induction motor rotor impedances S. Williamson/ M.A

A finite element based method for calculating cage

induction motor rotor impedances

S. Williamson/ M.A. Mueller," C.I. McClay,* D.R. Gersh*

"Department of Engineering, University of Cambridge, Cambridge,

UK

* Brook Hanson, St. Thomas' Road, Huddersfield, West Yorkshire,

1 Introduction

The per-phase equivalent circuit of the induction motor is employed univer-sally by industrial motor designers. Design programs routinely calculate theequivalent circuit parameters, and estimate performance using equivalent-circuit calculations. A corollary of this is that designers tend to think ofdesign changes in terms of the effect that they will have on the equivalent cir-cuit components with their effect on performance being consequential. Theaccurate calculation of those circuit components therefore assumes someimportance.

The components themselves are highly non-linear, varying due to both ironsaturation and skin effect in the embedded conductors. Their calculationcan be facilitated by the use of finite-element analysis, provided that analysismeets the general requirements of the design office. It must be transpar-ent to the user and must run very quickly on a workstation or personalcomputer. The former requirement arises because of the already-existingpressures on the designer, who will be skilled in the art of machine design,but probably not have the time (or inclination!) to drive complex generalpurpose FEA packages. The latter requirement arises because the designermay wish to make several design changes and review performance as partof the usual design procedure. The purpose of this paper is to describe howa FEA-based impedance calculation method, proposed by Williamson et al[1] [2] has been used to develop design-office oriented software which enablesrotor parameters to be calculated on-line or loaded into a look-up table foron-line use.

Transactions on Engineering Sciences vol 11, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533

328 Software for Electrical Engineering

2 Model

The basic model for calculating rotor impedances is shown in Fig. 1. Itconsists of a sector of the rotor, spanning one rotor slot pitch, together withthe associated portions of the shaft and the airgap. AD in Fig. 1 is theboundary between the shaft and the rotor laminations; BC is the inside ofthe stator surface, which is both smooth and infinitely permeable. Statorslotting is allowed for in the model by increasing the airgap length usingCarter's coefficient for the stator.

Rotor slot

Shaft

Figure 1: Single-slot rotor model

The presence of the rest of the rotor, together with its current distribution,is implied by enforcing an appropriate periodic boundary condition on theradial boundaries, AB and CD. If X and X' are two nodes which occupycorresponding positions on these radial boundaries, then their potentialsare related by

Xx = &'e (1)

where p is the number of pole-pairs and 7V& the number of rotor bars. The useof complex potentials implies sinusoidal time variation of electromagneticquantities, therefore magnetic saturation must be dealt with in a time-averaged sense. This is handled by using an ac reluctivity, which is derived


Software for Electrical Engineering 329

directly from dc magnetism characteristics, following the recommendationsof Luomi [3]. The solution begins by specifying the bar current /, and therotor (slip) frequency a;. The governing differential equation is

in which J<, is the mean bar current density, u is the bar conductivity, andAO is the mean value of A over the bar cross-section. The right-hand sideof eqn (2) permits the specified bar current to redistribute over the barcross-section under the influence of deep-bar effect. Outside the bar, theright-hand side of eqn (2) is zero, except in the shaft where it becomesjujviA, where a\ is the conductivity of the shaft material. Alternatively, itmay be assumed that no flux penetrates the shaft, in which case the curvedboundary AD in Fig.l becomes a flux line (Dirichlet) boundary.

In a real machine the rotor mmf is opposed by the stator mmf and it isnecessary to represent this in the model, to avoid excessive main flux-pathsaturation. This is achieved by placing an infinitely thin current sheet onthe stator surface (ie boundary BC in Fig.l) and iteratively adjusting themagnitude and phase of that sheet until the resulting airgap flux wave isjust sufficient to sustain the assumed rotor current against the calculatedrotor impedance. Using the flux-cutting rule

g/w- = 7(A6+jwZ,6) (3)

where / is the axial length of the machine and d is the diameter at which thefundamental flux density B is calculated. Rb and I/& are the effective barresistance and leakage reactance, allowing for the presence of the end-rings.It is therefore necessary to begin with approximately calculated values forRb and L&. Equation (3) is used to determine B, and the stator surfacecurrent density distribution adjusted until that value of B is obtained. Theelemental reluctivities are then 'frozen' (equivalent to linearising at a point)and the solution repeated with only the rotor bar current present. This lastsolution enables the bar impedance to be re-estimated, following the meth-ods described in references [1] and [2]. The new estimate of bar impedanceresults in an up-dated value for B (via eqn (3)) so that the stator surfacecurrent density must again be updated, and so on. This whole procedure isrepeated until convergence is achieved, usually judged in terms of % and



3 Industrial Implementation

3.1 Data Input and Mesh Generation

The performance of a cage induction motor is commonly specified in terms ofcertain key performance indicators, such as locked-rotor torque and current,pull-up and pull-out torque and full-load power factor and efficiency. Allof these indicators are influenced by rotor design, but some are particularlydependent on the shape of the rotor slots. This is because some slot shapestake particular advantage of deep-bar effect to produce high starting torquesand low starting currents. The designer will select one of these when thecustomer's specification calls for it. Each manufacturer of induction motorshas a limited repertoire of rotor slot shapes, which may be considered toform a 'pattern book'. A typical entry in this pattern book might be theBoucherot slot shown in Fig. 2.

