High Speed Solid Rotor Induction Machine: Analysis and Performances L. Papini1, C. Gerada1, D. Gerada2, Abdeslam Mebarki2 1
Faculty of Engineering, University of Nottingham, Nottingham, UK Research and Technology Department, Cummins Generator Technologies, Stamford, UK e-mail:
[email protected],
[email protected],
[email protected],
[email protected] 2
Abstract—The paper presents the design, analysis and testing aspects of high speed induction machines equipped with solid rotor. At first the theoretical background and design aspects of solid rotor for induction machines is presented considering electromagnetic, thermal and mechanical aspects and focusing on the assessment of end-region factor effects. The techniques are benchmarked against a 120 kW solid rotor induction motor designed for power generation application. Keywords — high speed, solid rotor, induction machine, multi-pysics
I. INTRODUCTION The increasing demand for high speed, direct drive systems is driving a significant research effort on high speed electrical machines. Induction machines are very competitive in the high speed range with respect to permanent magnet machines [1]. The performance and feasibility of the designed structure in high speed applications have to take into account the electromagnetic, thermal, mechanical aspects and the interaction between each other [1, 2, 3]. Thermal management [2] gain importance as well as the mechanical aspects in terms of stresses and rotor-dynamics [1, 4]. The maximum peripheral speed is limited by the mechanical properties of the material adopted for the rotating component [4, 5]. These limitations can be overcome by the adoption of high performance materials in terms of mechanical properties or by reducing the maximum stresses generated in the structure itself [5]. Solid Rotor topologies for induction machines operating at high speed have gained a lot of interest in the 1950s; after many years when this topology has been not considered as technologically interesting, in the 1990s, solid rotor structure has been re-considered for high speed applications [6]. Easy to manufacture, highly reliable, suitable to operate in harsh environment they found applications as compressors, turbocharger, pumping system, drilling system. Induction machines equipped with solid rotor take advantage of the simple structure of the rotating part to push the speed limit achievable. The electric and magnetic field share the same material path thus leading to a more complicate field and current density distribution that are highly affected by the phenomena located in the rotor end-region [7]. The material choice is a fine compromise
between electrical and mechanical characteristic. In this paper a multi-physics approach is described to analyse the performance of an induction machine equipped with solid rotor. II. SOLID ROTOR INDUCTION MACHINE Solid Rotor Induction machines (SRIM) result in a cheap and reliable alternative to caged induction machines thanks to their simple rotor structure. Although the principle of operation is identical to the other asynchronous machines, the magnetic field, current density, temperature and mechanical stresses in the rotor structure are peculiar and have to be carefully investigated to achieve a good design [2, 4]. Solid rotor induction machine are usually characterized by a very low power factor and high timespatial harmonic losses. The rotor structure does not filter any of the time-space harmonic featured in the main air-gap, therefore leading to high rotor losses and low efficiencies. The 3-dimensional physical distribution of the magnetic field can’t be neglected for solid rotor structure [7]. Usually over-lengthing the length of the rotor with respect the active length of the stator structure, the rotor end region is necessary as the closing path for the eddy currents. End-ring featuring low resistivity has been adopted to shorten the rotor structure but reducing the rotor mechanical strength.
(A)
(B)
(C)
Fig. 1 Solid Rotor topologies: (A) Smooth solid rotor; (B) Solid Rotor with Slitting; (C) Coated Solid Rotor
Different rotor topologies have been proposed with the aim of improving the electromagnetic performance although losing something on the mechanical and rotor-dynamic side. Solid rotor with smooth rotor structure [6], as sketched in Fig.1.A, or equipped with regular-modulated axial slitting [8] as shown in Fig.1.B are widely adopted. The axial slitting is performed with the aim of increasing the field penetration in the rotor structure. The rotor surface eddy current losses are reduced significantly but the slitting affects the rotor stiffness and leads to an increase of the
windage losses that at high speed become relevant. A thin sleeve of high conductive material has been proposed on the outer surface of the rotor structure as shown in Fig.1.C, with the aim of concentrating the eddy currents in the rotor surface and acting as a continuous ring of rotor bars [6, 9]. Mechanically robust, this machine features higher efficiencies but lower power factors with respect to the smooth solid rotor structure resulting from the bigger equivalent air-gap. The electromagnetic and mechanical characteristic of the material adopted for the rotor structure are important as they affect the field distribution within the rotor structure and have to withstand the maximum stresses featured on the rotor. Multi-physics approach is thus necessary when SRIM are analysed as the temperature distribution due to the losses in the stator and rotor structure affect both the resistivity of the material [2] as well as the mechanical stress and deformation distribution. In the following sections, the electromagnetic, thermal, mechanical and rotor-dynamics modelling of a Solid Rotor Induction machine are described. A 120 kW-30000 rpm induction machine for a waste heat recovery system is considered as a benchmark. III. ELECTROMAGNETIC MODEL In the past many different approaches has been adopted to investigate the performances of SRIM. Analytical methods [6, 10] are quick but they lose in accuracy as many assumptions are considered; the field distribution analytically calculated can be used to estimate the parameter of the equivalent circuit of SRIM [6, 11]; voltage-fed FEM simulations are accurate but time consuming; time-harmonic FEM simulations require an accurate modelling of the materials and solutions are sometimes difficult due to the non-linearity of the problem [8]; Multi-Layer Transfer Matrix (MLTM) methods [12] has been adopted with good results and quick computational time but relies on the boundary conditions that have to be known; Finite Difference Method (FDM) [13] is an accurate and reasonably fast technique to determine the field distribution in the rotor structure but the accuracy decreases when high number of space harmonics are considered. A good compromise between accuracy and computational time can be found adopting the Discrete Domain Method described in [14] and summarized in the following section. A. Discrete Domain Method (DDM) The solution domain consists mainly in the rotor structure that can be discretized in layers as for FDM and MLTM as shown in Fig 2.
