Influence of the skin effect and the form of slot on the starting characteristics of induction motor squirrel cage

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1 Influence of the skin effect and the form of slot on the starting characteristics of induction motor squirrel cage Zakari MADDI, Djamel AOUZELLAG, Toufik LADDI Laboratory Control of Renewable Energies, Faculty of Technology, University of Bejaia 06000, Algeria, Abstract This article presents the influence of the form of rotor slot on the startup characteristics of the induction motor IM). Three methods are presented allowing the taking into account of the real shape of the slot rotor and the highlighting of the skin effect, the first is an analytical method classic, the second is a numerical method based on the finite element MFE) under software FEMM and the last is a method of analysis of circuits based on the use of the equivalent circuits of a rotor bar in the form of stairs. Keywords- Start, form of slot rotor, skin effect, IM. I. INTRODUCTION The asynchronous machine with a static frequency converter or directly to the network is the more repented in the industry. The modeling of the dynamic behavior can be performed simply by equivalent circuits has fixed elements. This method does not take into account the frequency behavior of the machine [].The eddy currents in the rotor bars indeed induce large changes so that the inductance of the rotor resistance [2]. This phenomenon called skin effect is a physical phenomenon in diffusive character, well use for different geometric shapes of rotor slots in order to see its influence on IM starting characteristics. II. MODELING OF DEEP RECTANGULAR SLOT There are several forms of rotor slots rectangular, trapezoidal...), in the case of engines with deep slots, at start, the current is on the upper part of the slots, it follows that the rotor behaves as if the section the conductors being wound was reduced, resulting in an increase of the ohmic resistance of the rotor. Late start speed, current moves downward slot and is distribute nearly uniformly over the entire section of the beam. From the AC gap exists only to a depth called penetration depth given as follows: 2ρ ρ δ = = µ gω πµ gf ) With : Depth of penetration of the alternating field; ρ: Resistivity of the material of the cage; μ: Permeability of the material of the cage; f: Frequency of the alternating field; gω: Heartbeat of the current in the slots. As the speed increases, the slip g decreases and the current extends over the entire section of the bar, the resistance is diminished, and after the expression ), the depth of penetration increases up to embrace the entire surface of the bar, for the low slip of the nominal market. One thus gets, an engine with the resistance of the rotor varies by current movement [2]. We can therefore express the resistance and the leakage reactance in the following way: r 2 = r 2e + r 2), 2 f x 2 = n 2e + x 3) 2 f For the rectangular shape: r h sinh = = r0 δ cosh k n ) + sin δ δ) ) cos δ δ) ) sin ) ) ) n 3δ sinh δ δ = = n0 cosh cos δ δ and : Coefficients that take into account respectively for the increase of the resistance and of the decrease of the reactance '; h: Height of the bar [3]. III. NUMERICAL CALCULATION OF PARAMETERS OF ROTOR BARS The calculation of and a rectangular bar by the analytical solution seems relatively easy and gives exact solutions, born less it has a major drawback it does not take into account the non-linearity and distortion of the geometry. Another solution is needed to address this problem, in this case a numerical calculation MEF is performed, it is to mesh the space subdivide the area) element [3]. The mesh size can be formed of triangles or quadrilateral for areas axisymetrical or 2D and prisms or headers for areas 3D. A. Simulation of a rectangular bar 4) 5) ISBN:

2 Recent Advances in Mechanics, Mechatronics and Civil, Chemical and Industrial Engineering The simulation conditions, similar to that which the rotor bars are subject requires to properly attaching the boundary conditions. n our case, Neumann conditions are applied to the left and right sides of the bar, and Dirichlet conditions on the high and low sides, distribution of the field lines and the variation of the current density are presented by following figure [4]. Fig.2. Slot rotor divided into sub-elementary conductors The currents in the sub-conductors k and k+ are et In the steady state, the equation for the voltage is: 6) Or ΔΦk is the leakage flow circulating between the k and th k +)th sub-conductors. The flux density of F=25Hz F=3Hz F.M.M in the sub-driver k depends on the of connection of current calculated from the lower 7) part of the slot of the sub-driver k. 8) Where sub-driver. is the width of the slot to the position of the kth 9) By substituting 9) into 6), we get: 0) Then, one obtains: ) The resistance and inductance of the driver elementary k are: F=50Hz Fig.. Current densities and field lines in a rectangular bar under different frequencies, 3) III. MODELING OF SLOTS BY THE THEORY OF THE CIRCUITS If the initial value of the current I is not known, one can choose a value arbitrarily for example A), and the rest of the currents will be resolved according to the system of equations 4). Now we present another method of calculation of When the rotor bars are traversed by coefficients and currents comers of the starting frequency to said vacuum operation. The skin effect occurs it could be analyzed using the circuit theory; this method is very suitable for bar cross section of the rotor of arbitrary shape double cage, trapezoidal...). In this method, a solid conductor was divided into n layers imaginary or actual sub conductors. The height of the driver is the height of the sub-driver is. The width of the slots also width of driver can vary. This usually varies of to occurs when the skin effect of a bar mold under pressure to squirrel cage is evaluated. The length of the iron core is l [5]. ISBN: ) 26

