3-D Optimal Design of Induction Motor Used in

2076IEEE TRANSACTIONS ON MAGNETICS, VOL. 45, NO. 5, MAY 20093-D Optimal Design of Induction Motor Used in High-Pressure Scroll CompressorHokyung Shim1 , Semyung Wang1 , and Kwansoo Lee2Department of Mechatronics, Gwangju Institute of Science and Technology (GIST), Gwangju 500-712, Korea School of Mechanical Engineering, Hanyang University, Seoul 133-791, KoreaThis paper presents a 3-D optimal design, regarding both magnetic and thermal characteristics, for a three-phase induction motor used in a high-pressure scroll compressor. In the scroll compressor, the three-phase induction motor does play an important role of causing dynamic force, but generates high heat, which exerts negative influences on both lifetime and performance. Thus, it is necessary to design the scroll compressor considering two physical disciplines in order to improve the performance while protecting against overheating. A 3-D topology approach using a multiobjective function yields optimal design with higher torque and more efficient heat transfer. Index Terms—Induction motor, 3-D optimal design, topology approach.I. INTRODUCTIONUP TO NOW, a great deal of research on improving the efficiency of a variety of industrial compressors has been carried out in the field of single physics, mostly electromagnetics. Even when the magnetic characteristics of an electrical machine are solely improved for its efficiency, the total efficiency of the compressor might not be as high as expected. This is due to the fact that the machine generates high heat due to Joule loss which is transferred via the yokes of the rotor and stator and housing shell of the compressor. The induced heat raises the internal temperature of the compressor such that the heat has not only negative influence on the electromagnetic quantity, such as the torque and efficiency, but also leads to a short lifetime due to overheating. As a result, the performance of the compressor becomes unreliable. Thus, this paper presents a design approach regarding the thermal characteristics associated with the electromagnetic design of the high-efficiency machine used in the compressor. A study reported that the oil circulation rate of compressors is a crucial factor affecting the performance and cooling systems—too much oil in the refrigerant mixture causes poor heat transfer and too little oil yields excessive friction and bearing wear [1]. This study presented optimal designs of a single-phase induction motor for a rotary compressor using the topology optimization [2], in which the magnetic energy was maximized based on 2-D finite element analysis. However, the high-pressure scroll compressor which we deal with in this paper has severe thermal problems rather than electromagnetic quantities. Therefore, the thermal characteristics of the compressor must be analyzed to design a three-phase induction motor used to provide compressive power. Here, the 3-D topology optimization for the induction motor in the scroll compressor is implemented to improve the thermal characteristics while maximizing the magnetically induced torque. In conventional structural optimizing techniques, such as sizing or shape/configuration optimization, the design variablesdirectly control the exterior and interior boundary shapes of the structures. Shape optimization [3] is applicable to the design of electrical machines, but has severe limitations: computational cost due to mesh regeneration, tendency of convergence to local minima, and inconvenient compatibility for complex geometrical problems. Furthermore, shape optimization can not create new holes in the process of optimization. However, topology optimization which was introduced a few decades ago by Bendsoe and Kikuchi [2], is able to generate some holes in design domains. Topology optimization does not require a sophisticated initial design, but rather only requires enough geometric information to define boundary conditions. The topology optimization has been extended to various physical systems [4], [5] and has shown the possibility of designing practical machines [1], [6]. Topology optimization is based on numerical analysis such that a numerical model must be verified by experiments. Experiments were carried out for the torque profile along with revolving speed and temperature of the compressor. Since the induction motor with the squirrel cage in the rotor has difficulty in measuring the current of the rotor, FLUX2D, a commercial finite element code, is used for the nonlinear transient analysis. Now that the magnetic flux flows in 2-D due to laminated yokes, 2-D analysis is sufficient to analyze the magnetic quantities. However, once thermal analysis is taken into consideration, a 3-D model is inevitable because heat transfer is affected by the 3-D conduction and convection. Thus, a 3-D fluid analysis is performed using FLUENT with heat sources that are Joule heat of the stator and rotor computed from the nonlinear transient analysis. It is noted that the