Multi-Sector Windings For Bearing Relief E-Machine: Saturation and Cross Coupling Effects

The key driving elements for any electrical machine (EM) used in aerospace and other safety critical applications are the reliability, mass, volume, and efficiency. To reduce mass/volume while satisfying the power required, the option is to increase the EM speed. The possible failure of the mechanical bearings at high speed, the higher bearing losses, and the excessive rotor displacement at high speed are some of the design challenges encountered when the speed increases. The main objectives of applying this MSBRM concept is to achieve the magnetic levitation of the rotor to (a) overcome the bearing reliability issues, as well as eliminating the bearing friction, (b) apply online radial force control on the rotor to mitigate the rotor displacement/whirling, and (c) reducing requirements for maintenance and monitoring. In this paper, the multi-sector windings bearing relief electric machine (MSBRM) concept is applied to high torque density permanent magnet EM. The electromagnetic characterization and the radial force control concept of the highly saturated EM has been studied deeply in this paper.


I. INTRODUCTION
The Climate Change Act of the UK government (2008) [1]- [3], aim to reduce UK greenhouse gas emissions by at least 80%, compared to 1990 levels, by 2050. The adoption of zero emission policies set by governments is directly impacting automotive, aerospace, marine, and railway sectors. All the systems conventionally powered by combustion engines, chemical accumulators or fuel cells, will undergo drastic changes to meet the current and future regulations [4].
In accordance with these targets, aircraft industry is seeking for developing more fuel-efficient and environmentally friendly aircraft to alleviate pollution concerns and to comply with the goals of the 2050 strategy. The vision is to replace the traditional systems of the main generation and propulsion units, as well as the auxiliaries of the aircraft with electrical systems. In other words, the main target is to push towards more electric aircraft, both hybrid and full electric aircrafts. For both generation and propulsion systems, the electric machine requirements are very demanding in terms of power density (40kVA/kg by 2050) [5], speed, efficiency, thermal management, volume, mass, cost, and reliability.
The bearing element is one of the most critical components when dealing with high rotational speed and reliability of electrical machines. Referring to the literature review reported in [6], the magnetic levitation allows to remove the bearing mechanical friction, and reduce the monitoring and regular maintenance requirements.
The active magnetic bearings have the potential for applying the desired magnetic levitation. They are employed in several industrial and commercial applications such as compressors, flywheels, and generators, where high rotation speed is a requirement. Nevertheless, the use of the magnetic bearings generally lead to an increased overall length of the electric machine (EM), added weight, and higher cost of the drive. To this regard, multi-sector windings bearing relief electric machines (MSBRM) offer the advantage of generating both torque and radial suspension force using only a single stator structure. This reduces the volume and mass of the electric machine, and hence, maximizing the power-to-weight and power-to-volume ratio, as desired for the aerospace applications [7]- [11], and the Paris Agreement (2015) [12]- [18]. This paper studies the Bearing Relief EM (BREM) technology in the hybrid/electric aircraft. The bearing drag is reduced by applying radial force control on the