dimensional approximation of the field distribution. Fourier decomposition for both time-harmonics and spatialharmonics in the tangential direction is applied. The vector potential for the hth time-harmonic and the mth space harmonic results expressed as in (1). {̂
∑∑
}
where ωs,h,m is the slip angular speed for the mth spaceharmonic and the hth time-harmonic [14] and p is the number of pole pairs. The Helmholtz equation (2) remains valid for each kth layer as well as its general analytical closed form (3). ̂ is the complex permeability [6], ́ the modified resistivity of the material in which the kth layer consist in and is its slip angular speed. ̂ ̂
̂
̂ (̂
̂ ́
)
̂
̂
with ̂ the separation constant variable [14]. Eq (2) is valid in the region where the material is conductive. In the main air-gap of the machine the Laplace equation (4) has to be solved. The generic solution (5) requires the calculation of the integration constant to be univocally determined. ̂ ̂
̂
̂
The winding structure and the related magneto-motive-force resulting from the supply system that fed the machine is modelled by means an equivalent infinitesimal current sheet considered at the inner bore of the stator structure [15]. The integration constant for both solution (3) and (5) can be evaluated by means solving the linear algebraic matrix system resulting from the collection of the boundaryinterface condition between each layer based on the Maxwell’s equations. B. Equivalent Circuit The field solution in each of the sectors of the machine that are modelled [11] can be used to determine the values of the elements of the lumped parameter circuit that is shown in Fig. 3. Based on the power flow from the electrical supply to the mechanical shaft of the machine, each element of the circuit can be associated with a real-imaginary power component [11].
Fig. 2 Discrete Domain Method domain subdivisions
Considering a cylindrical reference frame system, the Helmholtz equation has to be solved considering a 2-
Fig. 3 Equivalent circuit model for Solid Rotor Induction Machine
The parameters result dependent on the operating slip and current values. The complex power flowing in the different part of the circuit can be evaluated and the equivalent circuit element values deduced. The solution of the circuit for a given voltage level can be used to determine the torque against slip characteristic of the induction machine.
Fig. 4a 3D Finite Element mesh adopted for the time-harmonic simulations
C. Model Validation A 2 pole induction machine has been investigated. The main machine characteristics are reported in Tab. I. TABLE I DESIGN SPECIFICATIONS continuous power rating power density supply voltage rated speed maximum speed peripheral speed rated torque
120kW 30MW/m3 400V 30 krpm 45 krpm >200m/s 38Nm
The rotor has been design with a structure that is overhanging with respect to the stator to allow enough material for the closing path of the eddy current flowing in the rotor structure. The SRIM model presented above is benchmarked against the solutions achieved by means Finite Element where time-harmonic solver is adopted. The solutions have been calculated for the different space harmonic separately considering only the fundamental supply time harmonic. The correction factor for the rotor resistivity has been estimated by means analytical simplified relations. A 3D FE model has been created with the aim to validate the correction coefficient [7, 8]. In fact the rotor resistivity is calculated according to (6)
Fig. 4b 3D-Eddy current density distribution in solid rotor structure
The factor has been calculated according to (6) considering the results of 2D and 3D simulations. Fig. 5 reports the end factor variation with respect the operative slip of the machine.