current profiles of the stator windings and the rotor bar are obtained from FLUX2D and thermal boundary conditions, i.e., convection coefficients, are estimated in 3-D FLUENT. Using those data, a 3-D optimal design for the magneto-thermal field is obtained by employing the topology approach which is incorporated with ANSYS. The adjoint variable method (AVM) is employed to derive sensitivity of design variables. It leads to the avoidance of numerous sensitivity calculations for each iteration. II. TOPOLOGY OPTIMIZATION Topology optimization aims to search for an optimum material distribution that maximizes or minimizes an objective func-Manuscript received July 30, 2008; revised January 15, 2009. Current version published April 17, 2009. Corresponding author: S. Wang (e-mail: smwang@gist.ac.kr). Color versions of one or more of the figures in this paper are available online at . Digital Object Identifier 10.1109/TMAG.2009.20137150018-9464/$25.00 © 2009 IEEESHIM et al.: 3-D OPTIMAL DESIGN OF INDUCTION MOTOR2077tion while satisfying given constraints. A general optimization problem takes the following form:and(1) is the objective function, is the state variable, where and is design variables which are the material density of each element in the solid isotropic microstructure with penalization (SIMP) method. Expressing each quantity of a governing and are the energy bilinear form and the system, load linear form, respectively. In the optimization, sensitivity is always used for a numerical search method to provide the optimizer with the best direction in the next iteration. This sensitivity is very important to efficiently obtain the final optimum. The adjoint variable method (AVM) is a unique alternative to the finite difference method (FDM). The FDM requires the same number of sensitivity evaluations to design variables. However, the AVM requires only an adjoint analysis in a single field, regardless of the number of design variables [7]. A. Electromagnetic Field The 3-D diffusion equation is derived from the set of Maxwell’s equations. By multiplying the virtual vector potential, , integrating it over the analysis domain, and applying the boundary conditions, the variational equation for the electromagnetic field is expressed by [8], [9] for all (2)(6)where is a parameter that controls the perturbation size and means small perturbation. Consider a measure of electromagnetic performance which is defined in an integral form (7)where and the subscripts are the directions of Cartesian coordinate system. The function is continuously differentiable with respect to its arguments. Taking direct differentiation of (7) yields the design sensitivity equation [7] (see Lemma 1 and 2)(8)The corresponding adjoint equation of the electromagnetic , is as follows: system to obtain the adjoint variable, (9)where is the magnetic vector potential and is the space of admissible vector potential. The magnetic energy bilinear form and the magnetic load linear form are described as follows:(3) (4)is the adjoint virtual vector potential which is held where in the space of . Equation (9) is rewritten by introducing an adjoint load as follows: (10)where is the permeability of materials and is the current density applied to the conductors, such as coil and rotor bar, and end-ring. Taking the variations of (3) and (4), respectively, yields, is applied Here, when the equivalent current source, to the system, adjoint responses are obtained under the same boundary conditions as the original analysis. Replacing (5) and (6) into (8) leads to topology design sensitivity as(5)(11)2078IEEE TRANSACTIONS ON MAGNETICS, VOL. 45, NO. 5, MAY 2009After integrating (15) by parts, applying the boundary conditions to (15) leads to a variational equation (19) where belongs in the space of admissible temperature . Con, and the load linear sequently, the energy bilinear form, , are defined as follows: form, (20)andFig. 1. Body subject to heat transfer.(21)In order to represent the porous material based on the SIMP method, the material interpolation functions describing the relationship between the material and design variables are defined as follows: (12) (13) (14) is the permeability of air and is the relative perwhere are the directions of meability of material, and subscripts Cartesian coordinate system, respectively. The indicates the material density distribution between the solid and void. The intermediate value of design variable are removed using penalty parameter in the SIMP method. B. Thermal Field A general equilibrium equation for the thermal field can be derived from the energy balance and Fourier’s law [10]. Applying Galerkin’s method with an approximation function, , a single governing equation becomes (15)respectively. For magneto-thermal analysis of the induction motor, the internal heat generation is computed by (22)where is the electric conductivity of materials. A thermal performance may be written in integral form as (23)Taking direct differentiation of (23) yields the design sensitivity