This is a repository copy of Analytical optimisation of electromagnetic design of a linear (tubular) switched reluctance motor.

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1 This is a repository copy of Analytical optiisation of electroagnetic design of a linear (tubular) switched reluctance otor. White Rose Research Online URL for this paper: Version: Accepted Version Proceedings Paper: Corda, J (05) Analytical optiisation of electroagnetic design of a linear (tubular) switched reluctance otor. In: Proceedings of the XVII International Syposiu on Electroagnetic Fields in Mechatronics, Electrical and Electronic Engineering (ISEF 05). XVII International Syposiu on Electroagnetic Fields in Mechatronics, Electrical and Electronic Engineering (ISEF 05), 0- Sep 05, Valencia, Spain.. ISBN Reuse Unless indicated otherwise, fulltext ites are protected by copyright with all rights reserved. The copyright exception in section 9 of the Copyright, Designs and Patents Act 988 allows the aking of a single copy solely for the purpose of non-coercial research or private study within the liits of fair dealing. The publisher or other rights-holder ay allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this ite. Where records identify the publisher as the copyright holder, users can verify any specific ters of use on the publisher s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by eailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. eprints@whiterose.ac.uk

2 ISEF 05 XVII International Syposiu on Electroagnetic Fields in Mechatronics, Electrical and Electronic Engineering Valencia, Spain, Septeber 0-, 05 ANALYTICAL OPTIMISATION OF ELECTROMAGETIC DESIGN OF A LINEAR (TUBULAR) SWITCHED RELUCTANCE MOTOR J. Corda University of Leeds, School of Electronic and Electrical Engineering, Leeds LS 9JT, United Kingdo e-ail: j.corda@leeds.ac.uk Abstract The paper presents a ethod for optiising radial proportions of the agnetic circuit of a linear tubular switched reluctance otor in ters of axiising the developed thrust fro a given achine volue at constant (rated) dissipation. The ethod is based on the use of analytically derived expressions for the achine s phase inductance in the position of full isalignent between phase saliencies and over teeth, and the flux-linkage vs. current relationship in the aligned position. The average thrust is calculated fro a coplete cycle of co-energy change. Introduction Linear switched reluctance (SR) otors, alike their rotary counterparts, are increasingly popular as variable-speed drives, due to siple brushless construction and absence of peranent agnets. A longitudinal cross-section of a linear 4-phase SR achine in tubular (cylindrical) for [] is shown in Fig. The achine consists of the transversely slotted inner part, referred here as the over, and the outer cylindrical assebly coprising a nuber of identical phase sets with casing and linear bearings, referred here as the stator. Each phase set consists of the ferroagnetic core fored of two discs, which are spaced by a ring-shaped back-of-core, and a solenoidal coil accoodated within the core. The phases are spaced fro each other by non-agnetic rings. The achine operates in conjunction with a power electronic converter [] which is coutated fro a position detector [], altogether foring a linear switched reluctance drive. stator pole disc back-of-core ring over Fig. Longitudinal cross-sectional view through the 4-phase linear tubular SR achine Fig. Set of agnetisation characteristics for range of positions Perforance prediction of a SR achine requires the input agnetisation data in the for of fluxlinkage/current/position characteristics, as shown in Fig. For a hypothetical achine, these characteristics are defined by geoetric paraeters, nuber of turns and B-H curve, and can be coputed by using widely available software packages for field analysis based on finite-eleent ethods (FEM). However for a typical SR achine with sall airgap and saturated agnetic

3 circuit the optiisation procedure based on FEM, which involves variations of several circuit paraeters through a large nuber of increental changes, consues a considerable tie. This paper outlines a rapid estiation of agnetisation characteristics which is useful in analytical odelling and optiisation of a linear tubular switched reluctance achine. Estiation of Phase Inductances The agnetic field pattern in a linear tubular SR achine is syetrical with respect to the achine longitudinal axis. This iplies that the cross-sectional area of a flux tube, and hence the flux-density, varies along the tube which requires appropriate adaptation in deriving the partial pereances copared to the ethod originally developed for a rotary SR achine [4]. The ethod is based on an approxiation based on splitting the phase flux into -D tubes of cylindrical and toroidal for, which are represented by straight-line and circular-arc segents in -D field plots as shown in Fig. (a) (b) Fig. Sketch of approxiate phase field paths in: (a) fully isaligned position and (b) aligned positions In fully isaligned position, the air parhs of the field lines are long and the f drop in the iron paths is sall copared to that of the air paths. Thus when deriving analytical expressions it is justifiable to assue that the iron is infinitely pereable which iplies that the field lines are perpendicular to the iron surface. In this position the inductance can be treated as independent of the excitation current. (The inductance in this position is here fro referred to as the iniu inductance.) In aligned position, the air paths of the field lines are short copared to path lengths in the core, which requires appropriate accounting for the..f. drop in the core. Furtherore the segents of the core which are closer to the longitudinal axis becoe agnetically saturated which iposes that non-linear aspect of B-H characteristic ust be included in the estiation of inductance. (The inductance in this position is here fro referred to as the axiu inductance.)

