A Novel Solution for Variable Speed Operations of Switched Reluctance Motors Abhinav A. Kalamdani Dept. of Instrumentation Engineering, R.V.College of Engineering, Bangalore, India, 560059 Email:
[email protected] Abstract— Switched Reluctance Motors(SRMs) have been a ray of hope for the challenging motion requirements of the world. There have been various attempts to build the converters and drives for the SRM. There has been a similar attempt here. Most of the applications of high torque machines require variable speed operations. The proposed drive topology is being designed considering the variable speed operation. The C-dump converter is modified for simplicity and practicability. The torque ripple has also been reduced to a great extent. The variable speed drive is being tested by using it for a Position Control system. Since the control signal is continuously varying, the robustness of the drive can be tested out clearly. The converter characteristics were simulated and an experimental setup was built to implement the control system. The simulation results have shown good current and torque profiles and the experimental results have shown a satisfactory tracking performance, thus proving that the proposed drive is performing well.
I. I NTRODUCTION Among the many types of motors, Switched Reluctance Motors (SRM) have come up as a very viable alternative for most applications against the conventional motors. SRMs are fast becoming a popular alternative to Induction Motors (IM) in the variable speed drive market. Though SRM is an old member of the family, due to electronics and complex control requirements, it did not come into usage. SRM has a simple and robust structure with no windings on the rotor of the machine and hence it is inherently less expensive. It has high torque-to-inertia ratio and the high starting torque without the problem of in-rush currents and its power density is comparable to that of an induction machine. The maximum operating speed and maximum rotor temperatures it can withstand are higher when compared to other machines of similar ratings. Each phase winding of the SRM is independent of the other phase windings and this makes the machine highly reliable. However there are a few limitations to SRM. The pulsed nature of torque production, which leads to torque ripple and acoustic noise. To have high torque-to-volume ratio, the air gap between the stator and rotor should be very small. This leads to less manufacturing tolerances and increased acoustic noise. A Switched Reluctance Motor (SRM) is an electric motor in which torque is produced by the tendency of its movable part to move to a position of least reluctance, which corresponds to the position of maximum inductance. It is a doubly salient, singly excited motor. That is, the SRM has salient poles on both the rotor and the stator, but only the stator poles carry
windings. The rotor tries to get to a position of minimum reluctance by aligning itself with the stator magnetic field. In the presence of a rotating magnetic field, the rotor tries to rotate along with the rotating magnetic field to always be in a position of minimum reluctance. Thus, exciting the stator phase windings of the motor in a particular sequence and consequently, controlling the rotating magnetic field, we can control the movement of the rotor. The number of poles on the stator and on the rotor is usually not equal. This is to avoid the eventuality of the rotor being in a state of producing no initial torque, which occurs when all the rotor poles are locked in with the stator poles. Here, the diametrically opposite stator pole windings are connected in series and they form one phase. Thus, the six stator poles constitute three phases. When the rotor poles are aligned with the stator poles of a particular phase, the phase is said to be in an aligned position. Similarly, if the inter-polar axis of the rotor is aligned with the stator poles of a particular phase, the phase is said to be in an unaligned position. II. M ATHEMATICAL M ODEL The mathematical model of the SRM is not consistent over a complete region, and the parameters are changing continuously hence showing highly non-linear characteristics. However segregation is done by considering only the operational region of the motor. The SRM motoring action is considered for mathematical analysis with general parameters characterizing the motion. The phase voltage equation in a Switched reluctance motor can be written as: dλ V = iR + (1) dt where, V is the DC bus voltage, i is the instantaneous phase current, R is the phase winding resistance and λ is the flux linking the phase coil. The flux linkage, mainly constituting of inductance L and current i is differentiated and the equation is given by (2). The inductance of the SRM is continuously changing with respect to the rotor position. The inductance profile is as shown in the figure (1) is considered to be linear. And hence it can be described by equation (3) for increasing inductance. di dL +i ω dt dθ dL θ L(θ) = Lu + dθ
V = iR + L
(2) (3)
performance and the cost of the drive are highly affected by performance of the converters. The phase independence and unipolar current requirement have generated a wide variety of converter topologies for SRM drives. Many different topologies have emerged with reduced number of switches and faster commutation time through continued research. The various converter topologies should be able to satisfy the following requirements, may be not completely but quite closely at least [2]. 1. Each phase of an SRM should be able to conduct independent of the other phases. 2. The converter should be able to demagnetize the phase before it steps into the generating region if the machine is operating as a motor and should be able to excite the phase before it steps into the generating region if operated as a generator. 3. The converter should be able to energize another phase before the off going phase has completely demagnetized. 4. The converter should be able to utilize the demagnetization energy from the off-going phase in a useful way by either feeding it back to the source or using it in the next conducting phase. 5. In order to make the commutation period small the converter should be able to demagnetize the off going phase in a very short time. 6. The converter should be able to freewheel during the chopping period to reduce the switching frequency.
