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\CopyrightLine[]{}{\textit{This is an open access article under the \textcolor{blue}{\underline{CC BY-SA}} license.}
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\author[1]{\bfseries Nicolás Toro-García}
\author[2]{\bfseries Yeison Alberto Garcés-Gómez}
\author[3]{\bfseries Fredy E. Hoyos}

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\affil[1]{Department of Electrical and Electronics Engineering and Computer Sciences,Universidad Nacional de Colombia - Sede Manizales, Colombia}
\affil[2]{Unidad Académica de Formación en Ciencias Naturales y Matemáticas, Universidad Católica de Manizales, Colombia}
\affil[3]{Facultad de Ciencias, Escuela de Física, Universidad Nacional de Colombia - Sede Medellín, Colombia}

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\title{Discrete and continuous model of three-phase linear induction motors considering attraction force and end-effects}
\shorttitle{Discrete and continuous model of three-phase linear induction motors considering attraction force and end-effects (N. Toro-García)}

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\journalname{International Journal of Power Electronics and Drive System (IJPEDS)}
\journalshortname{Int J Pow Elec \& Dri Syst}
\journalhomepage{http://ijpeds.iaescore.com}
\vol{x}
\no{x}
\months{May}
\years{202x}
\issn{2088-8694}
\DOI{10.11591}
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\textbf{Article Info}
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\begin{history}
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%%article info. editor's privilege
Received Jan 20, 2018

Revised Nov 9, 2018

Accepted Jun 10, 2018

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\begin{keyword} 
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Linear induction motors \sep Discrete time model \sep Continuous time model \sep Discrete control systems \sep Non-linear behaviors
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\begin{abstract}
\vspace{.3em}
%% Text of abstract
The continuous model of the linear induction motor (LIM) has been made considering the edge effects and the attraction force. Taking the attraction force into account is important when considering dynamic analysis when the motor operates under mechanical load. A laboratory prototype has been implemented from which the parameters of the equivalent LIM circuit have been obtained. The discrete model has been developed to quickly obtain computational solutions and to analyze non-linear behaviors through the application of discrete control systems. In order to obtain the discrete model of the LIM we have started from the solution of the continuous model. To develop the model, the magnetizing inductance has been considered, which reflects the edge effects. In the results, the model is compared without considering the edge effects or the attraction force with the proposed model.
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\emph{\textbf{Corresponding Author:}}
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Y. A. Garcés-Gómez,\\
Unidad Académica de Formación en Ciencias Naturales y Matemáticas,\\
Universidad Católica de Manizales,\\
Cra 23 No 60-63, Manizales, Colombia.\\
Email: ygarces@ucm.edu.co
}
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\section{Introduction}

The linear induction motor (LIM) was invented and patented more than a hundred years ago being impractical due to the difficulties in its construction by not being able to have small air space without roughness in addition to not being able to achieve good efficiency factors. Nowadays, technological advances have allowed the LIM to have greater importance, extending its use to important industrial and research applications \cite{cita0,cita1,cita2,cita3,cita4,cita5,cita6}.

Linear induction motors are three-phase AC devices that work by the general principles of electromechanical energy transformation like other induction motors and are constructed for to produce movement on a straight line. Although are named “Linear” the mathematical models are nonlinear and due to symmetrical missing in their construction is necessary to consider effects that not are present in the rotary electric machines.

When the topological characteristics of an electric machine are modified, such as the development of the LIM from a rotating electric machine, the operating conditions and design criteria differ. The magnetic circuit of the machine introduces new phenomena that cannot be fully explained with conventional theory. Consequently, old methods of analysis have been modified and, sometimes, new theories developed \cite{cita0,cita1,cita3}.

The use of linear motors for applications requiring linear motion eliminates gears and other mechanisms needed in rotary motors. The advantages of LIMs applied to linear motion are, among others, high acceleration and deceleration capacity, ability to work in hostile environments, avoid mechanical gears and transmissions, great ease of control of thrust and speed, existence of normal forces that can be used in levitation, low maintenance cost, low noise, great versatility in negotiating tight curves and steep slopes, ability to exert secondary force without mechanical contact, movement and braking independent of the terrain, and low pollution \cite{cita7,cita8,cita9,cita10,cita11}.

There is little work on sampling LIM dynamics; therefore, it is of great importance to investigate an accurate representation of the sampled data of the complete dynamics of linear induction motors, and to design slide controllers at discrete time \cite{cita6,cita11_1}.

With respect to the non-linear models of control strategies applied to LIMs, an in-depth review is made of in \cite{cita6,cita12,cita12_1}, also in the terms of the mathematical model.

For the develop of the model, the physical system was implemented to obtain the parameters of the LIM. The system is illustrated in Figure \ref{fig1}, where there are four important blocks: a controller board and an ASD (adjustable speed controller) that acts as the power source of the LIM, a power electronics drive board to power the DC motor that moves the LIM without mechanical load, and the LIM with the measurement system. The construction aspects of the LIM are well explained in \cite{cita6,cita13}.

\begin{figure}[h]
\begin{center}
\includegraphics{Imagen 1.pdf}
\end{center}
\caption{Physical system implemented to obtain the linear induction motor (LIM) model parameters.}
\label{fig1}
\end{figure}

This document is organized as follows. The second section presents the LIM model is developed considering the effects of the boundaries and considering the forces of attraction from the equivalent circuit. This model was discretized in order to validate the continuous model. In the third section we compare the model with and without end-affects and we present the conclusions of the work.

\section{LIM model considering attraction force and end-effects}

The dynamic model of the linear induction motor is analyzed by using the $d-q$ model of the equivalent electrical circuit with end effects included \cite{cita13}. The q-axis equivalent circuit of the LIM is identical to the q-axis equivalent circuit of the induction motor (RIM), i.e. the parameters do not vary with the end effects. However, the d-axis entry linoric currents affect the air gap flux by decreasing $\lambda_{dr}$. Therefore, the d-axis equivalent circuit of the RIM cannot be used in the LIM analysis when the end effects are considered.

