Comparison of Two Anisotropic Layer Models Applied to ... - IEEE Xplore

IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 50, NO. 4, JULY/AUGUST 2014

2533

Comparison of Two Anisotropic Layer Models Applied to Induction Motors Ruud L. J. Sprangers, Student Member, IEEE, Johannes J. H. Paulides, Senior Member, IEEE, Konstantin O. Boynov, Elena A. Lomonova, Senior Member, IEEE, and Jeroen Waarma

Abstract—A general description of the Anisotropic Layer Theory, which is derived in the polar coordinate system and applied to the analysis of squirrel-cage induction motors (IMs), is presented. The theory considers nonconductive layers, layers with predefined current density, and layers with induced current density. The electromagnetic field equations are solved by means of Fourier analysis. Furthermore, two different magnetic models for IMs are proposed, namely, the direct rotor current model and the indirect rotor current (IRC) model. The magnetic models are coupled to the single-phase equivalent circuit by means of an iterative algorithm, which also accounts for saturation of the main flux path. Finally, the calculation results are validated against results obtained from measurements on two benchmark motors. Comparison of the validation results shows that the IRC model is the more promising one. Index Terms—Anisotropic Layer Theory (ALT), harmonic modeling, induction motor (IM).

I. I NTRODUCTION

I

N 1954, a model for squirrel-cage induction motors (IMs) directly derived from Maxwell’s equations was proposed by Mishkin [1]. This model is based on the homogenization of the slotted regions of the motor by introducing anisotropic material parameters. Although the results were far from perfect, the work of Mishkin led to the general formulation of the Anisotropic Layer Theory (ALT) through a number of subsequent publications [2]–[6]. However, these publications mainly focus on formulating the ALT for Cartesian coordinates, even in the case of rotating induction machines [5], [6]. In 1996, Madescu et al. [7] implemented the ALT for a polar coordinate system for two types of layers, namely, nonconductive layers and layers with a predefined current density. In addition, Manuscript received August 3, 2013; revised November 5, 2013; accepted December 11, 2013. Date of publication January 14, 2014; date of current version July 15, 2014. Paper 2013-EMC-571.R1, presented at the 2013 IEEE International Electric Machines and Drives Conference, Chicago, IL, USA, May 12–15, and approved for publication in the IEEE T RANSACTIONS ON I NDUSTRY A PPLICATIONS by the Electric Machines Committee of the IEEE Industry Applications Society. R. L. J. Sprangers and E. A. Lomonova are with Eindhoven University of Technology, Eindhoven, The Netherlands (e-mail: [email protected]; [email protected]). J. J. H. Paulides is with Eindhoven University of Technology, Eindhoven, The Netherlands, and also with AEGROUP, Sprang-Capelle, The Netherlands (e-mail: [email protected]). K. O. Boynov is with Protonic Holland B.V., Zwaag, The Netherlands, and also with Eindhoven University of Technology, Eindhoven, The Netherlands (e-mail: [email protected]). J. Waarma is with Vostermans Ventilation B.V., Venlo, The Netherlands (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIA.2014.2300156

Fig. 1. Picture of the IM analyzed by the modeling tools.

Madescu et al. described an iterative algorithm to estimate the saturation level of the main flux path and used this model to calculate the induced electromotive force (EMF) as a function of the magnetizing current. The results of this approach show that this method is fairly successful, and it has been applied for IM optimization in [8]. In this work, first, the existing ALT in polar coordinates is extended with a solution for the electromagnetic field in layers with an induced current density. For all three layer types, a general solution for the electromagnetic field is derived by means of Fourier analysis [9]. Second, the obtained electromagnetic field solution is applied to the analysis of three-phase IMs with a squirrel-cage rotor, such as the one shown in Fig. 1. To this extent, two different magnetic models are proposed, namely, the direct rotor current (DRC) model and the indirect rotor current (IRC) model. Both magnetic models are coupled to the single-phase equivalent circuit model by means of an iterative algorithm. This algorithm is an extension of the procedure described by Madescu et al., and thus, main flux path saturation is accounted for. The main contribution of this paper is to identify whether the DRC model or the IRC model is more suited for IM analysis. Therefore, both models are implemented for two benchmark motor topologies and validated against finite element analysis (FEA) and measurement results. Finally, the validation results are used to compare the DRC model and the IRC model to determine which one is more suitable for (automated) design tools. II. M ETHODOLOGY To apply the ALT to an electromagnetic structure, the structure is divided into a set of layers. For example, a typical cross section for a low-power IM and its corresponding layer division are shown in Fig. 2. Three types of layers are distinguished, namely, nonconductive layers, predefined current density layers, and induced current density layers.

0093-9994 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

2534


Fig. 2. General IM geometry and its corresponding layer division.

