Chapter 8: Variable - Reluctance Machines and Stepping Motors

Module by: NGUYEN Phuc

• Variable-reluctance machines (VRMs) are perhaps the simplest of electrical machines. They consist of a stator with excitation windings and a magnetic rotor with saliency. Rotor conductors are not required because torque is produced by the tendency of the rotor to align with the stator- produced flux wave in such a fashion as to maximize the stator flux linkages that result from a given applied stator current. Torque production in these machines can be evaluated by using the techniques of Chapter 3 and the fact that the stator winding inductances are functions of the angular position of the rotor.
• Although the concept of the VRM has been around for a long time, only in the past few decades have these machines begun to see widespread use in engineering applications. This is due in large part to the fact that although they are simple in construction, they are somewhat complicated to control. The position of the rotor must be known in order to properly energize the phase windings to produce torque. It is the widespread availability and low cost of micro and power electronics that has made the VRM competitive with other motor technologies in a wide range of applications.
• By sequentially exciting the phases of a VRM, the rotor will rotate in a step- wise fashion, rotating through a specific angle per step. Stepper motors are designed to take advantage of this characteristic. Such motors often combine the use of a variable-reluctance geometry with permanent magnets to produce increased torque and precision position accuracy.

• VRMs can be categorized into two types: singly- salient and doubly-salient. In both cases, their most noticeable features are that there are no windings or permanent magnets on their rotors and that their only source of excitation consists of stator windings.
• To produce torque, VRMs must be designed such that the stator-winding inductances vary with the position of the rotor.
• Figure 8.1a shows a cross-sectional view of a singly-salient VRM, which can be seen to consist of a nonsalient stator and a two-pole salient rotor, both constructed of high-permeability magnetic material. In the figure, a two-phase stator winding is shown although any number of phases is possible.
• Figure 8.2a shows the form of the variation of the stator inductances as a function of rotor angle θm for a singly-salient VRM of the form of Fig. 8.1 a. Notice that the inductance of each stator phase winding varies with rotor position such that the inductance is maximum when the rotor axis is aligned with the magnetic axis of that phase and minimum when the two axes are perpendicular. The mutual inductance between the phase windings is zero when the rotor is aligned with the magnetic axis of either phase but otherwise varies periodically with rotor position.
• Figure 8.lb shows the cross-sectional view of a two-phase doubly-salient VRM in which both the rotor and stator have salient poles. In this machine, the stator has four poles, each with a winding. However, the windings on opposite poles are of the same phase; they may be connected either in series or in parallel. Thus this machine is quite similar to that of Fig. 8.1a in that there is a two-phase stator winding and a two-pole salient rotor. Similarly, the phase inductance of this configuration varies from a maximum value when the rotor axis is aligned with the axis of that phase to a minimum when they are perpendicular.
• Unlike the singly-salient machine of Fig. 8.1 a, under the assumption of negligible iron reluctance the mutual inductances between the phases of the doubly-salient VRM of Fig. 8.1 b will be zero, with the exception of a small, essentially-constant component associated with leakage flux. In addition, the saliency of the stator enhances the difference between the maximum and minimum inductances, which in turn enhances the torque-producing characteristics of the doubly-salient machine. Figure 8.2b shows the form of the variation of the phase inductances for the doubly-salient VRM of Fig. 8.lb.
• The relationship between flux linkage and current for the singly-salient VRM is of the form

Figure 1

Figure 2

Figure 3

• These inductances are periodic with a period of 18001800 size 12{"180" rSup { size 8{0} } } {} because rotation of the rotor through 180o180o size 12{"180" rSup { size 8{o} } } {} from any given angular position results in no change in the magnetic circuit of the machine.
• The electromagnetic torque of this system can be determined from the coenergy as

• For the double-salient VRM of Fig. 8. lb, the mutual-inductance term dL12(θm)/dθmdL12(θm)/dθm size 12{ ital "dL" rSub { size 8{"12"} } \( θ rSub { size 8{m} } \) /dθ rSub { size 8{m} } } {} is zero and the torque expression of Eq. 8.5 simplifies to

• Equations 8.6 and 8.7 illustrate an important characteristic of VRMs in which mutual-inductance effects are negligible. In such machines the torque expression consists of a sum of terms, each of which is proportional to the square of an individual phase current. As a result, the torque depends only on the magnitude of the phase currents and not on their polarity. Thus the electronics which supply the phase currents to these machines can be unidirectional; i.e., bidirectional currents are not required.
• Since the phase currents are typically switched on and off by solid-state switches such as transistors or thyristors and since each switch need only handle currents in a single direction, this means that the motor drive requires only half the number of switches (as well as half the corresponding control electronics) that would be required in a corresponding bidirectional drive. The result is a drive system which is less complex and may be less expensive.
• The assumption of negligible mutual inductance is valid for the doubly-salient VRM of Fig.8.1b both due to symmetry of the machine geometry and due to the assumption of negligible iron reluctance. In practice, even in situations where symmetry might suggest that the mutual inductances are zero or can be ignored because they are independent of rotor position (e.g., the phases are coupled through leakage fluxes), significant nonlinear and mutual-inductance effects can arise due to saturation of the machine iron.
• At the design and analysis stage, the winding flux-current relationships and the motor torque can be determined by using numerical-analysis packages which can account for the nonlinearity of the machine magnetic material. Once a machine has been constructed, measurements can be made, both to validate the various assumptions and approximations which were made as well as to obtain an accurate measure of actual machine performance.
• The symbol Ps is used to indicate the number of stator poles and PrPr size 12{P rSub { size 8{r} } } {} to indicate the number of rotor poles, and the corresponding machine is called a Ps/PrPs/Pr size 12{P rSub { size 8{s} } /P rSub { size 8{r} } } {} machine.
• Example 8.1 examines a 4/2 VRM.

Figure 4

Figure 5

• Example 8.1 illustrates a number of important considerations for the design of VRMs. Clearly these machines must be designed to avoid the occurrence of rotor positions for which none of the phases can produce torque. This is of concern in the design of 4/2 machines which will always have such positions if they are constructed with uniform, symmetric air gaps.
• It is also clear that to operate VRMs with specified torque characteristics, the phase currents must be applied in a fashion consistent with the rotor position. For example, positive torque production from each phase winding in Example 8.1 can be seen from Fig.8.4c to occur only for specific values of θm. Thus operation of VRMs must include some sort of rotor-position sensing as well as a controller which determines both the sequence and the waveform of the phase currents to achieve the desired operation. This is typically implemented by using electronic switching devices (transistors, thyristors, gate-turn-off devices, etc.) under the supervision of a microprocessor-based controller.
• Although a 4/2 VRM such as in Example 8.1 can be made to work, as a practical matter it is not particularly useful because of undesirable characteristics such as its zero-torque positions and the fact that there are angular locations at which it is not possible to achieve a positive torque.
• The analysis of VRMs is conceptually straightforward. In the case of linear machine iron (no magnetic saturation), finding the torque is simply a matter of finding the stator-phase inductances (self and mutual) as a function of rotor position, expressing the coenergy in terms of these inductances, and then calculating the derivative of the coenergy with respect to angular position (holding the phase currents constant when taking the derivative). The electric terminal voltage for each of the phases can be found from the sum of the time derivative of the phase flux linkage and the iR drop across the phase resistance.
• In the case of nonlinear machine iron (where saturation effects are important), the coenergy can be found by appropriate integration of the phase flux linkages, and the torque can again be found from the derivative of the coenergy with respect to the angular position of the rotor.
• Although VRMs are simple in concept and construction, their operation is some what complicated and requires sophisticated control and motor-drive electronics to achieve useful operating characteristics.

• Practical VRM drive systems (the motor and its inverter) are designed to meet operating criteria such as

• Low cost.
• Constant torque independent of rotor angular position.
• A desired operating speed range.
• High efficiency.
• A large torque-to-mass ratio.
• A compromise must be made between the variety of options available to the designer. Because VRMs require some sort of electronics and control to operate, often the designer is concerned with optimizing a characteristic of the complete drive system, and this will