Linear Machines for Long Stroke Applications—A Review <s> II. LINEAR INDUCTION MACHINES <s> A hybrid control system using a recurrent fuzzy neural network (RFNN) is proposed to control a linear induction motor (LIM) servo drive. First, feedback linearization theory is used to decouple the thrust force and the flux amplitude of the LIM. Then, a hybrid control system is proposed to control the mover of the LIM for periodic motion. In the hybrid control system, the RFNN controller is the main tracking controller, which is used to mimic a perfect control law, and the compensated controller is proposed to compensate the difference between the perfect control law and the RFNN controller. Moreover, an online parameter training methodology, which is derived using the Lyapunov stability theorem and the gradient descent method is proposed to increase the learning capability of the RFNN. The effectiveness of the proposed control scheme is verified by both the simulated and experimental results. Furthermore, the advantages of the proposed control system are indicated in comparison with the sliding mode control system. <s> BIB001 </s> Linear Machines for Long Stroke Applications—A Review <s> II. LINEAR INDUCTION MACHINES <s> The movement of a linear induction motor (LIM) causes eddy currents in the secondary conductor sheet at the entry and the exit of the primary core. The eddy currents of the sheet tend to resist sudden flux variation, allowing only gradual change along the airgap. Hence, the so-called 'end effect' causes not only the losses but also airgap flux profile variation changes depending on the speed. In this work, an equivalent circuit model of LIM is developed following Duncan's per phase model. It is then transformed into a synchronous reference frame which is aligned with the secondary flux. Also, a field orientation control scheme is developed which accounts for the end effect. The validity of the proposed LIM model is demonstrated by comparing experimental and simulated voltage current relations. With the proposed control scheme, the (secondary) flux-attenuation problem due to the end effect is shown to be resolved in the high-speed range. <s> BIB002 </s> Linear Machines for Long Stroke Applications—A Review <s> II. LINEAR INDUCTION MACHINES <s> This work looks at a conception of a roped elevator prototype, using a linear induction motor (LIM) as traction machine and hence a development of a linear drive. Thus the development of a specific inverter to feeding the linear motor, allows several ways to drive the elevator, through a suitable choice of the RMS voltage levels, an angular phase displacement and a fundamental frequency of the output voltage. As result tries to establish a new paradigm of the vertical conveying, through a new construction technique and arrangement of the system with counterweight allowing an increase of trustworthy, comfort and costless. <s> BIB003 </s> Linear Machines for Long Stroke Applications—A Review <s> II. LINEAR INDUCTION MACHINES <s> This study focuses on the development of a robust Petri-fuzzy-neural-network (PFNN) control strategy applied to a linear induction motor (LIM) drive for periodic motion. Based on the concept of the nonlinear state feedback theory, a feedback linearization control (FLC) system is first adopted in order to decouple the thrust force and the flux amplitude of the LIM. However, particular system information is required in the FLC system so that the corresponding control performance is influenced seriously by system uncertainties. Hence, to increase the robustness of the LIM drive for high-performance applications, a robust PFNN control system is investigated based on the model-free control design to retain the decoupled control characteristic of the FLC system. The adaptive tuning algorithms for network parameters are derived in the sense of the Lyapunov stability theorem, such that the stability of the control system can be guaranteed under the occurrence of system uncertainties. The effectiveness of the proposed control scheme is verified by both numerical simulations and experimental results, and the salient merits are indicated in comparison with the FLC system <s> BIB004 </s> Linear Machines for Long Stroke Applications—A Review <s> II. LINEAR INDUCTION MACHINES <s> A field-programmable gate array (FPGA)-based adaptive backstepping sliding-mode controller is proposed to control the mover position of a linear induction motor (LIM) drive to compensate for the uncertainties including the friction force. First, the dynamic model of an indirect field-oriented LIM drive is derived. Next, a backstepping sliding-mode approach is designed to compensate the uncertainties occurring in the motion control system. Moreover, the uncertainties are lumped and the upper bound of the lumped uncertainty is necessary in the design of the backstepping sliding-mode controller. However, the upper bound of the lumped uncertainty is difficult to obtain in advance of practical applications. Therefore, an adaptive law is derived