Low Thrust Research Articles

System-Level Assessment of a Partially Distributed Hybrid Electric Propulsion System

Abstract Hybrid electric propulsion system-based aircraft designs are paving the path toward a future greener aviation sector and thus, have been the major focus of the aeronautical community. The fuel efficiency improvement associated to such propulsion system configurations are realized at the aircraft level. In order to assess such benefits, a radical shift in the subsystem modeling requirements and of a conceptual-level aircraft design environment are necessary. This work highlights performance model development work pertaining to different hybrid electric propulsion system components and the development of a design platform that facilitates tighter integration of different novel propulsion system disciplines at the aircraft level. Furthermore, a serial/parallel partially distributed hybrid electric propulsion system is chosen as the candidate configuration to assess the potential benefits and associated tradeoffs by conducting multidisciplinary design space exploration studies. It is established that the distributed hybrid electric configurations pose the potential for aircraft structural weight reduction benefits. The study further illustrates the impacts of onboard charging during the low thrust requirement segments, quantitatively. The provision of onboard charging lowers the potential for block fuel savings, and improvement in battery specific energy can make it more promising, which is also dependent on the hybridization power level. It is established that distributed propulsion system configurations particularly benefit from a high aspect ratio wing structure, which manifests in high hybridization power levels. A high voltage level transmission system with more efficient electrical components enhances opportunities for achieving block fuel saving benefits.

Numerical Investigation on the Thrust Vectoring Performance of Bypass Dual Throat Nozzle

Modern aircraft and missiles are gradually integrating thrust vector control systems to enhance their military capabilities. Bypass Dual-Throat Nozzle (BDTN) control is a new fluidic thrust vectoring technique capable of achieving superior performance with large vector angles and low thrust loss. In this study, we analyzed the flow characteristics and performance parameters of BDTN by varying the bypass angle, nozzle convergence angle, and bypass width. The flow governing equations are solved according to a finite volume discretization technique of the compressible RANS equations coupled with the Renormalization Group (RNG) k-ϵ turbulence model for Nozzle Pressure Ratio (NPR = 2~10) to capture the significance of under-expanded and over-expanded jets. Results show that by decreasing the bypass angle from 90° to 35°, there is a 6% increase in vectoring angle while the vectoring efficiency is enhanced by 18%. However, a decrease of 3% in the thrust and discharge coefficients is also observed. When the convergence angle was increased from 22° to 37°, vectoring angle, discharge coefficient, and thrust coefficient increased by 2%, 1%, and 0.26%, respectively. Moreover, vectoring efficiency is also enhanced by 8% by reducing the convergence angle from 37° to 22°. Based on the investigated parameters, it is determined that nozzle convergence angle does not significantly influence thrust vectoring performance, however, bypass width and bypass angle have a significant effect on thrust vectoring performance.

Numerical analysis of propellers for electric boats using computational fluid dynamics modelling

In the maritime industry, propellers are the most commonly used form of propulsion and are core to the optimum performance of a ship. As marine propellers have complicated geometries, the flow around the propeller is also complicated. Generally, the performance characteristics of a propeller are determined and analysed by experiments like open water and self-propulsion scale model tests which are costly and time-consuming at the initial design stage. In this study, the CFD simulations were performed to evaluate propeller performance. Three Wageningen B-series propellers with varying Expanded Area Ratios (EAR) were modelled with respect to some design constraints, such as ship speed and rotational velocity. The performance of the hydrodynamic coefficients, thrust, torque and open water efficiency are then analysed. These characteristics are then validated against experimental data obtained from the Netherlands Ship Model Basin open water test in Wageningen and used to investigate the flow behaviour. The analysis considers the Multiple Reference Frame (MRF) model. This research aims to provide a well-founded framework for applying CFD in the analysis and selection of Wageningen B-series propellers, as well as investigating the relationship between EAR, flow behaviour, thrust coefficient, and torque coefficient for electric boats. The results showed that a lower thrust and torque coefficient manifested in improved flow behaviour, increasing efficiency by up to 62%. In addition, the outcome revealed that the lower expanded area ratio of 0.6 is more suitable for electric boats, creating a larger pressure difference of 1.079 MPa and generating extra potential thrust at the same advance ratio, which leads to greater open water efficiency.