Nonconductive layers do not conduct electric current. Therefore, such layers do not carry any predefined or induced current density. For example, the yokes and the air gap of an IM can be modeled by nonconductive layers. Furthermore, predefined current density layers conduct a predefined current density but cannot induce additional current density. For example, the stator tooth/coil layer and the rotor tooth/bar layer of an IM can be modeled by a predefined current density layer. Finally, induced current density layers do not conduct any predefined current density. However, a current density can be induced in the layer by a time-varying magnetic field. As the variation of the magnetic field depends on the speed at which the layer is moving, a constant angular velocity, i.e., ωm , will be considered. For example, the rotor tooth/bar layer of an IM can be also modeled by an induced current density layer. When a layer contains materials with different electromagnetic properties, for example, layers with teeth and coils, the materials are modeled by an equivalent material. The equivalent material takes the effect of the variation in material properties into account by the calculation of homogeneous but anisotropic equivalent material properties. For a layer with iron teeth and coil slots, the equivalent material properties are calculated as μrad = μθ = σ=

μtooth btooth + μslot bslot btooth + bslot

(1)

μtooth μslot (btooth + bslot ) μtooth bslot + μslot btooth

(2)

σtooth btooth + σslot bslot btooth + bslot

(3)

where μtooth is the iron tooth permeability, μslot is the slot permeability, σtooth is the iron tooth conductivity, σslot is the slot conductivity, btooth is the average tooth width, and bslot is the average slot width. Furthermore, μrad is the permeability in the radial direction, μθ is the permeability in the tangential direction, and σ is the conductivity of the equivalent material. To apply the ALT to IMs, a cylindrical coordinate system is most suitable. In this modeling tool, the following simplifying assumptions are applicable. • Structures are infinite and invariant in the axial direction. • Field quantities sinusoidally vary in time with angular frequency ω.

• All physical materials have linear, isotropic, homogeneous, and temperature-invariant electromagnetic properties. However, equivalent materials can be anisotropic. • The structure is periodic in the tangential direction. • The electric current density is constant within each slot. • The electric current flows solely in the axial direction. Under these assumptions, the cylindrical coordinate system can be reduced to a polar coordinate system, which will be used in the remainder of this paper. III. E LECTROMAGNETIC F IELD S OLUTION Here, a general description of the electromagnetic field solution for each layer type is derived. The derived expressions will be applied to IM analysis in this paper, but they can be applied for the analysis of other rotating machines as well. A. Electromagnetic Field Equation The solution of the electromagnetic field equations is found in each layer. With by solving the magnetic vector potential A the given assumptions, A can be written as a complex Fourier series as = Az uz = Re Aˆz (r)ej(ωt−pθ) uz (4) A where p denotes the number of pole pairs, and uz denotes the unity vector in the z-direction. The equation for the electromagnetic field in an anisotropic layer can be derived from two of the four Maxwell equations, i.e., =− ∇×E = J. ∇×H

∂B ∂t

(5) (6)

Applying the definition of the magnetic vector potential A =∇×A B

(7)

and the constitutive relation given by =μ ¯H B

(8)

¯ is the permeability tensor; in (6), the electromagnetic where μ field equation is found as =∇×μ = J ¯ −1 (∇ × A) ∇×H

(9)

SPRANGERS et al.: COMPARISON OF TWO ANISOTROPIC LAYER MODELS APPLIED TO INDUCTION MOTORS

¯ −1 is defined as where the inverse permeability tensor μ 1 0 ¯ −1 = μrad μ . 1 0 μθ

where (10)

For conductive layers, either stationary or moving with velocity vector v , the induced current is accounted for using Ohm’s law for moving regions, i.e., + v × B). J = σ(E

(11)

The resulting electromagnetic field equation for conductive layers, found by applying (11) to (9), is given by 1 . = − ∂ A + v × (∇ × A) ¯ −1 (∇ × A) ∇×μ σ ∂t

(12)

In this paper, only rotational movement with angular velocity ωm is considered, such that v = vθ uθ = rωm uθ . B. Nonconductive Layers For nonconductive layers, (9) becomes = 0. ¯ −1 (∇ × A) ∇×μ

(13)

With the given simplifying assumptions, (13) reduces to 1 ∂ 2 Anz 1 1 ∂ 2 Anz 1 1 ∂Anz + + =0 μθ ∂r2 r μθ ∂r r2 μrad ∂θ2

(14)

where superscript n denotes the nonconductive layer. Now, the separation of variables is applied as Anz (r, θ, t) = Rn (r)Θ(θ)T (t)

(15)

where the solutions of Θ(θ) and T (t) should comply with (4) and are given by Θ(θ) = e−jpθ T (t) = ejωt .

(16) (17)

To solve the radial part of the solution, i.e., Rn (r), (15)–(17) are applied to (14), which gives r2 Rn (r) + rRn (r) − α2 Rn (r) = 0 (18) where α = p μθ /μrad . The solution for (18) is given by Rn (r) = an rα + bn r−α .

