General Closed-form Analytical Expressions of Air-gap Inductances for Surfacemounted Permanent Magnet and Induction Machines

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2 General Closed-form Analytial Expressions of Air-gap Indutanes for Surfaemounted Permanent Magnet and Indution Mahines Ronghai Qu, Member, IEEE Eletroni & Photoni Systems Tehnologies General Eletri Company Bldg EP, Rm 110-B, One Researh Cir Niskayuna, NY 109, USA Thomas A. Lipo, Fellow, IEEE University of Wisonsin - Madison 557B Engineering Hall 1415 Engineering Drive Madison, WI , USA lipo@engr.wis.edu Abstrat General losed-form analytial expressions for airgap indutane of surfae-mounted PM and indution mahines are derived without the assumption of sinusoidally wound windings. The expressions are suitable for all single- or doublelayer, onentrated or distributed windings, and demonstrated to be idential with the winding funtion approah and share the same advantage that all the harmonis are inherently inluded. These expressions are straightforward analytial equations, whih are easily used by mahine designers. I. INTRODUCTION Air-gap indutane, defined as the indutane due to flux rossing the air gap, is one of the most important parameters for Permanent Magnet (PM) and indution mahine designs. The well-known indutane alulation method the winding funtion [1] has the advantage that all the harmonis are inherently inluded. It is often onvenient to alulate the indutane of a speial ase by using the winding funtion. However, a tedious solution of an integral is needed in the winding funtion method, whih is not very straightforward and diffiult to be used in omputer programs. Therefore, a general losed-form analytial expression that should be suitable for omputer programs, a neessary tool for modern mahine designs, is desirable, but has not been available. This paper will derive suh an equivalent analytial expressions whih are suitable for mahine designers, who usually use omputer programs to design mahines and ompare the indutanes of several different designs. To derive the analytial expressions for the air gap indutane, the following assumptions are made: The surfae-mounted magnets and spaers between them have essentially the same relative permeability, and The rotor and stator relative permeabilities are high, so that the rotor and stator bak iron relutane are negligible and no saturation exists, and The fringinffet at both ends of the windings is negligible. Based upon the assumptions above and the indutane expression of a single full pith oil, the oil self indutane, oil-to-oil mutual indutane, air-gap indutane per pole, mutual indutane between two oils in two adjaent poles, and phase air-gap indutane will be derived sequentially. The final expression is an analytial equation in terms of the mahine physial sizes and the winding distribution. Given the differene between PM mahines and indution mahines, the indutane expressions for PM mahines are modified to apply to indution mahines. A general losedform analytial expression of the air-gap indutane for indution mahines will be given. In addition, two simplified forms for the full pith windings and the onentrated full pith windings will be presented. The derived expressions and the winding funtion will be demonstrated by two examples to be idential and share the same advantage that all the harmonis are inherently inluded. Furthermore, these expressions are the straightforward analytial equations without any integral inluded, whih are desirable and easily used by mahine designers. II. GENERAL CLOSED-FORM EXPRESSIONS OF AIR-GAP INDUCTANCES FOR PM MACHINES A general losed-form expression of air-gap indutane of surfae-mounted PM mahines will be derived in this setion, then applied to indution mahines in the next setion. A. Coil Self Indutane Based upon the assumptions above, the indutane of a single full pith oil with N oil turns is found to be [] L goil = N oil µ r µ o Lτ p (1) (H PM + µ r ) where is the effetive length of air gap inluding the slottinffets, H pm the magnet height shown in Figure 1, L the axial length of the stator ore or rotors, τ p the pole pith, µ o permeability of free spae, and µ r the magnet relative permeability. Following the same derivation proess, the indutane of the oil with the short pith of slots an be expressed as

