Rev. Roum. Sci. Techn.– Électrotechn. et Énerg. Vol. 61, 1, pp. 18–21, Bucarest, 2016 NUMERICAL MODELING APPROACHES FOR THE ANALYSIS OF SQUIRREL-CAGE INDUCTION MOTOR CRISTINA MIHAELA GHEORGHE1, LEONARD MARIUS MELCESCU, TIBERIU TUDORACHE, EMIL MIHAI Key words: Induction motor, Magnetoharmonic model, Transient magnetic model, Experimental validation. This paper presents a comparative analysis in terms of results accuracy and computational effort of two numerical models used for the 2D finite element analysis of the three-phase squirrel-cage induction motor: the magnetoharmonic model and the transient magnetic model. The operating characteristics of an induction machine obtained by these models are presented in comparison with the experimental ones. The limitations and applicability of each numerical model are detailed and discussed. 1. INTRODUCTION One of the most important electric energy consumers in industry is the three-phase squirrel-cage induction motor (TPSCIM) . This motor is characterized by simple construction, robustness, long lifespan and low manufacturing and maintenance costs. TPSCIMs are built in a large range of rated powers from hundreds of watts to hundreds of kilowatts and they could be found in various applications as machine tools, fans, industrial or agricultural pumps, cranes etc. Fed from inverters, TPSCIMs could be used in electrical drive systems where the speed varies over a wide range. Being a reliable and a cheap device, the cost of the electricity consumed by a TPSCIM during its entire lifespan may be up to one hundred times larger than its acquisition price . Improvements of TPSCIM performance require finding out strategies to accurately model and analyze the machine operation regimes with a reasonable computation effort . The analysis of TPSCIM is typically done by using circuit models, field models or field-circuit models. Circuit models are easier to implement, the numerical results requiring less computation effort since these models are usually based on simplifying hypothesis. The circuit models are not able to take into account accurately special effects such as teeth harmonics, skin effect in eddy current regions, iron losses etc. Field models, usually based on finite element (FE) method, are more complex, more accurate and they are able to take into account more sophisticated phenomena but with the price of a higher computational effort. These models are implemented starting from electromagnetic field equations specific to the operation of the TPSCIM. Depending on the state of the studied machine, there are two types of numerical models that are typically used: (a) magnetoharmonic model used for the steady state ac magnetic analysis of TPSCIM and (b) transient magnetic model used for the transient analysis of the machine [2, 3]. Field-circuit models represent an extension of the field models and they allow the study of TPSCIM when supplied from a voltage source (when currents are apriori unknown). These models suppose to add an associated circuit model of the machine to its field model. This operation assumes that circuit equations are added to the existing field model equations, all of them being assembled in a single matrix [4−10]. This study aims at highlighting the applicability, limitations and performance features in terms of accuracy and com- putational effort of two numerical models used for the FE analysis of TPSCIM: magnetoharmonic model and transient magnetic model. 2. NUMERICAL MODELS FOR THE ANALYSIS OF TPSCIM A detailed analysis of TPSCIM able to evaluate the running characteristics of the machine cannot be done using circuit models. The machine efficiency, the electromagnetic torque ripples, the teeth harmonics, the magnetic non-linearity and skin effect in rotor bars are some aspects difficult or impossible to be considered by simple circuit models. To take into account all these effects the machine should be studied by field or field-circuit models, usually based on FE method. The FE analysis of radial flux TPSCIMs is carried out typically by 2D plane approaches. The most used models for such studies are the: the magnetoharmonic model and the transient magnetic model [2, 3]. 2.1. MAGNETOHARMONIC MODEL The magnetoharmonic model (steady state ac magnetic harmonic model) uses the complex magnetic vector potential as state variable . In 2D applications the partial differential equation governing this model, expresses in complex magnetic vector potential A is the following : curl[(1 / µ )curl A] = J s − jωσ A , (1) where µ is the magnetic permeability, A[0, 0, A(x, y )] is the complex magnetic vector potential, J s [0, 0, J (x, y )] is the complex image of source current density, σ is the electric conductivity of solid conductor regions, ω = 2πf is the angular frequency and the term ( − jωσ A ) represents the induced current density in solid conductor regions. The features of the magnetoharmonic model are as follows: (a) all material properties are considered linear, magnetic non-linearities of materials can be taken into account by energy equivalence methods (e.g. Flux by Cedrat) , (b) time-harmonic physical quantities (electrical and magnetic) are represented by their complex image for a given frequency value (i.e. supply voltage frequency), (c) current harmonics, teeth harmonics and electromagnetic torque ripples cannot be taken into account [2, 3]. This model is however eco- 1 “Politehnica” University of Bucharest, Faculty of Electrical Engineering, 313 Splaiul Independentei, 060042 Bucharest, Romania, E-mail: [email protected] 2 19 Numerical analysis of three-phase squirrel-cage induction motor nomical from computation point of view and it can be applied to estimate rapidly the TPSCIM running characteristics in steady state such as: torque-slip curve, current-slip curve, efficiency characteristic etc. Such a model may be ten times faster than a transient magnetic model used for the same application. 