2535

where Aˆnz (r) = Rn (r). The radial and tangential components of the magnetic flux density are calculated using (7) and are given by 1 ∂Anz ˆrn (r)ej(ωt−pθ) = Re B Brn (r, θ, t) = (21) r ∂θ n ∂Az ˆ n (r)ej(ωt−pθ) (22) = Re B Bθn (r, θ, t) = − θ ∂r

(23)

ˆ n (r) = − jp Aˆn (r). B r r z

(24)

The radial and tangential components of the magnetic field strength can be found in (21) and (22) using (8). C. Predefined Current Density Layer The current density in a predefined current density layer which is is described by the current density vector, i.e., J, given by J = Jz uz = Re Jˆz ej(ωt−pθ) uz . (25) The complex Fourier coefficient of the current density, i.e., Jˆz , can be calculated from winding analysis by means of winding factors [10], [11]. For a balanced three-phase winding, Jˆz is given by √ 6 2Irms Nph kw1 jφ ˆ Jz = e (26) As with rms phase current Irms , phase current angle φ, number of turns per phase Nph , fundamental winding factor kw1 , and excitation layer area As . For predefined current density layers, (9) reduces to the inhomogeneous differential equation given by 1 1 ∂ 2 Apz 1 ∂ 2 Apz 1 1 ∂Apz + + = −Jz μθ ∂r2 r μθ ∂r μrad r2 ∂θ2

(27)

under the given simplifying assumptions, where superscript p denotes the predefined current density layer. The homogeneous , is similar to the solution part of the solution for (27), i.e., Ap,ho z for (14) and is given by (r, θ, t) = Re Aˆp,ho (r)ej(ωt−pθ) Ap,ho (28) z z where Aˆp,ho = ap rα + bp r−α z

(19)

where an and bn are constants for nonconductive layers. The final solution for Anz in nonconductive layers is given by (20) Anz (r, θ, t) = Re Aˆnz (r)ej(ωt−pθ)

ˆ n (r) = − α an rα − bn r−α B θ r

(29)

and ap and bp are constants for predefined current density layers. For the nonhomogeneous part of the solution for (27), i.e., , (27) should hold after applying Apz = Ap,no . FurtherAp,no z z more, the solution should comply with the electric current density described by (25). As a result, the nonhomogeneous solution is found [7] as 2 j(ωt−pθ) (30) (r, θ, t) = Re Gr e Ap,no z where G = −Jˆz

μrad μθ . 4μrad − p2 μθ

(31)

2536


The final solution for Apz in predefined current density layers and Ap,no together and is given by is found by adding Ap,ho z z Apz (r, θ, t) = Re Aˆpz (r)ej(ωt−pθ) (32)

TABLE I B ENCHMARK M OTOR PARAMETERS

where Aˆpz (r) = ap rα + bp r−α + Gr2 . The radial and tangential magnetic flux density components are found as 1 ∂Apz ˆrp (r)ej(ωt−pθ) (33) = Re B Brp (r, θ, t) = r ∂θ ∂Apz ˆ p (r)ej(ωt−pθ) Bθp (r, θ, t) = − (34) = Re B θ ∂r where

ˆ p (r) = − jp ap rα + bp r−α + G B r r α 2G p α −α ˆ Bθ (r) = − a p r − bp r + . r α

(35) (36)

The radial and tangential magnetic field strength components are again found using (8). D. Induced Current Density Layers For the given simplifying assumptions and the assumed velocity vector v , it is found that = −vθ v × (∇ × A)

1 ∂Az ∂Az uz = −ωm uz . r ∂θ ∂θ

(37)

Furthermore, considering (4), it can now be found that ∂A = −jωAz uz + jpωm Az uz . (38) + v × (∇ × A) − ∂t

where Aîz (r) = ai Iα (βr) + bi Kα (βr).

The radial and tangential magnetic flux density components are found as 1 ∂Aiz ˆ i (r)ej(ωt−pθ) (44) = Re B Bri (r, θ, t) = r r ∂θ i ∂Az ˆθi (r)ej(ωt−pθ) Bθ,i (r, θ, t) = − (45) = Re B ∂r where ˆri (r) = − jp Aîz (r) B r î ˆ i (r) = − ∂ Az . B θ ∂θ

Applying (38) and the simplifying assumptions to (12), the electromagnetic field equation for induced current layers is found as 1 ∂ 2 Aiz 1 1 ∂ 2 Aiz 1 1 ∂Aiz + + = jσAiz ωind μθ ∂r2 r μθ ∂r μrad r2 ∂θ2

(46) (47)

The induced current density in the layer is calculated as (39)

where ωind = (ω − pωm ) denotes the angular frequency of the induced current, and superscript i denotes the induced current density layer. Using separation of variables and the solutions for Θ(θ) and T (t) given by (16) and (17), the equation for Ri (r) is found as r2 Ri (r) + rRi (r) − (α2 + β 2 r2 )Ri (r) = 0 (40) √ where β = jωind μθ σ. Equation (40) is a modified Bessel equation, for which the solution is given by Ri (r) = ai Iα (βr) + bi Kα (βr).

(43)

(41)

Iα (γ) and Kα (γ) are modified Bessel functions of the first and second kind, respectively; and ai and bi are constants for the induced current density layers. The final solution for Aiz in induced current layers is given by Aiz (r, θ, t) = Re Aîz (r)ej(ωt−pθ) (42)

Jzind = −jσAiz ωind .