to adapt the value of the lumped uncertainty in real time, and an adaptive backstepping sliding-mode control law is the result. Then, an FPGA chip is adopted to implement the indirect field-oriented mechanism and the developed control algorithms for possible low-cost and high-performance industrial applications. The effectiveness of the proposed control scheme is verified by some experimental results. With the adaptive backstepping sliding-mode controller, the mover position of the FPGA-based LIM drive possesses the advantages of good transient control performance and robustness to uncertainties in the tracking of periodic reference trajectories. <s> BIB005 </s> Linear Machines for Long Stroke Applications—A Review <s> II. LINEAR INDUCTION MACHINES <s> Although linear electrical motors (LMs) are increasingly used in industry to develop linear motion, they suffer from two major drawbacks: low efficiency and low power factor. These drawbacks cause high energy consumption and a rise in input current, and occupy transmission line capacity. We present a multiobjective optimization method to improve both efficiency and power factor, simultaneously. Our method uses an analytical model of the machine to calculate the efficiency and power factor. It allows us to investigate the effects of various motor specifications on the efficiency and the power factor. Motor parameters and dimensions can then be optimized by using a genetic algorithm in an appropriate objective function. The results show an enhancement in motor performance. We have used 2-D and 3-D time-stepping finite-element methods to evaluate the analytical results. A comparison of results validates our optimization method. <s> BIB006 </s> Linear Machines for Long Stroke Applications—A Review <s> II. LINEAR INDUCTION MACHINES <s> In rotating machines, no end effects occur because of their cylindrical geometry, which has no ends in the direction of movement. In linear motors, however, either the rotor or the stator has a finite length, and end effects occur because of the part having finite length entering or exiting the magnetic field. End effects occur as magnetic waves having attenuation, and play a major role in the performance of linear induction motors (LIMs), especially at high speeds. As the speed increases, efficiency decreases because of longitudinal end effects. This decrease in efficiency has a braking effect. In high-speed maglev devices, such as trains, the efficiency and performance of the LIMs become very important. We report on our investigation of end effects by the finite-element method. <s> BIB007 </s> Linear Machines for Long Stroke Applications—A Review <s> II. LINEAR INDUCTION MACHINES <s> This paper presents the analysis of single-sided linear induction motors (SLIMs) for application in various systems demanding direct linear motion. In order to investigate the characteristics of linear induction motors according to secondary conductor structures, five types of secondary slot shape structures, usually used in rotary motors, were set up. The equivalent circuit of a SLIM is proposed and the equivalent parameters are computed. In addition, the characteristics such as the air-gap flux density, thrust, normal force, force ripple, speed response, etc., are analyzed by 2-D FEM with an external circuit, and the results are compared. <s> BIB008 </s> Linear Machines for Long Stroke Applications—A Review <s> II. LINEAR INDUCTION MACHINES <s> Linear induction motors (LIMs) are the best solution to the problem of producing linear motion directly, but the low efficiency, low power factor, and longitudinal end effect (LEE) are their major problems. These problems become more baneful in high-power and high-speed applications. For example, in the transport industry, they lead to high energy consumption, high input current that occupies transmission line capacity, and phase unbalance. Thus, it is necessary to overcome these problems; furthermore, the decrease in motor weight results in lower cost and better performance of linear induction motors. In this paper, first we present an accurate equivalent circuit model (ECM). Then, we introduce a new factor, Klee, which can illustrate the end effect intensity in the LIM. At the next stage, we investigate the influence of motor design parameters on the efficiency, power factor, and end effect intensity. Then we develop several different muliobjective functions, which will be used to improve efficiency, power factor, end effect intensity, and motor weight. This procedure will be done by analytic method and by using the particle swarm optimization technique. Results will show the accuracy of the equivalent circuit model and the improvement of objective functions at the end of the optimization procedure. Two-dimensional finite-element analysis evaluates the results from the ECM. <s> BIB009 </s> Linear Machines for Long Stroke Applications—A Review <s> II. LINEAR INDUCTION MACHINES <s> Single-sided linear induction motors (SLIMs) have lately been applied in transportation