The end effects are not very noticeable in conventional induction motors. On the other hand, in LIM, these effects become increasingly relevant with the increase in the relative velocity between the primary and the linor. Therefore, the end effects will be analyzed as a function of the LIM’s velocity. The end effects behave differently at the entry and exit ends of the LIM. The currents induced in the linor at the entry end decay more slowly than at the exit end, due to its larger time constant.

\subsection{Equivalent Circuit for LIM}

In Figure \ref{fig2}(a) shows a conceptual construction of LIM. When the primary moves, the linor is continuously replaced by a new linoric region. This new linoric region tends to oppose to the sudden increase in the penetration of the magnetization flux allowing a gradual accumulation of the magnetization field density in the air gap. The arising of a new linoric region and its influence in the magnetic field modifies the LIM performance when compared to the traditional rotary induction motor (RIM) \cite{cita14,cita15,cita15_1}. Such flux variation along the motor length is illustrated in Figure \ref{fig2}(c).

\begin{figure}[h]
\begin{center}
\includegraphics{Imagen 2.pdf}
\end{center}
\caption{(a) Eddy current generation at the entry and exit of the air gap when the primary coil moves with velocity v. (b) Polarity and decaying profile of the entry and exit eddy currents. (c) Air gap flux profile.}
\label{fig2}
\end{figure}

As the primary coil set of the LIM moves, a new field penetrates into the reaction rail in the entry area, whereas the existing field disappears at the exit area of the primary core. Both generation and disappearance of the fields create the eddy current in the reaction rail. The eddy current in the entry grows very rapidly to mirror the primary current, nullifying the primary MMF and reducing the flux to nearly zero at entry \cite{cita16}. On the other hand, the eddy current at the exit generates a kind of wake field, dragging the moving motion of the primary core. The rise and decay of the secondary eddy current are controlled by the sheet leakage time constant $T_{lr}=\frac{L_{lr}}{R_r}$ and total secondary time constant $T_r=\frac{L_r}{R_r}$ respectively \cite{cita16}. Since $T_{lr}$ is small compared with $T_r$ the eddy current at the entry grows very rapidly to the primary current level, and then decay slowly (see Figure \ref{fig3}(b)).

Hereafter, we consider the problem of incorporating the end effects into an equivalent circuit model. Note from the previous argument that, the secondary eddy current generated by the end effect is in phase opposition to the primary current.

Since we align the reference frame with the reaction linor flux and call it d-axis, it results in $\lambda_{qr}=0$. It should be noted that as far as $\lambda_{qr}=0$ and $\lambda_{dr}$ does not change, the end effect does not play any role in equivalent circuit. Since $i_{q\varepsilon}=-i_{qs}$ the entry q axis eddy current keeps $\lambda_{qr}=0$ within small error bound. Hence, the q-axis equivalent circuit is identical to the case of the RIM. However, the d-axis air gap flux is affected much by the eddy current since d-axis entry eddy current, $i_{d\varepsilon}$, reduces $\lambda_{dr}$. Therefore, the d axis dynamic model of the RIM is not appropriate to that of the LIM.

\begin{figure}[h]
\begin{center}
\includegraphics[width=10cm]{Imagen 3.pdf}
\end{center}
\caption{(a) Effective air gap MMF and (b) eddy current profile in normalized time scale.}
\label{fig3}
\end{figure}

\subsection{Magnetizing Inductance Reflecting the End Effects}

As the primary moves, the primary MMF seen by the rail will be reduced at the entry and be reflected in the exit rail to maintain the air gap flux (continuity of flux). Specifically, the polarity of the entry eddy current is opposite to that of the exit eddy current, since they are, in nature, opposing to the generation and disappearance of the fields, respectively. Note that the entry eddy current has a longer decaying time constant than the exit eddy current, since inductance is larger in the air gap than in the open air. The profile of the eddy currents is plotted in Figure \ref{fig4} based on the normalized time scale \cite{cita16}.

\begin{figure}[h]
\begin{center}
\includegraphics[width=9cm]{Imagen 4.pdf}
\end{center}
\caption{The LIM equivalent circuits taking into account the end effects: (a) d-axis equivalent circuit, (b) q-axis equivalent circuit.}
\label{fig4}
\end{figure}

Noting that the d-axis entry eddy current decays with the time constant $T_r$, the average value of the d axis entry eddy current $i_{d\varepsilon}$ over the motor length is given by \ref{eq1}

\begin{equation}
    i_\varepsilon=\frac{i_{ds}}{T_v}\int_{0}^{T_v}{e^{-t/T_r}dt}
    \label{eq1}
\end{equation}

where $T_v=D/v$, and $D$, $v$ are the motor length and velocity, respectively. Note that $T_v=D/v$ is the time taken for the motor to traverse a point. Since the travel distance over the period $T_r$ is equal to $vT_r$ we can normalize the motor length by $vT_r$ such that \ref{eq2} \cite{cita17}.

\begin{equation}
    Q=\frac{vT_v}{vT_r}=\frac{DR_r}{(L_m+L_{lr})v}
    \label{eq2}
\end{equation}

Note that $Q$ is dimensionless but represents the motor length on the normalized time scale. On this basis, the motor length is clearly dependent on the motor velocity so that at zero velocity, the motor length is infinitely long. As the velocity rises, the motor length will effectively shrink. Using \ref{eq2}, equation \ref{eq1} can be rewritten as follows:

\begin{equation}
    i_\varepsilon=\frac{i_{ds}}{Q}\int_{0}^{Q}{e^{-x}dx}=i_{ds}\frac{1-e^{-Q}}{Q}
    \label{eq3}
\end{equation}

\noindent Therefore, the effective magnetizing current is reduced such that:

\begin{equation}
    i_{ds}-i_\varepsilon=i_{ds}\left[1-\frac{1-e^{-Q}}{Q}\right]
    \label{eq4}
\end{equation}

However, the reduction in the magnetizing current due to the eddy current can be accounted by modifying the magnetizing inductance such that:

\begin{equation}
    L'_{m}=L_m(1-f(Q))
    \label{eq5}
\end{equation}

\noindent where $f(Q)=(1-e^{-Q})/Q$ \cite{cita16}. As velocity tends to zero, $L'_m$ converges to $L_m$ i.e., the LIM dynamics becomes equivalent to the RIM dynamics as the end effect disappears. Fig. \ref{fig4} shows the effective air gap MMF and the eddy current profile in normalized time scale.

