A Dual ‐ Redundancy Two ‐ Phase Hybrid Stepping Motor for Satellite Antenna Drive System

: Stepping motors are highly preferred in spacecraft applications because of their simple structure, convenient control and lack of accumulated error. In satellite antenna drive systems, reli ‐ ability and fault tolerance have been paid more and more attention. Accordingly, to enhance the reliability of the electrical system in the field of aerospace, the redundancy technology, which can effectively improve the system reliability while reducing the requirements of components, was ap ‐ plied in the design of hybrid stepping motors (HSM), a dual ‐ redundancy two ‐ phase HSM which has two sets of stator windings was designed in this paper. The mathematical model of stepping motor and the specific structure parameters of the motor are given. The torque ‐ frequency charac ‐ teristic was regarded as the key to measure the performance of the motor proposed in this paper. The results of 3 ‐ D finite element modeling are presented. The corresponding performance experi ‐ ment on the designed motor was carried out and both the simulation results and the experimental results verified the validity and superiority of the dual redundancy HSM.


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 sim-ple 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.

Basic Equations
The voltage balance equation of two-phase HSM is where x u is the phase voltage, x r is the phase resistance, x i is the phase current, xx L is the self-induction and xy L is the mutual-inductance (x, y = a, b).
Based on the hypocycloid characteristics principle, the electromagnetic torque of HSM can be expressed as where f W 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 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 s L is 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  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 m  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 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 dm T 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 x1 T and x2 T are torque corresponding to two redundant stator windings.
Since mutual inductance between two redundant windings was ignored, it can be considered that x1 x2 T T  .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.

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-ofstep during operation or speed regulation is important, and the appropriate input frequency can be specified from the torque-frequency characteristic.

Design of Dual-Redundancy HSM
In order to ensure the reliability and safety of the system, redundancy technology is often used in the field of aerospace.A dual-redundancy motor can be divided into two forms according to sets of winding arrangement: the coaxial structure and the multiplewinding structure [31].
The two redundancies of the coaxial redundancy motor are independent of each other; there is no coupling between circuit and magnetic circuit because of complete isolation of magnetism and electricity.The stator and rotor are independent and connected with each other through rotary shaft.however, this structure has the disadvantage of the large volume and high power dissipation.
The multiple-winding structure placed two sets of windings in one motor.The two sets were electrically independent of each other; there was a coupling between the two spatial magnetic fields.Because two sets of windings were embedded in one core, the iron loss of the system was only that of one core, and the system efficiency was improved.The advantages over the coaxial structure are compact structure, small size, light weight and low power dissipation.
A multiple-winding structure dual-redundancy two-phase HSM was designed in this paper; the stator structure is shown as Figure 1.There were two sets of stator windings in parallel and 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 parameters of dual redundancy two-phase HSM were designed; the design specifications are shown in Table 1.