system traction drives, particularly in the intermediate speed range. This is because they have merits, such as the ability to exert thrust on the secondary without mechanical contact, high acceleration or deceleration, less wheel wear, small turning circle radius, and flexible road line. The theory of operation for these machines can be directly derived from rotary induction motors (RIMs). However, while the cut-open primary magnetic circuit has many inherent characteristics of the RIM equivalent circuits, several issues involving the transversal edge and longitudinal end effects and the half-filled slots at the primary ends need to be investigated. In this paper, a T-model equivalent circuit is proposed which is based on the 1-D magnetic equations of the air gap, where half-filled slots are considered by an equivalent pole number. Among the main five parameters, namely, the primary resistance, primary leakage inductance, mutual inductance, secondary resistance, and secondary inductance, the mutual inductance and the secondary resistance are influenced by the edge and end effects greatly, which can be revised by four relative coefficients, i.e., Kr, Kx, Cr, and Cx. Moreover, two-axis equivalent circuits (dq or αβ) according to the T-model equivalent circuit are obtained using the power conversion rule, which are analogous with those of the RIM in a two-axis coordinate system. The linear induction motor dynamic performance, particularly the mutual inductance and the secondary resistance, can be analyzed by the four coefficients. Experimental verification indicates that both the T-model and the new two-axis circuits are reasonable for describing the steady and dynamic performance of the SLIM. These two models can provide good guidance for the electromagnetic design and control scheme implementation for SLIM applications. <s> BIB010 </s> Linear Machines for Long Stroke Applications—A Review <s> II. LINEAR INDUCTION MACHINES <s> This paper deals with finding the optimal ratio of height and length of Single-Sided Linear Induction Motors (SLIM) using Finite Element Method (FEM) for magnetic field analysis coupled with optimal design methodology. For effective analysis, FEM is conducted in time harmonic field which provides steady state performance with the fundamental components of voltage and current. The ratio of height to length providing the required output power is obtained by Response Surface Methodology (RSM) and optimal values are presented by the variation in output power. When output power is small, the ratio is high and as the power increases, the ratio shows a converged value. Considering the general application of linear motors, using a small ratio can be limiting, however, the shape ratio for maximum thrust can be identified. <s> BIB011 </s> Linear Machines for Long Stroke Applications—A Review <s> II. LINEAR INDUCTION MACHINES <s> Regarding the different capabilities, linear motors have been employed in many applications in industry. Among different linear motors, single-sided linear induction motors (SLIMs) have been widely used due to their simplicity and low construction cost. However, these types of motors suffer from low efficiency and power factor and also, existence of so called end effect. The end effect produces braking force that deteriorates the performance of the motor, especially at high speeds. In this paper, analytical equations are derived for end effect braking force, efficiency, power factor, and output thrust. Employing the derived equations and considering all phenomena involved in the single-sided SLIM, a simple design procedure is presented and the effect of different design variables on the performance of the motor is analyzed. A multiobjective optimization method based on genetic algorithm is introduced to maximize efficiency and power factor, as well as to reduce primary weight and end effect braking force, simultaneously. All effective design variables are considered in optimization. The results show significant improvements in the objective function. Finally, the 2-D and 3-D finite element method is employed to validate the results obtained by the analytical method. <s> BIB012 </s> Linear Machines for Long Stroke Applications—A Review <s> II. LINEAR INDUCTION MACHINES <s> Both the static and the dynamic end effects of the linear induction motor (LIM) cause an increase of the difficulty in controlling with good performance the main electro-mechanical characteristics of the machine, in particular the thrust and the linear speed. These effects are related to asymmetries in the inductor structure with respect to the rotating counterpart, in both the longitudinal and in the transversal direction. This paper propose a technique for the compensation of the LIM static end effects by a synchronous controller in conjunction with a neural adaptive filter, exploited to extract the inverse sequence current component from the inductor current signature. This permits to compensate the static-end effects and thus to cover the static asymmetries of the LIM with respect to the rotating induction motor (RIM). <s> BIB013 </s> Linear Machines for Long Stroke Applications—A Review <s> II. LINEAR INDUCTION MACHINES <s> In this paper, design and optimization of a ladder-type single-sided linear induction motor (Ladder SLIM) for machine tool applications is investigated. High-speed linear induction machines suffer from the end-effect phenomenon, which can reduce the thrust and result in declined output characteristics. Although it is common to consider this phenomenon in high-speed applications, it is essential to take it into account in the design and analysis of low-speed low-air-gap linear machines. In addition, Ladder SLIMs have significant flux density ripples, and using skewed bars for secondary of the machine is a common solution for it. Therefore, providing required equations, an algorithm for designing a Ladder SLIM considering the end-effect phenomenon based on Duncan model is proposed, and effect of several design parameters on the output characteristics are analyzed. In order to find the optimum design parameters, a multiobjective optimization problem considering efficiency, power factor, and braking force due to the end effect is defined. This problem is optimized with two algorithms, interior point and genetic algorithms, and results are compared. Finally, the validity of the obtained solution is verified using a 2-D and 3-D time-stepping finite-element method. <s> BIB014 </s> Linear Machines for Long Stroke Applications—A Review <s> II. LINEAR INDUCTION MACHINES <s> In this paper, a study of the performance and experimental results for a low-speed linear induction motor (LIM) with toroidal winding is presented. The main idea behind the proposed winding is to provide better performance and efficiency in space-constrained applications where a LIM with much higher primary length in comparison with the primary width is required. The LIM with toroidal winding in this case enjoys the advantage of short end-connection length, which can provide more superior performance than a distributed winding linear induction motor (DWLIM). Also, as the motor is designed to operate at low speed, the output power, and thus efficiency, would be low, and then the performance of such machines cannot be evaluated in terms of output power or efficiency. However, an improvement in performance, and thus efficiency, may be achieved when the low-speed motor is fed from a low-frequency supply. This paper mainly focuses on the design optimization, prototyping, and performance evaluation of a low-speed LIM with toroidal winding, and also provides a detailed comparison between the LIM with toroidal winding and a DWLIM. <s> BIB015 </s> Linear Machines for Long Stroke Applications—A Review <s> II. LINEAR INDUCTION MACHINES <s> Linear induction machines dynamics may not be as fast as that of their rotary counterparts due to a larger air gap, a bigger leakage flux, and end effect. Direct thrust control can speed up their dynamics substantially with rather high pulsations, while vector control facilitates a smoother performance but slower dynamics. In this paper, an analogy between the two control methods, when they are applied to the linear machines, is proved first. The analogy is then used to justify combining the selected parts of the two control methods to come up with a control method that provides faster dynamics than vector control and smother performance than direct thrust control. The proposed control method is realized by getting rid of demanding parts of the two conventional control methods. End effect is also considered in the modeling and control system design. Extensive simulation results confirmed by experimental results prove improved motor performances under the proposed control method. The proposed control is particularly suitable for linear applications that need fast dynamics and smooth operation. <s> BIB016 </s> Linear Machines for Long Stroke Applications—A Review <s> II. LINEAR INDUCTION MACHINES <s> Electromagnetic characteristics of single-sided linear induction motors (SLIMs) will be changed when the primary is displaced sideways from the symmetrical position in the metro. The transverse force will be produced, and edge effect will increase. In order to reduce the influence of the transverse force on the stability of the vehicle guidance system, the construction of the ladder-slit secondary, which consists of the slit aluminum and solid back-iron plates, is proposed and analyzed. A 3-D finite-element model (FEM) of the linear induction motor with the ladder-slit secondary is developed. With the different transverse displacements, the air-gap flux density, eddy currents, and forces are calculated and compared with the linear induction motor with the flat-solid secondary. It shows that the ladder-slit secondary can reduce the transverse force and edge effect, as well as is a good method to achieve the transversal stabilization of the rail vehicle. <s> BIB017 </s> Linear Machines for Long Stroke Applications—A Review <s> II. LINEAR