1. Introduction
Because of the specificity of the work environment, the study of motors with high positioning accuracy and high reliability has become a research focus in the field of aerospace. In contrast with traditional motors, the stepping motor has the advantage of higher efficiency, power density and torque density [
1]. The stepping motor receives open-loop commands from the motor drive; the rotor position of the stepping motor is known simply by keeping track of the input step pulses [
2]. The open-loop control method of the stepping motor could avoid various position sensors that could reduce the reliability of the system. All the advantages make stepping motors widely used in various applications, such as space applications [
3].
The stepping motor could be divided into permanent magnet (PM), variable reluctance (VR) and hybrid stepping motor (HSM) according to the topology of the motor. Combing the advantages of PM and VR, the HSM could provide better efficiency and is widely used in control applications for industries [
4]. A new stator-permanent-magnet HSM(SHSM), in which some permanent magnets were applied to the stator structure, was designed in [
5]; the defects caused by axial magnetic field of rotor permanent magnet were eliminated. The new SHSM had a better performance but the higher production costs led to fewer applications. In recent years, many stepping motors with new structure have been designed [
6,
7,
8]. Ref. [
6] designed a power-optimal hybrid stepping motor with dimensional constraints for space applications. Through finite element analysis, the rotor teeth of the motor were gradually modified to obtain the ideal holding torque and detent torque. Ref. [
7] presented a new structure that contained a 3-section rotor and a 2-section stator with significant torque density increase.
The finite element method (FEM) is an important method to analyze the stepping motor [
8,
9,
10,
11]. The 2D finite element method is rarely used because of the radial and axial flux caused by axially magnetized permanent magnet. Accordingly, the 3D finite element analysis is a more commonly used method for analyzing stepping motors. Researchers used FEM to show the effects of different topologies and geometric designs, such as shapes of PM [
9], air gap [
10] and tooth geometries [
11]. A model transformation-based analysis method that analyzed the 2D equivalent model is presented in [
12]; with verified accuracy, this method was faster than the 3D FEM method.
In the wake of developments in technology in control systems, reliability and fault tolerance has been paid more and more attention. In the field of aerospace, the electric errors of the motor drives, including winding short-circuit and open-circuit fault, occurred, which led to serious consequences [
13]. Meanwhile, compared with the normal temperature and pressure environment on the ground, the space temperature span was huge and lacked air convection and conduction, which seriously affected the temperature field and stress field of the electric drive system; the drastic temperature changes led to the aging and life attenuation of materials and devices [
14]. Therefore, aero motors are more prone to encounter some failures. Improving the reliability of the motor of spacecraft has become a pressing matter.
The fault tolerance of the motor is increased by using materials with the flux tunable characteristics, which could be fully demagnetized in the fault condition [
15]. However, these materials are easy to demagnetize and are prone to problems in practical use.
The redundancy technology was adopted to improve the reliability and security [
16]. Redundancy technology is a technology to increase fault tolerance by adding the same components or systems. The redundancy control method of a traditional drive system runs multiple motors in parallel, the disadvantages of which are complex structure and high cost [
17].
Many new electric drive systems control methods with redundancy technology have been proposed and investigated [
18,
19]. Ref. [
18] proposed a new electric drive system based on a six-phase ten-pole dual-winding fault-tolerant permanent-magnet (DFPM) motor for aerospace applications. Two sets of three-phase full-bridge drive circuits with fault-tolerant control strategy were applied, which is a typical dual redundancy motor drive system. Based on the current fault-tolerant permanent magnet motor drive system, [
19] proposed a short-circuit fault-tolerant operation control strategy for the electrical flight control system under the four-quadrant condition. Ref. [
20] proposed an improved PWM modulation scheme which considered the modulation signal of the fault phase when the fault occurred. Ref. [
21] proposed a fault-tolerant strategy for the asymmetrical half-bridge converter that works as a driving circuit of a switched reluctance motor.
The redundancy technology has been applied to the design of many motors. Much focus has been placed on permanent magnet motors designed with redundancy structures [
22,
23,
24] for their great power density. The research on multiphase permanent magnet motors, such as six-phase eight-pole PM motors, began early; research on fault-tolerant control for PM motors began in the 1960s. The performance of redundancy structure PMSM and fault-tolerant structure PMSM were compared in [
25], where the simulation was carried out under normal and fault conditions, respectively, which showed that the two structures had good capabilities of fault isolation and restraining short-circuit current.
Dual stator winding induction motor possesses the potential to become a part of ac and dc micro grids as it is highly reliable, maintenance free, and economic [
26]. The dual-redundancy switched reluctance motor, which is characterized by design simplicity and easy maintenance due to the absence of windings and magnets, has been equally studied in the field of aerospace [
27].
At present, the research on the application of redundancy technology in the field of motors still gains much attention. In contrast with the motor previously studied [
22,
23,
24,
25,
26,
27], the stepping motor has exhibited a long service life, high precision, light quality, and simple driving circuit. The stepping motor has good prospects for application in high accuracy situations, such as satellite antenna drive system. Nevertheless, there are few achievements about the redundancy design of stepping motor.