\subsection{Equivalent Series Resistor Reflecting Rail Eddy Current Losses}

When the entry and exit eddy currents circulate in the rail, ohmic loss will be produced by $R_r$. Note that the $RMS$ value of the entry eddy current over the motor length is given by:

\begin{equation}
    i_{\varepsilon RMS}=\left[\frac{i^{2}_{ds}}{Q}\int_{0}^{Q}{e^{-2x}dx}\right]^{\frac{1}{2}}=i_{ds}\left[\frac{1-e^{-2Q}}{2Q}\right]^{\frac{1}{2}}
    \label{eq6}
\end{equation}

Hence, the loss caused by the entry eddy current is evaluated as \cite{cita17_1} in \ref{eq7}:

\begin{equation}
    P_{entry}=i_{\varepsilon RMS}^2 R_r = i_{ds}^2 R_r \frac{1-e^{-2Q}}{2Q}
    \label{eq7}
\end{equation}

Following the method in \cite{cita17_1}, we also evaluate the losses due to the exit eddy current by the time rate of change of the magnetic energy as it leaves the motor air gap. Note from \ref{eq3} that the total eddy current in the air gap is equal to $i_ds(1-e^{-Q})$. This current must be vanished at the exit rail during $T_v$ to satisfy the steady state condition in air gap flux. Hence, the loss due to the exit eddy current is given by \ref{eq8}:

\begin{equation}
    P_{exit} = \frac{L_r i_{ds}^2 (1-e^{-Q})^2}{2T_v} = i_{ds}^2 R_r \frac{(1-e^{-Q})^2}{2Q}
    \label{eq8}
\end{equation}

Adding (7) and (8), the total ohmic losses due to eddy current in the rail is given by This power loss can be represented as a serially connected resistor $R_r f(Q)$ in the magnetizing current branch. \ref{fig4} shows the final proposed equivalent Duncan’s circuits in which the end effects of the LIM are considered.

Duncan's circuit has been developed considering velocity and power loss. It supposes uniform winding and materials, symmetric impedances per phase and equal mutual inductances. It's based on traditional model of three-phase, Y-connected rotatory induction motor whit linear magnetic circuit in a synchronous reference system (superscript ``e'') aligned with the linor flux. Also only longitudinal end effects have been considered.

Duncan's model has been adopted in order to obtain a space state representation both continuous-time and discrete-time. Several techniques have been developed for non linear dynamics analysis in the state space.

Parameter $Q$, function $f(Q)$, Magnetizing Inductance Reflecting the End Effects and Equivalent Series Resistor Reflecting Rail Eddy Current Losses have been derived from circuit theory.

The $Q$ factor is associated with the length of the primary, and to a certain degree, quantifies
the end effects as a function of the velocity $v$ as described by equation \ref{eq9}.

\begin{equation}
    Q = \frac{DR_r}{L_r v}
    \label{eq9}
\end{equation}

Note that the $Q$ factor is inversely dependent on the velocity, i.e., for a zero velocity the $Q$ factor may be considered infinite, and the end effects may be ignored. As the velocity increases the end effects increases, which causes a reduction of the LIM's magnetization current. This effect may be quantified in terms of the magnetization inductance with the equation:

\begin{equation*}
    L'_m=L_m (1-f(Q))
\end{equation*}

\noindent where $f(Q)=\frac{1-e^{-Q}}{Q}$.

The resistance in series with the inductance $L'_m$ in the magnetization branch of the equivalent
electricalcircuitofthe $d-axis$, is determined in relation to the increase in losses occurring with the increase of the currents induced at the entry and exit ends of the linor. These losses may be represented as the product of the linor resistance $R_r$ by the factor $f(Q)$, ie, $R_r f(Q)$ \cite{cita18}.

From the $d-q$ equivalent circuit of the LIM, the primary and linor voltage equations in a stationary reference system aligned with the linor flux are given by:

\begin{equation}
    \begin{split}
    u_{ds} & = R_s i_{ds} + R_r f(Q)(i_{ds}+i_{dr}) + \frac{d\alpha_{ds}}{dt} \\
    u_{qs} & = R_s i_{qs} + \frac{d\alpha_{qs}}{dt} \\
    u_{dr} & = R_r i_{dr} + R_r f(Q)(i_{ds}+i_{dr}) + \frac{d\alpha_{dr}}{dt} +\frac{\pi}{\tau}v\lambda_{qr} \\
    u_{qr} & = R_r i_{dr} + \frac{d\alpha_{qr}}{dt} - \frac{\pi}{\tau}v\lambda_{dr}
    \end{split}
    \label{eq10}
\end{equation}

Due to the short-circuited secondary their voltages are zero, that is, $u_{dr}=u_{qr}=0$.

The linkage fluxes are given by the following equations:

\begin{equation}
    \begin{split}
    \lambda_{ds} & = L_s i_{ds} + L_m i_{dr} - L_m f(Q)(i_{ds} + i{dr}) \\
    \lambda_{qs} & = L_s i_{qs} + L_m i_{qr} \\
    \lambda_{dr} & = L_r i_{dr} + L_m i_{ds} - L_m f(Q)(i_{ds} + i{dr}) \\
    \lambda_{qr} & = L_r i_{qr} + L_m i_{qs}
    \end{split}
    \label{eq11}
\end{equation}

To develop a state space LIM model from \ref{eq10} and \ref{eq11} is necessary to combine both equations. Because $q-axis$ equivalent circuit of the LIM is identical to the $q-axis$ equivalent circuit of
the induction motor (RIM), the parameters do not vary with the end effects and so $\frac{d \lambda_{qr}}{dt}$ and $\frac{d i_{qs}}{dt}$ in \ref{eq12} remaind it equals to \ref{eq13} [6].