Structural optimization of engine nacelle fan cowl door hinge

To increase propulsive efficiency, future nacelle engines are projected to run at low specific thrust with large bypass ratios. Typically, this entails a larger fan diameter and nacelle, as well as increased drag and weight penalties. As a result, there is a need to construct more compact, nacelles’ components in comparison to present designs in order to minimize drag and weight. These designs are naturally more difficult, necessitating the use of a system to explore and identify the design area that is possible. Designing a nacelle is difficult due to the wide range of operational circumstances. This paper provides a multi-objective optimization technique for fan cowl door component of nacelle design utilizing an evolving genetic algorithm. A collection of geometry definitions utilizing Class Shape Transformations, automated simulation and analysis, a generic algorithm, assessments at various nacelle operating circumstances, and the addition of extra aerodynamic restrictions are all part of the unique framework. This framework was used to look into the nacelle design space for nacelle components. For the new nacelle component design challenge, multi-objective optimization was successfully shown, and the entire system was proved to allow the discovery of the feasible nacelle design space. Moreover, in this paper a series of fan cowl door hinge is designed from the scratch and optimized step by step for reducing the weight of the component without compromising for the stress capacity and load transformations within the allowable limit.

THRUST MEASUREMENTS OF THE RIT-10 THRUSTER WITH A NEW TYPE OF THRUST BALANCE

To study electric propulsion devices, accurate long-time measurements of the low thrust forces are necessary. For this purpose, a new, lightweight thrust balance was established. To evaluate the performance of this thrust balance, it was operated to perform measurements of the well-studied RIT 10 gridded-ion thruster. The new thrust balance and its performance were demonstrated, and a first estimate on accuracy and long-term stability was obtained. Although the setup was fully manufactured and calibrated, a final test using an electric thruster is still necessary. To study the behavior of the thrust balance, the RIT-10 thruster was operated for extended intervals on the DLR Electric Propulsion Balance thrust balance. The long-term stability in terms of thermal drift, effects of the fluid and electric interface, and effects of the vacuum and plasma environment onto the thrust balance were studied.

Models and Strategies for J 2 -Perturbed Orbital Pursuit–Evasion Games

This paper discusses the modeling and solving of orbital pursuit–evasion games (OPEGs) under J 2 perturbation. The optimal long-range maneuver method under J 2 perturbation is designed, and it is proved that the effect of eccentricity can be ignored when transfer times and the Δ V budgets are fixed. It is discovered that when the inclination between the initial and target orbit is equal and is between 10° and 25°, the whole maneuver process can be simplified to a fixed-inclination transfer. Subsequently, a long-term OPEG model is provided under the assumption of fixed inclination, zero eccentricity, and impulsive thrust. Winning conditions of OPEGs under J 2 perturbation are then carefully derived, with long-time OPEG ( J 2 dominated) and short-time OPEG (traditional) separated, and typical tactics of both sides formulated and verified. These studies are further extended to the “Arrival time matching game” for maintaining/avoiding resonant arrival time under J 2 perturbation, and the advantages and disadvantages of both sides are analyzed. The models and strategies obtained in this paper can be potentially used in practical applications of OPEGs, especially for evaders that have low thrust–weight ratios and are weak in traditional short-term OPEGs.

A Low Detent Force DS-PMSLM Based on the Modulation of Cogging and End Forces

This article presents a low detent force double-sided permanent-magnet synchronous linear motor (DS-PMSLM), which contains two flat-type stators and a mover. The mover is constituted of a fiberglass support plate and Halbach array permanent magnets attached on both surfaces of this plate, which provides a large thrust force and low mover mass for high dynamic response. For the sake of reducing the thrust ripple, a modulation method of the cogging and end forces is proposed based on the destructive interference. The even-order harmonics in cogging and end forces are reduced by adjusting the width of the side slots and the lengths of end teeth, respectively. In order to reduce the odd-order harmonics of the detent force, the cogging force and end force are modulated to reverse phases and identical amplitudes by changing the equivalent lengths of four end teeth. An integrated subdomain analytical model considering the cogging and end effects is built to optimize five structural parameters of the stators for low detent force. Finally, the prototype and testing platforms are set up and the experimental results validate that the proposed DS-PMSLM can achieve a low thrust ripple of less than 1.5 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$ \%$</tex-math></inline-formula> .