INDUCTION MACHINES <s> The linear induction motor (LIM) drive in linear metros suffers heavily from low efficiency due to its large air-gap length and the partial load conditions, where high loss appears in both the LIM and the inverter when a constant excitation current is generally required. Worse still, the end-effects, including both the transversal edge-effect and longitudinal end-effect, would lead to the decrease of magnetizing inductance and the increase of secondary resistance, resulting in extra loss and further deterioration of drive efficiency. To reduce the loss of the LIM drive, this paper proposes a novel loss model based loss minimization control (LMC) scheme for LIM drives. First, in the equivalent circuit and the mathematical model of LIM, four coefficients are introduced to evaluate the influence of the end-effects. Based on a thorough analysis of the LIM copper and core losses, together with the inverter conduction and switching losses, a novel integrated loss model of LIM drive is then developed, and an improved LMC scheme to obtain the optimal solution online is proposed to minimize the loss of the LIM drive. The proposed control method is successfully implemented in a 3-kW LIM drive prototype. The effectiveness of the proposed method is validated by the experimental results. <s> BIB018 </s> Linear Machines for Long Stroke Applications—A Review <s> II. LINEAR INDUCTION MACHINES <s> To incorporate the merits of simple and robust movers of induction linear motors and higher power factor and high efficiency of permanent magnet linear motors, a new long primary double-sided linear flux-switching permanent magnet (DSLFSPM) motor is proposed, investigated, and quantitatively compared with linear induction motors (LIM) for the electromagnetic launch system. First, the topology, operation principle, and electromagnetic performance of the DSLFSPM motor have been investigated. Then, in order to analyze the LIMs comprehensively, three LIMs have been designed based on the same overall dimensions and other key parameters. Finally, the electromagnetic performance of DSLFSPM motor and LIMs have been compared and summarized. The results show that the DSLFSPM motor has some merits such as higher efficiency, higher power factor, more robust mover, larger thrust force, and smaller force ripple than LIM motors. In order to validate the simulation results, a small-scale prototype of DSLFSPM motor has been built and measured. <s> BIB019 </s> Linear Machines for Long Stroke Applications—A Review <s> II. LINEAR INDUCTION MACHINES <s> In order to improve the performance of linear induction motors for urban rail transit, the ladder-slit secondary is designed, and the performance of linear induction motors with the ladder-slit and flat-solid secondary is calculated and compared. First, for the ladder-slit secondary, the slot combination, widths of the secondary conductor, and side bar are designed. Second, the finite-element model of the linear induction motor with the ladder-slit secondary is developed, and the influence of vertical forces on widths of secondary conductors is presented. Third, the eddy current and the transversal air-gap flux density are compared when the ladder-slit and flat-solid secondaries are applied, respectively. Finally, experiments are carried out by test lines to validate the calculated results of the force and efficiency. It shows that the ladder-slit secondary can reduce the edge effect and improve the performance. <s> BIB020 </s> Linear Machines for Long Stroke Applications—A Review <s> II. LINEAR INDUCTION MACHINES <s> In this paper, the influence of the flat and cap secondary, which are usually applied in the linear metro system on transverse forces of linear induction motors, is calculated and analyzed. First, the cap-aluminum plate and lamination back iron of the cap secondary caused relative difficulty in the analytical calculation. Then, the equivalent of the cap-aluminum plate and lamination back iron is presented, and two equivalent coefficients are obtained. The two types of the asymmetry are discussed and the different magnetic motive forces are presented. Second, the analytical model and the method of the linear induction motor are presented. Third, for a cap secondary in practice, the two equivalent coefficients are calculated and discussed. Then, the flat and cap secondary are applied, and the different transverse forces are calculated and analyzed. For two types of the secondaries, the different eddy currents and air-gap flux densities are compared. Finally, the test workbench is built and the analytical result of the transverse force is experimentally validated. It shows that cap secondary has less influence on the railcar when the transverse displacement occurs. <s> BIB021 </s> Linear Machines for Long Stroke Applications—A Review <s> II. LINEAR INDUCTION MACHINES <s> Linear induction motors (LIMs) have been extensively adopted in urban railway transportation given their nonadhesion thrust and simple stator structure. However, these