# Transient simulation of an electrical rotating machine achieved through model order reduction

- Laurent Montier
^{1, 2}Email author, - Thomas Henneron
^{3}, - Stéphane Clénet
^{1}and - Benjamin Goursaud
^{2}

**3**:10

**DOI: **10.1186/s40323-016-0062-z

© The Author(s) 2016

**Received: **7 December 2015

**Accepted: **24 February 2016

**Published: **22 March 2016

## Abstract

Model order reduction (MOR) methods are more and more applied on many different fields of physics in order to reduce the number of unknowns and thus the computational time of large-scale systems. However, their application is quite recent in the field of computational electromagnetics. In the case of electrical machine, the numerical model has to take into account the nonlinear behaviour of ferromagnetic materials, motion of the rotor, circuit equations and mechanical coupling. In this context, we propose to apply the proper orthogonal decomposition combined with the (Discrete) empirical interpolation method in order to reduce the computation time required to study the start-up of an electrical machine until it reaches the steady state. An empirical offline/online approach based on electrical engineering is proposed in order to build an efficient reduced model accurate on the whole operating range. Finally, a 2D example of a synchronous machine is studied with a reduced model deduced from the proposed approach.

### Keywords

Model order reduction Proper orthogonal decompostion Discrete empirical interpolation method Finite element method Overlapping finite element method Synchronous machine## Background

Modeling electrical machines using the the finite element method (FEM) has proved to be an efficient approach since it allows to solve magnetostatic and magnetodynamic problems with complex geometries. When the design of the machine is almost invariant along its depth, the use of a 2D FEM model is often recommended since it allows to consider only a decent number of unknowns. For more complex geometries however, a 3D FEM model may be required which leads to a huge computational cost. One may also mention the use of analytical or hybrid 2D/3D FEM models for specific designs which provide a significant speedup [1, 2]. The modeling of electrical machines beside the solution of partial differential equations requires taking into account the nonlinearities of the materials, the motion of the rotor and the coupling with electrical circuit equations.

In order to model the motion of the rotor, several methods have been developed in order to avoid any re-meshing process which can be computationally prohibitive. When the rotation speed is constant, the locked-step method is very powerful since the motion is considered by only permuting the unknowns at the surface of the rotor. When the rotation speed varies however, techniques like the moving band, the mortar FEM, or the overlapping FEM should be used [3].

To take into account the electrical environment of the machine, circuit equations should be coupled with a FEM model [4]. This approach leads however to a transient problem governed by a system of differential algebraic equations. Therefore, the use of an implicit time-stepping scheme, like the Backward Euler, is required in order to achieve stability. Moreover, mechanical equations can be coupled to the FEM model, usually in a *weak* sense by considering the different constant times with respect to the different equations [5, 6]. Therefore, the time-step of the numerical model must be chosen very small compared to the length of the transient state duration, especially when one uses circuit and mechanical couplings. In fact, the time-step is determined with respect to the smallest time constant which corresponds to the electric time constant whereas the time interval of the simulation is linked to the time constant associated with the mechanic equation. The ratio between the electrical and the mechanical time constants can be of several hundred leading to the need of a huge number of time-steps to simulate the transient state.

Finally, this approach enables to deal with nonlinear ferromagnetic materials. This point is relevant since most of the machines have their operating point located into the nonlinear area [4]. Therefore, a numerical scheme such as the fixed point method or the Newton-Raphson have to be implemented in order to convert a nonlinear problem into a sequence of linear problems.

Hence, the study of an electric machine requires to solve a nonlinear large-scale system for a high number of time-steps. All those points contribute to make this problem very expensive in terms of computational resources. Moreover, the aim of industrial applications often consists in reaching the steady state, which is the operating state of the electrical machine.

To overcome this issue, projection-based model order reduction (PMOR) appears to be a very interesting tool since it allows to dramatically reduce the number of unknowns. This approach consists in performing a projection of the original problem onto a reduced basis. This leads to a reduced system of equations with much less unknowns. Two subclasses of methods may then be distinguished: the *a priori* and the *a posteriori* ones.

With *a priori* approaches, the reduced basis is not known before the simulation: it is iteratively built from scratch. Then, the reduced solution is computed at each iteration until the approximated solution converges to the full model solution. The proper generalized decomposition is maybe the most famous *a priori* PMOR method and has successfully been applied to a large class of engineering problems[7–9]. Moreover, this method looks for a solution with a separable representation, allowing it to deal efficiently with multiparameter problems.