(48)

Finally, (8) is used to find the radial and tangential magnetic field strength components. IV. IM M ODELING A. Model Definition The cross section of the IM is divided into five layers, as shown in Fig. 2. Several relevant parameters of the IM cross section are given in Table I. The represented regions in the layer division are the following: 1) I: rotor yoke layer; 2) II: rotor slot layer; 3) III: air gap layer; 4) IV: stator slot layer; 5) V: stator yoke layer. Layers I, III, and V are assumed homogeneous and can be modeled as such. Layers II and IV, however, contain both magnetic iron and conducting material (copper or aluminium


2537

TABLE II L AYER D EFINITION OF THE DRC M ODEL AND THE IRC M ODEL

alloy). Therefore, these layers are modeled using equivalent material parameters, which are calculated using (1)–(3). The stator and rotor tooth tips have been included into the stator and rotor slot regions. Using the presented layer division, two different types of IM models are proposed, namely, the DRC model and the IRC model. An overview of the types of layers assigned for each of these models is shown in Table II. The main difference between the DRC and IRC models can be found in the definition of layer II. In the DRC model, layer II is defined as an induced current density layer. This means that the induced rotor current density is directly taken into account in the magnetic model. Because of the assumed infinite length of the magnetic model, end effects are not accounted for. Therefore, the equivalent conductivity of layer II is multiplied by a correction factor, i.e., ker = Rbar /Rr , to account for the end ring resistance in the DRC model. Here, Rbar is the rotor bar resistance, and Rr is the equivalent rotor bar resistance including the end rings [10]. Alternatively, in the IRC model, layer II is assigned as a predefined current density layer. The value of the rotor current is externally calculated by means of the electric equivalent circuit (EEC). This current is then translated into a current density and applied to layer II as a fixed parameter. The effect of the end ring resistance is included in the external circuit for the IRC model. B. Boundary Conditions The solutions for the electromagnetic field equations given in Section III can only be used when the coefficients an , ap , or ai and bn , bp , or bi are known for each layer. To solve these coefficients, boundary conditions are applied. For the IM models, the boundaries on which the boundary conditions are defined are given in Fig. 2 by r1 to r6 , where r1 = rshaft . On the boundaries between two adjacent layers, i.e., i and i + 1, continuous boundary conditions are applied. Assuming that there is no line current density on the boundaries, the continuous boundary conditions are given by Hθ,i+1 − Hθ,i |r=rboundary = 0 Br,i+1 − Br,i |r=rboundary = 0.

(49) (50)

Furthermore, it is assumed that the magnetic flux is completely confined within the yoke regions. This means that a Neumann-type boundary condition is to be applied on the outer boundary of the stator yoke and on the inner boundary of the rotor yoke. The specific Neumann-type boundary condition used here is given by Br |r=rboundary = 0.

(51)

Fig. 3. Single-phase EEC. TABLE III EEC PARAMETERS

The equations resulting from the boundary conditions can be combined to form an equation matrix of the form EX = Y

(52)

where X contains the coefficients an , ap , or ai and bn , bp , or bi for each layer. The equation is solved for X, and the coefficients of each layer are extracted. C. Electric Circuit Model Both the DRC and IRC models are coupled to an electric circuit model to calculate the excitation current(s). For the IRC model, both the stator and rotor currents are required, whereas for the DRC model, only the stator current is used. Assuming a balanced three-phase system, the single-phase EEC shown in Fig. 3 can be used to find the required currents. A description of the circuit parameters shown in Fig. 3 is given in Table III, where the inductance parameters are translated into reactances. The values given in this table are valid only for the nominal operating points of two benchmark motors, which will be introduced later. For each operating point, the calculation of the circuit parameters is embedded into a nonlinear solving algorithm. Methods to find values for Rs , Lσ,s , Rr , and Lσ,r based on analytical or empirical approaches are extensively described in the literature [10]–[14]. Finally, the slip, i.e., s, is defined as s=

ω − pωm . ω

(53)

The voltage across the magnetizing inductance Lm repre¯m,eec . However, the EMF can be sents the EMF phasor, i.e., E also calculated from the magnetic model. To do so, first, the ˆ p , is calculated in the middle of the air peak flux per pole, i.e., Φ gap as π

p ˆp = l Φ 0

ˆrgap rdθ = −2l(agap rαgap + bgap r−αgap ). B

(54)

2538


Fig. 5. μr (B) characteristics of the two lamination materials.

Fig. 4. Schematic overview of the iterative process.

¯m,alt Since the flux is assumed to vary sinusoidally in time, E can now be found from the magnetic model as ¯m,alt = − √1 Nph kw1 dφp (t) E dt √2 = jω 2lNph kw1 (agap rαgap + bgap r−αgap ). (55) Assuming that the magnetizing current phasor, i.e., I¯m , is known from the electric model, the magnetizing inductance Lm ¯m,alt as can be calculated from E ¯m,alt | |E Lm = . (56) 2π|I¯m | Furthermore, also the iron loss resistance, i.e., RFe , is calculated from the magnetic model [10]. It should be noted here that a loop is created since the electric model requires Lm and RFe to be known and the magnetic model requires the current phasor(s) to be known. An iterative procedure is required to solve this dependence. Furthermore, an iterative approach is also required to take the nonlinear B(H) characteristic of the lamination material into account. D. Algorithm for Nonlinear Solving It was shown in [7] that the ALT can account for global saturation in the main flux path of the motor by an iterative procedure. In this paper, the algorithm proposed in [7] is extended such that the interaction between the electric model and the magnetic model is accounted for. A schematic overview of the extended iterative procedure is shown in Fig. 4. Initially, the parameters for the EEC are calculated assuming infinitely permeable lamination material and RFe = ∞. The electric model is then analyzed, and the results are used to initialize the magnetic model. The relative permeability of the lamination material in each saturable layer, i.e., m, is initially set to μm r,k=1 = 1000. From here, the iterative loop is started. During the kth iteration step, the actual relative permeability , is calculated using the in each saturable layer, i.e., μm,act r,k

nonlinear μr (B) characteristic of the lamination material, for example, such as shown in Fig. 5. In the yoke layers, μm,act r,k is obtained from the tangential peak magnetic flux density, i.e., ˆ m , in the middle of the layer. Furthermore, in the tooth layers, B θ is obtained from the radial peak magnetic flux density, μm,act r,k ˆrm , in the middle of the layer. However, B ˆrm should be i.e., B corrected as most of the flux passed through the teeth, not the ˆ m , is slots. Therefore, the actual tooth flux density, i.e., B tooth found as m = Btooth

bd,m + bs,m ˆ m Brad bd,m

(57)

where bd,m is the tooth width, bs,m is the slot width in the ˆ m is the peak of the radial magnetic middle of the layer, and B rad flux density in the middle of the layer. The error between the m,act is calculated as initial relative permeability μm r,k and μr,k μm r,k (58) εm k = m,act − 1 . μr,k If εm k is larger than the required accuracy, i.e., εacc , the relative permeability of the lamination material in the respective layer is recalculated for the next iteration step, i.e., k + 1, as m,act m (59) − μm μm r,k+1 = μr,k + Cit μr,k r,k where Cit is the relaxation constant. Then, the magnetic model is updated with the new relative permeability values, and the inductance parameters of the electric model are recalculated. Finally, the electric model is analyzed with the new circuit parameters, and the magnetic model is rerun to close the iterative loop. Once all relative permeabilities have converged, it should be checked whether the electric model and the magnetic model predict the same EMF. The error between both EMF values is calculated as ¯m,eec | |E (60) − 1 . εemf = ¯ |Em,alt | If εemf is larger than the desired accuracy εacc , convergence has not been reached, and another iteration step is needed. Finally, when the EMF has converged, the analysis is finished. The described algorithm applies for the analysis of one single operating speed. However, it can easily be repeatedly applied for a series of operating speeds to generate motor characteristics.


2539

TABLE IV M AGNETIC VALIDATION C OMPARISON FOR M OTOR 1

E. Electromagnetic Torque Calculation A prediction of the developed electromagnetic torque, i.e., Tem , can be calculated either from the magnetic model or from the electric model. First, from the magnetic model, the torque is calculated by means of the Maxwell stress tensor (MST) [15]. By integrating the MST along a circumferential path in the middle of the air gap, the electromagnetic torque is found as MST = Tem

2πlαgap p [B1 A2 − B2 A1 ] μ0

(61)

where A1 = Re(agap ), A2 = Im(agap ), B1 = Re(bgap ), B2 = Im(bgap ), and r = r3 + (g/2). Second, from the electric circuit model, Tem is calculated as 1 cir 21 − s ¯ Rr Tem = 3|Ir | (62) s ωm

TABLE V M AGNETIC VALIDATION C OMPARISON FOR M OTOR 2

where ωm is the angular velocity of the rotor. It should be noted that, in both cases, the mechanical losses, i.e., Pmech , are still to be subtracted to find the mechanical output torque, i.e., Tout . The main origins of Pmech are bearing friction and windage. Methods to calculate these loss components are given in [10]. V. M AGNETIC M ODEL VALIDATION A. Benchmark Motors To validate the IRC and DRC models, two benchmark motors are implemented and analyzed. Table I gives an overview of some of the benchmark motor parameters. Both machines have approximately the same nominal output power and speed. However, their dimensions and magnetic lamination materials are different. Furthermore, both motors have skewed rotors, but the skew angle, i.e., αskew , of motor 2 is approximately twice as large as the αskew of motor 1. To obtain more reliable calculation results, the relative permeability of the magnetic materials, i.e., μr , is measured as a function of the magnetic flux density B using ring core samples of the materials. Fig. 4 shows the results of the measurements on both materials, referred to as M1 and M2. These measurements are performed on an MPG-200 (Brockhaus GmbH), which measures in compliance with the IEC 60404 norms. B. Comparison With Two-Dimensional FEA Simulation Results Both benchmark motors are simulated using a 2-D steadystate FEA model. These simulations are performed at four different operating points, namely, synchronous speed, i.e., nsyn , nominal speed, i.e., nnom , half the synchronous speed, i.e., nhalf , and zero speed (locked rotor), i.e., nlr . The results of the simulations are compared with the results of the presented models obtained at the same operating points. The comparison is made in terms of the radial and tangential peak magnetic flux ˆ gap , ˆ gap and B densities in the middle of the air gap layer, i.e., B r θ respectively. Additionally, also the predicted rms phase current,

i.e., Iph,rms , is compared. An overview of the comparison is shown in Tables IV and V. ˆ gap are in relatively good agreeThe predicted values of B r ment for both benchmark motors. In comparison, the IRC model seems to be slightly closer to the FEA results than the DRC model for large slips. However, the predicted value of ˆ gap are not in good agreement. Both the IRC and DRC models B θ ˆ gap with respect to the FEA results. Only for underestimate B θ motor 2, the DRC model seems to be in good agreement with ˆ gap . FEA for B θ Furthermore, Iph,rms is underestimated by the IRC model. This underestimation can be explained by local saturation effects, which are neglected in the IRC model. In addition, skewing effects are neglected in the 2-D FEA model. This causes the Iph,rms predicted by the 2-D FEA model to be slightly too large. On the contrary, Iph,rms is overestimated by the DRC model. To explain this overestimation, however, the distribution and amplitude of the rotor current density should be investigated. Comparisons of the current density distribution calculated at locked rotor by the IRC and DRC models are shown in Figs. 6 and 7, respectively. It is shown that, for the DRC model, the rotor current density amplitude is much larger than the stator current density amplitude, whereas this difference is much smaller for the IRC model. This illustrates that the rotor impedance is underestimated by the DRC model, despite of the correction factor ker . Furthermore, it is shown that the current density distribution calculated by the DRC model at locked rotor shifts in the direction of motion toward the air gap. This effect is also believed to be a cause of small discrepancies between the DRC calculation results and FEA results.

2540


Fig. 8.

Schematic overview of the measurement system.

Fig. 9.

Benchmark motor 1—no-load comparison.

Fig. 6. Current density distribution at locked rotor calculated by the IRC model for motor 2.

Fig. 7. Current density distribution at locked rotor calculated by the DRC model for motor 2.

VI. P ERFORMANCE P REDICTION VALIDATION

Fig. 10.

Benchmark motor 2—no-load comparison.

A. Measurement Setup The validation of the motor performance predicted by the presented models is done by means of comparison with measurement results, obtained for the two benchmark motors. These measurements are performed on a measurement setup, for which an overview is shown in Fig. 8. The tested IM is physically coupled to an eddy current brake, which acts as a load. Prior to a measurement, the motor is heated up to its nominal operating temperature. Then, the measurement is started by increasing and controlling the load torque, such that the speed reduces from synchronous speed to 30% of the synchronous speed in a stepwise manner. At each step, the phase voltages, i.e., Vph , phase currents, i.e., Iph , shaft speed, i.e., n, output torque, i.e., Tout , winding temperature, i.e., Twind , and ambient temperature, i.e., Tamb , are measured and stored for further analysis. A full measurement is performed within a few seconds, such that the temperature rise in the stator and rotor conductors is kept to a minimum.

B. No-Load Validation First, the performance of the motors at no-load operation is measured. The load is decoupled from the tested IM, and the phase current Iph is measured for different values of phase voltage Vph . Then, after the correction of Iph for the mechanical and iron losses, the magnetizing current, i.e., Im , is obtained. Similar characteristics are also calculated using the IRC and DRC models. In these calculations, the phase voltages and motor operating speeds obtained from the measurements are applied, to keep the simulations as close to the measurements as possible. Comparisons of the measured and calculated results for both benchmark motors are shown in Figs. 9 and 10. It is shown that a good agreement is obtained for both motors. Additionally, it is also shown that the IRC and DRC models produce nearly the same results. This is expected since the rotor current is close to zero at no-load operation, and the models are thus essentially equal for this operating point.


Fig. 11. Benchmark motor 1—IRC torque comparison.

Fig. 12. Benchmark motor 1—IRC current comparison.

2541

Fig. 15. Benchmark motor 1—DRC torque comparison.

Fig. 16. Benchmark motor 1—DRC current comparison.

Fig. 13. Benchmark motor 2—IRC torque comparison. Fig. 17. Benchmark motor 2—DRC torque comparison.

Fig. 14. Benchmark motor 2—IRC current comparison.

C. Performance Characteristics Validation To validate the IRC and DRC models under loaded operating conditions, performance characteristics of the motors are measured as a function of operating slip. Similar characteristics are computed using the IRC and DRC models. Then, a comparison is made in terms of output torque Tout and phase current Iph .

Fig. 18. Benchmark motor 2—DRC current comparison.