rotor. This yields to reduce the bearing friction losses and increase the EM efficiency. Consequently, the BREM allows to run the generator at higher speeds whilst mitigating the rotor dynamics problems. This yields to use smaller air gap, which lead to higher efficiency and facilitate the thermal management of the heat extraction. Fig. 1 shows the overall control process applied to the EM in order to achieve the bearing relief operating mode. The simultaneous measurements of the rotor displacements in both x and y axes are fed to a simple PID controller, as in [7], [19]. Then, the reference values of the required forces to return the rotor back to its centre of rotation are estimated by the PID controller. These forces signals are imported to the radial force controller (RFC) controller besides to the reference operating torque. This RFC is based on the Psuedo inverse matric approach presented in [6], [7]. The outcomes 978-1-7281-9945-0/20/$31.00 ©2020 IEEE of the RFC are the reference values of the independent multiple phase currents which fed to the traditional current controller of the EM. Similar RFC algorithm has been used in [6] for low torque density surface mounted permanent magnet (PM) EM. The torque density of the studied BREM in [6], [7], [19] is about 20% of the one investigated in this paper [11], [20]. High torque density BREM means highly saturated EM, yields to strong nonlinear magnetic characteristic. Thus, the aim of this paper is to fill this gap with a detailed investigation of the electromagnetic characterisation of the highly saturated BREM designed for aerospace applications. In other words, it studies the bearing relief strategy for high torque density EM.
The paper is organized as follows: Section II summarises the mathematical model of the RFC algorithm using the pseudo inverse matrix approach. This section highlights the dependency of the RFC on the electromagnetic characterisation of the EM. Section III discusses the outcomes of electromagnetic characterisation of the highly saturated BREM highlighting the magnetic nonlinearities, which were missed in the literature [6], [7], [19]. Section III determines the impact of the computed nonlinear electromagnetic characteristics on the accuracy of the feedforward RFC and the radial force capability of the EM. Finally, section V concludes the analyses reported in this paper.
II. RADIAL FORCE CONTROL Both electromagnetic torque and the radial force acting on the rotor of the MSBRM are referred to as the mechanical outputs of the MSBRM. These mechanical output terms can be determined by supplying the different winding sectors placed in the stator by independent ns phase currents. As ns is the total number of winding sectors. In addition, the electromagnetic characteristic of the EM itself plays an important role on the production of those mechanical outputs.
The mathematical presentation of the mechanical outputs as a function of the currents and EM characteristics has been presented in [7]. However, a linear electromagnetic characteristic of the investigated EM has been assumed in [7], as it has low torque density and saturation level. In other words, the steel laminations magnetic permeability has been set to infinity. This assumption can not be applied in the case of high torque density saturated electric machine, as the actual B-H curve of the lamination needs to be considered in order to calculate the flux density in the different parts in the EM.
This section summarizes the mathematical model of the RFC considering the electromagnetic nonlinearities of the studied MSBRM. In addition, it investigates the impact of these nonlinearities of the saturated MSBRM on the accuracy of the RFC.