2.2. TRANSIENT MAGNETIC MODEL The transient magnetic model allows the study of the TPSCIM in dynamic regimes such as: motor start up, motor acceleration and deceleration etc. The partial differential equation of the transient magnetic model in 2D applications, expressed in magnetic vector potential A is as follows [2, 3]: curl[(1/µ )curlA] = J s − σ ∂A , ∂t (2) where µ is the magnetic permeability, A[0, 0, A(x, y, t )] is the magnetic vector potential, J s [0, 0, A(x, y, t )] is the current density, σ is the electric conductivity of solid conductors. Some important features of the transient magnetic model are as follows: (a) it is able to consider the rotor-stator relative motion of the TPSCIM, the electromagnetic field problem being solved step by step in time domain, (b) it can take into account the magnetic non-linearity of the magnetic cores, (c) the time variation of physical quantities (electrical and magnetic) may be arbitrary (i.e. time-harmonic or not), (d) it can take into account the current harmonics, the eddy currents induced in the rotor bars and the electromagnetic torque ripples [2, 3]. The major drawback of this model is the important computational effort. The transient magnetic model is used frequently for the analysis of electrical machines when operating at constant speed, at constant torque or in transient regimes. In order to the model dynamic regimes of the TPSCIM the kinematic equation should be added to the field-circuit model of the machine: J dΩ = Te − TL − FΩ , dt (3) where J is the inertia moment of the rotating parts, Ω is the angular speed, Te is the electromagnetic torque of the motor, TL is the load torque and F is the viscous friction coefficient. 2.3. CIRCUIT MODEL ASSOCIATED TO THE FE MODEL. FIELD-CIRCUIT COUPLING In order to analyze a TPSCIM by using eq. (1) and (2) we must know the current density values in the stator coils ( J s ) and rotor bars ( − jωσ A ). Unfortunately the quantities that are typically known for a voltage supplied TPSCIM are the voltage and frequency values, the currents being apriori unknown since they depend on several factors such as: machine geometry, load torque, material properties. Therefore in order to solve the under determination in eq. (1) or (2) additional equations able to link the current density values and the magnetic vector potential values are needed. These supplementary equations are usually generated by coupling the circuit model of the machine, Fig. 1, to the field models presented in Sections 2.1 and 2.2, obtaining thus field-circuit models. Such coupled models can be therefore used to analyze the voltage supplied TPSCIM in various operation regimes, the stator/rotor currents resulting accordingly by solving the circuit and field model equations. Fig. 1 – Circuit model of the TPSCIM associated to the FE model. In Fig. 1 we can identify the following circuit components: (VU, VV, VW) – voltage sources supplying the machine, (RU, RV, RW) – stator resistances per phase, (LσU, LσV, LσW) – stator end winding inductances per phase analitically calculated (LσU = 8.6 mH), (RFe-U, RFe-V, RFe-W) – phase resistances modeling the iron losses, (BU, BV, BW) – coils of the three-phase windings whose inductances are implicitly computed by FE analysis, Q1 – macro-circuit used to model the squirrel cage (each rotor bar is modeled as a solid conductor and each interbar ring section is modeled by a series connected resistor and inductor). 3. MAGNETOHARMONIC AND TRANSIENT MAGNETIC ANALYSIS OF A TPSCIM The analysis of the TPSCIM was carried out by using the magnetoharmonic and the transient magnetic field-circuit models implemented in the professional software package Flux®. A comparison of the results obtained by the two models with the experimental ones is made and comments are added. 3.1. MAIN DATA OF STUDIED TPSCIM. COMPUTATION DOMAIN AND BOUNDARY CONDITIONS The studied TPSCIM is characterized by the main technical data presented in Table 1. Table 1 Main technical data of the studied TPSCIM Rated mechanical power [kW] Rated phase voltage [V] Rated phase current [A] Rated supply frequency [Hz] Rated torque [Nm] Internal diameter of the stator core [mm] External diameter of the stator core [mm] ρcage (115°C) [Ωm] Rated speed [rpm] Rated rotor slip [%] Rated power factor Number of stator slots (Z1) Number of rotor bars (Z2) Length of stator/rotor core [mm] Phase stator resistance [Ω] Electric steel laminations type 2.2 220 5.4 50 15.15 100 160 5.07·10−8 1420 5.33 0.78 36 28 90 2.28 M600-50A 20 Cristina Mihaela Gheorghe et al. Fig. 2 – Electromagnetic field computation domain and regions with different physical properties. 