Aligned Position
Lmax
Phase Inductance Un−Aligned Position
Lmin θon
Fig. 1.
θoff
Rotor Position
Inductance Profile of SRM
where, ω is the rotor speed, θ is the rotor angular position, and L(θ) is the instantaneous phase inductance and dL dθ being the slope for change of inductance with rotor position, which is typically constant over a maximum range. The term iω dL dθ is the back emf generated by the motoring action. It very much depends on the sign of inductance slope. A positive slope generates positive back emf hence generating a positive and useful torque. However if the current persists in the coil even after the slope becomes negative, the back emf becomes negative and generates negative torque opposing the motion. The rate of flow of energy can be obtained by multiplying the voltage with current and can be written as: V i = i2 R + Li
dL dL + i2 ω dθ dθ
(4)
A. Simplified C-Dump Converter
or
d 1 1 dL P = i2 R + ( Li2 ) + i2 ω (5) dt 2 2 dθ The first term of the above equation represents the rate of increase in the stored magnetic field energy while the second term is the mechanical output. Thus, the instantaneous torque can be written as: 1 dL (6) T = i2 2 dθ hence we see that torque is proportional to square of the current, so the direction of the torque is independent of the polarity of the current flowing in the coil. Hence it can be inferred here that the sign of the inductance slope decides the direction of the torque and the motoring action.
The Simplified C-Dump Converter is proposed here. The topology of the converter is as shown in the figure (2). The topology is a modified version of Mir’s Energy Efficient C-Dump Converter II [3]. The node C0 is the common point of the 3-phase coil. The other nodes are C1, C2, C3 all connected to the collectors of the power switches Z1, Z2 and Z3 respectively. The diodes D1, D2, D3 are connected such that they provide a path for the current to flow after the corresponding power switch is turned off. The dump capacitor Cd is connected to the n-point of the diodes. As the energy dumping takes place in the capacitor, a voltage is built up across the capacitor, which tends to be around the supply voltage Vdc. This voltage has polarity opposite to the direction of the flow of the current. Hence it provides a negative voltage to decay the residual energy faster and store it in the form of charge in the dump capacitor.
III. SR M OTOR D RIVE The SRM is not a self commutated and driven motor. It needs a very high performance drive system. The drive consists of a converter for delivering the power, and a commutation module with a rotor position sensor for sensing the relative position of the rotor with the stator phase. This provides the the commutation signal for the converter. There is a facility in the switching logic for direction control of the motor. The converter consists of a facility for electronic voltage control.
The diode D4 provides the path for free-wheeling. It also dumps the energy in the capacitor, back into the upcoming phase thus improving the time response of the rise of the current in the coil. As it goes on in [3], the voltage across the dump capacitor is regulated at Vdc.
One of the main aspects of the research in switched reluctance motor drives has been the converter design. The 2
dL di dL ωt) + iR + iω (7) dθ dt dθ Where Vi is the input voltage, and α is the duty cycle of the PWM. Considering the duty cycle and the speed to be constant, the phase current is given by the equation (8). = (Lu +
Fig. 2.