equation(24) where , , and are the thermal conductivity, temperature, and internal heat generation rate per volume, respectively. There are a variety of boundary conditions in thermal analysis. The area of the known temperature, , heat flow, , and heat transfer by convection, , are taken into consideration, as illustrated in Fig. 1 (16) (17) (18) is the convection coefficient and where perature of the known environment. is the bulk tem-where the adjoint response, , is obtained from the corresponding adjoint equation of the thermal field (25) Equation (25) indicates that the thermal adjoint response is obtained from the original model with differently specified conditions: the bulk temperature must be set to 0 while the conduction and convection are maintained. Once some parts of solid domain are changed to the void in the process of optimization, heat exchange by the convection occurs. This means that both the conduction and convection areSHIM et al.: 3-D OPTIMAL DESIGN OF INDUCTION MOTOR2079taken into consideration for design changes. Here, the material interpolations for the thermal field are as follows: (26) (27) (28) (29) where the subscript initial means initial value of its property. III. VERIFICATION OF ANALYSIS MODEL It is impossible to measure current flowing in the rotor bar because the three-phase induction motor has the squirrel cage in the rotor. In this paper, Joule heat in the stator and rotor is estimated by using FLUX2D, a commercial finite element code, in which nonlinear transient analysis is carried out. Since the magnetic flux takes place in a 2-D laminated core, 2-D analysis is sufficient to analyze the magnetic quantities. Heat generated in the coils and yokes is transferred toward the outside. Here, conduction and convection lead to exchange heat within each component of the compressor. The 3-D simulation is inevitable to depict heat flows in the compressor. FLUENT is used to simulate the fluid flows with a 3-D compressor model. It is stressed that an objective of the fluid analysis is to obtain the heat convection coefficients. The numerical analyses are verified by experiments. A. 2-D Electromagnetic Analysis Using Finite Element Method The three-phase induction motor used in the scroll compressor yields a mechanical output of 9.5 Hp at a rated speed of 3469 rpm, when input voltage of 380 V with a frequency of 60 Hz is applied. The combination ratio between the stator slots and rotor slots is 0.86. The finite-element model to be analyzed has only half of the total geometry due to a symmetric condition. Fig. 2 shows the analysis model meshed by finite element. To drive the motor, an external circuit operated by a voltage source was constructed with a squirrel cage circuit. From the nonlinear transient analysis, torque and current profile were obtained at a rated speed of 3469 rpm, which is a target speed to be investigated in optimization. Additionally, performances of the motor at different speeds such as 3427, 3448, and 3461 rpm were computed to be compared with experiments for verification. Fig. 3 shows the magnetic flux contour lines and magnetic flux density when the rotor speeds up to the rated RPM. Using (22), Joule heat generated in the induction motor is estimated in Table I. B. 3-D Fluid Analysis Using Finite Volume Method 3-D fluid analysis is performed using FLUENT with heat sources that include the Joule heat of the stator and rotor computed in the previous section. The scroll compressor, as illustrated in Fig. 4 is composed of three main structures: compression part, induction motor, and bearing support. In this section, a 3-D analysis model is constructed for the fluid domain, and Fig. 4 (right) shows the directions of fluid flow. The boundary conditions are given as shownFig. 2. Finite-element half model of the induction motor. (a) Analysis model. (b) Finite-element model.Fig. 3. Plots of magnetic flux lines (left) and magnetic flux density (right).TABLE I JOULE HEAT OF INDUCTION MOTORin Fig. 5. The mass flow rate and temperature in the inlet and fluid pressure in the outlet were obtained from experiments. For outer surface of the housing shell, 5.56 W/m K were applied based on Nusselt number of vertical cylinder in the natural convection [10]. Since fluid flows were mostly weak right below the compression part and around the bearing support, thermal isolation is imposed on the top and bottom surface of the 3-D analysis model. Table II shows the physical properties of the motor and housing shell (compressor shell), and Table III presents the properties of the working fluid. Due to the yokes (stator and rotor) stacked with insulated lamination, the stator and rotor have anisotropic thermal conductivity. Fig. 6 describes the history of fluid flows in the compressor. Highly compressed gas in the scroll compression part comes down to the induction motor. Due to the revolving motor at2080IEEE TRANSACTIONS ON MAGNETICS, VOL. 45, NO. 5, MAY 2009TABLE III PHYSICAL PROPERTIES OF WORKING FLUIDFig. 4. Structure of the scroll compressor (left) and fluid flow (right).Fig. 6. Path history of fluid flow: (a) 100 step, (b) 300 step, (c) 1000 step, and (d) 7000 step.Fig. 5. Boundary conditions for fluid analysis.TABLE II PHYSICAL PROPERTIES OF COMPONENTS IN THE COMPRESSOR3469 rpm, the fluid whirls around the upper side. The pressure difference between the upper and lower side of the motor circulates the fluid up and down. Fig. 7 shows the temperature distribution inside the compressor when the fluid circulation reaches a steady state. The temperature of the lower side reached up to 118 C. Figs. 8 and 9 present the temperature of the stator windings and rotor conductors, respectively. It is found that the temperature of the upper side of the structures is lower than the lower side because high velocity of theFig. 7. Temperature distribution inside the compressor.fluid in the upper side promotes the exchange of the heat generated in the motor.SHIM et al.: 3-D OPTIMAL DESIGN OF INDUCTION MOTOR2081Fig. 10. Experimental schematic for electromagnetic quantities.TABLE IV COMPARISON BETWEEN NONLINEAR TRANSIENT FEA AND EXPERIMENTFig. 8. Temperature distribution of the stator windings.Fig. 11. Experimental schematic for thermal characteristics.Fig. 9. Temperature distribution of the rotor bar and end-ring.From the fluid simulation, the convection coefficients for all surfaces of the structures are obtained from the thermal equilibrium (30) where the subscript solid and fluid indicate the solid (such as the stator and rotor) and the working fluid, respectively. The convection coefficients in the air-gap, i.e., the gap between the stator and rotor, are 180.9 W/m K for the stator and 155.3 W/m K for the rotor. The end-turn of the stator is 322.1 W/m K for the upper and 290.8 W/m K for the lower. C. Experiments The objectives of the experiments were as follows: 1) to verify the electromagnetic performances of the induction motor such as torque and current of the stator windings and 2) to check the temperature distribution of the compressor at rated speed of the motor and the pressure of the outlet. A driving motor operated by the universal inverter transmits revolving force to the induction motor. Using an encoder and adynamometer, the evolving speed and electromagnetic quantities are detected, as shown in Fig. 10. When the induction motor was tested experimentally, the efficiency peaked at the speed 3469 rpm. Table IV presents a comparison between the torques of the nonlinear transient analyses and experiments. Here, the finite-element model is verified in four different RPMs. As presented in Fig. 11, a calorimeter is utilized to be equivalent to practical operating conditions. Electrical signals obtained from K-type thermocouples are converted to digital signals in the hybrid recorder. In order to store the transient temperature, a computer-based data acquisition system coupled by means of a General Purpose Instrumentation Bus (IEEE488/GPIB) is used in experiments on thermal characteristics. The several thermocouples are attached to the surface of the stator and the shell. To verify the thermal source of Joule heat, the thermocouples are embedded in the windings of the stator and the end-turn parts in Fig. 12. In the scroll compressor test, it took more than three hours to reach a steady state. The maximum temperature was measured up to 115 C in the lower end-turns where the highest heat was generated. Table V shows a temperature comparison between experiments and fluid analysis for eight locations which are illustrated in Fig. 13. The errors are below 4%. IV. OPTIMIZATION OF THREE-PHASE INDUCTION MOTOR An objective in this paper is to design an optimal stator shape in terms of the effects of both the electromagnetics and heat transfer. For the electromagnetics, the magnetic energy generated in the air-gap is maximized since the induced torque2082IEEE TRANSACTIONS ON MAGNETICS, VOL. 45, NO. 5, MAY 2009Fig. 12. Pictures of thermocouples embedded in the motor. (a) Top view. (b) Bottom view. Fig. 15. Applied current in stator windings. TABLE V TEMPERATURE COMPARISON BETWEEN EXPERIMENT AND FLUID ANALYSISThe topology optimization problem takes the form Energy Initial Energy Nodal Temp Initial Nodal Temp subject to (31)Fig. 13. Top (left) and side (right) view for temperature measurement.Fig. 14. Target nodes for initialization of 3-D optimization.is calculated by the derivative of the energy. In the meanwhile, specified nodal temperature is maximized for improvement of thermal characteristics. Since maximum nodal temperature means the thermal resistance between the input (heat source) and output (target node) is minimized, the main heat flow is intentionally manipulated in the motor. From an engineering viewpoint, the directions of heat dissipation must be fixed to radiate to the outside. Here, 12 target nodes are predefined as shown in Fig. 14. Fig. 14 presents the initial design domain and design strategy for optimization.bounded to for all . Here, is a weighting factor, is the