself with the stator attaining a minimum reluctance path for the magnetic field, and maximum inductance. In the unaligned state the reluctance is maximum and the inductance is minimum. The continuous excitation of the phases produces a rotating magnetic field and the rotor rotates according to magnetic field synchronously.
Introduction (contd.) There are several disadvantages of SRM: - The pulsed nature of torque production, which leads to torque ripple and acoustic noise. - Higher torque-volume ratio needs a small air gap between stator and rotor, which leads to increased acoustic noise and less manufacturing tolerances. - The motor is not self-commutated, it needs a converter and a commutation controller. - The presence of high inductances increases the current rise and decay time thus giving higher negative torques.
Mathematical Model The mathematical model is inconsistent over the entire operation. The parameters change continuously throughout and hence showing a great deal of non-linearity. However segmenting the operation regions is helpful to analyze the motor. The phase voltage equation is given by: where V is DC bus voltage, i is the phase current, R is coil resistance and is the flux linking the coil.
Mathematical Model (contd.) The flux linkage consists of inductance L and current i and equation (1) is differentiated giving (2), where is speed and is rotor position. The inductance is continuously varying with position as shown in the figure according to equation (3), where L u is unaligned inductance.
Mathematical Model (contd.) In equation (3) the term L() is the instantaneous inductance at that rotor position. In equation (2) the term is the slope of the inductance profile and the term is the back-emf generated by the motoring action. The power equation of the motor is given by equations (4) and (5): The equation (5) shows the power consumed by motor. The power is lost in resistance, inductance and the remaining energy is the mechanical output. The third term is equivalent to P mech = T., T is the torque and is given by: The sign of the slope of inductance profile decides the direction of torque.
SR Motor Drive The SRM is not a self-commutated motor, it needs drive consisting of a converter for delivering the power and a commutation module. One of the major research domains of SRM has been the converters, since they decide the performance, efficiency, controllability and cost of the SR drive. The phase independence and uni-polar current characteristic have encouraged the development of various converter topologies. The drive should be able to satisfy the following requirements: - Each phase should be able to conduct independent of other phases. - It should demagnetize the phase before it steps into generating region. - It should energize the next phase before the off-going phase demagnetizes. - The demagnetization energy should be used up either by feeding it to the source or the upcoming phase. - The demagnetization of the phase should be very fast, to reduce the commutation time. - The free-wheeling facility should be provided during chopping mode.
Simplified C-Dump Converter The Simplified C-Dump Converter is a modified version of Sayeed Mir et. al. s Energy Efficient C-Dump Converter II.
Simplified C-Dump Converter (contd.)
Simplified C-Dump Converter (contd.) The diodes D1, D2, D3 are connected such that they provide a path for current to flow when the corresponding power switch is turned off. The demagnetization current flows into the dump capacitor Cd, building a voltage slightly more than Vdc. The polarity of the voltage across Cd is opposing the flow of demagnetizing current, hence it pulls down the current faster and hence tending to build up the charge, hence the voltage across Cd faster, and decreases decay time. During demagnetization the diode D4 is reverse biased since the voltage at n-node i.e. Vdc is higher than that at p-node i.e. capacitor voltage. When capacitor voltage exceeds Vdc, D4 is forward biased and the energy flows through D4 and dumps in the next excited phase which is the low impedance path and decreasing the rise time. When capacitor voltage drops below Vdc, it is now ready to provide high negative voltage for faster demagnetization. This cycle provides an in-built configuration for regulating the capacitor voltage.
Simplified C-Dump Converter (contd.) The converter has no issues considering a bidirectional operation. The power switches are driven by Pulse Width Modulation (PWM), this gives a facility to vary the voltage and hence the current resulting in the variation of the speed and the torque. The duty cycle specifies the percentage of the maximum voltage provided by the source. During PWM operation, the Cd is charged and the diode D4 is forward biased, thus providing a freewheeling path for the current when the power switch is off. The switching frequency of the PWM has to be high since the decay of current is faster in this converter due to the presence of dump capacitor. The rise in the current during PWM is defined by equation: where is the duty cycle of the PWM signal
Simplified C-Dump Converter (contd.) Considering the duty cycle and hence the speed to be constant, the phase current is specified by: Design of the dump capacitance is done by considering the diode D4 to be reverse biased, since it is during this instant the dump capacitor plays a major role and experiences all the stress. Demagnetization dynamics is defined by: which can be broken down to: where Cd is the dump capacitance and Vci is the initial capacitor voltage
Simplified C-Dump Converter (contd.) The decaying phase current is given by equation: where I is the final current reached The current decay time is given by the equation: The change in capacitor voltage is given by: Using Td and i(t) equations the simplified equation is: This equation gives the expression for Cd:
Commutation and Direction Control The commutation of the SRM is achieved using the stator-rotor relative position sensor as shown in the figure. The resolution of the sensor is step angle of the SRM. The signals from the opto-couplers are decoded and based on the excitation state table, the commutation commands are fed to the base drives of the power switches. The excitation state table is decided based on the direction of rotation and controlled by the direction bit.