" Vi 1− i(t) = R + ω dL dθ
ωt dL − 1+ Lu dθ
Simplified C-Dump Converter
−
1+
R ωt dL dθ
#
(8)
Design of the dump capacitor is carried out based on the voltage built across the capacitor with absence of the diode D4 as in [3]. The voltage balance equation during the commutation or when the power switch is OFF can be given by: Z dλ 1 i dt + Vci = 0 (9) + dt Cd
The power switches are driven by PWM operation for controlling the input voltage to the phases, hence controlling the current, and varying the torque and the speed. This converter topology is well suited for bi-directional operation of the motor, and hence can be used in control and variable speed applications.
Where λ is the phase flux linkage, Cd is the dump capacitor value, i is the phase current and Vci is the initial capacitor voltage. Neglecting saturation, the equation can be expressed as: Z dL 1 di + L + iω i dt + Vci = 0 (10) dt dθ Cd
The working of the converter goes on with two modes of operation. In the first mode, the switch Z1 is ON, hence energizing the first phase. The current starts building up. In the second mode when Z1 goes OFF, the residual energy starts dumping in the capacitor Cd and if the voltage across the capacitor increases beyond Vdc, then the diode D4 gets forward biased and the current starts freewheeling through diodes D1 and then D4. When the commutation takes place next phase is energized and the current builds up in the coil. In the meanwhile if the dump capacitor voltage has gone above Vdc, then the capacitor starts discharging through D4 into the excited phase, thus supplying the same energy which the phase had dumped in it during mode two. This operation conserves some amount of residual energy and uses the same during excitation of the upcoming phase. However there is a small disadvantage with this working. During PWM operation when the switch Z1 has gone OFF, the current in the phase decays down faster into the capacitor Cd, than just freewheeling into the coil, this actually makes the necessity for having higher switching frequency for PWM.
The back emf term iω dL dθ will rapidly decrease towards zero is zero in aligned position. A design equation since dL dθ for Cd will be derived neglecting the back emf term and considering the inductance at aligned position. The equation for the decaying phase current is given by (11). r Cd t √ i(t) = Vci sin + La La Cd t (11) + Icos √ La Cd
The time required (Td) by the phase winding to discharge completely into the dump capacitor can be derived from the equation (11) and is given by: r ! p I La (12) Td = La Cd tan−1 Vci Cd
Where I is the phase current at the beginning of commutation. The change in voltage across the dump capacitor (∆Vc ) is given by: Z Td 1 ∆Vc = i dt (13) Cd 0
During the commutation when the phase is energized, the PWM operation starts and while in this mode of operation, the average current builds up till the desired level which is specified as the percentage of the maximum current that is permitted. This percentage is specified as the duty cycle to the PWM. The voltage equation for building up of the current in the phase is given by equation (2). The inductance during this mode is changing continuously according to the equation (3). Considering the motor speed to be constant, θ = ωt and by substituting equation (3) in (2) the equation describing the rise of the phase current is given as (7).
Substituting for i and Td in (13) and simplifying: r La ∆Vc = Vci2 + I 2 − Vci Cd
(14)
The dump capacitor can now be expressed as: Cd = I 2
Vi = αVdc 3
La 2
(∆Vc + Vci ) − Vci2
(15)
However, in this case the speed and the duty cycle are also changing. The input to the system is the voltage, that is the duty cycle which now becomes the control input to the system. Considering the mechanical part of the system, the torque equation (6) can be used for modeling. The general rotational dynamics of the system are given by equation (16). d2 θ dθ +B (16) 2 dt dt where J and B are the inertia and the viscous co-efficient of the motor system. Tm is the torque generated by the motor. The equation (6) gives the relation of the current with the torque. Substituting for Tm in (16) the electro-mechanical system equation (17) is obtained. Tm = J
Fig. 3.
