The DQ Analysis of a Variable Reluctance Stepper Motor

David J. Kish. M.S.E.E. , Purdue University. August 1089. The DQ Analysis of a Variable Reluctance Stepper'. Motor. Major Professor: Paul Krause Variable reluctance stepper motors are commonly used in applications where incremental motion is desired. Recently, these motors have been Used in situations which call for a large stepping rate; so large in fact, that the motor hums continuously. Although the equations which describe the stepper motor in this mode are decoupled, they are not time invariant. In order to eliminate the time varying terms, Park’s transformation is applied to the motor equa­ tions. It is found that this transformation does not render the system equa­ tions time invariant, so an averaging technique is employed which does provide the desired time invariance. The averaged model is evaluated by comparing its dynamic response with that of a derailed simulation of the motor in a variety of operating modes. It is found that the averaged model does not adequately represent the motor dynamics under most operating conditions. CHAPTER 1 . ;; ■ '• INTRODUCTION'. ; This thesis deals with Stepper motors. In general, there are two types of stepper motors, the permanent magnet motor and the variable reluctance motor. The latter type is ths subject of this thesis. The name "stepper motor" is appropriate since there are a discrete number of locations at which the rotor will come to rest when the windings of the motor are supplied by a dc source. When the dc source is applied to the; different windings successively, the motor steps from position to position. Stepper motors have often been used in conjunction with digital control systems. If each discrete rotor position is represented by a binary number, interfacing the digital output of the controller with the motor is not difficult. Positioning systems Which use stepper motors can be quite precise, in fact, these motors have been used to position radar antennas to within 30 seconds of arc of a desired location. Another application which takes advantage of both the positioning capabilities of the motor as well as its inherent digital system compatibility is a computer disc drive. Here, the disc head is accurately moved from sector to sector in order to read the information stored on the disc. Stepper motors have certain practical features which make them particularly attractive to industry in terms of both manufacturing and applications. First, stepper motors are relatively easy to assemble. Because these motors are designed with "teeth" on the stator, it is not difficult to wind the stator. From an economic standpoint, step motor drivesystems can be designed which use only one power semiconductor per machine phase. Also, control systems can be quite simple, as the motor’s positioning capabilities do not require a closed-loop controller. Like induction motors, stepper motors d.o not have slips rings or brushes which are features of dc and synchronous machines. For this reason, stepper motors can be used in hostile, possibly explosive environments where the use of a motor with brushes is not advisable. For the most part, these motors have been used in the positioning applications for which they are particularly suited. Recently, stepper motors have been used in both open and closed loop systems for continuous speed operation. In this mode, stepper motors can produce more power per given volume than Can an induction motor with about the same power factor. Rotor heating is reduced in the stepper motor since the machine is not designed to have rotor currents flowing. The familiar torque speed curve of an induction machine is quite useful in the design of systems using this motor. This curve can be predicted analytically [1],[2]; however, in the case of the stepper motor, such curves are often obtained experimentally. Transfer function formulations are helpful in system design; in fact, the transfer function of a dc motor is often used as a practical example of a transfer function in control systems texts [3]. A corresponding simple transfer function for the stepper motor does not exist. When stepper motors are operated in the continuous mode (also referred to as high frequency operation),a velocity oscillation is superimposed on the synchronous speed of the motor. These oscillations can reduce the torque produced by the motor and cause a loss of synchronism. This loss of synchronism is referred to as parametric instability. A means of analysis which would provide an expression for a torque speed curve while also showing the parametric instability of the motor would be useful. Perhaps equally useful is a technique by which the machine equations can be couched to permit linearization and a transfer function formulation. Reference frame theory has been used for such purposes with both induction and synchronous machines, and it will be applied in the analysis which follows. This thesis is divided into five additional chapters. Chapter 2 discusses the operation of a variable reluctance stepper motor and presents the basic equations which are used in its analysis. In Chapter 3, the equations which were derived in Chapter 2 are transformed to the dq reference frame, and certain simplifying assumptions are made. Chapter 4 and Chapter 5 present simulation results for the complete and simplified models of the motor. Finally in Chapter 6, simulation results are summarized, and recommendations on the use of the simplified model are made. CHAPTER 2 DESCRIPTION OF A VARIABLE RELUCTANCE STEPPER MOTOR ^ '. '/;'2.1vIntroduction. -'VThe machine to be analyzed is a two pole, three stack, variable reluctance stepper motor. This particular device has two rotor teeth and two stator teeth. Fig 2.1 shows a cross section of a three stack variable reluctance machine. While this figure shows the physical construction of a variable reluctance stepper motor, it is more complicated than the one under analysis due to a greater number of stator and rotor teeth. In this chapter, a general explanation of motor Operation will be provided, and an expression for step length is obtained. Next, an electrical model of the motor will be developed by examining one phase (stack) of the motor. The mechanical equations which describe the motor will be explained, and a state variable representation of the motor will be obtained. 2.2 Motor Operation As its name implies, the operation of this motor depends upon the variable reluctance path the device provides to the flux developed by current in each phase winding. In a coupledelectro-mechanical system, a force is developed in order to minimize the reluctance path seen by the flux linkages. A familiar application of this principle is an ordinary electromagnet. When current flows in the magnet windings, a force is developed which "picks up" a, piece of metal. Originally, there was a high reluctance air-gap between the electromagnet and the metal. When the metal and magnet are touching, there is very little air gap, rather, a low reluctance path through the metal exists for the flux linkages. The force generated by the magnet acted to minimize the reluctance in the system. Examination of the geometry of the stepper motor shows there is a minimum and maximum reluctance position. Fig 2.2 shows the minimum reluctance rotor position for the motor, while the maximum reluctance position is shown in Fig 2.3. When the windings are energized, the flux must crossi a large air gap in the maximum reluctance position. In the minimum reluctance position, the flux is for the most part contained within the stator back iron and the rotor. Since a force (torque) is developed to minimize the reluctance path, it is expected that a torque to move the rotor from the position shown in Fig 2.3 to that of Fig 2.2 exists. If the rotor is positioned in the minimum reluctance position, such a torque should not exist. This is what occurs in practice. Assume the windings iu stack A have a current flowing in them. The rotor will move to the minimum reluctance position and remain there. Now the minimum reluctance position for stack A is not the minimum reluctance position for stack B or stack C. If the current flowing in stack A is shut off, and at the same time, a current is made to flow in the windings of stack B, -the rotor will move to the minimum reluctance position of stack B. The rotor has just "stepped" from one minimum reluctance ppsition to another. An expression for the distance between these minimum reluctance positions is desired. The tooth pitch (TP), i.e., the angular distance between two rotor teeth, is defined as TP 27T (2.2-1) where RT represents the number of rotor teeth. If each stack is excited in turn, e.g., as, bs, cs, as the rotor will move one tooth pitch. The step length (SL) then is given by SL = 27T rt" n (2.2-2) The symbol N in EQ (2.2-2) represents the number of stacks in the machine. So far, step operation has been discussed for a no-load condition on the machine. A. brief explanation of motor operation with a load torque follows. If a mechanical load is attached to the rotor while a set of windings is energized, the motor will develop a torque to try to move the rotor to the minimum reluctance position. The load will oppose this motion, and the rotor will come to rest at a position 8, where When the next winding in the excitation sequence is energized, the rotor will move to a position displaced from the new minimum reluctance position by