A dual-redundancy HSM design scheme was proposed in this paper, which has a double stator windings structure with a single rotor. The detailed design parameters of the motor are given. Experimental results and finite element analysis results are presented in this paper.
The dual-redundancy HSM proposed in this paper had double reliability and could produce twice the torque in contrast with ordinary HSM. The results show that the influence of redundancy coupling was small; the two sets of windings did not have a negative impact on the motor performance. In addition, the motor had the advantages of small volume, light weight, and simple structure. It has good application value in aerospace and other high reliability applications.
2. Mathematical Model
2.1. Basic Equations
The voltage balance equation of two-phase HSM is
where
is the phase voltage,
is the phase resistance,
is the phase current,
is the self-induction and
is the mutual-inductance (x, y = a, b).
Based on the hypocycloid characteristics principle, the electromagnetic torque of HSM can be expressed as
where
Wf is magnetic energy storage and
θ is position angle of the rotor. The electromagnetic torque is further obtained as
For the dual-redundancy HSM proposed in this paper, the basic formula is different. The stator structure is shown as
Figure 1. There were two sets of stator windings in parallel, called Redundancy1 and Redundancy2; each winding was controlled by a corresponding independent drive. The two sets of windings were placed in the same slot; there was an insulating layer to separate the two windings.
The dual-redundancy HSM has two working modes: dual channel mode and single channel mode. The formula will be slightly changed when the motor works in dual channel mode, i.e., the two windings are energized simultaneously.
The new voltage balance equation of the dual-redundancy HSM is
where
and
are the phase voltage of Redundancy1 and Redundancy2.
and
are the resistance,
and
are the current,
and
are the self-induction,
and
are mutual-inductance of the same redundancy,
and
are the mutual-inductance between redundancy1 and redundancy2.
The electromagnetic torque is
Normally, the mutual inductance of HSM is small and can be ignored. In this section, the mutual inductance between two redundant windings is ignored, too. The equivalent circuit of one phase of HSM can be regarded as an R-L circuit, as shown in
Figure 2.
The voltage of one phase can be expressed as
where
Ls the self-induction, the first item of (6) is resistance voltage drop, the second item is the potential induced by the change of flux linkage in phase winding caused by the change of rotor position, the third item is the back EMF,
e(
θ) which was caused by the change of current.
The motor working in single channel mode can be regarded as a conventional HSM; in the case of two stator phases, the back EMF can be approximated by a sinusoidal function [
28] as shown in
where
is the maximum magnetic flux,
p is the number of pole pairs.
The currents are independent of each other in generating torque. The torque generated by the separate phases are as follows
where
ia,
ib are the phase currents.
The total torque is complemented by reluctance torque component called the detent torque
Td which result from the saliency of the dented rotor, the formula is
where
is the maximum detent torque.
The electromagnetic torque produced by a two-phase HSM is equal to the sum of the torque generated by the separate phases and the detent torque:
When the motor works in single channel mode, i.e., one winding of the two sets of redundant windings of the motor works alone, the electromagnetic torque of the motor can be described by (10). When the dual-redundancy HSM works in dual channel mode, the torque can be described as
where
and
are torque corresponding to two redundant stator windings.
Since mutual inductance between two redundant windings was ignored, it can be considered that . Simultaneously, the detent torque of the stepping motor is much smaller than the phase torques. Thus, theoretically, the torque of the motor working in dual channel mode is twice the torque of an ordinary motor. In practice, factors such as magnetic circuit coupling and coupling between redundancies may affect the performance of the motor, which will be tested by simulation and experiment.
2.2. Torque-Frequency Characteristics of Stepping Motor
The torque-frequency characteristic is one of the most important performance targets of stepping motors. When the frequency of the control pulse increases gradually, the torque of the stepping motor drops [
29]. The main reason for the torque drop is that the control winding is inductive, which can delay the change of current.
Generally, the applied pulse voltage is a rectangular wave. When the control pulse frequency is low, the power on and power off time of each phase winding are long. The rise and fall of current in the winding can reach a stable value, and its waveform is close to the rectangular wave. During the power on time, the average value of current is large, and the average torque generated by the motor is also large.
When the pulse frequency increases, with the time constant of the circuit remaining unchanged, the current waveform is quite different from the rectangular wave. The average value of the current during the energization time decreases, and the average torque generated by the motor decreases. The current will decrease faster with the further increase of frequency, which greatly reduces the average torque.
In addition, when the pulse frequency increases and the rotor speed increases, additional rotating EMF is generated in the control winding, which lead to electromagnetic damping. The electromagnetic torque will be further decreased due to the effect of electromagnetic damping [
30].
When the stepping angle and subdivision ratio are determined, the rotational speed is proportional to the frequency, so the torque-frequency characteristic of the HSM can be approximately obtained by obtaining its torque-speed characteristic. The curve is shown in the
Figure 3.
The torque-frequency characteristic is the basis of stepping motor selection. The maximum static torque can be obtained from the curve. More importantly, avoiding out-of-step during operation or speed regulation is important, and the appropriate input frequency can be specified from the torque-frequency characteristic.