Low Resolution Position Sensor for Commutation
i2 dL d2 θ dθ =J 2 +B (17) 2 dθ dt dt This electro-mechanical relation is used to obtain the expression for phase current i in terms of the speed and acceleration. di is also obtained subsequently. The derivative of the current dt These are given by the equations (18) and (19). " 2 # 12 d θ 2 dθ i = dL J 2 + B (18) dt dt dθ " 2 #− 12 d θ dL dθ di J 2 +B = 2 · dt dθ dt dt 3 d2 θ d θ · J 3 +B 2 (19) dt dt
The voltages Vci and ∆Vc are considered to be in accordance with the stated modes of operation. Choosing the values around the Vdc is needed and the approximate value of the dump capacitance is computed. B. Commutation And Direction Control The SRM is an externally commutated motor. This task is being accomplished by the position sensor. The configuration of the sensor is such that commutation occurs at every step angle of the motor, and the signal is sent to the converter for switching the phase. Which phase has to be excited is determined by the unique signals at every phase. Hence three unique signals have to be generated for three phases to be excited sequentially. A low resolution optical encoder is used to sense the rotor position with respect to the stator as shown in figure (3). This position information is decoded and based on the excitation state table, the commutation commands are given to the base drives of power switches. A direction control bit is configured, and is used in the bidirectional operations. This drive topology for the SRM is being designed for a typical variable speed drive application. The performance of this variable speed SRM drive is tested by using it for a simple position control application.
The step input to the system is given with required position specified as θd and the actual position is measured as θa and the tracking error is e given by equation (20).
The multi-position proportional control law is described by the equation (21). Kp1 |e| for |e| > δ y= (21) Kp2 |e| for |e| ≤ δ
Where Kp1 and Kp2 are the different gains, u is the control signal, δ is the error band for which the different gains are defined. As specified earlier the control signal is the input to the system and the duty cycle α is the input, hence the relation (22) is obtained. α(t) = u(t) (22)
IV. P OSITION C ONTROL The simple feedback control is being implemented here. The SR motor block is preceded by the power converter which provides the interface between the controller and the motor. The position of the motor shaft is sensed and fed back to the error generator. The tracking error is computed and fed to the controller. The controller acts upon the error and generates the corresponding control signal which is sent to the power converter. The control law being used is a simple Multi-Position Proportional controller. Recalling the motor voltage equation (7), the control signal can analyzed.
In the equation (7) the sides are interchanged and the output dynamics of the system are obtained when the control signal is fed as the input to the system. This is defined by the equation (23). dL di Lu + ωt + iR + dθ dt dL = Vi + iω dθ = α(t)Vdc
Vi = αVdc = (Lu +
di dL dθ ωt) dt
(20)
e = θ d − θa
+ iR + iω dL dθ
= u(t)Vdc 4
(23)
di are defined by where the phase current i and derivative dt equations (18) and (19) in terms of speed and acceleration. Hence the position control is achieved.
configured for very low torque ripple within a band of 10% of the average torque as shown in the figure (7). The capacitor value being chosen based on the computation is 1000µF for an input DC voltage Vdc of 20V . The current profile at higher speed for the same Vdc but a higher duty cycle is quite triangular in shape but however the symmetry of the rise and decay of the phase current and a perfect overlapping have generated quite lower torque ripple. Thus the simulations for the Simplified C-dump Converter have shown that the torque profile is quite acceptable at lower speeds and also the higher speeds. The presence of the torque ripple during lower speeds is distinctly seen and the proposed converter tends to solve this problem by showing very low ripples. However at higher speeds, the presence of torque ripples is not felt much, although the Simplified C-dump converter shows lower ripples even in this range.
V. R ESULTS
Fig. 4.