Modified Primary Flux Linkage for Enhancing the Linear Induction Motor Performance Based on Sliding Mode Control and Model Predictive Flux Control

This paper proposes a modified primary flux linkage and an improved weighting less model predictive control for linear induction motors (LIMs) to enhance the drive system in terms of linear speed response, wide speed range, efficiency, and computation time. Sliding mode controller is presented in this work to get quick response instead of the use of the PI controller. A weighting less model predictive flux control (MPFC) is employed to eliminate the weighting factor and reduce the computation time. Furthermore, the optimum value of the primary flux linkage is calculated to guarantee higher efficiency under the operation of maximum thrust per ampere, loss minimization control and wider speed range in the field weakening region. The FCS-MPFC uses only the primary flux in the cost function independent on the weighting factor. Moreover, simplified calculation process can be executed greatly in the αβ-coordinates without transformation matrix, where the end-effect is fully taken into consideration. In comparison with the PI controller under different conditions, the proposed control method can achieve faster dynamics with lower thrust ripple, computation time, and so on. Comprehensive simulation and experimental results based on one prototype with 3 kW linear induction machine have verified the advantages of the proposed strategy in this work.

The effect of lumbar spinal manipulation on biomechanical factors and perceived transient pain during prolonged sitting: a laboratory-controlled cross-sectional study

BackgroundSpinal manipulation has been shown to affect muscle activity, posture, and pain. To date, no studies have examined the effect of manipulation on biomechanical factors during sitting. Therefore, the purpose of this study was to investigate the immediate effect of lumbar spinal manipulation on trunk muscle activation, spine posture and movements, and perceived ratings of transient pain in asymptomatic adults during prolonged office sitting.MethodsTwenty healthy adults were recruited for a single laboratory session that included a standardized office sitting/data entry protocol (120 min total, 3 blocks of 40 min). Data were collected between July and August 2012. The first block (baseline) was immediately followed by two experimental blocks. Prior to the start of each experimental block, participants were transferred to a therapy plinth and placed side lying (right side down), and a random presentation of either a control or high velocity low amplitude thrust directed at L4/L5 was delivered. Continuous measures of muscle activity, spine posture, and spine movements were recorded throughout the sitting trials. Perceived transient pain was measured by visual analogue scale at 10-min intervals (including immediately before and after the randomized maneuvers).ResultsThere were no significant differences in spine or pelvic posture or perceived back pain following either the manipulation or control maneuvers. Significantly reduced muscle activity and increased shifts of the lumbar spine angle were identified in the block following manipulation compared to both baseline and post control blocks.ConclusionsSpinal manipulation does not appear to have an immediate impact on spine or pelvic posture in healthy adults but does appear to reduce muscle activity and increase spine movement during sitting. Future work should replicate this study with a larger population in a field setting. It may be worthwhile to explore the implication of reduced muscle activation and increased spine movements during prolonged sitting for office workers that receive manipulations or mobilizations during their workday.

A Non-Permanent Magnet DC-Biased Vernier Reluctance Linear Machine with Non-Uniform Air Gap Structure for Ripple Reduction

Thrust ripple and density greatly impact the performance of the linear machine and other linear actuators, causing positioning control precision, dynamic performance, and efficiency issues. Generalized pole-pair combinations are difficult to satisfy both the thrust and ripple for double salient reluctance linear machines. In this paper, a DC-Biased vernier reluctance linear machine (DCB-VRLM) is proposed to solve the abovementioned issues. The key to the proposed design is to reduce the ripple and enhance the thrust density with non-uniform teeth by utilizing and optimizing the modulated flux in the air gap. To effectively verify the proposed design, the DCB-VRLMs with different winding pole pairs and secondary poles are compared. The 12-slot/10-pole combination is chosen to adopt a non-uniform air gap structure. Moreover, the energy distribution of AC/DC winding is studied and optimized to further enhance the performance of the proposed DCB-VRLM. The results indicate that the DCB-VRLM with the non-uniform air gap has a lower thrust ripple, better overload capability, and higher thrust density, which confirms its superiority in long-stroke linear rail transit and vertical elevator applications.