\begin{equation}
    \begin{aligned}
    &\frac{d i_{q s}}{d t}=-\left[\frac{R_{s}}{\rho L_{s}}+\frac{1-\rho}{\rho T_{r}}\right] i_{q s}-\frac{L_{m} \pi}{\rho L_{s} L_{r} \tau} v \lambda_{d r}+\frac{L_{m}}{\rho L_{s} L_{r} T_{r}} \lambda_{q r}+\frac{1}{\rho L_{s}} u_{q s}\\
    &\frac{d i_{d s}}{d t}=-\left[\frac{R_{s}}{\rho L_{s}}+\frac{1-\rho}{\rho T_{r}}\right] i_{d s}+\frac{L_{m}}{\rho L_{s} L_{r} T_{r}} \lambda_{d r}+\frac{L_{m} \pi}{\rho L_{s} L_{r} \tau} v \lambda_{q r}+\frac{1}{\rho L_{s}} u_{d s}\\
    &\frac{d \lambda_{q r}}{d t}=\frac{L_{m}}{T_{r}} i_{q s}+\frac{\pi}{\tau} v \lambda_{d r}-\frac{1}{T_{r}} \lambda_{q r}\\
    &\frac{d \lambda_{d r}}{d t}=\frac{L_{m}}{T_{r}} i_{d s}-\frac{1}{T_{r}} \lambda_{d r}-\frac{\pi}{\tau} v \lambda_{q r}\\
    &\frac{d v}{d t} \quad=\frac{K_{f}}{M}\left(\lambda_{d r} i_{q s}-\lambda_{q r} i_{d s}\right)-\frac{B}{M} v-\frac{F_{L}}{M}
    \end{aligned}
    \label{eq12}
\end{equation}

\begin{equation}
    \begin{array}{l}
    {\frac{d i_{q s}}{d t}=-\left[\frac{R_{s}}{\rho L_{s}}+\frac{1-\rho}{\rho T_{r}}\right] i_{q s}-\frac{L_{m} \pi}{\rho L_{s} L_{r} \tau} v \lambda_{d r}+\frac{L_{m}}{\rho L_{s} L_{r} T_{r}} \lambda_{q r}+\frac{1}{\rho L_{s}} u_{q s}} \\
    {\frac{d \lambda_{q r}}{d t}=\frac{L_{m}}{T_{r}} i_{q s}+\frac{\pi}{\tau} v \lambda_{d r}-\frac{1}{T_{r}} \lambda_{q r}}
    \end{array}
    \label{eq13}
\end{equation}

The RIM electrical torque in an arbitrary reference frame is giving by [19], and modifying it with relation $v=\tau \frac{\omega_1}{\pi}=2 \tau f_1$  we obtain following LIM thrust force.

\begin{equation*}
F_{e}=\frac{3}{2} \frac{\pi}{\tau \omega_{r}}\left[\omega\left(\lambda_{d s} i_{q s}-\lambda_{q s} i_{d s}\right)+\left(\omega-\omega_{r}\right)\left(\lambda_{d r} i_{q r}-\lambda_{q r} i_{d r}\right)\right]
\end{equation*}

in a stationary reference frame $(\omega=0)$ the thrust force becomes:

\begin{equation}
F_{e}=\frac{3}{2} \frac{\pi}{\tau}\left[\lambda_{q r} i_{d r}-\lambda_{d r} i_{q r}\right]
\label{eq14}
\end{equation}

clearing $i_{dr}$ from $\lambda_dr$ in \ref{eq11}

\begin{equation}
i_{d r}=\frac{\lambda_{d r}-L_{m}(1-f(Q)) i_{d s}}{L_{r}-L_{m} f(Q)}
\label{eq15}
\end{equation}

clearing $i_qr$ from $\lambda_qr$ in \ref{eq11}

\begin{equation}
i_{q r}=\frac{\lambda_{q r}-L_{m} i_{q s}}{L r}
\label{eq16}
\end{equation}

Substituting $i_{dr}$ and $i_{qr}$ into \ref{eq14} results in:

\begin{equation}
F_{e}=\frac{3}{2} \frac{\pi}{\tau} \frac{L_{m}}{L_{r}}\left[\lambda_{d r} i_{q s}+\frac{f(Q)}{L_{r}-L_{m} f(Q)} \lambda_{q r} \lambda_{d r}-\frac{1-f(Q)}{L_{r}} \lambda_{q r} i_{d s}\right]
\label{eq17}
\end{equation}

Then space state mechanical equation is giving by \ref{eq18}

\begin{equation}
\frac{d v}{d t}=\frac{K_{f}}{M}\left[\lambda_{d r} i_{q s}+\frac{f(Q)}{L_{r}-L_{m} f(Q)} \lambda_{q r} \lambda_{d r}-\frac{1-f(Q)}{1-\frac{L m}{L_{r}} f(Q)} \lambda_{q r} i_{d s}\right]-\frac{B}{M} v-\frac{F_{L}}{M}
\label{eq18}
\end{equation}

Considering short-circuited linor circuit $(u_{dr}=0)$ and solving for $\frac{d\lambda_{dr}}{dt}$ gets

\begin{equation}
\frac{d \lambda_{d r}}{d t}=-\frac{R_{r}(1+f(Q))}{L_{r}-L_{m} f(Q)} \lambda_{d r}-\frac{\pi}{\tau} v \lambda_{q r}+\frac{R_{r}\left(L_{m}-L_{r} f(Q)\right)}{L_{r}-L_{m} f(Q)} i_{d s}
\label{eq19}
\end{equation}

Substituting the first equation of \ref{eq11} into first equation of \ref{eq10} results:

\begin{equation}
\begin{aligned}
u_{d s}=&\left[R_{s}+R_{r} f(Q)-L_{m} \frac{d f(Q)}{d t}\right] i_{d s}+\left[L_{s}-L_{m} f(Q)\right] \frac{d i_{d s}}{d t} \\
&+L_{m}[1-f(Q)] \frac{d i_{d r}}{d t}-L_{m} \frac{d f(Q)}{d t} i_{d r}
\end{aligned}
\label{eq20}
\end{equation}