4 The following set of paraeters is used to define the agnetic core geoetry: Stator outside diaeter (d o ); Inner diaeter of the back-of-core ring (d i ); Stator pole disc width (v); Spacing between adjacent phases (u); Mover outside diaeter (d ); Mover diaeter in the slot (d s ); Mover pole tooth width (t ); Mover slot width (s ); Radial air-gap length between the stator and the rotor poles (g ) The axial width of the stator back-of-core ring is given by: The radial width of the coil is taken as: w = (t + s) v z = (d i d ) / g Miniu Inductance In fully isaligned position the flux is represented by nine tubes (coponents) and their field paths in the longitudinal cross-sectional plane are shown in Fig a. The points on the excited stator pole (shown on the figure s right side) where field lines diverge are defined by the condition of equal agnetic field strengths along the two diverging field lines. Figure 4 shows detail with the tube of flux-linkage. The radial distance of the diverge point A fro the over pole surface is denoted with sybol k and it is deterined as follows. k Fig.4 Detail with the tube of flux-linkage The field lines AP and AA'A'' are linked respectively with the fractions ( k / z) and ( k /(4w z)) of the total nuber of turns N per phase. The axial distance between the stator and over pole tips can be expressed as = (s v) /, and the lengths of field lines l and l 9 can be expressed as (k + (k t)) / and w. Using the equation for agnetic field strengths along lines AP and AA'A'' k ( ) NI f f 9 k NI 4wz i.e. ( ) l l9 z w (k t) the radial distance of point A fro the over pole surface is found as being ()

5 k= z+ + t wz t t z ( + t ) 4z () The average field strength along an eleentary flux path at radius x is proportional to the fraction of the nuber of turns linked with the path, and inversely proportional to the overall length of the path in the air (which is twice the length shown in the sketch of Fig 4, due to the syetry of return path), i.e. NI x /4 H ( ) () x w z (Segent A'A'', which is sall in coparison with the overall length AA'', is ignored.) The average cross-sectional area of eleentary flux tube is expressed as d x da = ( + ) dx (4) So the flux-linkage of the tube is obtained as k x x d 0 H ( ) N da 0 [( ) N] I ( ) 4wz 4 w z x ( A ) dx (5) N 0 w z k - k - I - ( wz 6 k - + d ) k - k - d ( + d ) + ln 80 8 Figure 5 shows detail with the tube of flux-linkage. Referring to Fig a above alost the entire nuber of turns is linked with any eleentary field path within the tube and the average field strength, taking the syetrical return path into account, is siply expressed as H NI Thus the flux-linkage of the tube is obtained as N H d g) (8) ( 0 N I ( d ) (9) 0 g k (6) (7) Fig.5 Detail with the tube of flux-linkage

6 Figure 6 shows detail with the tube of flux-linkage. Here too the entire nuber of turns is linked with any eleentary field path within the tube. The flux-linkage of the tube is obtained as Fig.6 Detail with the tube of flux-linkage v / NI d x 0N ( g ) x dx (0) v/ v 0N I [( d g)ln ] () Referring to Fig a the tubes 4 and 5 are respectively identical with tubes and 4. Hence 4 () 5 () Other coponents of the flux-linkage are derived in analogous anner with the above. It should be noted when deriving a cobined expression for the flux-linkage coponents 6, 7 and 8, that the eanating and returning field paths are syetrical for inner phases, whereas they are asyetrical for the outer phases. (Each inner phase is surrounded by adjacent phases on both sides, whereas each outer phase is surrounded by adjacent phase on one side only.) The iniu inductance is found as L in 9 j I j (4) Maxiu Inductance In aligned position, shown in Fig b, the f drop in the ferroagnetic core cannot be ignored as well the ipact of agnetic saturation on the value of inductance at different levels of the excitation current. Due to the fringing effect at pole tips, the effective cross-sectional area of the flux tube in the airgap is bigger than the area of the pole surface. To ake soe allowance for this effect, the field line which leads fro the point of diverge on the stator pole side is approxiated by a circular arc of radius r such that the length of the seicircle is equal to the spacing between adjacent phases. This enables approxiating the effective pole width in the airgap through Carter s expression, i.e.