As for the *a posteriori* methods, the reduced basis onto which the problem is projected, is determined before the simulation. Several procedures allow to generate this basis. To name but a few, the Arnoldi–Lanczos which proceeds by solving the problem in the harmonic domain while the proper orthogonal decomposition (POD) combined with the snapshot method requires well-chosen solution of the full model [10, 11]. Those two methods have often been used to study problems in engineering in very different fields such as interconnected circuits [12] or for modeling aerodynamic problems [13]. Finally, the reduced basis (RB) method proposes a complete reduction framework which allows to build efficiently a reduced basis and an error estimator, through a greedy algorithm. The RB method is also particularly adapted to multiparameter problems.

All those methods have proved to be extremely robust and efficient for linear problems. However, applying them in the nonlinear case appears to be much more challenging. For instance, the Arnoldi–Lanczos procedure which relies on harmonic solutions cannot be transposed easily to the nonlinear case since it requires nonlinear multiharmonic solver. In mechanics, the PGD has been successfully applied to nonlinear problems when coupled with the LATIN method (LArge Time INcrement method) [14]. Recently, it has been applied in electromagnetic to solve nonlinear magnetostatic problems [15]. Until now however, taking into account the motion with the PGD remains an open problem preventing the modeling of an electrical machine. Nevertheless, the POD allows to deal quite easily with nonlinearities but requires additional calls to the full model, cancelling out partially its advantages [16]. However, the POD approach can be combined with the (Discrete) empirical interpolation method (DEIM) which is an interesting way to avoid the calls to the full model [17, 18]. The POD–DEIM has been successfully applied in many fields of engineering problems such as electrical networks [19], fluid mechanics [20] or reservoir simulations for petroleum industry [21].

Although the PMOR approaches are quite recent in the field of electromagnetics, they have already been successfully applied on electromagnetic devices such as electrical machines in the nonlinear case without considering both electrical and mechanical coupling simultaneously [6], conducting plates [22] or nonlinear three-phased transformers [23].

In this article, a POD–DEIM approach is developed in order to study a synchronous machine. The numerical model takes into account the nonlinear behaviour of ferromagnetic materials, circuit and mechanical equations. This enables to study the start-up of a machine until it reaches the steady state. To build an efficient reduced model valid on the whole operating range of the electrical machine, an empirical “offline/online” approach is used. This is based on the typical tests of electrical devices (at no load and in short-circuit) [23].

First, the numerical model of the nonlinear magnetostatic problem based on the vector potential formulation coupled with electrical and mechanical equations is presented. To take into account the motion of the rotor, the overlapping finite element method [24] is introduced.

Secondly, the proper orthogonal decomposition approach, which allows to project the full model in a reduced-basis of small size, is combined with the (Discrete) empirical interpolation method in order to compute nonlinear terms efficiently. Furthermore, an empirical offline/online approach based on *a posteriori* electrical engineering knowledge is presented.

Finally, a synchronous machine is studied with the reduced model which will be compared to the full model in terms of accuracy and calculation time.

## Numerical modeling of an electrical machine

### Nonlinear magnetostatic field problem

Let us consider a nonlinear magnetostatic field problem in a domain *D* of boundary \(\Gamma \) composed of a synchronous machine. We assume that the domain *D* is divided into two parts: a static subdomain and a rotating subdomain \({D}_{\theta }\) of boundary \(\Gamma _\theta \). The nonlinear behaviour of the ferromagnetic materials of the rotor and the stator is considered. The domain \(D_{NL}\) of boundary \(\Gamma _{NL}\) such that \(\Gamma _\theta \cap \Gamma _{NL} = \varnothing \) gathers all the region with non linear materials. Four stranded inductors are considered. Only one inductor belongs to the subdomain \({D}_{\theta }\) and the others to \(D {\setminus } D_{\theta }\). We denote by \(i_j\) the current associated with the coil of the *j*th phase and \(i_0\) the current flowing into the inductor of the rotor.

*j*th stranded inductor.

*D*is discretized using a 2D mesh composed of

*N*nodes. In the 2D case, the vector potential \(\varvec{A}\) is assumed to be orthogonal to the plane of the study of normal \(\varvec{n}_\perp \): \(\varvec{A}(\varvec{x}) = A(\varvec{x}) \varvec{n}_{\perp }\). Thus, \(\varvec{A}\) is expressed as linear combination of nodal shape functions (Fig. 1). Let \(X_{A,k}\) denote the value of \(A (\varvec{x}) \) on the

*k*th node. We introduce \(\varvec{X}_A\) the vector of size

*N*of entries \((X_{A,k})_{k = 1 \ldots N}\). Then, applying the Galerkin method to solve (8) leads to the following non linear system of equations.

*N*.

### Motion through the overlapping method

Basically, the overlapping method consists in computing interactions between \(D_{\theta }\) and \(D {\setminus } D_{\theta }\) inside \(D_r\) [24, 25]. This can be done by extending the finite element functions of each domain onto \(D_r\) by ensuring continuity properties. Since \(D_r\) is a non-meshed domain, no extra unknowns are added to the problem.

### Circuit and mechanical coupling

The coupling of the rotating nonlinear magnetostatic problem with circuit and mechanical equations are studied in this section.

#### Circuit coupling

*R*and inductances

*L*. In these conditions, the currents \(\{ i_k ~,~ k = 1 , \ldots , 3 \}\) are three new unknowns of the problem. Then, circuit equations must be added to the initial problem:

*k*th inductor. The magnetic fluxes are express as a function of the vector potential such that:

*t*. Then, by introducing \(\varvec{X}\) the new unknown vector such as \( \varvec{X} = [ \varvec{X}_A(t); i_1(t); i_2(t); i_3(t)]\), the numerical model can be written from (10)–(12) as the following system of differential algebraic equations:

#### Mechanical coupling

*J*the inertial momentum of the rotor,

*f*a friction constant and \(T_{Mech}\) the load. The computation of the electromagnetic torque \(\varvec{T}_{EM}(\varvec{A})\) derives from the virtual work principle. It can be written as a quadratic function of \(\varvec{A}\) [26]. In a discrete setting, one can write:

*N*. Applying an explicit Euler scheme on (15) and using (16) leads to the discrete mechanical equation:

### Numerical solution

*k*th time-step, the residual vector \(\varvec{R}^k\) of size \(N+3\) is defined such that:

*k*th time-step is:

*j*th linear problem from the NR method reads:

## Model order reduction methods

The nonlinear coupled problem (14)–(17) has a high computational cost. Indeed, solving the nonlinear equation (14) requires a high number of unknowns and several Newton-Raphson iterations for each time-step. Moreover, adding the mechanical equation (17) to the problem makes the simulation time interval much larger versus the time-step \(\tau \) defined from the smallest time constant \(\tau _{Elec}\) related to the electrical behaviour of the rotating machine. Moreover, the electromagnetic torque \(T_{EM}\) has a very high oscillating frequency. Therefore, we propose to apply a MOR approach to the problem (14)–(17) through the proper orthogonal decomposition combined with the (Discrete) empirical interpolation method.

### Proper orthogonal decomposition

The POD approach allows to significantly reduce the number of unknowns of the equation system. Indeed, the POD belongs to the projection model order reduction methods which consist in seeking an approximation of the large-scale solution \(\varvec{X}_r\) into a small reduced basis \(\varvec{\Psi }\).

#### Projection-based model order reduction

*m*, with \(m << N\). \(\varvec{\Psi }\), the reduced basis, is a rectangular matrix in \(\mathbb {R}^{(N+3)\times m}\).

*m*reads:

#### Determination of the reduced basis through the proper orthogonal decomposition

The proper orthogonal decomposition, introduced in 1967 for fluid mechanic applications [28], is one of the most used model order reduction method [10, 11]. This approach requires snapshots, i.e. solutions of the problem for different parameter values, in order to build a reduced basis. Those snapshots can be computed numerically or extracted from experimental data.

*l*snapshots \(\varvec{S} = \{ \varvec{X}_A^1, \varvec{X}_A^2, \ldots , \varvec{X}_A^l \} \in \mathbb {R}^{N \times l}\), one has to perform upon it a singular value decomposition (SVD). Let

*r*be the rank of the matrix \(\varvec{S}\) with \(r < l\) and \(r < N\), then the SVD of the snapshot matrix is written:

*r*which contains the so-called singular values \(\{ \sigma _i, ~ i = 1 \ldots r \}\) such that \( \sigma _1 > \sigma _2 > \cdots > \sigma _r > 0\). Thus, the reduced basis which allows to approximate \(\varvec{X}_A\) is \(\varvec{\Psi }_A = \varvec{U}_1\).

#### Truncation of the reduced basis

Truncation plays an essential part on methods like the POD which are based on the SVD. Indeed, we have seen that the SVD allows to find the rank *r* of the snapshot matrix \(\varvec{S}\) in (25). However for numerical problems, the snapshot matrix \(\varvec{S}\) is likely to be full rank since its singular values are never equal to the spurious zero because of the numerical noise.

Therefore, it is often advised to truncate the basis, i.e. to reduce the size of the reduced basis, according to a truncation criterion. In the literature, several methods have been proposed.

*p*first modes amongst

*l*which correspond to singular values larger than a user-defined small constant \(\epsilon \):

*r*. However, due to numerical computations and noise, this equality does not hold and \(\varvec{M}_{\perp }\) is not diagonal:

*p*modes but produce extra-diagonal terms for the last modes:

*less orthogonal*its corresponding mode is, which implies that the less relevant the mode is. Therefore, we propose to use the following truncation approach, based on the value of:

#### Structure preserving approach

*N*, only the unknowns \(\varvec{X}_A\) are approximated in a reduced basis. Therefore, the full reduced basis \(\varvec{\Psi }\) from (21) is written:

### (Discrete) Empirical interpolation method

Although the number of unknowns has been dramatically reduced though the POD approach, the computational cost of the nonlinear term \(\varvec{G}_{NL,r}(\varvec{\Psi } \varvec{X}_r^k)\) remains prohibitive. Indeed, one has to express first this quantity in the full basis, and secondly, project it onto the reduced basis. Therefore, we propose to apply the (Discrete) empirical interpolation method which allows to compute some nonlinear terms only on DEIM-chosen points, and then to interpolate the remaining entries of \(\varvec{G}_{NL}\).

## Applications

A 2D model of a nonlinear synchronous generator is studied. The geometry and the mesh of the domain are presented in the Fig. 1. The rotor is driven by a constant torque \(\varvec{T}_{Mech}\) and is supplied with a direct current \(i_0\). Moreover, the motion of the rotor is taken into account through the mechanical coupling (17) and the overlapping finite element method. Since the time constant of the mechanical equation (17) is much larger than the one arising from the circuit equation (11), a high number of time-steps is required to reach the steady state. In our example, we have \(n_t = 10^4\) time-steps. The 2D mesh of this machine is made of 23,492 triangles and 12,097 nodes.

### Model order reduction strategy

In order to build an accurate reduced system, a collection of snapshots has to be chosen in such a way that they contain most of the different behaviours of the system. For our problem, we impose that at least, the snapshots collect a full mechanical period so that the reduced basis contains an entire rotation. Thus, applying a “direct” reduction approach where one collects the snapshots during the first *m* time-steps and then begin the reduced scheme at the \((m+1)\)th time-step may lead to a high computational cost. Indeed, in our settings, a full mechanical period requires approximately 2000 time-steps.

Moreover, applying a greedy procedure to construct *m* basis vectors over the full time interval would require solving *m* times the reduced system on every time-step, which is also very resource-demanding.

However, in the field of electrical engineering, a synchronous machine is easily characterized through two simple tests procedures: a no-load and a short-circuit test. During those tests, there are no more mechanical coupling: the rotation speed is set to a constant value \(\Omega _0\). Thus, the electromagnetic torque \(T_{EM}\) of high oscillating frequency is no more related to the problem, and allows to take a quite important time-step in order to compute a full mechanical rotation with a reduced number of snapshots (in our case 122). Furthermore, the short-circuit test considers the case where both the value of *R* and *L* is zero in the circuit equation (11) whereas the no-load case consists in not taking into account the circuit coupling. Therefore, the only parameter left in those two test cases is the rotation angle \(\theta \).

### Offline procedure

The \(2 \times 122=244\) snapshots obtained from the two tests procedures are concatenated in order to build the two reduced basis \(\varvec{\Psi }\) and \(\varvec{\Psi }_{NL}\). The truncation has been applied according to the orthogonality criterion (“Truncation of the reduced basis” section) with \(\epsilon = 10^{-7}\). Finally, the two reduced basis \(\varvec{\Psi }\) and \(\varvec{\Psi }_{NL}\) are respectively of size 71 and 160.

Figure 7 shows the interpolation points selected by the (D)EIM algorithm. One can see that the (D)EIM points are located around the air gap for the stator and the rotor. This is relevant since the magnetic field magnitude varies much more on the surface of the rotor and also on the stator tooth tips than inside the rotor and in the stator yoke and teeth.

### Online procedure

Once the reduced system has been built, the online computation involving the mechanical and the circuit coupling is performed by applying a constant torque \(T_{Mech} = 47 N.m\) in (15) on \(10^4\) time-steps.

#### Verification

#### Adaptivity of the reduced basis

*R*and

*L*in the circuit equation (11). The evolutions of the currents obtained from the reduced model and the full model associated with the stranded inductors of the stator are presented on Figs. 9, 10, 11, 12 and 13. We can see that the currents from the reduced model are close to the references. This implies that the reduced system is dynamic in the sense that it adapts to values of

*R*and

*L*for which no snapshots were computed. By taking into account the snapshots computation, the speedup obtained with model order reduction is approximately 35.

## Conclusions

The POD approach has been combined with the (D)EIM in order to study an electrical machine. The numerical model is based on a nonlinear magnetostatic problem through the vector potential formulation coupled with circuit and mechanical equations. The nonlinear problem has been solved by using the Newton-Raphson approach and the motion of the rotor with the overlapping finite element method. In order to obtain an efficient reduced model on the whole operating range of electrical machines, an empirical “Offline/Online” approach has been developed. With the proposed approach, the start-up of a synchronous machine until it reaches the steady state has been studied. On this example, the reduced model appears to be more efficient with respect to the computation cost than the reference model especially when the time-step is very small. In terms of accuracy, the global quantities can be approximated with a low number of unknowns and thus, the computation time is significantly reduced. Our future works will focus on POD-(D)EIM error estimators in order to build robust and adaptive reduced systems of electrical machines.

## Declarations

### Acknowledgments

The four authors have developed the theory part. LM has carried out the numerical implementation and has drafted the manuscript. TH, SC and BG have supervised the different studies and the corrections of the draft. All authors read and approved the final manuscript.

### Acknowledgements

None.

### Competing interests

The authors declare that they have no competing interests.

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## Authors’ Affiliations

## References

- Ho SL, Fu WN. A comprehensive approach to the solution of direct-coupled multislice model of skewed rotor induction motors using time-stepping eddy-current finite element method. IEEE Transact Magn. 1997;33(3):2265–73.View ArticleGoogle Scholar
- Demenko A, Dolinar D, Hameyer K, Nowak L, Zawirski K, Krebs G, Tounzi A, Piriou F, Pauwels B, Willemot D. Design and study of a linear actuator using an analytical approach and the 3D FEM. Int J Comput Math Electr Electr Eng. 2007;26(4):1005–16.View ArticleMATHGoogle Scholar
- Demenko Andrzej, Hameyer Kay, Nowak Lech, Zawirski Krzysztof, Shi Xiaodong, Le Menach Yvonnick, Ducreux Jean-Pierre, Piriou Francis. Comparison of slip surface and moving band techniques for modelling movement in 3D with FEM. Int J Comput Math Electr Electr Eng. 2006;25(1):17–30.View ArticleMATHGoogle Scholar
- Taibi Saoudi, Tounzi Abdelmounaïm, Piriou Francis. Study of a stator current excited vernier reluctance machine. Energy Convers IEEE Transact. 2006;21(4):823–31.View ArticleMATHGoogle Scholar
- Henrotte F, Nicolet A, Hedia H, Genon A, Legros W. Modelling of electromechanical relays taking into account movement and electric circuits. IEEE Transact Magn. 1994;30(5):3236–9.View ArticleGoogle Scholar
- Albunni MN, Rischmuller V, Fritzsche T, Lohmann B. Model-order reduction of moving nonlinear electromagnetic devices. IEEE Transact Magn. 2008;44(7):1822–9.View ArticleGoogle Scholar
- Henneron Thomas, Clénet Stéphane. Model order reduction of quasi-static problems based on POD and PGD approaches. Eur Phys J Appl Phys. 2013;64(02):24514.View ArticleGoogle Scholar
- Nouy Anthony. A priori model reduction through proper generalized decomposition for solving time-dependent partial differential equations. Comput Methods Appl Mech Eng. 2010;199(23):1603–26.MathSciNetView ArticleMATHGoogle Scholar
- Chinesta F, Leygue A, Bordeu F, Aguado JV, Cueto E, Gonzalez D, Alfaro I, Ammar A. PGD-based computational vademecum for efficient design, optimization and control. Arch Comput Methods Eng. 2013;20(1):31–59.MathSciNetView ArticleGoogle Scholar
- Anthanasios AC, Sorensen DC, Gugercin S. A survey of model reduction methods for large-scale systems. Contemp Math. 2001;280:193–220.MathSciNetView ArticleMATHGoogle Scholar
- Jolliffe I. Principal component analysis. Wiley; 2002.
- Odabasioglu A, Celik M, Pileggi LT. PRIMA: passive reduced-order interconnect macromodeling algorithm. In Proceedings of the 1997 IEEE/ACM international conference on Computer-aided design. IEEE Computer Society. 1997;58–65.
- Bui-Thanh T, Damodaran M, Willcox K. Proper orthogonal decomposition extensions for parametric applications in compressible aerodynamics. AIAA paper. 2013; 4213.
- Ladeveze P, Passieux JC, Neron D. The latin multiscale computational method and the proper generalized decomposition. Comput Methods Appl Mech Eng. 2010;199(21):1287–96.MathSciNetView ArticleMATHGoogle Scholar
- Henneron T, Clenet S. Application of the PGD and DEIM to solve a 3D non-linear magnetostatic problem coupled with the circuit equations. Magn IEEE Transact. 2015; 99.
- Galbally D, Fidkowski K, Willcox K, Ghattas O. Non-linear model reduction for uncertainty quantification in large-scale inverse problems. Int J Numer Methods Eng. 2010;81(12):1581–608.MathSciNetMATHGoogle Scholar
- Barrault M, Maday Y, Nguyen NC, Patera AT. An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus Mathematique. 2004;339(9):667–72.MathSciNetView ArticleMATHGoogle Scholar
- Chaturantabut S, Sorensen DC. Nonlinear model reduction via discrete empirical interpolation. SIAM J Sci Comput. 2010;32(5):2737–64.MathSciNetView ArticleMATHGoogle Scholar
- Hinze M, Kunkel M. Discrete empirical interpolation in POD model order reduction of drift-diffusion equations in electrical networks. In scientific computing in electrical engineering SCEE 2010. Springer; 2012. p. 423–31.
- Ştefănescu R, Navon Ionel Michael. POD/DEIM nonlinear model order reduction of an ADI implicit shallow water equations model. J Comput Phys. 2013;237:95–114.MathSciNetView ArticleMATHGoogle Scholar
- Ghasemi M, Yang Y, Gildin E, Efendiev Y, Calo VM. Fast multiscale reservoir simulations using pod-deim model reduction. In SPE reservoir simulation symposium, 2015.
- Pierquin A, Henneron T, Clénet S, Brisset S. Model-order reduction of magnetoquasi-static problems based on POD and Arnoldi-based Krylov methods. IEEE Transact Magn. 2015;51(3):1–4.View ArticleGoogle Scholar
- Henneron Thomas, Clenet Stephane. Model order reduction of non-linear magnetostatic problems based on POD and DEI methods. IEEE Transact Magn. 2014;50(2):33–6.View ArticleGoogle Scholar
- Tsukerman IA. Overlapping finite elements for problems with movement. IEEE Transact Magn. 1992;28(5):2247–9.View ArticleGoogle Scholar
- Krebs G, Henneron T, Clenet S, Le Bihan Y. Overlapping finite elements used to connect non-conforming meshes in 3-D with a vector potential formulation. IEEE Transact Magn. 2011;47(5):1218–21.View ArticleGoogle Scholar
- Ren Z, Razek A. Local force computation in deformable bodies using edge elements. IEEE Transact Magn. 1992;28(2):1212–5.View ArticleGoogle Scholar
- Vassent E, Meunier G, Foggia A, Reyne G. Simulation of induction machine operation using a step by step finite element method coupled with circuits and mechanical equations. IEEE Transact Magn. 1991;27(6):5232–4.View ArticleGoogle Scholar
- Lumley J. The structure of inhomogeneous turbulent flows. Atm Turb Radio Wave Prop. 1967; 166–178.