Clearing $i_{dr}$ from $\lambda_{dr}$ in \ref{eq11}

\begin{equation*}
i_{d r}=\frac{1}{L_{r}-L_{m} f(Q)} \lambda_{d r}-\frac{L_{m}(1-f(Q))}{L_{r}-L_{m} f(Q)} i_{d s}
\end{equation*}

and substituting into \ref{eq20} results

\begin{equation*}
\begin{aligned}
u_{d s}=&\left[R_{s}+R_{r} f(Q)-\frac{\left(L_{r}-L_{m}\right)^{2}}{\left(L_{r}-L_{m} f(Q)\right)^{2}} L_{m} \frac{d f(Q)}{d t}\right] i_{d s} \\
&+\left[L_{s}-L_{m} f(Q)-\frac{L_{m}^{2}(1-f(Q))^{2}}{L_{r}-L_{m} f(Q)}\right] \frac{d i_{d s}}{d t} \\
&+\frac{L_{m}\left(L_{m}-L_{r}\right)}{\left(L_{r}-L_{m} f(Q)\right)^{2}} \frac{d f(Q)}{d t} \lambda_{d r}+\frac{L_{m}(1-f(Q))}{L_{r}-L_{m} f(Q)} \frac{d \lambda_{d r}}{d t}
\end{aligned}
\end{equation*}

substituting $\frac{d\lambda_{dr}}{dt}$ \label{eq19} in last term into above equation we obtain:

\begin{equation*}
\begin{aligned}
u_{d s}=&\left[R_{s}+R_{r} f(Q)-\frac{\left(L_{r}-L_{m}\right)^{2}}{\left(L_{r}-L_{m} f(Q)\right)^{2}} L_{m} \frac{d f(Q)}{d t}+\frac{R_{r} L_{m}(1-f(Q))}{L_{r}-L_{m} f(Q)} \frac{\left(L_{m}-L_{r} f(Q)\right)}{L_{r}-L_{m} f(Q)}\right] i_{d s} \\
&+\left[L_{s}-L_{m} f(Q)-\frac{L_{m}^{2}(1-f(Q))^{2}}{L_{r}-L_{m} f(Q)}\right] \frac{d i_{d s}}{d t} \\
&+\left[\frac{L_{m}\left(L_{m}-L_{r}\right)}{\left(L_{r}-L_{m} f(Q)\right)^{2}} \frac{d f(Q)}{d t}-\frac{R_{r} L_{m}\left(1-f^{2}(Q)\right)}{\left(L_{r}-L_{m} f(Q)\right)^{2}}\right] \lambda_{d r}-\frac{L_{m}(1-f(Q))}{L_{r}-L_{m} f(Q)} \frac{\pi}{\tau} v \lambda_{q r}
\end{aligned}
\end{equation*}

Solving for $\frac{di_{ds}}{dt}$

\begin{equation}
\begin{aligned}
\frac{d i_{d s}}{d t}=& \frac{\left[R_{s}+R_{r} f(Q)\right]\left[L_{r}-L_{m} f(Q)\right]^{2}-L_{m}\left(L_{r}-L_{m}\right)^{2} \frac{d f(Q)}{d t}+R_{r} L_{m}[1-f(Q)]\left[L_{m}-L_{r} f(Q)\right]}{\left[L_{S} L_{r}-L_{s} L_{m} f(Q)-L_{r} L_{m} f(Q)-L_{m}^{2}+2 L_{m}^{2} f(Q)\right]\left[L_{m} f(Q)-L_{r}\right]} l_{d s} \\
&+\frac{L_{m}\left(L_{m}-L_{r}\right) \frac{d f(Q)}{d t}-R_{r} L_{m}\left[1-f^{2}(Q)\right]}{\left[L_{s} L_{r}-L_{s} L_{m} f(Q)-L_{r} L_{m} f(Q)-L_{m}^{2}+2 L_{m}^{2} f(Q)\right]\left[L_{m} f(Q)-L_{r}\right]} \lambda_{d r} \\
&+\frac{L_{m}[1-f(Q)]}{L_{s} L_{r}-L_{S} L_{m} f(Q)-L_{r} L_{m} f(Q)-L_{m}^{2}+2 L_{m}^{2} f(Q)} \frac{\pi}{\tau} v \lambda_{q r} \\
&+\frac{L_{r}-L_{m} f(Q)}{L_{s} L_{r}-L_{S} L_{m} f(Q)-L_{r} L_{m} f(Q)-L_{m}^{2}+2 L_{m}^{2} f(Q)} u_{d} s
\end{aligned}
\label{eq21}
\end{equation}

Grouping the state equations and changing the index d and q by $\alpha$ and $\beta$ respectively, and omitting the primary and secondary (linor) indexes because the voltages and currents are with respect to primary and the fluxes are with respect to secondary, we obtain \ref{eq22}:

\begin{equation}
\begin{aligned}
&\frac{d i_{\beta}}{d t}=-\left[\frac{R_{S}}{\rho L_{S}}+\frac{1-\rho}{\rho T_{r}}\right] i_{\beta}-\frac{L_{m} \pi}{\rho L_{S} L_{r} \tau} v \lambda_{\alpha}+\frac{L_{m}}{\rho L_{S} L_{r} T_{r}} \lambda_{\beta}+\frac{1}{\rho L_{S}} u_{\beta}\\
&\frac{d i_{\alpha}}{d t}=\frac{\left[R_{S}+R_{r} f(Q)\right]\left[L_{r}-L_{m} f(Q)\right]^{2}-L_{m}\left(L_{r}-L_{m}\right)^{2} \frac{d f(Z)}{d t}+R_{r} L_{m}[1-f(Q)]\left[L_{m}-L_{r} f(Q)\right]}{\left[L_{S} L_{r}-L_{S} L_{m} f(Q)-L_{r} L_{m} f(Q)-L_{m}^{2}+2 L_{m}^{2} f(Q)\right]\left[L_{m} f(Q)-L_{r}\right]} i_{\alpha}\\
&+\frac{L_{m}\left(L_{m}-L_{r}\right) \frac{d f(Q)}{d t}-R_{r} L_{m}\left[1-f^{2}(Q)\right]}{\left[L_{S} L_{r}-L_{S} L_{m} f(Q)-L_{r} L_{m} f(Q)-L_{m}^{2}+2 L_{m}^{2} f(Q)\right]\left[L_{m} f(Q)-L_{r}\right]} \lambda_{\alpha}\\
&+\frac{L_{m}[1-f(Q)]}{L_{s} L_{r}-L_{s} L_{m} f(Q)-L_{r} L_{m} f(Q)-L_{m}^{2}+2 L_{m}^{2} f(Q)} \frac{\pi}{\tau} v \lambda_{\beta}\\
&+\frac{L_{r}-L_{m} f(Q)}{L_{S} L_{r}-L_{S} L_{m} f(Q)-L_{r} L_{m} f(Q)-L_{m}^{2}+2 L_{m}^{2} f(Q)} u_{\alpha}\\
&\frac{d \lambda_{\beta}}{d t}=\frac{L_{m}}{T_{r}} i_{\beta}+\frac{\pi}{\tau} v \lambda_{\alpha}-\frac{1}{T_{r}} \lambda_{\beta}\\
&\frac{d \lambda_{\alpha}}{d t}=-\frac{R_{r}(1+f(Q))}{L_{r}-L_{m} f(Q)} \lambda_{\alpha}-\frac{\pi}{\tau} v \lambda_{\beta}+\frac{R_{r}\left(L_{m}-L_{r} f(Q)\right)}{L_{r}-L_{m} f(Q)} i_{\alpha}\\
&\frac{d v}{d t}=\frac{K_{f}}{M}\left[\lambda_{\alpha} i_{\beta}+\frac{f(Q)}{L_{r}-L_{m} f(Q)} \lambda_{\beta} \lambda_{\alpha}-\frac{1-f(Q)}{1-\frac{L_{m}}{L_{r}} f(Q)} \lambda_{\beta} i_{\alpha}\right]-\frac{B}{M} v-\frac{F_{L}}{M}\\
&\frac{d x}{d t}=v
\end{aligned}
\label{eq22}
\end{equation}

where $v$ is the mover linear velocity; $\lambda_\alpha$ and $\lambda_\beta$ are the $d-axis$ an $q-axis$ secondary flux; $i_\alpha$ and $i_\beta$ are the $d-axis$ and $q-axis$ primary current; $u_\alpha$ and $u_\beta$ are the $d-axis$ and $q-axis$ primary voltage; $T_r=\frac{L_r}{R_r}$ is the secondary time constant; $\rho=1-\frac{L_m^2}{L_s L_r}$ is the leakage coefficient; $K_f=\frac{3}{2}\frac{\pi L_m}{\tau L_r}$ is the force constant; $R_s$ is the winding resistance per phase; $R_r$ is the secondary resistance per phase referred primary; $L_m$ is the magnetizing inductance per phase; $L_r$ is the secondary inductance per phase referred primary; $L_s$ is the primary inductance per phase; $F_L$ is the external force disturbance; $M$ is the total mass of the mover; $B$ is the viscous friction and iron-loss coefficient; $\tau$ is the pole pitch; $D$ is the primary length in meters; $Q=\frac{DR_r}{L_r v}$ is a factor related to the primary length, which quatifies the end effects as a function of the speed and $f(Q)=\frac{1-e^{-Q}}{Q}$ is the factor related to the losses in the magnetization branch. To discretize the state LIM model with end effects we use the backward difference method \cite{cita20} and finally we obtain an approximate discrete time version of the LIM model \ref{eq23} taking into account end effects.

\begin{equation}
\begin{aligned}
&i_{k+1}^{\beta}=i_{k}^{\beta}-\left[\frac{R_{S}}{\rho L_{S}}+\frac{1-\rho}{\rho T_{r}}\right] T i_{k}^{\beta}-\frac{L_{m} \pi}{\rho L_{S} L_{r} \tau} T v_{k} \lambda_{k}^{\alpha}+\frac{L_{m}}{\rho L_{S} L_{r} T_{r}} T \lambda_{k}^{\beta}+\frac{1}{\rho L_{S}} T u_{k}^{\beta}\\
&i_{k+1}^{\alpha}=i_{k}^{\alpha}+\frac{\left[R_{s}+R_{r} f(Q)\right]\left[L_{r}-L_{m} f(Q)\right]^{2}-L_{m}\left(L_{r}-L_{m}\right)^{2} \frac{\Delta f(Q)}{T}+R_{r} L_{m}[1-f(Q)]\left[L_{m}-L_{r} f(Q)\right]}{\left[L_{s} L_{r}-L_{s} L_{m} f(Q)-L_{r} L_{m} f(Q)-L_{m}^{2}+2 L_{m}^{2} f(Q)\right]\left[L_{m} f(Q)-L_{r}\right]} T i_{k}^{\alpha}\\
&+\frac{L_{m}\left(L_{m}-L_{r}\right) \frac{\Delta f(Q)}{T}-R_{r} L_{m}\left[1-f^{2}(Q)\right]}{\left[L_{S} L_{r}-L_{S} L_{m} f(Q)-L_{r} L_{m} f(Q)-L_{m}^{2}+2 L_{m}^{2} f(Q)\right]\left[L_{m} f(Q)-L_{r}\right]} T \lambda_{k}^{\alpha}\\
&+\frac{L_{m}[1-f(Q)]}{L_{s} L_{r}-L_{s} L_{m} f(Q)-L_{r} L_{m} f(Q)-L_{m}^{2}+2 L_{m}^{2} f(Q)} \frac{\pi}{\tau} T v_{k} \lambda_{k}^{\beta}\\
&+\frac{L_{r}-L_{m} f(Q)}{L_{s} L_{r}-L_{s} L_{m} f(Q)-L_{r} L_{m} f(Q)-L_{m}^{2}+2 L_{m}^{2} f(Q)} T u_{k}^{\alpha}\\
&\lambda_{k+1}^{\beta}=\lambda_{k}^{\beta}+\frac{L_{m}}{T_{r}} T i_{k}^{\beta}+\frac{\pi}{\tau} T v_{k} \lambda_{k}^{\alpha}-\frac{1}{T_{r}} T \lambda_{k}^{\beta}\\
&\lambda_{k+1}^{\alpha}=\lambda_{k}^{\alpha}-\frac{R_{r}(1+f(Q))}{L_{r}-L_{m} f(Q)} T \lambda_{k}^{\alpha}-\frac{\pi}{\tau} T v_{k} \lambda_{k}^{\beta}+\frac{R_{r}\left(L_{m}-L_{r} f(Q)\right)}{L_{r}-L_{m} f(Q)} T i_{k}^{\alpha}\\
&v_{k+1}=v_{k}+\frac{K_{f}}{M} T\left[\lambda_{k}^{\alpha} i_{k}^{\beta}+\frac{f(Q)}{L_{r}-L_{m} f(Q)} \lambda_{k}^{\beta} \lambda_{k}^{\alpha}-\frac{1-f(Q)}{1-\frac{L m}{L_{r}} f(Q)} \lambda_{k}^{\beta} i_{k}^{\alpha}\right]-\frac{B}{M} T v_{k}-\frac{F_{L}}{M} T\\
&x_{k+1}=x_{k}+v_{k} T
\end{aligned}
\label{eq23}
\end{equation}