Robust control of a hub-spoke tethered formation system of microsatellites using Hamilton-Jacobi inequality

The problem of controlling a rotating hub-spoke tethered formation system in low Earth orbit is considered, in which microsatellites are located radially around the central spacecraft (hub) and connected to it by tethers (spokes). To analyze the dynamics of the tethered system, a mathematical model is developed in the orbital coordinate system by Lagrange method, in which the central spacecraft is regarded as a rigid body. In the proposed control scheme, the spin motion of the central body is regulated by the control moment, and tether deployment control law is proposed by robust approach, which is carried out by regulating the tether tensions and low thrusts acting on the microsatellites. The robustness and stability of the system are investigated using Lyapunov theory and Hamilton-Jacobi inequality, which is used to determine the robustness index of the control system. The results of numerical calculations are presented, which confirm that the proposed control scheme is effective when taking into account periodic gravitational perturbations, external perturbations and perturbations associated with uncertainty in the initial states of the system and with the rotation of the central body.

Application of homotopy perturbation method to the radial thrust problem

The dynamics of a spacecraft propelled by a continuous radial thrust resembles that of a nonlinear oscillator. This is analyzed in this work with a novel method that combines the definition of a suitable homotopy with a classical perturbation approach, in which the low thrust is assumed to be a perturbation of the nominal Keplerian motion. The homotopy perturbation method provides the analytical (approximate) solution of the dynamical equations in polar form to estimate the corresponding spacecraft propelled trajectory with a short computational time. The accuracy of the analytical results was tested in an orbital-targeting mission scenario.

Thrust Ripple Suppression for Linear Induction Machines Based on Improved Finite Control Set-Model Predictive Voltage Control

In this article, two kinds of finite control set-model predictive voltage control (FCS-MPVC) methods have been proposed for linear induction machine to eliminate the cumbersome selection of the weighting factor (WF), achieve lower ripples of both thrust and flux, and reduce the heavy calculation steps. The first method, FCS-MPVC I, uses one voltage vector (VV) during the whole control period, and the second method, FCS-MPVC II, uses two VVs. Moreover, the proposed methods use a single cost function without any WF. This function consists of the errors between primary voltage references with their predicted values in the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">αβ</i> -axis. Depending on the distance between the reference VV (RVV), as calculated using the deadbeat control method, and all candidate VVs (CVVs), the first proposed method, FCS-MPVC I, can select the nearest VV to the RVV as the optimal VV (OVV), which can reduce the redundant evaluation work greatly. Meanwhile, the second proposed method, FCS-MPVC II, applies two VVs, active VV and null VV, to adjust the duty ratio effectively. To avoid the iterated evaluation for all CVVs, one simplified search order excluding unsuitable CVVs has been proposed to determine the optimal OVV, which selects only one active VV from the two adjacent active VVs in the sector owning the RVV. Afterward, based on a prototype of 3-kW arc induction machine with a large radius, extensive simulation and experiments have fully demonstrated that the proposed methods can get much lower thrust ripples and less calculation steps effectively, compared with those of conventional finite control set-model predictive thrust control strategy.

Very low thrust station-keeping for low Earth orbiting satellites

Traditional station-keeping for Earth observation satellites with chemical thrusters generally involves maneuvers every couple months that are able to change significantly the semi-major axis and the inclination. These strategies do not scale down to very low thrust level (a few hundreds of μN) electrical thrusters. This paper presents both in-plane and out-of-plane strategies that spread corrections over very long arcs and discretize them to tiny maneuvers every couple orbits, taking into account mission-constraints on maneuvers locations. These strategies scale up to medium thrust strategies, filling the gap between propulsion technologies. The out-of-plane strategy although features a new no-deadband property and controls the full orbital momentum. All strategies allow control very close to the reference (a few hundreds meters in osculating parameters) and very low cost.