The comparisons between the IRC calculation results and measurement results for both benchmark motors are shown in Figs. 11–14. It is shown that for operating slips up to 0.4, cir and Iph are in good agreement the predicted values of Tout with the measurement results. For operating slips larger than

2542


0.4, it is shown that the error in the computed characteristics increases. This is particularly true for benchmark motor 2. In addition, the torque calculated by the MST method does not match the measurement results in general. This illustrates that the accuracy of the predicted radial and tangential magnetic field distributions in the air gap is not accurate enough for the MST method. Additionally, the comparisons between the DRC calculation results and measurement results are shown in Figs. 15–18. From cir is in these comparisons, it is shown that the computed Tout good agreement with the measurements. However, the phase current is largely overestimated, which makes the validity of the computed torque questionable. Only for small slips, up to 0.15, does the DRC method seems to be reliable. Finally, it should be noted that the presented models are relatively fast. Performance characteristics consisting of 100 operating points are calculated by the IRC model in approximately 7 s, whereas the DRC model takes approximately 13 s on a modern computer. In comparison with the finite element method, which is relatively slow, the short calculation time is an advantage when many different operating points or designs have to be analyzed.

VII. C ONCLUSION It has been shown in Section VI-B that both the DRC and IRC models are in good agreement with measurement results under no-load operating conditions. This indicated that the ALT can properly model the global nonlinear behavior of the main magnetic flux path. Under loaded operating conditions, it is observed that the IRC model gives the best agreement with respect to measurements. The results are in good agreement with measurement results up to a slip of 0.4, whereas the error increases for larger values of slip. This mismatch is believed to be caused by effects that are currently not included in the IRC and DRC models, such as saturation of the leakage flux paths [16], [17] and interbar currents [18]. For the DRC model, it is observed that the region in which the predicted phase current agrees with the measurements is much smaller than that for the IRC model. On the other hand, the predicted torque does show good agreement with measurements. However, it can be argued whether or not this conclusion is genuine, as the torque is predicted for an overestimated value of the phase current. From the comparison results, it can be concluded that the IRC model appears to be the more suitable model for IM analysis. In future work, the IRC model could be further improved by inclusion of currently unaccounted effects, such as leakage path saturation and interbar current. Furthermore, the magnetic model could be extended by including higher spatial harmonics [6] and extension to a 3-D model [6], [19]. However, these extensions will also increase the calculation time of the model. In its current form, the relatively short calculation time and good accuracy in the low-slip operating region of an IM make the IRC model very suitable for automated design tools. Therefore, future extensions should be made with care such that this conclusion remains valid.

R EFERENCES [1] E. Mishkin, “Theory of the squirrel-cage induction machine derived directly from Maxwell’s field equations,” Quart. J. Mech. Appl. Math., vol. 7, no. 4, pp. 472–487, 1954. [2] A. L. Cullen and T. H. Barton, “A simplified electromagnetic theory of the induction motor, using the concept of wave impedance,” Proc. Inst. Elect. Eng., Pt. C, Monographs,, vol. 105, no. 8, pp. 331–336, Sep. 1958. [3] J. Greig and E. M. Freeman, “Travelling-wave problem in electrical machines,” Proc. Inst. Elect. Eng., vol. 114, no. 11, pp. 1681–1683, Nov. 1967. [4] E. M. Freeman, “Travelling waves in induction machines: Input impedance and equivalent circuits,” Proc. Inst. Elect. Eng., vol. 115, no. 12, pp. 1772–1776, Dec. 1968. [5] S. Williamson, “The Anisotropic Layer Theory of induction machines and induction devices,” IMA J. Appl. Math., vol. 17, no. 1, pp. 69–84, 1976. [6] S. Williamson and A. Smith, “Field analysis for rotating induction machines and its relationship to the equivalent-circuit method,” Proc. Inst. Elect. Eng., vol. 127, no. 2, pp. 83–90, Mar. 1980. [7] G. Madescu, I. Boldea, and T. J. E. Miller, “An analytical iterative model (AIM) for induction motor design,” in Conf. Rec. 31st IEEE IAS Annu. Meeting, 1996, vol. 1, pp. 566–573. [8] G. Madescu, I. Boldea, and T. J. E. Miller, “The optimal lamination approach to induction machine design global optimization,” IEEE Trans. Ind. Appl., vol. 34, no. 3, pp. 422–428, May/Jun. 1998. [9] B. L. J. Gysen, K. J. Meessen, J. J. H. Paulides, and E. A. Lomonova, “General formulation of the electromagnetic field distribution in machines and devices using Fourier analysis,” IEEE Trans. Magn., vol. 46, no. 1, pp. 39–52, Jan. 2010. [10] J. Pyrhonen, T. Jokinen, and V. Hrabovcova, Design of Rotating Electrical Machines. Hoboken, NJ, USA: Wiley, 2008. [11] I. Boldea and S. Nasar, The Induction Machines Design Handbook. Boca Raton, FL, USA: CRC Press, Taylor & Francis, 2010. [12] A. Boglietti, A. Cavagnino, and M. Lazzari, “Algorithms for the computation of the induction motor equivalent circuit parameters—Part I,” in Conf. Rec. IEEE IECON, Orlando, FL, USA, Nov. 2008, pp. 2020–2027. [13] A. Boglietti, A. Cavagnino, and M. Lazzari, “Algorithms for the computation of the induction motor equivalent circuit parameters—Part II,” in Conf. Rec. IEEE IECON, Orlando, FL, USA, Nov. 2008, pp. 2028–2034. [14] O. Butler and T. Birch, “Comparison of alternative skew-effect parameters of cage induction motors,” Proc. Inst. Elect. Eng., vol. 118, no. 7, pp. 879– 883, Jul. 1971. [15] B. L. J. Gysen, E. A. Lomonova, J. J. H. Paulides, and A. Vandenput, “Analytical and numerical techniques for solving Laplace and Poisson equations in a tubular permanent-magnet actuator: Part I. Semi-analytical framework,” IEEE Trans. Magn., vol. 44, no. 7, pp. 1751–1760, Jul. 2008. [16] B. Chalmers and R. Dodgson, “Saturated leakage reactances of cage induction motors,” Proc. Inst. Elect. Eng., vol. 116, no. 8, pp. 1395–1404, Aug. 1969. [17] L. Monjo, F. Córcoles, and J. Pedra, “Saturation effects on torqueand current-slip curves of squirrel-cage induction motors,” IEEE Trans. Energy Convers., vol. 28, no. 1, pp. 243–254, Mar. 2013. [18] D. Dorrel, T. J. E. Miller, and C. Rasmussen, “Inter-bar currents in induction machines,” IEEE Trans. Ind. Appl., vol. 39, no. 3, pp. 677–684, May/Jun. 2003. [19] K. J. Meessen, B. L. J. Gysen, J. J. H. Paulides, and E. A. Lomonova, “General formulation of fringing fields in 3-D cylindrical structures using Fourier analysis,” IEEE Trans. Magn., vol. 48, no. 8, pp. 2307–2323, Aug. 2012.

