# Analysis of Longitudinal End Effects in a Single Sided Linear Induction Motor Using Stream Function Approach

Abstract-This paper presents the analysis of longitudinal end effects in a single sided linear induction motor leading to the calculation of fields and currents in the rotor sheet and propulsion and levitation forces. Such problem can be tackled by vector potential or stream function approach. The present paper adopts the second approach. A second order partial differential equation has been formulated for calculation of stream function or current density in the rotor sheet. As the end effects have not been considered in the direction of width of the rotor sheet, the said formulation merges to a second order PDE in one dimension i.e. the length of the rotor. For solution of such problem suitable boundary conditions have been imposed based on the fact that current cannot escape the rotor sheet. Based on the numerical calculations for current and flux density in the rotor sheet, the important performance figures such as Propulsion and levitation force have been calculated. These performance figures have been expressed as normalized quantities and plotted against magnetic Reynolds number. Such plots become helpful for a designer to design a realistic Transportation system using magnetic levitation, taking finite length and finite width effect of the rotor sheet in consideration. Index Terms—SLIM, vector potential, stream function, infinitely long rotor, Reynolds number. Introduction

T he analysis presented in [3] takes into account the finite width effects of the rotor in a SLIM while [4] deals with the effects of both finite length and finite width of the rotor. In this literature, the longitudinal end effects of the rotor are exclusively considered to give a better understanding of the effects due to discontinuity of the rotor in the longitudinal direction without being bothered about the discontinuity of the rotor in the transverse direction. In this work, both stator and rotor are considered to be infinitely wide; the stator winding as usual is considered to be infinitely long while the rotor is assumed to be of finite length extending from x=0 to x=L and lying at a certain height ‘h’ from the stator surface. The coordinate system is fixed with respect to the rotor. Since only z-component of the stator and rotor current exists, only Az component is present. Az does not vary with z but is a function of x, y and time. This literature presents method for calculation of Fields and currents in the rotor sheet. Calculations for propulsion and levitation forces shall also be presented subsequently. Formulation for vector Potential, A

With reference to fig. 1, if the elemental current at source point (x, h) in the sheet is diz ,the vector potential A at (x’, y’) (taking the effect of the stator in iron boundary into count) can be expressed as Ax’,y’=K1-μ02π [logr1+logr2]diz (1) Where r1=2[x’-x2+y’-h2]

r2=2[x’-x2+y’+h2]

and K1 is an arbitrary constant. The elemental source current diz can be expressed as diz= Jzr d dx J z r= rotor current density in amp/m2 and‘d’ is the thickness of the rotor sheet. J z r, in turn can be expressed as J z r = σ E z r E z r= electric field intensity in the sheet, σ=conductivity of sheet Curl E=-dBdt and B=curl A it can be shown that

E=-dAdt+∇ψ (2) Where ψ is a scaler potential.For infinitely wide rotor ∇ψ=0 Hence E=-dAdt

Ezr=-∂Az(t)∂t=-j s ωsAzt (3) diz=-j s ωs σd×Azt dx (4) ‘s’ is the rotor slip

It can be seen from eqn. (4) that the elemental rotor current ,diz...

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