where $v_{k}=v(k T)$ is the mover linear velocity; $\lambda_{k}^{\alpha}=\lambda_{\alpha}(k T)$ and $\lambda_{k}^{\beta}=\lambda_{\beta}(k T)$ are the $d-axis$ an $q-axis$ secondary flux; $i_{k}^{\alpha}=i_{\alpha}(k T)$ and $i_{k}^{\beta}=i_{\beta}(k T)$ are the $d-axis$ and $q-axis$ primary current; $u_{k}^{\alpha}=u_{\alpha}(k T)$ and $u_{k}^{\beta}=u_{\beta}(k T)$ are the $d-axis$ and $q-axis$ primary voltage; $T_r=\frac{L_r}{R_r}$  is the secondary time constant; $\rho=1-\left(\frac{L_{m}^{2}}{L_{s} L_{r}}\right)$ is the leakage coefficient; $K_{f}=\frac{3}{2} \frac{\pi L_{m}}{\tau L_{r}}$ is the force constant; $R_s$ is the winding resistance per phase; $R_r$ is the secondary resistance per phase referred primary; $L_m$ is the magnetizing inductance per phase; $L_r$ is the secondary inductance per phase referred primary; $L_s$ is the primary inductance per phase; $F_L$ is the external force disturbance; $M$ is the total mass of the mover; $B$ is the viscous friction and iron-loss coefficient; $\tau$ is the pole pitch; $D$ is the primary length in meters; $Q=\frac{D R_{r}}{L_{r} v_{k}}$ is a factor related to the primary length, which quantifies the end effects as a function of the speed; $f(Q)=\frac{1-e^{-Q}}{Q}$ is the factor related to the losses in the magnetization branch and $\frac{\Delta f(Q)}{T}=\left.\frac{d f(Q)}{d t}\right|_{t=k T}$.

\section{Results}

Figure \ref{fig5} shows the end effects on mover velocity, fluxes and currents. Figures 6, 7 and 8 show the system \ref{eq22} behavior when the frequency of input voltage vary. The steady state velocity is a periodic wave in all cases, but when the fed frequency is lower, higher output frequency components appear. Phase portraits in subfigures \ref{fig6b}, \ref{fig6c}, \ref{fig7b}, \ref{fig7c}, \ref{fig8b} and \ref{fig8c} with attractive limit cycles are shown.

\begin{figure}[H]
\centering
  \subfloat[]{
	\begin{minipage}[c][1.2\width]{0.4\textwidth}
	   \centering
	   \includegraphics[scale=0.9]{Imagen 5a.pdf}
	\end{minipage}}
\hspace{1cm}
  \subfloat[]{
	\begin{minipage}[c][1.2\width]{0.4\textwidth}
	   \centering
	   \includegraphics[scale=0.9]{Imagen 5b.pdf}
	\end{minipage}}
\hspace{1cm}
  \subfloat[]{
	\begin{minipage}[c][1.2\width]{0.4\textwidth}
	   \centering
	   \includegraphics[scale=0.9]{Imagen 5c.pdf}
	\end{minipage}}
\hspace{1cm}
  \subfloat[]{
	\begin{minipage}[c][1.2\width]{0.4\textwidth}
	   \centering
	   \includegraphics[scale=0.9]{Imagen 5d.pdf}
	\end{minipage}}
	\vspace{.7em}
\caption{Mover velocity, velocity differences, currents and fluxes resulting from the model simulation using ODE45 function of Matlab, taking into account end effects in model of LIM Equation \ref{eq22} and without end-effects model Equation\ref{eq3}, \linebreak (a) Mover velocity of LIM with and without end-effects, (b) Velocity difference vs mover velocity without end-effects, (c) $\beta-axis$ With and without end-effects currents, (d) $\beta-axis$ With and without end-effects fluxes}
\end{figure}

\begin{figure}[H]
\centering
  \subfloat[]{
	\begin{minipage}[c][1\width]{0.3\textwidth}
	   \centering
	   \includegraphics[scale=0.8]{Imagen 6a.pdf}
	\end{minipage}}
  \subfloat[]{
	\begin{minipage}[c][1\width]{0.3\textwidth}
	   \centering
	   \includegraphics[scale=0.8]{Imagen 6b.pdf}
	\end{minipage}}
  \subfloat[]{
	\begin{minipage}[c][1\width]{0.3\textwidth}
	   \centering
	   \includegraphics[scale=0.8]{Imagen 6c.pdf}
	\end{minipage}}
	\vspace{.7em}
\caption{LIM behavior with $30 Hz$ input frequency. Mover velocity and phase portraits of some state variables, (a) Mover velocity of LIM with end-effects, (b) Phase portrait of mover velocity vs $i_\alpha$, (c) Phase portrait of $\lambda_\alpha$ vs $i_\alpha$}
\end{figure}