7 v eff v ( ) r v ( ) u (5) where r = u / and sybol denotes Carter s coefficient given by r g r arctan ln (6) g 4r g As the radial airgap length g is uch saller than the pole width, it is justifiable to assue that in aligned position the entire flux passes through effective airgap area. A further siplification is ade by splitting the agnetic circuit into a series of the following eleents: one half of the stator yoke (ring) cross-section A y = (d o d i ) / 4 length L y = v / + w stator pole disc treated as two series coponents coponent : cross-section A s = v eff d i length L s = (d i d + g) / 4 coponent : cross-section A s = v eff (d i + d + g) / length L s = (d i d + g) / 4 air-gap cross-section length A g = v eff (d + g) g rotor pole (tooth) cross-section A r = v eff d length L r = (d d s ) / fraction of the over body (shaft) cross-section A b = [(d +d s ) / ] / 4 length L b = v / + w The relationship between the flux-linkage and current is expressed iplicitly through the f equation NI Bg = ( g H 0 s L s H s L s H y L y H b Lb H Finding the value of flux-linkage for a given current requires an iterative approach in solving Eq (7). To avoid iterative procedure, the value of current is being found for a given flux-linkage, i.e. the value of flux density in each eleent of the series can be calculated for different values of fluxlinkage and the corresponding value of the field strength can be obtained using the B-H characteristic of the agnetic core aterial. Values of inductance at different levels of flux-linkage are found by dividing the flux-linkage with the corresponding current. Table gives an indication how the calculated values of inductance in fully isaligned and aligned positions copare with values obtained by easureents and FEM coputations. The results are related to the prototype achine having the nuber of turns per phase N = 40 and the following geoetric paraeters of the core constructed fro ild steel: Stator outside diaeter, d o = 80 ; Inner diaeter of the back-of-core ring, d i = 74; Stator pole disc width, v = 4 ; Spacing between adjacent phases, u =.5 ; Mover outside diaeter, d = 40 ; Mover diaeter in the slot, d s = 8; Mover pole tooth width, t = 4 ; Mover slot width, s = 6 ; Radial air-gap length between the stator and the rotor poles, g = 0.. r Lr) (7)

8 Table : Coparison of easured and estiated results Current [A] Measured values Analytically calculated Coputed by FEM L in [H] L ax [H] L in [H] L ax [H] L in [H] L ax [H] Measured values are obtained by applying the ethodology described in [] which allows the presence of agnetic saturation and eliinates the ipact of eddy currents, i.e. it provides easureent of pure inductance. Other ethods bases on applying d.c. voltage step or pulse, which also allow presence of agnetic saturation, do not eliinate entirely the ipact of eddycurrents. Optiisation of Radial Proportions The electroagnetic circuit optiisation ais to deterine the proportions of the agnetic circuit which provides developent of axiu force fro a given volue of the core defined by the outer diaeter and length. The variation of radial paraeters is being considered here. The axial paraeters such as the over pitch and spacing between phases are deterined by the nuber of phases and overall length, and the ratio pole width / over pitch is kept constant (4/0). The inner diaeter of the back-of-core ring is deterined by the condition that the flux density in the ring atches the ean value of flux density in the pole disc. The radial airgap length is kept at the iniu size which is practically achievable with slide bearings ade out of bronze. So the two reaining internal paraeters the over outside diaeter (d ) and slot diaeter (d s ), which defines the slot depth, are considered as variables. The variations of either the over diaeter or the slot diaeter affect the level of agnetic saturation. Furtherore the phase coil resistance is affected by the variation of the over diaeter and therefore the optiisation is conducted under condition of constant coils dissipation of rated level. Under such a condition, it can be assued that the radial paraeter variations within the investigated range have insignificant effect on the heat transfer, i.e. the rated teperature rise can be considered unaffected. When the achine operates in a dual-phase ode of excitation, each phase coil is energised over a half-cycle which is equal to a linear displaceent of a half of the over pitch, i.e. the four phase coils (A, B, C, D) are excited with currents of ideally square wavefors, which are utually shifted in phase by a quarter of the over pitch length. The four phase currents produce thrust coponents with wavefors which are overlapping each other alternatively through the pattern AB, BC, CD, DA. The average linear force (thrust) per phase can be expressed as change of co-energy per cycle of linear displaceent, i.e. F ph di where denotes the over pitch length. Variation of radial paraeters affects the flux-linkage vs. current characteristics and hence the change of co-energy per cycle of linear displaceent. So the thrust axiisation is equivalent to the axiisation of the area between the two extree characteristics of flux-linkage vs. current (corresponding to aligned and fully isaligned positions) and the line of constant current as illustrated in Fig 7. (8)

9 Fig.7 Illustration of co-energy area defining the average force per phase Diagra of Fig 8 relates to the optiisation of radial proportions of the 4-phase linear switched reluctance achine of the volue defined by the outside core diaeter and length as in the prototype achine (80 and 06 ) under excitation with phase currents of 'square' wavefor having agnitude of. A. The geoetry with over diaeter in the range between 8 and 40, and slot diaeter between and 4 is capable of developing the thrust of around 55 N. Fig.8 Optiisation of radial proportions of the 4-phase linear SR achine with outside core diaeter 80 and length 06, under excitation of. A References [] J. Corda and E. Skopljak, "Linear switched reluctance actuator", IEE Publication No.76, Oxford, Sept.99, pp [] J. Corda J and S. M. Jail, Experiental deterination of equivalent circuit paraeters of a tubular switched reluctance achine with solid steel agnetic core, IEEE Transactions on Industrial Electronics, Vol.57, No., Jan.00, pp 04-0 [] J. Corda J and B. Ouhab, "Discrete position sensing for a linear switched reluctance otor", IEE Publication No.475, London, Sept. 000, pp 7-0 [4] J. Corda and J. M. Stephenson, "Analytical estiation of the iniu and axiu inductances of a double-salient otor", International Conference on Stepping Motors and Systes, Leeds, Sept.979, pp 50-59