Fig. 5 End Factor (
where is the correction factor determined as the ratio of the power losses evaluated by means of 3D FE time harmonic solution with respect the losses evaluated with the 2D FE solver and it is a function of the slip frequency. In Fig 4a the mesh adopted for the 3D model is shown and in Fig 4b the eddy current distribution in the rotor structure evaluated by means the 3D FE time harmonic solver when the machine is operating at rated slip is shown. The simulations has been performed adopting the non-linear solver and the fundamental of the DC magnetization curve of the material (Fe52-S355J2) has been implemented to fit the calculations requirement when time-harmonic solver is used.
against slip at fixed current
The correction factor has been adopted in the analytical model described in Section III.A. The spatial harmonics have been computed. In Fig.6a the eddy current density in the rotor due to the fundamental component of the spatial harmonic decomposition of the m.m.f is shown while in Fig. 6b is shown the effect of the higher spatial harmonics summed together.
Fig. 6a Eddy current density distribution due to the fundamental spatial harmonic of the m.m.f.
A. Finite Difference Method (FDM) The temperature distribution can be calculated by solving the steady-state Fourier equation (7) in a representative sector of the machine where stator, air-gap and rotor have to be modelled. ̇ The solution domain has been discretised with a distribution of nodes in a half-slot section of the machine. The adopted mesh is reported in Fig. 8
Fig. 6b Eddy current density distribution due to the higher order spatial harmonic of the m.m.f.
In Fig 7 the vector potential distribution along the radius coordinate of the rotor structure evaluated by means the 2D FE simulation is compared with the one achieved by means the DDM numerical-analytical method and good agreement is shown.
Fig. 8 Thermal Finite Difference mesh distribution
The discretized form of (7) that has been implemented to solve the temperature distribution within the domain under investigation results as (8) (
Fig. 7 Comparison of vector potential distribution obtained by means 2D finite element with respect the one calculated with DDM.
)
(
)
̇
where ̇ is the heat source, is the specific conductance, is the mesh step which results constant is a regular mesh is adopted and is the temperature of the hth node. The heat source distribution consist in copper losses for the region that models the stator conductors evaluated according to the Joule’s Law ( ); for what concerns the air gap, the windage losses are implemented. An empirical calculation of the windage losses is achieved levering on the simple structure of the structure under investigation and neglecting the effects of the slot opening. According to the dimensionless quantities theory widely adopted in fluid-dynamics computations, the windage losses are estimated as (9)
IV. THERMAL MODEL The thermal management is an important aspect to take into account when high speed rotating speed is achieved. In literature many approaches have been proposed to predict the temperature distribution within the machine structure such as lumped parameter model [16], Finite Difference Method (FDM) [17, 2], Finite Element Methods (FEM) [18]. The analytical formulation to estimate the windage losses are considered levering on the simple rotor geometry; stator power losses are taken into account by considering the Joule losses developed in the stator winding structure; the power losses in the rotor structure can be evaluated by means of the eddy current density distribution in the rotating structure. The Finite Difference Method (FDM) described in this paper results in a good compromise between accuracy and computational efficiency as well it can be closely coupled with the electromagnetic computation previously described [2].
where is the mass density of the fluid, the rotational mechanical speed, the rotor outer radius, L the axial rotor length and is the losses coefficient that can be found in [19]. The heat source for the rotor structure that has been modelled in a layer form as for the electromagnetic computation (consistent rotor mesh for both thermal and electromagnetic computations) is evaluated for each defined layer as expressed in (10), levering on the eddy current density distribution calculated by means the analytical model [14] ̂
̂
where ̂ is the current density and modified resistivity hth layer [2, 14].
is the
B. Model Validation A Finite Element model of the machine under investigation is adopted to benchmark the FDM solution for the temperature distribution in the machine. The temperature distribution is shown in Fig. 9 and a direct comparison of the temperatures in the rotor structure is instead presented in Fig. 10. Good agreement of the calculation with respect to FEA is shown.
where is the Poisson number. A consistent set of boundary conditions can be defined considering that the stress value has to be finite when and considering that the radial component of the stress has to be null at the external radius ( . The analytical solution of (12) results then as (13.a) and (13.b) ( (
Fig. 9 Temperature distribution achieved by means FDM
) )(
)
In Fig. 11 the tangential and radial stress distribution for the Solid Rotor structure when rotating at 50 krpm are shown. The maximum stress that is generated within the rotor structure has to be compared with the yield strength of the rotor material [22].