In that figure all of the key dimensions are specified in terms of algebraicvariables. If the designer wishes to use this slot it will be necessary to choosenumerical values for these variables. For example, the set of values given inTable 1 will produce the slot shown in Fig. 3.

Table 1: Boucherot slot dimensions

dOdld2d3d4d5

218 mm0.5 mm5 mm2 mm22 mm80 mm

awlw2w3w4

7.5°5 mm

1.2 mm2.3 mm2.3 mm

The 'pattern book' approach holds the key to automatic mesh generation.Each slot shape in the book is assigned a subroutine that fits a primitivemesh to create a rudimentary single-slot model. The co-ordinates of thenodes in the primitive mesh are expressed in terms of the same algebraicvariables that are used to specify the slot (in Fig. 2) together with cer-tain others that can be taken from the usual machine data (airgap length,Carter's coefficient for the stator). When the designer inputs the numericaldata for all of the machine design variables, the primitive mesh is generatedautomatically to suit that data set. The primitive mesh for a Boucherotslot having the dimensions specified in Table 1, is shown in Fig. 4a.

The primitive mesh is sufficient to define the shape of the rotor slot, butis not suitable for an accurate field solution. The slot bridge region, inparticular, experiences very high saturation due to rotor leakage flux andtherefore requires a fine mesh, as does the top of the slot itself. Fig. 4b



Figure 2: Pattern book entry for Figure 3: Boucherot slot for dimen-Boucherot slot sions of Table 1

shows an expanded view of this part of Fig. 4a to illustrate the poor qualityof the primitive mesh. It is clearly necessary to refine this mesh. Theauthors accomplish this using an a posteriori technique based on the workof Biddlecombe et al [4] and Hahn et al [5]. To summarise the method: theprimitive mesh is used to obtain a non-linear field solution from which anestimate of the error in calculated stored energy is made for each element.The mesh in the region of those elements with highest error is then refinedusing a bisection algorithm proposed by Rivara [6]. This bisection is followedby a Delaunay triangulation to improve the quality of the elements. Thefield is then re-solved and the procedure repeated until the required accuracy



Figure 4a: Coarse mesh for Figure 4b: Expanded view of coarse meshBoucherot slot

Figure 5a: Refined mesh for Figure 5b: Expanded view of refined meshBoucherot slot



is obtained.

Two practical points are worth recording at this juncture. Firstly, when themesh is refined along a curved material boundary, care must be taken to en-sure that new boundary nodes are placed on the real (ie physical boundary)not on the most recent polygonal approximation to it. This point may beillustrated with respect to Fig. 5b, which is a close-up of the top of the slotin the refined mesh shown in Fig. 5a. Fig. 5a is the refined version of Fig.4a, and Fig. 5b the refined version of Fig. 4b. Comparison between theslot 'heads' in Fig. 4b and 5b shows that in Fig. 5b, the extra nodes havebeen placed on the circular arc that defines the head of the real slot, andnot along the straight lines that join the boundary nodes in Fig. 4b. Thesecond point is that the operating condition that is simulated in the fieldsolution used for this purpose should correspond to the maximum frequency(locked-rotor) condition, to ensure an adequate refinement of the mesh inthe skin-effect limited conductor at the top of the slot. Mesh refinement,like mesh generation, is arranged to be totally transparent to the user, whoonly needs to identify the slot type and to provide the necessary dimensionaldata.

3.2 Assimilation into a Design Program

The single-slot rotor model enables rotor bar resistance and leakage reac-tance to be determined for any slip frequency and rotor bar current. Italso enables rotor magnetising inductance to be determined. This is themagnetising inductance that would be obtained if the stator were infinitelypermeable (Williamson and Robinson [7]). The model is fast enough to useon-line with a modern workstation. In general, some means will be neededfor the calculation of the stator parameters of the equivalent circuit (Ri andXi) and the stator magnetising inductance. The latter is the magnetisinginductance that would be obtained if the rotor iron where infinitely per-meable. These component values, together with those calculated for therotor, give the complete equivalent circuit, which may then be solved todetermine the stator and rotor currents. If the rotor current obtained fromthis solution differs from that used in the single-slot model to determinethe rotor circuit components, then those components must be recalculatedusing this latest value for rotor current. The newly-calculated equivalentcircuit components are then reinserted into the equivalent circuit and thecurrents recalculated. This whole procedure is repeated until satisfactoryconvergence is achieved.

