Induction machines are being used in many modern applications as these machines have features of simple assembly, low moment of inertia, and high initial torque with low ripple factor. These induction motors are used in heavy-duty applications by developing different control schemes [1]. The most important application of the LIM is the traction system, as the use of LIM in the traction system facilitated the technical design by direct drive [2]. A direct drive system requires an electric motor with high torque in a specified space. LIM in a proper shape and provided with variable frequency meets the required performance features by a driving system in electric traction. For many modern and heavy-duty applications such as mechanical manufacturing, robotics, CNC machines, and traction systems, linear motors are considered a better choice than rotary machines as they offer superior output and efficiency in modern applications [3–7]. The working principle of a linear motor is simple: the magnetic flux of the moving part is synchronized with the stationary section, which converts electromagnetic energy to linear motion, and the load is directly fixed with the mover. The invention of linear machines eliminated many tools such as ball and screw, belt and pulley, and other turning arrangements which were used for the conversion of rotary motion to linear motion, as well as losses involved in this conversion process [8–11]. Additionally, linear machines provide speed precision with improved efficiency and are easy to construct as the idea is simple: cut the rotary motor radially and set it flat. The working rule for linear and rotary induction motors is the same, but the air gap is more extensive in LIM, and the mover is shorter for a track, resulting in end effects [12].

The dynamic performance, precise speed control, and trust force are significant, especially for heavy-duty applications. This research aims to develop SVPWM based control scheme, considering end effects, for speed and thrust force control of LIM, which will be applicable for a different speed range of low to high-speed applications. For this purpose, LIM’s working principle and construction are explored in depth. The dynamic model is developed in MATLAB Simulink after performing detailed mathematical modeling. Regarding the LIM, one of the objectives of research work is to establish the electrical parameters of the LIM to predict the response of dynamics associated with this system. LIM description in the dynamic response requires interaction between electrical and mechanical subsystems. For this purpose, governing differential equations representing the different phenomena of the electromechanical system is considered. A reliable simulation model is developed that describes the LIM dynamics derived from mathematical modeling, and for this purpose, the specifications of the LIM are chosen based on the previous study [2,13]. Relevant parameters are attached in Annexure, and motor specifications considered in this research are presented in Table 1. A valid SVPWM-based control scheme is implemented, and an optimized set of parameters are achieved in MATLAB Simulink.

Based on the developed dynamic model and integrated SVPWM inverter, the next step to control the electromechanical system is designing and tuning the PI controller. The proposed scheme contains three parts, a dynamic model of LIM, an SVPWM modulator-based inverter, and the PI controllers. Precise control of LIM parameters, including speed and thrust force, is given, verified by the simulation results.

In most applications related to control machine drives, transient response analysis of the electrical machines is critical, and a dynamic d-q model is developed for this purpose [14]. The d-q model of the equivalent electrical circuit, considering end effects, is used to present the dynamics for LIM and complete steps to develop the dynamic model are represented in Fig. 1 [15].

In the dynamic model, q-axis equivalent circuit for LIM is the same as of the rotary induction machine (RIM), and the reason for this similarity is that the parameters are independent of the end effects. For the d-axis equivalent circuit of LIM, end effects are reflected in the equivalent circuit of RIM. So, the development of an appropriate dynamic model of LIM needs consideration of fluxes induced due to moving parts named as end effect. The end effect at the starting and ending points from the secondary is reflected as a factor f(Q), mathematically given as in Eq. (1): (1)f(Q)=1−1−e−QQ

The value of Q in (1) is given in Eq. (2): (2)Q=dTr.v=d (LrRr).v=dRr(Lm+Lir).vwhere Rr represents resistance and Lr indicates the inductance of the rotor, respectively, d gives the length of primary in meters, v shows the linear speed of the primary in m/s [16].

For dynamic modeling, 3-φ stationary frame of reference (as-bs-cs) variables are changed to 2-φ stationary frame of reference (ds-qs) variables, and after that, 2-φ variables of stationary frame of reference are changed to synchronously rotating frame of reference [17].

After considering end effects, the first step for dynamic modeling is the transformation of axes a-b-c in the ds-qs stationary frame of reference, as explained in Fig. 2, and the conversion matrix for this transformation is represented in Eq. (3): (3)[vsqsvsdsvsos]=[cos⁡θsin⁡θ1cos⁡(θ−2π3)sin⁡(θ−2π3)1cos⁡(θ+2π3)sin⁡(θ+2π3)1][vasvbsvcs]

Synchronously rotating axes (de−qe) rotate at synchronous speed ωe concerning stationary axes (ds−qs) and θe represents the angle between them. The relation between synchronous speed and the angle is expressed as θe=ωe t. Voltages on stationary axes (ds−qs) are resolved into synchronously rotating (de−qe) frames are represented in Fig. 3 and written mathematically in matrix form in Eq. (4): (4)[veqsveds]=[cosθesinθesinθecosθe][vsqsvsds]

