IEEE Industry Applications Society Annual Meeting New Orleans, Louisiana, October 5-9, 1997

Design Optimisation of an Axially Laminatedl Synchronous Reluctance Motor

C.E. Coates, D. Platt and B.S.P. Perera Department of Electrical and Computer Engineering

University of Wollongong, NSW 2522 Australia Phone: +61 42 21 3065 Fax: +61 42 21 3236 E-mail: coco@st.elec.uow.edu.au

Abstract‘ - The rotor saliency of the axially laminated synchronous reluctance motor (SynRM) produces a square airgap flux density distribution. If a square current distribution interacts with this flux the SynRM generates its optimal torque. This paper extends previous work on the magnetic modelling of the SynRM. A design model is proposed which incorporates all rotor and stator dimensions. Saturation effects have also been included in the model. The key dimensions affecting SynRM performance are identified as rotor radius, rotor pole pitch, rotor steel : steel + fibre ratio, airgap, stator current density and stator slot opening. An iterative algorithm is determined for the design optimisation process. This algorithm is simple enough to be implemented in a spreadsheet environment.

The model is used to design a 5kW motor with an optimal torque / unit mass ratio. The designed motor has a nine phase concentrated winding to approximate the ideal square current excitation. Finite element analysis and a static load test on the constructed machine confirm the model’s predictions, In addition, optimal values for the key motor dimensions, over a range of motor sizes, are determined and discussed.

NOM EN C LATU R E

Average surface area of rotor laminations (m2) Average surface area of stator slot pitch (m2) Airgap flux density (T) Effective airgap (m) Rotor magnetic field intensity (Alm) Stator magnetic field intensity (Alm) Stator current density distribution (Nm) Motor stack length (m) Mean direct axis length (m)

279

Stator slot pitch (rn) Airgap radius (m) Quadrature axis reluctance per metre (ANVblm) Rotor steel : steel + fibre ratio Yoke depth (m) Angular displacernent from direct axis of rotor (rad) Rotor pole pitch (cad) Permeability of free space, 471 x IO- ’ (H/m) Quadrature axis flux (Wb)

I. INTRODUCTION

To date efforts to optimise the design of the axially laminated synchronous reluctance motor (SynRM) have generally focused on one or two key rotor variables [ I -41. This paper seeks to consider the entire magnetic circuit of the machine as a whole. All rotor and stator dimensions are included in the design model as it is acknowledged that they all can limit or enhance the machine’s performance. For example, the iron in the rotor must be matched to the iron in the stator teeth as they both carry similar magnetic flux.

The rotor saliency of the SynRM produces a ‘square’ airgap flux density distribution. If a square current distribution interacts with this flux the SynRM generates its optimal torque. Recent work has shown that adding a third harmonic component to the MMF distribution can raise the torque per RMS ampere of the machine [5 - 61. This idea is carried to its logical conclusion in the design model by assuming the motor is excited by an ‘ideal” square current distribution. In practical terms this requires the designed machine to have more than three phases.

0-7803-4067-1 /97/$10.00 0 1997 IEEE.

II. MAGNETIC MODELLING

The magnetic circuit design model was developed from previous work by Ciufo [7]. He determined an expression for quadrature axis reluctance, given the geometry of the machine. Significantly, the derivation of his expression assumes two mechanisms by which flux flows in the quadrature axis. One is through the rotor body and the other following a zigzag path across the airgaps. His final expression was

Ciufo uses this value to calculate the average airgap flux density distribution. His calculation assumes no quadrature axis flux passes out from the rotor pole sides and ignores saturation effects. However, his results are shown to closely match those predicted using finite element analysis.

Figure 1 shows a typical profile of the airgap flux density obtained using Ciufo’s model. In this instance a two-pole machine is assumed with sufficient excitation to cause saturation in the iron if it had been allowed for. The distribution can be thought of as an average flux density produced as a result of the direct axis excitation. At either end of the pole face the flux density rises or falls due to the flow of quadrature axis flux. Quadrature axis flux is concentrated here as this path offers lower magnetic reluctance than through the relatively large airgap at the pole sides.

1.6 1.4 1.2

- 1 t ~ 0.8

0.6 0.4 0.2

1

1 O L - 7 , I , I , , I I I I I I I

-1.4 -1.0 -0.7 -0.3 0.0 0.3 0.7 1.0 1.4

Angular displacement (rad)

Fig. 1. Typical airgap flux density distribution in a two-pole SynRM using Ciufo’s model.