The means by which the equivalent circuit components for the stator arecalculated may be conventional or finite-element. A suitable finite-elementmodel for the stator. which uses an arc corresponding to a phase-band ofthe stator is described in reference [7].



Manufacturers of small machines generally employ a fixed range of lamina-tions, producing machines of different ratings by varying stack length, wind-ing design and end-ring dimensions. In these circumstances, some economymay be afforded by computing characteristic curves for each rotor lamina-tion. This is done by calculating the rotor resistance and leakage inductanceon a 'per bar per metre' basis, over a typical range of rotor frequencies (say0.1 Hz to 50 Hz) and mean current densities (say 0.5 Amm~^ to 20 Amm~^).Typical results are shown graphically in Fig. 6 and Fig. 7. Once generatedin the form of a look-up table, such characteristics may be used repeat-edly in conjunction with a suitable two-dimensional interpolation routine inalmost any induction motor design software.

This approach must be used with caution because inappropriate combina-tions of current and frequency will produce apparently spurious results. Tounderstand this, we return to eqn (3), which at low frequency approximatesto

o r / ,\Bu, = I— (4)

At low frequency and constant current, therefore, the airgap flux is inverselyproportional to the frequency. If w is therefore made very small, the cor-responding value of B will be very large, producing excessive saturationand alarmingly low values for the leakage inductance. The problem arisesbecause an unrealistic operating point has been demanded. In practice, ina real induction motor, the rotor bar current, /, will fall to zero as w getssmall, so that the airgap field does not rise. An unrealistically high no-loadrotor current could only be obtained with an unrealistically high airgapfield, and this is the situation that occurs when a low frequency and highcurrent is demanded.

To avoid this apparent anomaly, it is necessary only to reject combinationsof w and / that correspond to unfeasibly large airgap fields. This may con-veniently be checked using eqn(4) with B (the rms value of the fundamentalfield in the airgap) constrained to be no higher than, say, 0.8T.

4 Conclusions

The authors have described how the single-slot model for the rotor of a cageinduction motor may be used in a design environment. The key feature ofuser transparency is maintained by employing a 'pattern book' approachwhich requires slot-defining and meshing subroutines for each slot. Themodel may be incorporated for on-line use in a design package, or usedoff-line to generate look-up tables for repeated use.



le-03

le-03

8e-04 -

6e-04

4e-04 h

2e-04

0.5Amm ~5Amm""lOAmm"15Amm""20Amm~

0.0 10.0 20.0 30.0 40.0Slip frequency (Hz)

50.0

Figure 6: Typical variation in rotor bar resistance for a Boucherot slot

3e-05 \

2e-05

2e-05

le-05

0.5 Amm ~1 Amm2 Amm5 Amm"10 Amm15 Amm"20 Amm

10 20 30Slip frequency (Hz)

40 50

Figure 7: Typical variation in rotor bar inductance for a Boucherot slot



5 Acknowledgements

The authors are grateful for the support of Brook Hanson Ltd and GECAlsthom Large Machines Ltd.

References

[1] S. Williamson and M.C. Begg. Calculation of the bar resistance andleakage reactance of cage rotors with closed slots. IEE Proceedings,Pf.B, 132(3):125-132, May 1985.

[2] S. Williamson and M.J. Robinson. Calculation of bar resistance andleakage reactance of cage rotors. Proceedings of 3rd IEE Electrical Ma-chines and Drives Conference, London, pages 122-126, November 1987.

[3] J. Luomi, A. Niemenmaa, and A. Arkkio. On the use of effective reluctiv-ities in magnetic field analysis of induction motors fed from a sinusoidalvoltage source. International Conference Electrical Machines, Munich.,pages 706-709, 1986.

[4] C.S. Biddlecombe, J. Simkin, and C.W. Trowbridge. Error analysis infinite element models of electromagnetic fields. IEEE Transactions on

,s, 22(5) :81 1-813, 1986.

[5] S. Hahn, C. Calmels, G. Meunier, and J.L. Coulomb. A posteriori errorestimate for adaptive finite element generation. IEEE Transactions on

a, 24(1):315-317, 1988.

[6] M.C. Rivara. Algorithms for refining triangular grid suitable for adaptiveand multigrid techniques. International Journal for Numerical Methodsin Engineering, 20:745-756, April 1984.

[7] S. Williamson and M.J. Robinson. Calculation of cage induction motorequivalent circuit parameters using finite elements. IEE Proceedings,Pf.B, 138(5) :264-276, September 1991.


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induction motor rotor impedances S. Williamson/ M.A