Re-entry high speed commercial air vehicles: Improving the environmental eligibility at take-off and approach accounting the flight safety restrictions

In addition to the flight safety of re-entry high speed commercial air vehicles their environmental eligibility come into particular important issue due to environmental requirements for commercial subsonic aviation are becoming more stringent. During the re-entry take-off and approach cycles the vehicles are a subject to almost the same type of environmental requirements as the commercial subsonic aviation. The requirements are primarily concern the limitations of community noise and engine exhaust emissions levels, as well as the sonic boom level (for commercial supersonic aviation). Meeting the requirements may significantly complicate the development, certification and further operation of the vehicles. The paper discusses one of the potential ways of increasing the acoustic eligibility of the vehicles during take-off and approach cycles at using the some number of turbofan engines as part of the propulsion system. The way is based on the creation of a special control of turbofan engines thrust/power (so called low noise thrust management) accounting the provision of flight safety restrictions. The assessment of the acoustic impact (community noise) of the vehicle is carried out taking into account the contribution of major noise sources of turbofan engines, i.e. the fan and exhaust jet. The impact of two parameters of turbofan engines such as the bypass ratio and the number of turbofan engines was evaluated. The variation of these parameters was performed under the condition meeting the specified mission requirements in term of the required field (runway) length and the approach speed. The estimations shown that using the created low noise turbofan engine thrust management at the take-off and approach may significantly increase the acoustic eligibility of the vehicles using turbofan engines.

Hydrodynamic characteristics of a new thrust vectoring ducted propeller with slotted nozzle

A thrust vectoring ducted propeller with slotted nozzle is a new approach for vector propulsion that can produce lateral forces and torques, leading to the ability of manipulating the autonomous underwater vehicle (AUV) for various movements. In this work, the geometric model of the thrust vectoring ducted propeller is designed based on slotted nozzle. The nozzle of the ducted propeller is slotted and then divided into several pieces. Divided auxiliary nozzles are deflected to redirect the outflow, achieving thrust vectoring. The key feature parameters of the slotted nozzle are extracted. The performance of the thrust vectoring ducted propeller is evaluated using Computational Fluid Dynamic (CFD) method. Different parameters of the slotted nozzle are set to evaluate the influence on the thrust vectoring capability of the ducted propeller. Through comparison, a thrust vectoring ducted propeller model is determined. Hydrodynamic characteristics including thrust vectoring performance, scale effect and vertical-horizontal interactions of the thrust vectoring ducted propeller are analyzed. It is shown that the ducted propeller with slotted nozzle at full capacity has a higher vector ratio and lower thrust loss than ideal tilted thruster with a maximum tilt angle of 40°. For high Reynolds (Re) number working conditions, all forces and torques including vector thrust can be nondimensionalized into hydrodynamic coefficients related to advance ratio J. At lower Re number working conditions, thrust vectoring capability improves in a certain extent. Interactions between perpendicular directions are caused by the counterclockwise rotation of the propeller. The pattern is quite clear and can be quantified. The performance of the thrust vectoring ducted propeller is verified through towing tank experiment. All the results show that the ducted propeller with slotted nozzle can be considered controllable with good thrust vectoring capability.

MEASUREMENT OF THE REGRESSION RATE IN A HYBRID ROCKET MOTOR BY ACQUIRING THE HELMHOLTZ FREQUENCY IN THE COMBUSTION CHAMBER

Low thrust values obtained with a hybrid rocket motor (HRM) are a consequence of the difficulty in quickly mixing the fuel and oxidizer, which is characterized by a low regression rate of the fuel grain. Therefore, the measurement of this parameter is of great importance in studies that aim at solutions for this deficiency in HRM. Several studies calculate a reliable value of the average regression rate over time by measuring the total mass of fuel before and after each burn. A method to measure instantaneous regression rate is by acquiring the Helmholtz resonance frequency in the combustion chamber. This work uses a piezoelectric pressure transducer to obtain the Helmholtz frequency mode of the combustion chamber in a laboratory scale test bench with high-density polyethylene (HDPE) and gaseous oxygen, and is based on the principle that this frequency is inversely proportional to the square-root of the chamber volume. With the chamber volume variation, the port diameter of the grain variation is obtained. In conclusion, the calculated variation of port diameter agreed well with the correlation for average regression rate, determined from mass loss during operation.