Fig. 11 Tangential and radial stresses distribution within the rotor structure
VI. ROTOR-DYNAMIC MODEL Fig. 10 Comparison of FEM and FDM rotor temperature distribution against radius.
V. MECHANICAL MODEL A critical aspect that is taken into account in the design of high speed machines deals with the mechanical stress distribution and rotor-dynamic aspects. These aspects are not negligible when the peripheral speed of the rotating component becomes high [4]. The mechanical integrity of the rotor depends on the maximum mechanical stresses that the structure can withstand, therefore high tensile strength material has to be adopted. Due to the simple rotor structure, the stresses distribution can be analytically estimated solving the strain plain stress equations in the rotor domain. Combining the constitutive, congruence and equilibrium equations [20, 21] and assuming a strain stress distribution, the relation (11)
The rotor-dynamic aspects are relevant in high speed applications to avoid operating the machine at rotational speed that might be dangerous to the structure itself. The knowledge of the rotor behaviour at different rotational speed is an important index of the capabilities and performance of the machine at different speeds. The bending mode shape of the rotor can be estimated with Finite Element simulations (2D or 3D) [23] or with numerical-analytical techniques based on simplified model of the rotor allowing quicker estimation of the rotor performances [24]. Mainly focusing on the z-r plane, the axial development of the rotor structure can be modelled with step functions. The rotor is therefore discretized with the aim of take into account all the rotor discontinuities. The main equation that has to be solved is the 4th order PDE (partial differential equation) expressed as a function of the transversal displacement Φ with the curvilinear coordinate ζ and time as (14) (
where is the mass density, the rotational speed, the tangential component and the radial component of the stress. Combining (11) with the constitutive equations, the expression of the stresses distributions can be simplified as in (12) [21]
)
The solution of (14) can be expressed as the product of two functions dependent each on a single variable (15).
Considering only the spatial dependent function, the governing equation for the rotor-dynamic problem results
expressed as (16) for the nth element assuming that the rotor structure is discretized in its axial development and considering that the elastic beam equation results valid in each sector [24].
mechanical aspects are described. 3D FE simulations are performed to estimate the end-region factor to be adopted in the analytical calculations. Further works can be done to benchmark this methodology with different rotor structures and validate the results with experiment. REFERENCES [1]
[2] Fig. 12 Discretized rotor model
The discretization is performed with the aim of considering grouped part of the rotor that features the same inertia, material and external radius as shown in Fig. 12 (
[3]
) [4]
Its solution of (16) results in a general form as (17) where An, Bn, Cn, Dn are the integration constant that can be found by means imposing a consistent set of boundary conditions and the variable is expressed as (18)
[5]
[6]
The solution of the problem can be found by means ofcollecting the boundary condition in a matrix formulation and the problem of finding the rotor critical frequencies consists in solve the homogenous matrix problem where A is the matrix of coefficient and x the vector collecting the integration constant [24].
[7]
[8]
[9]
[10]
Fig. 13 Determinant of the matrix of coefficient with respect the rotational speed of the rotor.
The values of ω which leads to a null determinant of the matrix A results as the critical frequencies. Fig 13 shows the trend of the determinant of the matrix A for the rotor under investigation. The first critical frequency is found at around 47 krpm. VII. CONCLUSION A multi-physics analysis for Solid Rotor induction machine is presented. General methodologies to compute the electromagnetic performances and evaluate the thermal and
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BIOGRAPHIES Luca Papini received his Bachelor degree (Hons.) and Master degree (Hons.) in Electrical engineering in 2009 and 2011, respectively, both from the University of Pisa, Italy. In 2011 he spent six months at the University of Nottingham, UK, as a visiting student, developing analytical and numerical models for a new Vernier motor for his Master thesis. From June to November 2011 he collaborated with the Department of Energy Engineering, University of Pisa, as a research assistant. He is now working towards its Ph.D. with the Electric Motors and Drives Group, University of Nottingham. His main research interests are high speed, high power density electric machines and machine control. David Gerada received the B.Eng.(Hons.) degree in electrical engineering from the University of Malta, Msida, Malta in 2007 and the PhD degree in high speed electrical machines from the University of Nottingham, Nottingham, UK in 2012. Since 2007 he has been with the Research and Technology Department at Cummins Generator Technologies, Stamford, UK, where he is currently an Electromagnetic Design Engineer and Cummins Innovation Centre Technical Leader. His
research interests include traction machines, high speed induction and permanent-magnet machines, and multi-domain optimization of electrical machines. Chris Gerada obtained his PhD in Numerical Modelling of Electrical Machines from the University of Nottingham, England in 2005. He subsequently worked as a Researcher at The University of Nottingham, working on high performance electrical drives and on the design and modeling of electromagnetic actuators for aerospace applications, where he was appointed as a Lecturer in electrical machines in 2008, an Associate Professor in 2011, and a Professor in 2013. He has been the Project Manager of the GE Aviation Strategic Partnership since 2006. His core research interests include the design and modeling of high-performance electric drives and machines. Prof. Gerada was awarded a Royal Academy of Engineering Senior Research Fellowship supported by Cummins in 2011. He is an Associate Editor of the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS and an executive member of the management board of the U.K. Magnetic Society and the IET Aerospace Technical and Professional Network