A. The Mathematical Model
The mathematical representation of the EM mechanical outputs (radial forces and torque) can be described as where ϑ e is the electrical angular position of the rotor (ϑ e = pϑ m ),Ī αβ is an array that represents the stationary reference frame current components s i α and s i β of the three stator winding sectors, and K E (ϑ e , s γ) are the electromagnetic coefficients/characteristics of the MSBRM. Splitting the radial force into two components in xy plane; the mechanical output array can be represented as where F x (ϑ e ) and F y (ϑ e ) are the forces in x and y axis, respectively.

B. E-machine Electromagnetic Characterisation
The electromagnetic torque coefficient and force coefficients of the s th winding sector can be computed by feeding the current to that sector. Then, the overall torque and force coefficients of the MSBRM can be achieved by bringing the coefficients of all sectors together in a single matrix as in (3) The symmetry of the three sectors of the stator windings, allows us to derive the overall coefficient matrix from the coefficients of the first sector only. This can be achieved by exploiting the rotation matrix R as in (4) As a consequence, the coefficients matrix can be rearranged as where R( ns γ) can be computed as and 1 K E (ϑ e , s γ) consists of torque and forces in x and y axes and coefficients of the first winding sector, as shown below Solving (1), the reference current values can be obtained. However, inverting the rectangular matrix K E is complex task. Thus, pseudo inverse of K E has been applied as Finally, the reference current signalsĪ * αβ can be computed asĪ * 1) Torque coefficient of 1 st sector: The overall electromagnetic torque has been computed by varying the current amplitude, as shown in Fig. 2. In this case, all currents are fed into the model, including the ones of all three sectors. It is noted that the three sets of the three phase windings are in phase. Thus, the overall torque can be computed considering the equivalent to a single sector three machine. Consequently, the electromagnetic torque resulting from each winding sector can be obtained by dividing the overall torque by the number of sectors (3 sectors in our case), thanks to the symmetry of the three windings sectors.
Hence, the torque coefficient of the s th sector can be computed by dividing the electromagnetic torque of that sector by it's current. The electromagnetic torque has a linear relation versus the current, as it can be noted in Fig. 2. Consequently, the torque coefficient is varying in a  narrow range, as shown in Fig. 3. In other words, the torque coefficient has been assumed as a constant value, as 0.11 Nm/A (rated condition).
2) Force coefficient of 1 st sector due to I α : Different to the torque coefficients, which are based on the q-axis current only, the force coefficients in x and y axes are computed based on both I α and I β . Focusing the flux in the α -axis direction, both forces in x-axis (F xα ) and y-axis (F yα ) are evaluated as shown in Fig. 4. This concentrated α -axis flux is generated by the I α of the first sector windings. After that, the first sector force coefficients due to the α current can be computed as shown in Fig. 5. It can be noted that the coefficient K xα is highly affected by the magnetic saturation in the EM iron at high current values, whereas, the coefficient K yα is almost linear for a wide range of current values.
3) Force coefficient of 1 st sector due to I β : Analogously, the forces F xβ and F yβ are computed by concentrating the flux in the β-axis of the first winding sector. They are shown in Fig. 6. Once again, the magnetic saturation affects K xβ has a negligible impact on the K yβ , as can be noted from Fig. 7.

III. ANALYSIS RESULTS
The coefficients vector of the first winding sector, reported in (7), has been computed in the previous section. Then, the   full coefficient matrix of the studied BREM is computed as reported in (5). Setting the desired values of the EM torque and Radial force, (9) can be solved for obtaining the independent nine phase currents of the three winding sectors.  Different sets of the desired mechanical outputs of the EM have been imported to the radial force control algorithm. Then, the estimated currents for a complete electric period of the rotor rotation have been computed. These current waveforms have been imported to the finite element model of the studied EM. The evaluated radial forces by FE analysis are shown in Fig. 8. It can be note that the average value of the radial forces over one turn has a good agreement with the reference values imported to the control algorithm. In addition, it can be noted that there are oscillations in the forces waveforms around their desired values during the rotor rotation. This will be compensated simultaneously by the PID controller shown in [7]. Fig. 9 and Fig. 10 show the electromagnetic decoupling between the force and torque production of the MSBRM. The radial force has not been affected by applying the torque, as in Fig. 9. Furthermore, the torque waveform has not been affected by the production of the radial force.
However, the values of the torque and radial forces reported in Fig. 9 and Fig. 10 are low, i.e, the MSBRM works at the unsaturated condition reported in Fig. 4 and Fig. 6. The next two subsections discuss the nonlinearity effects on the accuracy and the decoupling between the torque and radial force production of the studied MSBRM.