LIMs undergo unfavorable power factor and efficiency considering their copper and eddy losses. Recently, linear flux-switching permanent-magnet motors (LFSPMs) have attracted the attention of researchers due to their high power density, simple structure, easy heat dissipation, high efficiency, and high power factor. However, so far, no quantitative comparison exists between LFSPMs and LIMs of the size 1:1 to prove the feasibility of LFSPMs to be used in railway transit systems. Therefore, a comparative analysis between the two motors is conducted in this paper. First, this paper investigates the LIMs for railway transit using the finite element method (FEM). Second, the LFSPMs for railway transit is designed and optimized. Third, the electromagnetic performance of the two motors is compared and analyzed through the FEM. Finally, a small-sized prototype of the LFSPMs is constructed to validate the FEM results. It is concluded that LFSPMs offer a favorable thrust force, have high efficiency, and high power factor but have large normal force and force ripple. Therefore, applying LFSPM requires a reliable supporting device and a shock absorber. <s> BIB022 </s> Linear Machines for Long Stroke Applications—A Review <s> II. LINEAR INDUCTION MACHINES <s> This paper proposes an analytical model to design and investigates the double-sided linear induction machine with a ladder-slot arrangement employed on the secondary mover. Emphasizing the longitudinal end effect, the equivalent circuits are based on Duncan's model modified by electromagnetic field analysis. By defining the ratio of ladder width to slot width (the ladder–slot ratio) equivalent circuit parameters, including the magnetizing inductance and resistance, and the secondary resistance, are estimated using formulae linked to the dimensions and geometry of the primary and secondary. The analytical model is proven to be effective through measurements made on an experimental machine. Magnetizing inductance and resistance estimates are shown to be predicted accurately using the analytical method, and the analytical method is further simplified for rapid and efficient practical design. The major contribution of this research is a practical method to optimize the design of large-scale drives using linear induction motors with a ladder–slot arrangement. <s> BIB023 </s> Linear Machines for Long Stroke Applications—A Review <s> II. LINEAR INDUCTION MACHINES <s> Since the linear induction motors working with a laterally asymmetric secondary is very common, this paper develops a complete equivalent circuit model considering the asymmetry to predict thrust, vertical and transversal forces. First, this paper proposes an improved equivalent circuit, which is able to consider the single sided linear induction motor (SLIM) operating condition with a laterally asymmetric secondary and an exponential correction factor, which is used to describe the decay due to the asymmetry. Then the correction factors to the circuit components, i.e. extra magnetizing branch, primary leakage inductance and magnetizing current are presented. Four coefficients to calculate the traction characteristic of a LIMs are then computed: attractive and repulsive force coefficient for vertical and lateral forces, respectively. Also, the thrust, vertical and transversal forces are calculated by the corrected equivalent circuit in a range of rated speed is presented. Finally, these calculated results are verified by measurements carried out on a linear motor test rig. <s> BIB024
Linear induction machines are one of the most common linear machine types nowadays. They are most widespread in railway transportation, where thanks to the non-adhesion VOLUME 8, 2020 thrust capability, LIMs allow climbing higher gradients than the conventional systems BIB017 , BIB021 , BIB022 , BIB018 . Linear elevators and electromagnetic launch systems are also other applications where the implementation of LIMs has been studied BIB019 , BIB003 . Their main advantages are their simplicity, robustness and low initial cost BIB019 , BIB023 , BIB022 . The absence of PMs also makes them inherently immune to irreversible PM demagnetisation, so their reliability is a remarkable feature BIB023 . However, much like in the case of rotary machines, the efficiency and power factor of LIMs still remain low BIB006 , BIB022 , BIB018 , BIB007 . The losses of LIMs are even increased when compared to induction RMs due to the end effects, especially at high speed. When the drive is moving, new flux must be generated in the front side of LIMs. This induces eddy currents in the secondary that create an opposing magnetic field, which results in a braking force. A similar thing happens in the rear part of the machine, where the flux of LIMs should be vanished. Here, the reduction of flux generates eddy currents on the secondary that aim to maintain the flux at previous levels. Thus, an attractive force is generated in the rear part of the machine, which also contributes to the total braking force. This effect only happens when the machine is moving, and is known as dynamic end effect. It is obvious that, apart from the extra loss generated by the end eddy currents, thrust capability of LIMs is also reduced due to the dynamic end effect, because the machine has to deal with an inherent braking force BIB014 , BIB002 , BIB013 . There are three main fields where research is being focused for LIMs: Design BIB006 - BIB022 , modelling BIB009 , BIB012 ,  - BIB024 and control BIB004 , BIB005 , BIB001 - BIB018 . Regarding LIM design, there are three main secondary types that are being investigated. Sheet secondary LIMs are the most simple configuration BIB006 , BIB009 , BIB020 . A sheet secondary LIM is shown in Fig. 4 . The secondary of these machines consists of a conductor sheet (typically aluminium) and the secondary back iron. The secondary of sheet LIMs gets even more simplified when adopting dual-side configuration, because the back iron is not needed anymore, thus consisting only of the conductive plate BIB019 . Despite their simplicity, sheet secondary configurations present an increased air gap length. The flux needs to get through the secondary sheet before entering the back iron, which reduces the flux linkage and thrust capacity of sheet secondary LIMs, as well as their power factor. In this sense, the performances of LIMs can be enhanced by employing a cage secondary BIB008 or a ladder-slit secondary BIB014 , BIB020 , BIB017 , BIB023 . However, these secondary structures can negatively affect the thrust force ripple BIB014 , and the complexity of the manufacturing process is also increased BIB020 . Different shapes of secondary cages are analysed and compared to a sheet secondary in BIB008 . Lee et al. conclude that the shape of the secondary can affect to the values of the equivalent circuit's parameters. When the thrust forces generated by a sheet secondary and different shaped cage secondary configurations are compared at rated slip, the results show that cage secondary type LIMs can generate higher thrust force. Besides, the phase currents are also reduced in the whole slip range. However, cage LIMs have to deal with a higher normal force. A ladder slit secondary machine is designed in BIB020 for an urban rail transit application. The results show that both thrust and efficiency can be enhanced if the ladder structure is employed instead of the traditional sheet structure. Similar results are obtained in BIB017 , where the transverse forces generated by sheet and ladder-slit secondary machines are also compared. The authors conclude that the ladder structure can help in laterally stabilizing and safely operating trains, because the transverse forces are mitigated. Reference BIB014 shows a design optimisation algorithm for ladder LIMs, based on a multiobjective multivariable optimisation technique. Toroidal windings are also found to be much superior to conventional distributed windings in BIB015 , when specified constraints include tight space requirements. Both thrust and efficiency are improved with this type of windings. Nonetheless, the toroidal winding must be manufactured by hand, thus, the manufacturing process gets more complicated this way. Optimisation techniques have also been proofed to be useful in the design of sheet secondary LIMs. In BIB009 the authors select particle swarm optimization method, as the search algorithm for the optimal design. A combination of optimal Latin-hypercube design and progressive quadratic response surface methodology is employed to optimise a single-sided LIM in BIB011 . The optimisation of the efficiency has also been assessed control-wise. An integrated inverter and LIM loss model is proposed in BIB018 . Hu et al. reduce the losses of the system by 3-4 %, which can reduce the energy consumption significantly in linear metro systems. The control has also been used to compensate the influence of the end effect  , BIB013 . A combined vector and direct thrust control is proposed in BIB016 . The experiments show that the proposed control can operate a LIM smoother than the direct thrust control, and the dynamic performance of the conventional vector control is improved with the new solution. Modelling of LIMs is another important field for both design and control perspectives. The search for a precise but simple enough equivalent circuit as the one for induction RMs has been assessed in many research papers BIB009 , BIB012 ,  - BIB024 . Duncan's is a well known equivalent circuit. It assumes an exponential increase of the air gap flux density, and proposes a simple function for analysing the end effect in the T-model induction RM equivalent circuit. Based also in the T model, Xu et al. BIB010 propose an equivalent circuit that by four relative coefficients takes account of both end and edge effects. The equivalent circuit is also given in dq and αβ frames. In spite of all the research that has been developed around LIMs, their efficiency, power factor, and dynamic braking force make them unsuitable for many applications.