Position Control The performance of the variable speed drive can be tested out by using it for position control application. The varying control signals are among the highly dynamic signals since the tracking error, which itself is continuously varying, makes the speed also varying. The control law being implemented here is the Multi-position proportional control with position feedback. The mathematical analysis of the control signal can be done by analyzing the dynamics of the motor and the drive. The drive and the electrical analysis can be done by the voltage equation: Here the duty cycle becomes varying and hence the speed. The mechanical rotational dynamics gives the equation: where J and B are inertia and friction coefficient of the rotor, Tm is motor torque
Position Control (contd.) Using the torque equation the electromechanical interface equation is: The expressions for current and its derivative in terms of speed and acceleration are obtained using the above relation: The tracking error e is obtained by difference of desired position (d) and actual position (a). The control signal u is generated by using the control law: Kp1 and Kp2 are gains, is the error band.
Position Control (contd.) The computed control signal is fed to the PWM generator: When the duty cycle is changed, the electromechanical dynamics of the motor also change and they are given by the equation: Referring to the equations (18) and (19) the current and its derivative in the above equation can be replaced by the speed and acceleration expressions. Since the dynamics of the motor are highly non-linear, the simplification of the above equation is difficult. The behavior of the control operation can be better seen by the experimental results.
Results The simulation and experimental results have been provided to prove the analyses and the claims. The SRM prototype that was used for simulation and experimentation was with following characteristics: - Geometry: 6-stator and 10-rotor poles - No. of Phases: 3 with opposite coils connected in series - Phase Resistance: 16 - Unaligned Inductance: 30mH - Aligned Inductance: 35mH - Maximum Current Carried by Coils: 2.0A - Step Angle: 12 - Inductance Gradient ( ): 0.024H/rad - Max. Holding Torque: 0.05N-m Considering these parameters the input voltage Vdc was chosen as 20V. The dump capacitance value was computed and fixed to 1000F.
Results (contd.) The simulation of the current profile and the torque profile at a speed of 500RPM is as shown in the figures. The symmetrical overlapping of the current curves for six commutations shows a really good torque profile with highly reduced torque ripples. The torque ripple is around the band of 10%.
Results (contd.) The simulation of the current profile and the torque profile at a speed of 2000RPM is as shown in the figures. The symmetrical overlapping of the current curves for six commutations shows a really good torque profile with acceptable torque ripples.
Results (contd.) Firstly an experiment was carried to with the converter, with and without the dump capacitor. Referring the figure, the torque speed characteristics show an increase in the speed for same load torque when the dump capacitor was used, and same power being consumed. The inference is that the efficiency of the drive has improved. Efficiency Computations: Load Torque = 0.6 N-mm Vi = 8 V, Iph = 0.016 A, Pe = 0.136 W 1 = 125 rad/s 2 = 185 rad/s Pm1 = 0.0625 W Pm2 = 0.0925 W E1 = 46% E2 = 68% Increase in Efficiency = 48%
Results (contd.) The drive system was built for the SRM and the control system was also setup. The DAQ card was used to provide an interface with the PC and the drive. The high resolution position sensor was fitted to provide the good amount of information for control. The figure shows the block diagram of the implementation of the system.
Results (contd.) The position control was carried out by implementing the control law on MATLAB. The magnitude of the control signal was given to the input of the PWM module and the sign of the control signal was decoded and given as command signal to the direction bit. The controller gains were tuned iteratively for a range of 5 to 2000. The results showed a very acceptable response with good rise time, low steady state error of about ±2 about the final position. A slight overshoot prevailed, but however the other characteristics were impressive. The figures show a step input response for 180 and 1000 final position values.
Results (contd.) Step Response for 180
Results (contd.) Step Response for 1000
Conclusions The results have shown the performance of the proposed variable speed drive for the SRM, to be greatly acceptable. The proposed drive is suitable mainly for automotive applications where the robustness of the motor comes into picture and the issues of efficiency also need to be addressed. The reduced torque ripples can encourage better solutions for many of the control applications. High amount of non-linearity in the motor and its drive makes it very subtle and there is hesitation for usage, hence a good amount of work needs to be done.
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