A key objective of the design process is to fully utilise the machine’s iron at rated conditions. This means the iron in both the stator and rotor should be on the limit of saturation. For this reason, Ciufo’s model had to be extended to include saturation so it would form a valid basis for the design model. These calculations are included in Appendix A.

In summary, Figure 2 shows a typical average airgap flux density distribution obtained when iron saturation effects are included into the model. A two-pole machine under similar conditions to those in Figure 1 is assumed. Figures 2a and 2b show the direct and quadrature axis contributions to airgap flux density distribution, respectively. Significantly, when the components are combined in Figure 2c, the peak where quadrature axis flux previously added to direct axis flux is removed. This corresponds to the point where the rotor and stator iron is first driven into saturation

1 0.8 1

I E 0.6 $ 0.4

O.* 0 i -1.4 -1.1 -0.7 -0.1 0.3 0.8 1.2

Angular displacement (rad)

t

-0.6 -0.4 1 Angular displacement (rad)

0.4 Approximation /

o.2 0 L- -1.4 -1.1 -0.7 -0.1 0.3 0.8 1.2

Angular displacement (rad)

Fig. 2. Airgap flux density distributions with iron saturation effects with (a) direct axis excitation (b) quadrature axis excitation (c) combined direct and quadrature axis excitation.

The design model requires a simple approximation to the airgap flux density distribution so that machine torque performance can be quickly calculated. To this end, a piecewise linear approximation is suggested.

280

Figure 2c shows this approximation as applied to the calculated airgap flux density distribution. The approximation assumes the direct axis excitation sets up an average flux density across the pole face. All quadrature flux flows through the final stator tooth located at the end of the rotor pole face. Having established this simple approximation the torque produced by the machine can be calculated as the cross product of flux density and current.

Select rotor radius

Ill. DESIGN OPTlMlSATlON

/ /

1

The goal of the design optimisation process was to produce the largest torque / unit mass. (Mass was defined as the sum of the rotor and stator iron that makes up the magnetic circuit plus the stator windings. It did not include the motor frame or shaft.) To this end, the machine iron should be fully utilised at rated conditions as previously detailed.

Optimise steel : fibre ratio

Considering these requirements, some of the dimensions in the SynRM are dependent on others. For example, the ratio of the stator tooth width to tooth plus slot width should be equal to the ratio of rotor iron to iron plus fibre. This means that the rotor and stator tooth iron will reach saturation at the same point since they both carry the same flux. Other dependent dimensions are slot depth, yoke depth and stack length. Slot depth is set so that there is sufficient direct axis excitation to fully flux the rotor iron when the excitation current is equal to the allowable continuous current. Yoke depth is set to allow sufficient return path for the direct axis flux. Stack length is restricted by the allowable temperature rise in the windings due to resistive losses.

\ \

Thermal considerations play a very important part in the design model. The model assumes the heat generated by the resistive losses in the stator winding is primarily dissipated through natural convection from the exposed area of the windings at the stator ends. A value for the acceptable rise in winding temperature above ambient is chosen, given the insulation characteristics of the winding. This thermal limit effectively sets the continuous rating of the machine.

The key independent dimensions in the SynRM were identified as rotor pole pitch, rotor radius, rotor steel : steel + fibre ratio, airgap, stator winding current density and stator slot opening. These are the values that are determined by the optimisation algorithm. There were four other dimensions also identified as independent. They were rotor lamination thickness, number of poles, number of stator slots and stator tooth tip thickness. Values for these are selected on the basis of practical limitations in either the machine

Figure 3 shows in block diagram form the algorithm used to optimise the SynRM design. This process optimises the design for a given rotor radius. It consecutively considers, each key variable finding its optimal value in isolatilon. The algorithm repetitively cycles through all the key variables until the performance index (torque / unit mass) converges to an optimal value. While optimising each key dimension a subroutine is called that sets the dependent dimensions to appropriate values. The entire algorithm is simple enough to be implemented in a spreadsheet environment that supports macro routines.

i (,1..)

Set Jq = Jd 0 Set yoke depth to carry d-axis flux I

Set rotor length to

Compute torquelmass I

i"> Finish

Fig. 3. Block diagram of design optimisation algorithm.