A. Saturation Impact of the RFC
Since the coefficients K xα and K xβ are significantly changing for increasing current values, they affect the accuracy of the RFC. Therefore, this section deals with the identification of an approach for the selection of these coefficients.
Indeed, there are two options for selecting the coefficients K xα and K xβ . The first option is to consider the unsaturated value of each coefficient, i.e. the slope of the linear part of the curve F x [I x ]. The second option is to consider the value of the coefficient for a given current level in the Fig. 11. Impact of the coefficients on the RFC accuracy. saturated region. This second option is equivalent to consider the apparent inductance and therefore can be called apparent force coefficient. Both options has been investigated setting wide range of reference radial forces.
Referring to low reference forces values, which lies in the linear part of Fig. 4 and Fig. 6, option 1 gives accurate results as shown in Fig. 9, Fig. 11, and Fig. 12. On the contrary, option 2 overestimates the radial force, as it takes into account the magnetic voltage drop of the saturated region, which does not exist in the linear part of the curve. This leads the RFC to overestimate the current values, and hence, higher radial forces. This criteria has been highlighted in the dotted lines in Fig. 4 and Fig. 6.
Once again, referring to high reference forces values lies at the saturated part of the curve. It concluded that option 1 has satisfactory accuracy, as it underestimate the force in the range of 10%. Besides, option 2 still overestimates the force by higher values.
This study focuses on applying radial force control not only for bearing relief, but also to compensate the unbalanced magnetic pull (UMP) resulting from the rotor eccentricity. The UMP has been computed at eccentricity value equal to 50% of the air-gap length. It has been concluded that, the maximum force required to compensate the UMP and the rotor weight is about 300 N (100N by each sector). Next subsection discusses how the torque loading will affects accuracy of the RFC although the needed forces are within the unsaturated part. Fig. 12 shows the impact of the high torque values on the radial force control. The radial force has been achieved accurately without loading the EM (with zero torque). Nevertheless, when the rated torque is produced (60 Nm), the radial force is reduced from 200 Nm to 180 N (10% less). This can be called the cross coupling saturation effect. Since the torque is set to high value, the q-axis currents of the Fig. 12. Cross coupling effect between torque and radial force production.

B. Cross Saturation Impact of the RFC
EM will be increased significantly, as shown in Fig. 13, while, the d-axis currents of the EM don't see any significant change, as shown in Fig. 14. As a consequent, the machine becomes highly saturated and hence the magnetic voltage drop in the laminations affects the radial force production. To understand in depth both saturation and cross saturation effects on torque and force production, a series of simulations has been carried out and the results are shown in Fig. 15. It can be noted that, the saturation effect on the force accuracy is negligible for forces under 300 Nm. In addition, the cross saturation effect increases when the torque is higher than 40 Nm. Furthermore, the cross coupling effect reduces the force by 10% at 60 Nm for all level of force production, as the magnetic voltage drop results from the 200 A q-axis current is constant for all forces values.

IV. CONCLUSION
This paper descried the overall process of the MSBRM concept. In addition, it highlights the nonlinearities of the high torque density MSBRM and their effects on the accuracy of the radial force controller. The electromagnetic decoupling between the torque and radial force production has been proved at low electromagnetic saturation level. In addition, the electromagnetic coupling at the high saturation operating condition has been investigated. A proper selection for the electromagnetic coefficients of the highly saturated MSBRM has been discussed. Finally, a clear understanding of the analysed EM nonlinearities on the RFC accuracy has been concluded. A wide operating range of the MSBRM has been involved. The presented analyses help to estimate the effect of the EM non linearities on the transient performance of the control scheme of the multi-sector bearing relief electric machine. Currently, He is a visiting Research Fellow at the PEMC group, Department of Electrical and Electronic Engineering, Nottingham University, UK, as well as Electric machine design specialist at the research and innovation department, Romax technology, UK. His research activities deal with novel and high efficiency electric machines design for automotive and aerospace applications. Besides, He has strong experience in analytical and finite element modelling of different electrical machine topologies.

V. BIOGRAPHIES
Dr Hanafy is the Winner of the IET premium award 2018. He has more than 25 publications in peer reviewed international journals and conferences. His current research interests are improving the electromechanical system interaction (electric motor and the Gearbox, etc..), design electric machines with less NVH excitations for the advanced propulsion systems, and analytical modelling of various electrical machines. In addition, He has an interest in the radial force control of the bearingless permanent magnet machines for both automotive and aerospace applications. Barry James received his two Masters degrees in Manufacturing Engineering from Clare College, Cambridge. He first joined Romax Technology in 1995 and has worked in a number of positions, including Engineering Manager and Chief Engineer. He currently is Chief Technical Officer and Head of Innovation and Research, leading the development of new technical methods for a wide range of industries across the world. He has a strong interest in a wide range of engineering disciplines (mechanical, electrical, control etc.) and how they can be linked together and applied to different industries (aerospace, automotive, renewables).