Mixed-integer GA optimization for the tracking control of a formation of small satellites equipped with multi-constrained electric thrusters

A formation mission scenario is considered, consisting of small satellites (Chasers) that must keep a reference trajectory with respect to a larger and non-cooperating Target satellite. The Chaser must perform its formation guidance and control tasks within the framework of its thruster constraints: the case of a low thrust electrical engine is here considered. The constraints of such a thruster are relevant to number and orientation of thrusters, the admissible thrust levels and minimum and maximum ON time. The control algorithm developed to satisfy all these constraints is a model predictive control based on a mixed-integer genetic algorithm optimization, which provides a fast estimate of the necessary scheduled actions. The algorithm takes J2 and drag perturbations into account; at the scope a differential orbital parameters linear relative dynamics model has been used, while the real-world dynamics is represented by a high precision orbital propagator. The developed orbital controller is particularly effective when trajectory keeping is required; different relative configurations, like Helix or Train formations, are taken as benchmark trajectories. The results show promising and robust performance in counteracting differential perturbations, even in the case in which the Target satellite performs scheduled orbital maneuvers which are not timely communicated to the Chaser satellite.

Оптимізація перельотів між низькими орбітами з суттєвою різницею довгот висхідних вузлів

At present, satellite systems, each comprising hundreds of satellites, are, and are to be, deployed in low orbits. In addition, existing satellite systems are replenished. There has appeared a trend towards the development of modular satellites, which will lead to the development of easy-to-maintain spacecraft consisting of many small structural modules with standardized interface mechanisms. To extend the life of all these systems and reduce their maintenance cost, it is advisable to develop a system for their maintenance. Despite the relatively large number of works on the rendezvous problem, this problem is considered in a somewhat simplified formulation, which is not sufficient for spacecraft servicing in low orbits. As a rule, the consideration is limited to coplanar rendezvous problems in an impulse formulation. In real conditions, rendezvous maneuvers in low orbits are nontrivial. As is known, the orbital parameters of low-orbit spacecraft may differ significantly: the difference in the longitude of ascending nodes (LAN) may reach tens and even hundreds of degrees. Because of this, the energy consumption for an orbital plane change becomes unacceptably high for modern service spacecraft. This energy consumption can be reduced by using the precession of the line of nodes due to the non-centrality of the Earth's gravitational field. A waiting maneuver of a service spacecraft in a well-chosen orbit makes it possible to eliminate the mismatch between the LANs of the service spacecraft’s parking and destination orbits, thus significantly reducing the orbital transfer energy consumption. However, the long wait time of the service spacecraft in its parking orbit significantly increases the total orbital transfer time. The aim of this article is to develop a mathematical model of bicriteria optimization of a transfer of a service spacecraft with a low constant thrust engine between low near-circular orbits with significantly different LANs. This problem is solved by averaging the service spacecraft’s dynamics equations over a fast parameter and using a genetic algorithm of global Pareto optimization. The novelty of the results obtained lies in a formulation of a bicriteria optimization problem and the development of a mathematical model for choosing an optimal service spacecraft parking orbit. The mathematical model developed may be used in planning service spacecraft transfers between low near-circular orbits with significantly different LANs.

Design Criterion for Dual-Modular Permanent Magnet Linear Synchronous Motor With Long Stroke and High Flexibility

Permanent magnet linear synchronous motor (PMLSM) has a wide application prospect in transmission systems of automatic production, logistics, and other fields. However, the existing PMLSM has the problems of poor flexibility and large thrust fluctuation adapted to load and stroke variation. In order to solve this problem, a dual-modular PMLSM is proposed. The motor can be flexibly configured with multiple-unit-primary and multiple-unit-secondary, and achieve low thrust fluctuation. The matching criterion of unit-primary (UP) and secondary is established to reveal the relationship between the slot/pole combination and the winding utilization. The rules of the average thrust varying with the different UP and unit-secondary (US) lengths are illustrated. Then, the matching criterion of different UP is proposed, and the optimization method of the spacing between UP ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ) is obtained by establishing the end force model. The design standard of the optimal <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> is obtained by combining the two factors of minimum detent force and maximum thrust. The results are verified by the simulation and experiments on a 3UP-96US dual-modular PMLSM circular transmission line system and its prototype.