IV. 5kW MACHINE

A four pole, 5kW motor was designed with an optimal torque / mass ratio of 2.98 Nm/kg. This promised a significant improvement over conventional induction motors in this size range. Typically, a 5kW induction motor has a torque / unnt mass ratio in the range of 1.2 to 1.8 Nm/kg. The desiigned motor has a nine phase concentrated winding to enable the appropriate MMF distributions to be generated. Appendix B contains a full description of the machine's dimensions.

construction or stator phase windings. 28 1

A finite element analysis was performed on the 5kW SynRM to confirm the design model’s predictions. Figure 4 shows the airgap flux density distribution of the machine, as predicted by the design model, compared to the finite element analysis results. The flux density distribution generated in the finite element analysis contains variations due to stator slot effects. However the design model does adequately represent the average airgap flux density values.

37 ~

35 - h

E 3 3 - 5 +j 31 -

29 -

27, I-

1.5 Firrte aMlysis resrlts

Deslgnmdel prediction

-$8 -0.6 -0.5 -0.3 -0.1 0.1 0.2 0.4 0.6 d6

Angular displacement (rad)

-0.5

Fig. 4. Airgap flux density distribution in 5kW machine

The problem of “cogging torque” has been previously noted in SynRM’s [ I ,2]. To determine the extent of this problem the torque variation was calculated, using finite element analysis, as the edge of the pole face moved across one tooth pitch. Figure 5 shows the variation in torque in this situation to be less than 3% which was considered to be m i t e acceptable. It is possible to reduce this further by appropriate variation of the phase currents with rotor position.

25 I r , , 0 1 2 3 4 5

Angular displacement (deg)

Fig. 5. Variation of SynRM torque over one tooth pitch

Figure 6 shows the sensitivity of the torque / mass ratio of the 5 kW SynRM to the key independent design variables as predicted by the design model. In

each instance the analysis assumes the other dependent dimensions remain appropriately sized with respect to the dimension being varied. The torque / mass ratio is most sensitive to rotor pole pitch, rotor steel : steel + fibre ratio and airgap.

V. RESULTS

Designs have been performed for a range of machine sizes (1 - 100kW). Figure 7 shows the optimum values of the key machine dimensions as the motor size is varied. Of particular interest is the rotor pole pitch and rotor steel : steel + fibre ratio. Results indicate advantages associated with large rotor pole pitches (approaching 180 electrical degrees) and steel : steel + fibre ratios between 0.45 - 0.47. These results differ to other published values which have indicated smaller pole pitches (120 electrical degree) and larger steel : steel + fibre ratios (0.6 to 0.7) to be desirable [I ,2,4].

The difference lies in the stator current distributions assumed for the optimisation. In a three phase machine sinusoidal current distributions are present and the edge regions of large pole faces are not fully utilised. Under these conditions it is necessary to put more iron in the rotor to maximise the direct axis flux in the useful central region of the pole face. By lifting the restriction of sinusoidal currents it has been possible to utilise the whole pole face creating better use of the machines iron.

Also of note is the optimal airgap, which is relatively small. For the 5kW motor the optimal airgap was determined to be 0.3”. This is approximately half of that typically encountered in a similarly sized induction motor. In an induction motor the rotor heats up due to the currents present in it. As a consequence, allowance for temperature rise is made in the choice of bearings. Given this restriction on the bearing type a lower limit on the possible airgap is set in the induction motor design. The SynRM rotor carries no current and is subject to no internal heating. Thus, the smaller airgaps proposed are achievable.

VI. CONCLUSIONS

A design model has been proposed for the SynRM. The model assumes square current distributions on the machine’s stator. This ensures that the machine’s torque production is optimal. It also leads to motor designs with large pole pitches, where a greater portion of the machine iron is utilised at rated conditions. The square current distributions are obtained by increasing the number of phases on the stator. In addition, it is shown that it is advantageous to have relatively small airgaps in an optimally designed SynRM. These airgaps are feasible since

282

-

I-" 0 1 " " I ' ' ' ' I -

1.30 1.34 1.38 1.42 1.46 1.50

Pole pitch (rad)

m .- '0 1.5 - i?