Ruud L. J. Sprangers (S’13) was born in Waspik, The Netherlands, in 1986. He received the M.Sc. degree in electrical engineering from Eindhoven University of Technology, Eindhoven, The Netherlands, in 2011, where he is currently working toward the Ph.D. degree. His research mainly focuses on (semi)analytical modeling and design of high-efficiency motors.


Johannes J. H. Paulides (M’03–SM’13) received the B.Eng. degree from Avans University of Applied Sciences, Hertogenbosch, The Netherlands, in 1998 and the M.Phil. and Ph.D. degrees in electrical and electronic engineering from The University of Sheffield, Sheffield, U.K., in 2000 and 2005, respectively. From 2005 to 2009, he held several postdoctoral positions with Eindhoven University of Technology, Eindhoven, The Netherlands, where he is currently a Part-Time Assistant Professor, focusing on the more sustainable society. He is also the Owner of AEGROUP, Sprang-Capelle, The Netherlands, a number of small- and medium-sized enterprises which, among other activities, are prototyping and producing electrical machines. His research interests include all facets of electrical machines, particularly linear and rotating permanent-magnet excited machines for automotive and highprecision applications.

Konstantin O. Boynov was born in Moscow, Russia, in 1975. He received the M.Sc. degree in electromechanical engineering from Moscow State Aviation Institute, Moscow, in 1999 and the Ph.D. degree in electrical engineering from Eindhoven University of Technology, Eindhoven, The Netherlands, in 2008. He is currently a Research and Development Engineer with Protonic Holland, B.V. Zwaag, The Netherlands, and a part-time Assistant Professor with the Department of Electrical Engineering, Eindhoven University of Technology. His research interests are focused on control and design of electrical machines.

2543

Elena A. Lomonova (M’04–SM’07) received the M.Sc. (cum laude) and Ph.D. (cum laude) degrees from Moscow Aviation Institute (State University of Aerospace Technology), Moscow, Russia, in 1982 and 1993, respectively, both in electromechanical and control systems. During 1982–1987, she was with Astrophysics, Moscow. During 1987–1997, she was with the Electromechanical and Control Systems Department, State University of Aerospace Technology (MAI), where she was active in research, education, and industrial projects. In 1998, she joined Delft University of Technology, Delft, The Netherlands. In 2000, she joined Eindhoven University of Technology, Eindhoven, The Netherlands, where she became a full-time Professor in March 2009. Her chair focuses on fundamental and applied research on enabling energy conversion theory, methods, and technologies for high-precision, automotive, and medical systems. She has authored/coauthored over 150 scientific publications and is a holder of more than ten patents. Her research interests include various facets of advanced mechatronics, electromechanics, and electromagnetics, including rotary electrical machines and drives and linear and planar actuation systems.

Jeroen Waarma was born in Eindhoven, The Netherlands, in 1985. He received the B.Sc. and M.Sc. degrees in electrical engineering from Eindhoven University of Technology, Eindhoven, in 2008 and 2011, respectively. Since 2011 he has been with Vostermans Ventilation B.V., Venlo, The Netherlands, working on the development of induction motors.