Block Diagram of the Experimental Setup
The Simplified C-dump Converter with direction and the input voltage control was built for the given SRM prototype. The block diagram for the setup is as shown in the figure (4). The IGBTs were used as power switches for the switching of the phases. The voltage and current handling capacities of these devices are well suited for SRM drives and the maximum switching frequencies of these devices have made the PWM operation possible. The commutation and direction control logic was built using the simple logic gates in integration with the low resolution position sensor. Hence the costing of the whole SRM drive is well within the lower range. An experiment for the load torque characteristics was performed on the drive. The experiment was performed without the dump capacitor and also with the dump capacitor. The figure (9) shows the results of the experiment. It is clearly seen that for same applied load torque, the speed reading for the drive with dump capacitor was much higher than the one for without the dump capacitor. This result shows that the efficiency of the drive has also increased considerably. The control system was built for the SRM drive. An optical encoder capable of reading up to 0.36◦ was used. The data acquisition hardware used for the setup was Labjack U12. The control algorithm was written on Matlab which communicated
The designed system is evaluated based on the simulation and the experimental results. The above built model of the Simplified C-dump Converter is simulated to obtain the current profiles, at lower and higher speeds for 500RPM and 2000RPM, and these are shown in figures (5) and (6) respectively. Based on the current profiles, the torque is also computed and the torque profiles for 500RPM and 2000RPM are shown in figures (7) and (8) respectively. The duty cycle being the control signal, is changed for obtaining various speeds right from zero to the maximum speed. The experimental SRM prototype that is being used has the following characteristics. Geometry- Stator poles: 6 , Rotor poles: 10 Number of Phases: 3 with opposite connected in series Coil Resistance: 16 Ω Unaligned Inductance: 30 mH Aligned Inductance: 35 mH Maximum Current carried by coils: 2.0 A Step Angle: 12◦ −1 Inductance Gradient( dL dθ ): 0.024 H rad Maximum Holding Torque: 0.05 N m
Fig. 5.
Current Profile at 500RPM
The current profile for six commutations at lower speed, shows a perfect overlapping at the instant of commutation. The rise time of the current and the decay time are perfectly
Fig. 6.
5
Current Profile at 2000RPM
VI. C ONCLUSIONS
with the DAQ card and the card provided the control signals consisting of the direction and the PWM duty cycle commands. The experiment was carried out at no load conditions. The controller was tuned iteratively for lower and higher step inputs from as low as 5◦ to as high as 2000◦. The results obtained were acceptable. The figures (10) and (11) show the step responses for the command angles of 180◦ and 1000◦. A slight overshoot was prevalent in all the responses, however the settling time and steady state errors were acceptable. The steady state error was confined to a band of ±2◦ about the command angle. Since the torque ripple shown at lower speeds was less, the speeds were limited to the lower range.
The results shown by the simulations and the experiment carried out, have proved that the proposed SRM drive topology is performing well for variable speed drives especially for automobile applications, where slight torque ripples are acceptable. The implementation of the position control of the SRM has proved the performance of the proposed drive topology, and also the implemented control system can be used in places where not very robust dynamic performance of position control is required. The mathematical model of the control is highly non-linear, which tends to divert the main motive of attaining an adequate control performance. But the experiment revealed that even with a simple control algorithm, the performance was acceptable. R EFERENCES
Fig. 9.
[1] P.C.Sen, Principles of Electric Machines And Power Electronics, 3rd ed. John Wiley, 2001. [2] Michael T. DiRenzo,Switched Reluctance Motor Control – Basic Operation and Example Using the TMS320F240, Application Report, SPRA420A - February 2000. [3] Sayeed Mir, Iqbal Husain, and Malik E. Elbuluk,Energy-Efficient CDump Converters for Switched Reluctance Motors, IEEE Trans. Power Electronics, vol. 12, no. 5, September 1997. [4] Mohammed S Arefeen,Implementation of a Current Controlled Switched Reluctance Motor Drive Using TMS320F240, Application Report: SPRA282 – September 1998.
Fig. 7.
Torque Profile at 500RPM
Fig. 8.
Torque Profile at 2000RPM
Fig. 10.
Step Response for 180 deg
Torque Speed Characteristics for With and Without dump capacitor
Fig. 11.
Step Response for 1000 deg
6
[5] Yinghui Lu, Instantaneous Torque Control Of Switched Reluctance Motors, A Thesis Presented for the Master of Science Degree The University of Tennessee, Knoxville, August 2000. [6] R. Krishnan, R. Arumugam, J. Lindsay, Design Procedure for Switched Reluctance Motor, IEEE Trans. Industry Applications, Vol. 24, No.3, May/June 1988, P. 456-461. [7] Borka, J., K. Lupan, L.Szamel, Control aspects of Switched Reluctance Motor Drives, Industrial Electronics, 1993. Conference Proceedings, ISIE’93 - Budapest., IEEE Intl. Symp., 1-3 June 1993, P. 296 - 300.
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