Transient Magnetic Field Analysis
The finite element model of the double-redundancy HSM was established in this chapter.As there were both axial and radial magnetic fields in the stepping motor, the magnetic path was distributed in three dimensions.The mutual inductance between the two redundancies will affect the performance of the motor when working in dual channel mode.The numerical analysis and experimental verification were necessary to master the variation law of internal parameters and to test dynamic performance.
In this paper, the 3D finite element model of the motor was established for simulation.The specific specifications of the motor are shown in Table 2.
The model of the motor was established by AutoCAD and Maxwell 3D according to the parameters in the Table 1.The rotor was divided into two sections; there was magnetic steel between the two sections of rotor.The two sections of rotor were staggered by half tooth pitch while modeling.Different from the conventional HSM, the stator slot was divided into two groups and put into two independent windings.Figure 4a shows the mesh division.The mesh division near the tooth was denser, where the change of magnetic state is the most complex and the magnetic energy is the most concentrated.The 3D static solver in Magnet was used to analyze the model to verify the rationality of motor structure and parameter design; the result is as shown in Figure 4.It can be seen from Figure 4 that the magnetic flux density of the motor was relatively small.The rotor was analyzed; the results are shown in Figure 5.As excepted, the magnetic flux density near the tooth layer is relatively large.Figure 6 shows the flux density distribution in the air gap; the magnetic density reached the maximum value when the teeth of stator and the teeth of the rotor were aligned.The peak value of the air-gap flux density exhibited appropriate value, with nearly 0.6 T to 0.7 T. Figure 7 shows the flux density distribution of the tooth.It can be seen that the magnetic field distribution of the motor is reasonable.Speed parameters were added to the motor magnet for dynamic analysis of the magnetic flux density distribution at different speeds of the motor working in dual-channel mode, as is shown as Figure 8.It can be seen from the simulation results that the magnetic density of the tooth was high.When two redundant windings were both excited, the magnetic density distribution of the main part of the motor was not large, which is acceptable.At different speeds, the distribution of electromagnetic field in the motor is reasonable.The two redundant windings had little effect on magnetic field distribution of the motor; now, it is necessary to verify whether they had a great impact on the dynamic performance of the motor.The performance indexes of the motor of the simulation results were analyzed.
The simulated load torque was 9 N•cm.To obtain the torque-speed characteristic of the motor, the output torque data at different speeds were measured and are shown in Table 3, where Torque1 is the torque produced by one winding and Torque2 is the torque produced by both the two redundant windings.The torque-speed characteristics curve of the motor is shown in Figure 9.  Figure 9 shows that the torque decreased as the speed increased and torque deceased sharply when the speed was above 125 rpm, which gave the reference value of appropriate working frequency.In addition, the Torque2 was approximately twice as large as Torque1, that is, the motor could provide greater torque when working in dual channel mode.This is consistent with the previous theoretical analysis, which showed that the coupling between the two redundancies had little effect on the torque in the design model of the motor.
The output torque of the motor could be increased by doubling the stator windings.A double-stator HSM was designed in [32]; the results demonstrated that there was an increase in the motor's torque output, but the torque was far less than twice that of a conventional HSM.Concurrently, the larger volume of the motor limited its application.In contrast, the motor proposed in this paper had great advantages in generating greater torque with smaller volume.
The fault tolerance of the motor was also improved.Even if the faults occurred in one stator winding, the fault-tolerant structure could ensure the continuous operation of the motor.The stepping motor and its driving circuit are simple, in contrast with the dual motor driving servo system [33] and other multi-level inverter fed electric machine drive systems [25,34], the dual-redundancy HSM has almost equivalent reliability.By comparison, it has outstanding advantages of small volume, light weight and simple structure.

Experimental Results
A prototype of the dual-redundancy HSM designed in this paper is manufactured and tested in this chapter.The MAGTROL experimental system was used to test the proposed motor.The experimental platform is shown in Figure 10.The torque and speed measured are shown in Table 4 and the curve is shown in Figure 11.As is shown in Figure 11, the experimental results were compared with the simulation results in the previous chapter.It was observed that the experimental results of torque-speed characteristics are in good agreement with 3D FEA simulation results.
The motor proposed in this paper has twice the reliability of ordinary motor in the case of a small volume difference.Even if one stator winding or drive circuit in the motor stops working due to fault, the redundancy design can ensure that the motor still has good dynamic performance.In contrast with the dual-motor parallel redundant system for satellite antenna drive system, the motor proposed in this paper has smaller volume and lighter quality, which makes it very suitable for aviation applications.The contrasts with other redundant schemes are shown in Tables 5 and 6.Meanwhile, it was confirmed that the coupling between redundancies had little effect on the output.The motor could produce approximately twice the torque of a conventional motor; the ability for load-supporting has improved greatly.

Conclusions
Due to the special environment of space, motors for satellite antenna drive systems are more prone to encounter some failures, which will lead to serious consequences.The importance of fault tolerance in satellites should never be overemphasized.To improve the fault tolerance of the stepping motor, this paper designed a dual-redundancy twophase hybrid stepping motor and experimental verification and simulation are implemented.The torque-speed characteristic, which is regarded as the key characteristic to measure the performance of a motor, is presented, it shows the output torque and speed range and optimum operating frequency of the motor.
The simulation and experimental torque-frequency results are in great agreement.It has been shown that the motor proposed in this paper has lightweight volume, excellent performance and reliability.It would be highly advantageous to apply the motor in satellite antenna drive system and other high reliability aerospace applications.

Figure 1 .
Figure 1.The stator structure of the dual-redundancy HSM.

Figure 2 .
Figure 2. The equivalent circuit of one phase.

Figure 4 .
Figure 4. Mesh shape and distribution of magnetic flux density.(a) Mesh division of the dual-redundancy HSM.(b) Magnetic flux density of the dual-redundancy HSM.

Figure 5 .
Figure 5. Magnetic flux density distribution of the rotor.(a) 3D magnetic flux density distribution of the rotor.(b) Magnetic flux density distribution of the rotor cross section.

FluxFigure 6 .
Figure 6.Magnetic flux density of the air gap.

Figure 10 .
Figure 10.Experimental platform for the prototype of the motor.

Figure 11 .
Figure 11.The experimental results and the simulation results.

Table 3 .
Output torque at different speeds.

Table 5 .
Contrast with dual-motor redundant system.

Table 6 .
Contrast with multiphase permanent magnet motor.