1.45 - Y 'E. 1.4 ~

2 3.2 -1

0.19 0.23 0.27 0.31 0.35 0.39

Airgap (mm)

0.35 0.39 0.43 0.47 0.51 0.55

Steel : steel + fibre

Fig. 6. Sensitivity of machine performance to design parameters

5 1.35 L---T---,- I , I I I I , , R o o o - f o o o o o o o o o

- 3 - L n " t - c O l n ~ ; ~ c 9 ~ ~

Rotor radius (mm)

0.012 2 0.01 s 0.008 0.006 0.004 .f 0

U) - - ~ ~ - i r i - r

bo 6" 2" 9" ,," ,!," ,%"

Rotor radius (mm)

4.30 4.50 4.70 4.90 5.10 5.30

Current density ( A h " 2 )

+ 0.47

1: 0.45

2 0.44 o

Rotor radius (mm)

, , , ,IyjI / ::::; - 0.002 g

0.001 F -I- 0 3

bo 6' "? 9" ,," ,$" ,$"

Rotor radius (mm)

t- -t+*f--+

Rotor radius (mm)

bQ 6Q 2% 9i ,%Q ,9Q *Q

Fig. 7. Optimum machine dimensions as motor size is varied

283

the SynRM rotor circulates no currents and is not subject to internal heating.

The validity of the design model has been demonstrated in the nine phase 5 kW machine design. In this instance finite element analysis and practical results have confirmed the model’s predictions.

APPENDIX A

SynRM MODELWITH SATURATION EFFECTS

Ciufo’s model of the SynRM can be modified as shown in figure 8 [7]. Differential equations that describe the airgap flux density and quadrature axis flux can be formed by considering MMF loops 1 and 3 and continuity of flux in areas 2 and 4.

L

Stator

I . _ . _ . _ . _ . _ _ _ . _ . _

Rotor

J’ J1

I - ’, L! 4

Fig. 8. SynRM model

These equations are;

MMF in loop 1:

+ m(e) + 4q (e)m, = o

Continuity of flux in area 2:

x-e Ir-e IJ(B)RdB = JH,(O)RdB e e

Continuity of flux into area 4:

Initial and boundary conditions:

(4)

(5)

Figure 9 shows the B-H curve assumed for the iron. The iron is ideal until the flux density reaches 1.7T. At this point the curve has slope 5b. With this approximation the above set of equations can be solved numerically. in the instance where square current excitation is applied to the stator. Figure 2 shows a typical airgap flux density distribution obtained when square current excitation is applied to the stator.

-40000 -20000 1; ;“ 20000 40000

B (TI

Fig. 9. B-H curve for iron

MMF in loop 3: 284

APPENDIX B

5kW SynRM DIMENSIONS

Number of poles Rotor pole pitch Rotor radius Rotor length Rotor steel : steel + Airgap Number of slots Number of phases Yoke depth Slot depth Slot opening

= 4 = 1.44 radians = 64 mm =51 mm

= 0.3 mm = 36 = 9 = 20 mm = 22 mm = 0.6 mm

fibre ratio = 0.45

REFERENCES

[ I ] I . Boldea, Z.X. Fu, and S.A. Nasar, “Performance Evaluation of Axially - Laminated Anistropic (ALA) Rotor Reluctance Synchronous Motors”, IEEE Transactions on Industry Applications, Volume 30, No. 4, pages 977-985, July / August 1994.

[2] T. Matsuo, and T.A. Lipo, “Rotor Design Optimisation of Synchronous Reluctance Machine”, IEEE Transactions on Energy Conversion, Volume 9, No. 2, pages 359-365, June 1994.

[3] J.D. Law, A. Chertok, and T.A. Lipo, “Design and Performance of Field Regulated Reluctance Machine”, IEEE Transactions on Industry Applications, Volume 30, No. 5, pages 11 85-1 191, September / October 1994.

[4] D.A. Staton, T.J.E. Miller, and S.E. Wood, “Maximizing the Saliency Ratio of the Synchronous Reluctance Motor”, IEE Proceedings-B, Volume 140, No. 4, July 1993.

[5] J.S. Hsu (Htsui), S.P. Liou, and H.H. Woodson, “Peaked-MMF Smooth Torque Reluctance Motors”, IEEE Transactions on Energy Conversion, Volume 5, No. 1, pages 104-109, March 1990.

[6] H.A. Toliyat, M.M. Rahimian, and T.A. Lipo, “A Five Phase Reluctance Motor with High Specific Torque”, IEEE Transactions on Industry Applications, Volume 28, No. 3, pages 659-667, May / June 1992.

[7] P.P. Ciufo, D. Platt, and B.S.P. Perera, “Magnetic Circuit Modeling of a Synchronous Reluctance Motor”, Australian Universities Power Engineering Conference, Volume I , pages 37-42, September 1994.

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