3 In this way, all the currents of sub-conductors of the bar are determined. The total current of the bar is: 5) After you have obtained the currents from 4), the rest of calculation will be as follows: The resistance of the bar taking into account the skin effect is given by: 6) The coefficient, which takes into account the increase in resistance of the bar, is: 7) ) numerical solution 2) With skin effect. : resistance of the bar without taking into account the The coefficient of the conductivity of dispersion of the slot : taking into account the skin effect is given by the following expression: 8) The coefficient of the conductivity of dispersion of the slot without taking into account the skin effect is given by the following expression: : 9) With Sum of sections of elementary conductors. The coefficient which takes into account the decrease of the conductivity of dispersion of the slot is given by the following formula: 20) IV. RESULT OF SIMULATION OF THE THREE METHODS The curves of developments of parameters and sont obtained from the simulation results, they are represented as follows: 2,6 2,4 2, The results obtained by the three analytical methods, digital and that of theory of the circuits are almost similar the relative error maximum is estimated at 0,075 % ) of or the validity of use of these last two methods for other forms of notches of geometry more complex. IV.. APPLICATION FOR THE DIFFERENT FORMS OF SLOTS A. Slot trapezoidal in shape The curves of developments of settings and for a slot trapezoidal in shape from two numerical method and theory of circuits analytical) are as follows: Fig.4. Evolution of,8.45,6,4,2 Analitical solution ) 2) Fig.3. Evolution of Fig.5. Evolution of ISBN:

4 Fig.6. Evolution of B. Slot of trapezoidal shape reverse Fig.9. Evolution of Fig.7. Evolution of Fig.0. Evolution of C. Slot to form double cage The results of the simulation for a double notch gage is as follows: V. DYNAMIC MODELING OF THE IM The above results have allowed us to observe the skin 5 5 T [N.m] With skin effect Trapezoidal Trapezoidal inverted Double slots Without skin effect Industrial Fig.8. Evolution of The figures below represent, respectively, evolutions of the and for parameters a slot of inverted trapezoidal shape Fig.. Various torque ISBN:

5 Speed [rad/s] With skin effect 40 Trapezoidal Tapezoidal inverted 20 Double slots Without skin effect Industrial Time [s] Fig.2. Evolution of speed effect for deferent forms of rotor slots and use it in a model of MI of average power 5kW) by means of resistance r 2 and the leakage reactance x 2 defined screen, has obtained the results shown in Figures below: V.. INTERPRETATION OF RESULTS According to the results obtained for the different forms of slots rotor blade rows, it was found that the latter to a significant influence on the characteristics of starting the machinevery good improvement of the starting torque, as well as the startup time), the forms of slot which tend to reduce the width of the bar in the direction of the slot reverse trapezoidal, double cage) have a better starting torque and a smaller time, since the current tends to flow on the upper part of the driver starting torque and a smaller time, since the current tends to flow on the upper part of the driver. These forms of slots further increases the resistance of the rotor, unlike in the case of rectangular slot the width is constant, the increase of the resistance is due only to the effect of skin, the torque developed by the engine during start-up, is more important than the one developed if account is not taken of the latter, whereas for the case of the trapezoidal slot, the section is large where the current density is high, therefore it is in the direction of reducing the effect of skin. VI. CONCLUTION We studied the influence of slot shape on the startup characteristics of IM 5kW and 2p=2 pole), exploiting the phenomenon of skin effect using three methods discussed earlier to determine the coefficients that take into account the increase resistance 'and the decrease in the leakage reactance rotor to various forms of rotor slots. We found that the skin effect and more important in the cage and double-reverse trapezoid notch and offer improved torque and low startup time since the current tends circulated on the upper part of the driver REFERENCES [] G. D. GRENIER, F. LABRIQUE, H. BUYSE et E. MATAGNE, «Electromechanical, energy converters and actuators", Editions Dunod, Paris, 200. [2] P. BARRET, Electrical Machines, theory and implementation", Ellipses Editions Marketing S.A., [3] MR. KOSTENKO, L. Piotrovsky, "Electrical Machines", Tome II, Editions Mir, Moscow, 979. [4] S. CANAT, "Contribution to the dynamic modeling of the asynchronous machine to cage",phd thesis of the Institut national polytechnique of Toulouse [5] J.PYRHÖNEN, T.JOKINEN et V.HRABOVCOVÀ Design of rotating electrical Machines, Wiley, ISBN: