By R. Belmans
Endwinding reactance
The endwinding inductance is a lumped parameter that cannot be computed by a 2D approach. It is also addressed as the endwinding leakage reactance, because the flux linked to this part of the winding is a leakage field. Theoretically, it is possible to compute the endwinding leakage from the difference between the measurements and the computed results using the flux linkage approach. The endwinding leakage inductance can be determined from the comparison of measurements and twodimensional finite element computation employing:
with 8 the measured andthe computed torque angle. As the endwinding leakage reactances are typically small compared to the main reactances (410%), this requires a high accuracy of the measured data. Furthermore, for the comparison of measurements with computed operating points, the excitation current, in particularare computed from the measured data, resulting in the analysis of a slightly different operating point. The measurement error is amplified.
Another possibility is the determination of the endwinding leakage inductance via the stored magnetic energy in a 3D model of the endwinding region (Fig. 10.5a). Taking advantage of symmetries in the machine, only one pole (or multiples of poles) have to be modelled. The inductance is determined from the stored magnetic energy by:
withthe number of poles included in the 3D analysis, 2p the number of poles of the machine. While the postprocessing is rather simple, the generation of the 3D model requires considerable interactive time and computer resources.
A linear, magneto static analysis can be performed, with only one coil system excited. The face of the nonmodelled iron core towards the end section of the machine is constrained in order to generate the effect of infinite permeability of the iron core (Fig. 10.5b) and to reduce the model size. The remaining boundaries are symmetry boundaries, and thus the flux is enclosed. This is an imperfect modelling of the shielding effect of induced currents in the aluminium or cast iron frame of the machine. Furthermore, the accuracy depends on the quality of the discretisation.
Induction motor
The finite element model used for the induction motor analysis is usually twodimensional (Fig. 10.6) and describes a part of the crosssection of the motor. Threedimensional models are applied to compute particular details of the machine such as the endwinding parameters.
Fig. 10.5. a) FEM mesh of the endwinding of the studied machine with a double layer, chorded winding and b) schematic of the applied boundary conditions using a 2D field solution on a planar cut in the axial direction.
Fig. 10.6. Cross section, winding layout and 2D mesh of the studied 4pole induction motor.
Nonlinear timeharmonic problem It is essential for the induction motor analysis to consider both effects, saturation and eddy currents, simultaneously. This requires a computationally expensive transient solution or can be approximated using a timeharmonic solution in combination with a modified magnetisation curve (method ot ettective reluctivities) or a combination of static and timeharmonic solutions.
Effective reluctivity
In this method, a timeharmonic solution is recommended. To consider the nonlinear characteristic of the ferromagnetic material, the reluctivities are adapted in an iterative process of successive timeharmonic solutions. Using Newton iterations yields in a faster nonlinear timeharmonic solver; a relaxation approach is possible as well. Here, the reluctivities are adapted following the iteration scheme:
is the relaxation factor.
It is assumed that the magnetic field strength is sinusoidal varying To consider the AC excitation of the magnetic field and thus the time dependent reluctivities, instead of using the regular BH characteristic, the values of an effective magnetisation curve is chosen.
To calculate the effective characteristic it is assumed that the stored magnetic energy over one period of H must be equal to the energy by using the effective characteristic.
Another possibility for determining the effective magnetisation characteristic is to choose the average value over one period.
Combination of timeharmonic and static solution
An alternative approach to determine the element reluctivities for a nonlinear timeharmonic problem consists in employing a magneto static solution. With the static solution, the saturation of the operational point is approximated. To define the static problem, the exciting currents and voltages have to be imposed. Their values can be obtained by a timeharmonic solution equipped with reluctivities reached in the static computation. This approach represents an iterative process (Fig. 10.7).
Problem definition The timeharmonic problem is defined by:
withthe angular frequency of the problem. In an induction machine the stator currents vary with the stator frequencywhile the rotor currents vary with slip frequency:
Fig. 10.7. Combined static and timeharmonic solutions to consider nonlinear timeharmonic problems.
Slip frequency
Using imposed currents at slip frequency with given phase angles can define the problem
If a voltagedriven problem with rotor frequency has to be defined, some changes are recommended to obtain a correct solution. The terminal voltage has to be transferred by multiplying by the slip s and the conductivity of the stator winding by the factor Ms .
Stator frequency
Defining as a function of the stator frequency, the problem is:
The solutions obtained with rotor slip frequencyand conductivity deliver the same solution as the problem defined by the stator frequencyand transferred conductivityThe latter approach is preferable because only one parameter, the rotor conductivity, has to be changed. No additional changes are necessary. The rotor resistance increases by the decreased conductivity, but the rotor reactance increases with the same factor due to the higher frequency. The product of frequency and conductivity determines the induced currents. Losses in the rotor are a factor 1 Is considered too low.
External circuits
To form the shortcut loops of a rotor cage winding, a lumped parameter model representing the endring impedances (Fig. 10.8) has to be coupled to the 2D FEM model. Zn is the endbar impedance and Zr the impedance of the endring. The resulting external circuit equations are simultaneously solved with the field problem.
Fig. 10.8. Coupled FEM – external circuit rotor model.
The parameter of the external circuit must be calculated analytically before the field computation or computed by a numerical method.
Endring parameter During load operations, the endring resistance has the most significant influence on the motor behaviour. Although the resistance of a ringsegment J?R is much smaller than the bar resistance (about 1 % of the bar resistance), its influence can be up to 20 – 30 %.
The equivalent bar resistanceis the resistance to be used to include the Joulelosses in the ring when calculating the rotor losses due to the bar currents.is the number of rotor bars and p is the number of pole pairs. To calculate the ring resistanceof the entire ring:
with
is applied.is the inner diameter of the ring and t the thickness. Current redistribution in the endring is not considered. The endbar resistance is given by:
with S the crosssection If the lengthof the endbar. To describe the influence of the endring, the inductance:
is used. Withthe bar length,is the length of the iron lamination, is the pole pitch diameter in the middle of the ringfor
(10.29) expresses the influence of the endring per bar. To compare the analytical formula with the 2D and 3D calculations, expression (10.29) is multiplied by
to refer it to the ring, as done in (10.25) and multiplied by in order to obtain an expression for the ring inductance Lr:
The value of the ring inductance of the motor under consideration using (10.30) is
2D/3D computed endregion parameters Twodimensional FEM model For the 2D finite element calculation, a model is made of the axial crosssection of the motor endregion (Fig. 10,9). A 400 kW traction motor is modelled. The accurate modelling of the crosssection is difficult to obtain since it contains a number of different materials with different, often unknown and generally anisotropic magnetic properties. Also for the correct modelling of the boundary between bearings and frame or bearings and shaft, questions arise as to whether to model them as good or bad magnetically permeable. To overcome these questions, a number of models are built using different boundary conditions and materials in order to find the influence of each component of the endregion.
Fig. 10.9. Axial crosssection of a squirrelcage induction motor endregion.
Two extreme situations are considered, one where the boundary of the frame, bearing and shaft is considered to be a flux line (Dirichlet condition), the other where the boundary is considered perfectly magnetically permeable. The finite element problem is described as axissymmetric. A unit current is considered to flow in the endring. The problem is defined as timeharmonic, neglecting saturation. The following figures (Fig. 10.10, Fig. 10.11) show flux plots for some of the models considering the different boundary conditions and material Drooerties.
Fig. 10.10. Only the air around the endring is modelled and considered being a flux line.
The inductance values obtained from the computations Fig. 10.10 and Fig. 10.11 are 0.35When all boundaries are considered to be magnetically conducting, only the air surrounding the ring has to be modelled (Fig. 10.12).
Fig. 10.11. Ferromagnetic stator and rotor iron considered.
Fig. 10.12. Outline of the finite element model to study the magnetically conducting situation.
Linein Fig. 10.12 is constrained to be a flux line. It is necessary to have at least one piece of the boundary considered to be a flux line. If not, there is no physical interpretation for the problem. Examining the solutions of this model, a large variation in the inductance is found when the distance d is varied. The model requires d being equal to half the core length. Only then isab a realistic flux line. The inductance value obtained in this case is 13.05 (J.H. This is a large difference when compared to the other extreme situation where all material boundaries are considered to be a flux line (Fig. 10.10). Therefore, from the 2D approach it is not obvious which value for the ring inductance has to be used as lumped parameter in the coupled finite elementcircuit model.
The 2D approach has some additional drawbacks:
• The correct excitation of the ring via the bars can not be accounted for in a 2D axissymmetric approximation. Therefore, the inductance of the barends outside the iron core are not included in the calculations.
• By exciting the ring in an axissymmetric problem definition, a flux through the shaft is introduced. Since only the leakage part of the ringinductance is needed, the coupling with the stator endwinding has to be considered. It is obvious that all flux through the frame links both endring and endwindings. Therefore, it is part of the mutual inductance and not part of the leakage components.
To conclude, separating the mutual and leakage components can be performed only using a threedimensional model of the details in the endregion.
Threedimensional FEM model
Fig. 10.13. Threedimensional material mesh generated by extrusion in axial direction.
The 3D model is built using an extrusionbased mesh generator. Due to symmetry, only one fourth of one endregion has to be modelled. The 3D model consists of a material mesh and a set of coil meshes required for the currents. Both meshes are generated separately allowing a different extrusion direction for material mesh and coil meshes. It can be noticed that a part of the iron core is modelled as well (Fig. 10.13).
The stator endwinding is not incorporated in the material mesh. The endwinding is modelled as a set of currentdriven coils in air. This is feasible since current redistribution due to skin effect is negligible in the stranded stator endwinding. In the endring and the rotorbars, skin effect cannot be neglected. Therefore, they are considered in the material mesh. Fig. 10.14 shows part of the material mesh (endring and barends).
Referring to Fig. 10.14, it is obvious that the generation of the stator endwinding coils requires a more complex extrusion procedure when compared to the modelling of the material mesh. Therefore, the building of such complex models using extrusion techniques is only possible if material and coil meshes can be built separately. Because the statorend winding is not modelled in the material mesh, the coil meshes have to represent the actual endwinding geometry as accurately as possible. This is not required for the coil meshes used for the excitation of the endring and barends. Because the endring and barends are modelled in the material mesh, the coil meshes for exciting them only have to be inside the materials and to provide a path for the current to flow. The current occupies the full material available considering skin effects.
Fig. 10.14. Material mesh of endring and barends and coil meshes of the stator endwinding.
Both stator and rotor coils are defined as current driven. In the 3D model, 9 full ringsegments and 2 halfsegments are present (Fig. 10.13, Fig. 10.14). Therefore, 11 rotor coils are required for the current excitation. Twentytwo currentdriven coils represents the stator winding. Only 2 endwinding coils are completely inside the model; the other 20 coils are cut off at the boundaries of the model (Fig. 10.14). Fig. 10,15 shows two of the coils used for the endwinding excitation, the two coils which are completely inside the model (coil 1 and coil 2), together with three other coils which are cut off at the boundary of the model. When referring to the crosssection of Fig. 10.13, coil 1 occupies the upper half of the first stator slot (the slot in the upper left corner) and the lower half of slot 11; coil 2 occupies the upper half of the second slot and the lower half of slot twelve. In the real motor, each of the stator coils contains four turns. The currents for both rotor and stator coils are obtained from a twodimensional finite element analysis.
Five inductances can be calculated:
endring inductance 