3 because usually the load torque is considered as constant while the motor speed may have ripples as a result of the kinematic equation of the motor where several quantities interferes (e.g. moment of inertia, electromagnetic torque). However the computational effort for the imposed torque model is much greater than for the imposed speed model, since the first model supposes a transient mechanical and a transient electromagnetic regime while the second one assumes only a transient electromagnetic regime. A question that still arises refers to the precision of these two models and this aspect will be concluded by a comparative numerical analysis for different speed and torque values represented under the form of a mechanical characteristic of the machine. The obtained results in Fig. 4 show negligible differences between the mechanical characteristics obtained by the two transient magnetic models (the results are in good agreement also with those obtained by the magneto-harmonic model). That is why the transient magnetic analysis in the following studies will be carried out using the imposed speed model because it is more economical. Fig. 3 – Finite element mesh. 3.2. COMPARISON BETWEEN MAGNETOHARMONIC AND TRANSIENT MAGNETIC MODELS The field-circuit analysis of TPSCIM using the magnetoharmonic model was carried out for different slip values in the range s ∈[0 ÷ 0.08] (i.e. from ideal no-load to 150% overload where the rated slip is sn = 0.053), and different running characteristics were determined. The same characteristics of the machine were computed by the transient magnetic model. The transient magnetic analysis of the TPSCIM could be performed in two ways: (a) by imposing the load torque value (i.e. imposed torque model) or (b) by imposing the rotational speed value (i.e. imposed speed model). If the ripples of the electromagnetic torque are small (i.e. very small speed ripples) both models should provide practically the same results. In most cases the first model (imposed torque model) is closer to reality than the second one (imposed speed model) Fig. 4 – Comparison between the imposed speed and imposed torque models. 50 Transient Model Harmonic Model Experimental 45 40 35 Me [Nm] In Fig. 2 is shown the 2D plan-parallel electromagnetic field computation domain of the studied machine and in Fig. 3 the finite element mesh. Due to the symmetries the computation domain of TPSCIM is reduced to ¼ of its cross-section (one magnetic pole). The boundary condition imposed on the stator outer surface is of Dirichlet type, A = 0 (i.e. nill normal component of the magnetic flux density on that frontier, B ⋅ n = 0 ). The physical symmetries are taken into account by imposing anticyclic type boundary conditions on the two radial boundaries of the computation domain. 30 25 20 15 10 5 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 slip [-] Fig. 5 – Torque-slip curves. The comparison between the magnetoharmonic and the transient magnetic (imposed speed variant) models of TPSCIM is carried out for different motor speed and torque values and the results are represented as electromagnetic torque/slip characteristic, Fig. 5, and running characteristics shown in Figs. 6−8. These figures, that include also experimental results, illustrate the dependence of power factor, the stator current and the efficiency on the level of motor loading (P2/P2n output power/rated power). 4 Numerical analysis of three-phase squirrel-cage induction motor computation effort associated to the magnetoharmonic model that proved to be up to 60 times smaller than that of transient magnetic model. Thus the magnetoharmonic model is well adapted for a rapid evaluation of machine running characteristics. If particular aspects are studied such as teeth harmonics, torque ripples, transient mechanical regimes of the machine, the transient magnetic model is more adapted. 1 Transient Model Harmonic Model Experimental 0.9 0.8 0.7 cos( φ) 21 0.6 0.5 0.4 0.3 4. CONCLUSIONS 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 P2/P2n [-] Fig. 6 – Power factor characteristics. The main purpose of this study was to evaluate two FE based modelling methods used for the analysis TPSCIM: the magnetoharmonic model and the transient magnetic model. The accuracy of results and the computational effort were evaluated, the numerical results being also experimentally validated. From the perspective of results accuracy, both studied models are suitable for the analysis of the TPSCIM above the breakdown speed. In terms of computational effort, the magnetoharmonic model is much more suitable because the results are obtained much faster (up to 60 times). For the analysis of the TPSCIM in transient regimes or around the starting point (where high current harmonics may appear), the transient magnetic model is more suitable, despite the computational effort. ACKNOWLEDGEMENTS Fig. 7 – Input current characteristics. The work of Ph.D. Eng. Cristina Mihaela Gheorghe was supported by Project SOP HRD – PERFORM /159/1.5/S/ 138963. 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