cross-sectional area of element, is the thickness of element, is the allowable volume ratio compared to the initial volume , and is the density variable. is the lower bound of the densities, introduced to prevent the singularity of the equilibrium problem. An optimization controller developed in C++ language manages the entire process and evaluates the design sensitivity during each iteration. The sequential linear programming (SLP) algorithm is used to compute the design changes. The optimization procedure is as follows. Step 1) Compute the nonlinear transient electromagnetic analysis using FLUX2D. Step 2) Compute the 3-D fluid analysis at given speed using FLUENT where the Joule heat of the stator and rotor obtained from Step 1) are applied. Step 3) Compute the objective functions in 3-D model using ANSYS where current profiles of the conductors and position of the revolving rotor, and convection coefficients are applied. Step 4) Calculate sensitivities derived in Section II. Step 5) Determine the optimal pattern using SLP Step 6) Update the material properties and convection coefficients. Step 7) Check convergence. If the ratio of the multiobjective value between the current and previous iteration is less than 0.001%, it stops. Step 8) If the ratio of the shape change is over 10%, go to Step 1). Otherwise go back to Step 3). It is noted that for the optimization process, the current profiles of the stator windings and the rotor bar are obtained from 2-D electromagnetic analysis, and thermal boundary conditions, i.e. convection coefficients, are predicted in 3-D fluid simulation. The Pareto optimality conditions are investigated to find good compromises satisfying two objectives. Here, 0.85 is usedSHIM et al.: 3-D OPTIMAL DESIGN OF INDUCTION MOTOR2083Fig. 18. Comparison between original and optimal designs.TABLE VI COMPARISON BETWEEN ORIGINAL AND OPTIMAL DESIGNFig. 16. Optimal patterns of each current case: (a) A case, (b) B case, (c) C case, (d) D case, (e) E case, and (f) F case.Fig. 17. Global optimal pattern of stator yoke.topology sensitivity equations for each field are derived employing the adjoint variable method by using the continuum approach. In order to obtain current profiles for the stator windings and rotor bar, a 2-D nonlinear transient electromagnetic analysis was carried out. Using Joule heat, 3-D fluid flow was simulated using FLUENT. Experiments were conducted to verify the results of the numerical analysis. Based on thermal boundary conditions estimated by the fluid analysis, 3-D optimization leads to an optimal design yielding better torque and improved thermal characteristics. APPENDIX Lemma 1: Letfor the weighting factor. Among the three-phase sinusoidal currents shown in Fig. 15, six instantaneous points are critical for the rotational magnetic field since those points induce the half of a revolution of the rotor with identical interval. Each case provides different topology since each instantaneous point of the current profile yields diverse magnetic field and temperature distributions. Therefore, the each optimal topology pattern is obtained using the magneto-thermal optimization problem and the average is taken for a global topology optimum. Fig. 16 illustrates 3-D optimal patterns in the case that each of the threephase currents is applied. (a)–(f) in Fig. 16 corresponding to A–F of Fig. 15 are converged at 20–22 iteration. Fig. 17 presents a global optimal pattern of the stator yoke. Fig. 18 compares the shape change between the original and optimal design. It is found that the stator volume of the optimal design is 2.1% less than the original design. The effectiveness of the optimal design is identified by reanalysis of the numerical model and is compared with the performance of the original design, as shown in Table VI. V. CONCLUSION In this paper, 3-D topology optimization regarding electromagnetics and heat transfer is performed to design a three-phase induction motor used in a high-pressure scroll compressor. Thewhereis the 3-D coordinate, andLemma 2: To take advantage of the adjoint equation, (9) at becomes:2084IEEE TRANSACTIONS ON MAGNETICS, VOL. 45, NO. 5, MAY 2009Taking total variation of both sides of (2) at toleadsThe energy bilinear form is symmetric in its arguments such that two equations above are identical. ConsequentlyACKNOWLEDGMENT This work was supported in part by LG Electronics, Inc., and by a Korea Research Foundation Grant funded by the Korean Government.REFERENCES [1] S. Wang, J. Kang, and J. Noh, “Topology optimization of induction motor of rotary compressor,” IEEE Trans. Magn., vol. 40, no. 3, pp. 1591–1596, Mar. 2004. [2] M. Bendsoe and O. Sigmund, Topology Optimization—Theory, Methods and Application. Berlin, Germany: Springer, 2003, p. 112. [3] J. Sokolowski and J. P. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis, ser. Springer Series in Computational Mathematics. Berlin, Germany: Springer, 1992, vol. 10. [4] H. Shim, S. Wang, and K. 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