Vice versa in Eq. (5): (5)[vsqsvsds]=[cosθesinθe−sinθecosθe][veqsveds]

For dynamic modeling in synchronously rotating reference frame q-axis and d-axis stator flux linkages [18] are shown in Figs. 4 and 5, respectively, which are represented in Eqs. (6)–(8): (6)λds=Lls.ids+ Lm(1−f(Q))(ids+idr)=(Ls−Lmf(Q)) ids+(Lm(1−f(Q))idr (7)λqs=Lls.iqs+Lm(iqs+iqr)= Ls iqs+Lm iqr (8)Total flux linkages of stator side=(λds)2+(λqs)2

From Figs. 4 and 5 q-and d-axis flux linkages for the rotor are represented mathematically in Eqs. (9)–(11): (9)λdr=Llr. idr+Lm(1−f(Q))(ids+idr)=(Lr−Lm f(Q)) idr+(Lm(1−f(Q))ids (10)λqr=Llr iqr+Lm(iqs+iqr)= Lr iqr+Lm iqs (11)Total flux linkages of rotor side=(λdr)2+(λqr)2where, Rr and Lr are rotor resistance and inductance, respectively, d represents the primary length in meters and v represents the primary linear speed in m/s. From Figs. 4 and 5, q-and d-axis air gap flux linkages are given in Eqs. (12)–(14): (12)λqm=Lm(iqs+iqr) (13)λdm=Lm(ids+idr) (14)Total flux linkages of air gap=(λdm)2+(λqm)2

With reference to Figs. 4 and 5 the Equation for q-axis and d-axis stator voltages are represented in matrix form as given in Eq. (15): (15)[vqsvdsvqrvdr]=[(Rs+sLs)ωe(Ls−Lmf(Q))sLm−ωe(Lm−Lmf(Q))−ωeLsRs+sLs+Rrf(Q)−ωeLm sLm+Rrf(Q)−sLmf(Q)sLm(ωe−ωr)(Lm−Lmf(Q))Rr+sLr(ωe−ωr)(Lm−Lmf(Q))−(ωe−ωr)Lm+Rrf(Q)sLm−sLmf(Q) Rrf(Q)(ωe−ωr)LrRrf(Q)(ωe−ωr)][iqsidsiqridr]

The matrix equations are transformed for Simulink modeling and are written as Eqs. (16)–(19): (16)iqs=1Ls∫ vqsdt−RsLs∫ iqsdt−ωeLs(Ls−Lm f(Q))∫ idsdt−LmLsiqr−1Lsωe(Lm−Lm f(Q))∫ idrdt (17)ids=1Ls∫vdsdt+ωe∫iqsdt−(Rs+Rrf(Q))Ls∫idsdt+ωeLmLs∫iqrdt−LmLsidr−LmLsf(Q))idr+RrLsf(Q)∫idrdt (18)iqr=1Lr∫vqrdt--LmLriqs--(ωe−ωr)(Lm−Lmf(Q))1Lr∫idsdt−(ωe−ωr)(Lr−Lmf(Q))Lr∫idr.dt−Rr∫iqrdt  (19)idr=1Lm(1−f(Q))∫vdrdt−[--(ωe−ωr)Lm+Rrf(Q)]1Lm(1−f(Q))∫iqsdt+[Lr−Lmf(Q)]1Lm(1−f(Q))ids+Rrf(Q)Lm(1−f(Q))∫idsdt+(ωe−ωr)LrLm(1−f(Q))∫iqrdt−RrLm(1−f(Q)∫idrdt

To develop a complete dynamic model, the mechanical part of the machine is also reflected in modeling. If Fm is mechanical thrust force, J represents rotor inertia, F shows the value of friction constant and vm gives mechanical speed than the following given differential equation represents a mechanical system of induction motor model as given in Eqs. (20)–(21): (20)ddt vm=12H(Fe−Fvm−Fm) (21)vm=12H∫(Fe−Fvm−Fm)dt=1J∫(Fe−Fvm−Fm)dt 

The expression for thrust force and the motion equation are given as in Eq. (22): (22)Fe=K(λdriqs−λqrids)= mdvdt+ Dv +FL 

In the above Equation, K=3pπ4τp where K represents force constant, p gives no. of pairs, τp indicates pole pitch, D shows viscous friction and iron loss coefficient, and m is the mass of the LIM. By placing values, the thrust force is estimated as follows in Eq. (23): (23)Fe=32πτpp2(λdriqs−λqrids)

An overall dynamic model of LIM is given in Fig. 6, developed by integrating different Simulink modules.

SVPWM is preferred for PWM implementation over other PWM techniques because of its advantages of improved fundamental output voltage, upgraded harmonic spectrum, and easy implementation in digital signal processors and microcontrollers [19]. SVPWM also provides instantaneous control of switching states and the flexibility for selecting vectors to achieve the balance of the neutral point. It is also possible to yield output voltages nearly for any average value by utilizing the closest three vectors, and this process gives the best spectral performance. SVPWM-based control is provided for LIM, and the main benefit of induction motor control with an SVPWM-based inverter is its improved performance and increased drive lifetime. These advantages motivate the SVPWM-based control scheme for managing electric vehicle-related machines as the proposed control scheme is provided for railway traction systems, and electric vehicles need high starting torque [20,21].

The space vector pulse width modulator is designed based on the SVPWM scheme, and its key parameters are given in Table 2. It comprises seven principle blocks as shown in Fig. 7. The first block is a three-phase generator that provides three sinusoidal waveforms by variable frequency and amplitude, and these three signals are at 120 degrees phase difference. This block has inputs of frequency and voltage required by the inverter. The low-pass bus filter eliminates fast transients from the DC bus voltage calculations. This filtered voltage computes the voltage vectors delivered to the motor [22]. The alpha-beta conversion block transforms variables from the three-phase reference frame to the two-phase reference frame. The plane is allocated into six different sectors separated by 60 degrees. The vector sector finds the sector of the plane having voltage vectors [23]. The switching time calculator gives the timing of the applied voltage vector. The voltage vector lies in the block input sector. The timing sequence from the switching time calculator and ramp signal from the ramp generator is delivered to the gate logic block. This block triggers the inverter switches based on comparing the ramp and the gate timing signals.

When utilizing an average-value inverter, the gates logic block is deactivated, and the inverter leg PWM duty cycles are allotted through the switching time calculator. In this mode, the Space Vector Modulator block outputs the duty cycles of the various pulses but not the pulses themselves. When used in space vector modulation mode, these duty cycle signals are expected by the average-value Three-Phase Inverter block [24]. The complete Simulink block of SVPWM is shown in Fig. 8.

Several approaches are being used to improve controller design and tuning performance [25–27]. An analytical methodology is followed to design a speed controller utilizing the transfer function in which flux linkages are kept fixed and transformed into the separately excited dc motor drive. The speed controller is designed with a systematic optimal approach by using the transfer function as this technique retains the speed-controller design’s consistency for all ac and dc drives [28].

The transfer functions of various subsystems are derived to obtain the block diagram of the indirect vector-controlled induction motor drive. These subsystems mainly include an induction machine, inverter speed controller, and transfer functions for feedback. The final block diagram of the induction motor drive is completed by integrating the subsystems in which the torque-current feedback loop and induced emf feedback loop are overlapped. Block-diagram reduction techniques are used to overcome this overlap in which the inner current loop is made independent of the motor mechanical transfer function, and this method offers a simpler construction of the current controller [29–32].

The proposed work is completed in three stages. In the first stage, mathematical modeling of a LIM is done in a step-by-step manner. The parameters for the respective machine are given in Table 1, and the presented model was tested in the Simulink environment for the same machine. The simulated LIM exhibited a satisfactory response regarding the torque and speed characteristics. After developing the LIM model, the SVPWM block is developed for the control scheme of LIM, and the output of the respective block is verified. The next step covers the PI controller tuning, using the transfer function of the various subsystems for indirect vector-controlled LIM. The developed dynamic model, SVPWM block, and PI controller with derived parameters are integrated into the Simulink environment as presented in Fig. 11, and Simulation results verify its applicability and robustness. The actual speed of LIM is compared with the step input taken as a reference speed value to get the error signal ultimately transmitted to the PI controller. The controller produces a speed control signal fed to the SVPWM generator as the magnitude of reference vector. The output of SVPWM generator is three modified sinusoidal waveforms of variable frequency and amplitude. These three signals have phase difference of 120°.

This research develops a novel framework in MATLAB Simulink by integrating three subsystems: an electromechanical model of linear induction motor, an SVPWM-based inverter, and a PI controller. This work aims to achieve precise LIM speed control for low and high-speed applications considering end effects. The developed electromechanical model helps improve the vector control scheme’s transient performance. The model reflects LIM’s final effect, which rises with speed. This effect must be reflected in high-speed control.

The SVPWM generates inverter bridge pulses based on dynamic model command values. SVPWM enables instantaneous switching state control and vector selection flexibility to balance the neutral point. This approach delivers the optimum spectrum performance for generating output voltages for any average value. SVPWM-based inverters are used to manage speed and torque with full dynamic control for high-speed applications.

Integrated controllers have been designed for a six-switch, three-phase inverter loop. The proposed framework is output by altering low and high-speed reference values. The reference value for low-speed performance is 5 m/s, and for high speed, it is 100 m/s. The simulation output motor achieved the reference speed in a few seconds and thrust force less than 500 N, which is acceptable. The proposed framework adjusted the end effects that dominate at high speed. Precise speed control and thrust force prove the framework’s success. Based on its performance, the suggested framework is highly recommended for electric vehicle control; because electric vehicles require high initial torque; thus, this yields the requisite torque at a very high speed, and its control with accommodation of end effects is a significant concern.