endwinding inductance 

mutual inductance between endring and endwinding 

endring leakage inductance 

endwinding leakage inductance 
Fig. 10.15. Coil meshes representing the endwinding model.
The inductances are calculated based on the stored energy in the model. To determine both leakage components and the mutual inductance, three problems have to be solved: one having only the stator coils excited, one having only the rotor coils excited and one with both stator and rotor coils excited. Considering the stored energy in these problems to bethe following equations can be given:
where
Since the rotor values are referred to the stator,The valueis less than(the negative sign has to be applied in the expression forif the flux caused by the rotor excitations opposes the stator flux. If the rotor flux supports the stator flux,is larger than Under regular conditions, the rotor flux opposes the stator flux, resulting inThe division by eight or four in (10.31) considers that only one fourth of the endregion or one fourth of the endring is modelled.
In the case where only the mutual inductance has to be calculated, it is sufficient to solve two problems, the first with the rotor flux opposing the stator flux, and the second with the rotor flux supporting the stator flux. The difference of both stored energy values is only a function of the mutual inductance and the applied currents. Introducing the endring leakage referred to the stator,the different inductances can be calculated by:
Because the threedimensional calculations performed are assumed to be linear, the number of calculations is reduced. The calculations with only the rotor or only the statorcoils excited, are performed only once with a unit current. From both calculations,are obtained. Using (10.31), the values for the stored energyfor the actual currentsare obtained. Therefore, for each studied slip value, only one additional calculation is required with both rotor and stator coils excited.
Some results of the calculations of the different inductance components are collected in Table 10.1. It can be noticed that both mutual and leakage components are strongly slip dependent. It can be stated thatduring noload operation, the full stator endwinding inductancerepresents a leakage reactance. Therefore, the endwinding leakage inductancevaries from 0.49 mH (at noload) to 0.28 mH (at standstill). The mutual inductance increases when the slip increases; the endwinding leakage inductance and the endring leakage inductance decrease with the same amount. Furthermore, the endring leakage referred to the stator,is found to be of the same value as the end
winding leakage inductanceThis already is an indication that the influence of theend ring leakage is not negligible at all load situations.
Table 10.1. Calculated inductance components for different slip values.
inductance 

slip 

[mH] 
IK] 

0.34 
1.42 
100 

0.49 
0.49 
0.49 

0.46 
0.46 
0.46 

0.13 
0.19 
0.21 

0.33 
0.30 
0.25 

0.36 
0.30 
0.28 
To predict the behaviour of the induction machine accurately, particular attention to the calculation of the resistance of the endregion of the squirrelcage is recommended. It was assumed to have uniformly distributed current in the endbar and endring (10.26). Computations using a 3D model result in a higher accuracy, and point out that the assumption of a uniformly distributed current is not valid (Fig. 10.16). It can be noticed that eddy current effects are present in the parts of the bars located inside the iron core and that a nonuniform current density distribution is computed in the endrings.
Fig. 10.16. Current density distribution in the endring region at startup.