\begin{figure}[H]
\centering
  \subfloat[]{
	\begin{minipage}[c][1\width]{0.3\textwidth}
	   \centering
	   \includegraphics[scale=0.8]{Imagen 7a.pdf}
	\end{minipage}}
  \subfloat[]{
	\begin{minipage}[c][1\width]{0.3\textwidth}
	   \centering
	   \includegraphics[scale=0.8]{Imagen 7b.pdf}
	\end{minipage}}
  \subfloat[]{
	\begin{minipage}[c][1\width]{0.3\textwidth}
	   \centering
	   \includegraphics[scale=0.8]{Imagen 7c.pdf}
	\end{minipage}}
	\vspace{.7em}
\caption{LIM behavior with $60 Hz$ input frequency. Mover velocity and phase portraits of some state variables, (a) Mover velocity of LIM with end-effects, (b) Phase portrait of mover velocity vs $i_\alpha$, (c) Phase portrait of $\lambda_\alpha$ vs $i_\alpha$}
\end{figure}

\begin{figure}[H]
\centering
  \subfloat[]{
	\begin{minipage}[c][1\width]{0.3\textwidth}
	   \centering
	   \includegraphics[scale=0.8]{Imagen 8a.pdf}
	\end{minipage}}
  \subfloat[]{
	\begin{minipage}[c][1\width]{0.3\textwidth}
	   \centering
	   \includegraphics[scale=0.8]{Imagen 8b.pdf}
	\end{minipage}}
  \subfloat[]{
	\begin{minipage}[c][1\width]{0.3\textwidth}
	   \centering
	   \includegraphics[scale=0.8]{Imagen 8c.pdf}
	\end{minipage}}
	\vspace{.7em}
\caption{LIM behavior with $120 Hz$ input frequency. Mover velocity and phase portraits of some state variables, (a) Mover velocity of LIM with end-effects, (b) Phase portrait of mover velocity vs $i_\alpha$, (c) Phase portrait of $\lambda_\alpha$ vs $i_\alpha$}
\end{figure}

\section{Conclusions}

The continuous model of the linear induction motor (LIM) has been made considering the edge effects and the attraction force. A laboratory prototype has been implemented from which the parameters of the equivalent LIM circuit have been obtained. The discrete model has been developed to quickly obtain computational solutions and to analyze non-linear behaviors through the application of discrete control systems. In order to obtain the discrete model of the LIM we have started from the solution of the continuous model. To develop the model, the magnetizing inductance has been considered, which reflects the edge effects. The model is compared with and without considering the edge effects. The edge effects show a decreasing in the mover velocity at steady state due to lose power in the magnetization. In the primary current there is an increase due the power requirements to supply the magnetic loses. The phase portraits were developed shown high frequency components in velocity mover with low frequencies in the supply currents.

%%-----------------------------------------------

\section*{Funding}
This research received external funding from ``Fondo Nacional de Financiamiento para la Ciencia, la Tecnología, y la Innovación, Fondo Francisco José de Caldas con contrato No 213-2018 con Código 58960.'' Programa ``Colombia Científica''

\section*{Acknowledgement}
This work was supported by the Universidad Nacional de Colombia, Sede Medellín, grupo de investigación en Procesamiento Digital de Señales para Sistemas en Tiempo Real under the projects HERMES-34671 and HERMES-36911, the Grupo de Investigación en Recursos Energéticos GIRE at the Universidad Nacional de Colombia, and the Universidad Católica de Manizales with the Grupo de Investigación en Desarrollos Tecnológicos y Ambientales GIDTA. The authors thank the School of Physics for their valuable support given to the conduction of this research. This research paper corresponds to ``programa reconstrucción del tejido social en zonas de pos-conflicto en Colombia del proyecto Modelo ecosistémico de mejoramiento rural y construcción de paz: instalación de capacidades locales,'' and is financed by the ``Fondo Nacional de Financiamiento para la Ciencia, la Tecnología, y la Innovación, Fondo Francisco José de Caldas con contrato No 213-2018 con Código 58960.'' Programa ``Colombia Científica''.

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\end{thebibliography}

%\section*{BIOGRAPHIES OF AUTHORS} 
%% make sure to add \vspace{-.7em} before \begin{biography}
%\vspace{-.7em} 

%\begin{biography}[{\includegraphics[width=2.5cm,height=4cm,clip,keepaspectratio]{Author's_Photo}}]
%\textbf{Firstname Lastname} %% Affiliation and educational background.
%is a graduate research assistant at Diagnostics Research Group with Master of Engineering from Universiti Teknologi Malaysia (2012). He obtained Bachelor Degree in Electronics Engineering from Institut Teknologi Bandung (Indonesia) in 2009.

%% Research interests, teaching, professional experiences, etc.
%His researches are in fields of electronics, digital systems, ultrasound sensor, signal processing, and biomedical instrumentation. Recently, system's application on ultrasound exposimetry has been tackled.

%% Related activities and services.
%He is affiliated with IEEE as student member. In IJECE, IAES journals, and other scientific publications, he has served as invited reviewer. Besides, he is also involved in NGOs, student associations, and managing non-profit foundation.

%% Additional info.
%Further info on his homepage: http://diagnostics.my/imamul
%\end{biography}

%\begin{biography}[{\includegraphics[width=2.5cm,height=4cm,clip,keepaspectratio]{Author's_Photo}}]
%\textbf{Firstname Lastname} %% Affiliation and educational background.
%is a graduate research assistant at Diagnostics Research Group with Master of Engineering from Universiti Teknologi Malaysia (2012). He obtained Bachelor Degree in Electronics Engineering from Institut Teknologi Bandung (Indonesia) in 2009.

%% Research interests, teaching, professional experiences, etc.
%His researches are in fields of electronics, digital systems, ultrasound sensor, signal processing, and biomedical instrumentation. Recently, system's application on ultrasound exposimetry has been tackled.

%% Related activities and services.
%He is affiliated with IEEE as student member. In IJECE, IAES journals, and other scientific publications, he has served as invited reviewer. Besides, he is also involved in NGOs, student associations, and managing non-profit foundation.

%% Additional info.
%Further info on his homepage: http://diagnostics.my/imamul

%\end{biography}

\end{document}