3 1 oil i j t +1 t +i t +j Φ i g Magnets H PM Rotor bak iron Fig. 1. Mutual oupling between two oils L g = N oil µ r µ o L K τ τ p () (H PM + µ r ) where the oil pith fator K τ is defined as K τ =y( y) () The oil pith ratio y is defined as the ratio of the oil pith to the pole pith and is given by y= (4) mq where is the oil pith measured in number of slots, m the mahine phase number, and q the slot number per phase per pole. For full pith oils, K τ is equal to 1. The pole pith measured in number of slots, τ ps, defined in the (5) will be used in the following setions. τ ps = mq (5) B. Coil-to-oil mutual indutane Mutual indutane is defined in terms of the flux linked by one oil due to the urrent in another. Air gap mutual indutanes between oils are a funtion of the relative plaement of the oils and therefore are a funtion of the slot number q per phase per pole. As shown in Figure 1, onsider any two oils, oil i and oil j (i< j q< ), with the oil pith, Φ i is the air gap flux reated by urrent flowing in oil i. This flux ouples to oil j in suh a way that (ji) of the j i flux is oupled in one diretion and of the flux is τ ps oupled in the opposite diretion. Thus, the net flux from oil i linking oil j is (ji) j i. Sine the self indutane τ ps of oil i is linearly related to the flux reated by oil i, the ratio of the air gap mutual and self indutane is (ji) j i τ ps. i.e., M gij =[ (ji) j i ]L τ ps τ gi = (1 ji j i ) L τ ps τ g (6) Due to the symmetry, all the oil self indutanes are equal to eah other, and to L g. C. Phase indutane per pole. For eah pole, there are q oils in series, and mutual indutanes exist between any two oils. Then the total phase indutane per pole must be L pole =ql g + M g1 + M g1 + M g M g1q + M g + M g4 + + M gq Substituting (6) yields L pole =ql g + q(q1)l g O L g [ q q M N (7) M + M g(q-1)q + 1 ] (8) 1 q1 L g [ τ ps τ ps τ ps 1 q τ ps τ ps τ ps M N 1 + ] τ ps Noting that (q1)+(q)+(q)+ +(q)+(q1)= q(q 1) (9) 6 and after some manipulation, (8) redues to L pole = [q q(q 1) q(q 1) (τ ps ) ] L g (10) Combining () and (10) yields L pole = [ q q(q 1) q(q 1) 6 6(τ ps ) ] N oil µ r µ o L K τ τ p H PM + µ r (11)

4 τ ps τ τ ps 0.5 ps + k τ ps +i i + i τ ps + + k Φ i Magnets Rotor bak iron Fig.. Mutual oupling between two oils in two adjaent poles (ase A) τ ps τ ps 0.5 τ ps + k τ ps + i i +i Φ i τ ps + + k Magnets Rotor bak iron Fig.. Mutual oupling between two oils in two adjaent poles (ase B) This expression denotes the indutane of q oils distributed in q slots, eah with N oil turns and a pith of slots. D. Mutual indutane between two oils in two adjaent poles. For double layer windings, the mutual indutane between two oils, one of whih is on one pole and the other on an adjaent pole, should be alulated before the phase indutane an be found. There is no mutual indutane between any oils on different poles for sing-layer ase sine these kind oils have at least a 60 degree phase shift. Considering the flux Φ i produed by oil i (1 i q) onsisting of the top layer ondutors in slot i and the bottom layer ondutors in slot +i, as shown in Figure, one half travels through its one side on whih oil τ ps +k (1 k q) is, and one half through the other side. Coil i and oil τ ps +k, whih is onsisted of the top layer ondutors in slot τ ps +k and the bottom ondutors in slot τ ps + +k, are in the two adjaent poles. For the ase shown in Figure, +i is larger than τ ps +k. The flux Φ i ouples to oil k in suh a way that ( +i)(τ ps +k) of the flux is oupled in one diretion and half the flux is oupled in the opposite diretion. Thus the net flux linking oil k is Φ ik = 1 Φ i (+i)(τ ps +k) Φ τ i =[ 1 (+i)(τ ps +k) ]Φ τ i (1) Sine the self indutane of oil i is linearly related to the flux reated by oil i, the mutual indutane between two oils in two adjaent poles is related to the oil self indutane by

5 M ppik =[ 1 (+i)(τ ps +k) for 0.5 ( +i) ]L τ g (τ ps +k)>0 (1) Given the relationship of ik q-1< mq = τ ps =τ ps τ ps τ ps for m >1 (14) the ondition of ( +i)(τ ps +k) 0.5, whih is equivalent to ik τ ps 0.5, is always met. Sine the fluxes reated by oil i and oil τ ps +i, whih is in the third pole or oil i itself for the pole mahine, share the area between +i and τ ps +i, one additional ondition is needed to make (1) valid. It is τ ps + + k ( + i ) τ ps (15) whih is equivalent to k i (16) For mahines, is usually greater than τ ps and hene greater than mq, while the maximum value of k i is q-1. That implies ondition (16) is always valid for multiphase mahines (m>) sine mq > q-1. These disussions keep (1) valid for multiphase mahines (m >). For the ase shown in Figure, +i is smaller than or equal to τ ps +k, so that only 1 [(τ ps+k) ( +i)] of the flux τ ps ouples to oil k. Thus the net flux linking oil k is Φ ik =[ 1 (τ ps+k)( +i) ]Φ τ ps τ i (17) Similarly, the mutual indutane between two oils in two adjaent poles in this ase is related to the oil self indutane by M ppik =[ 1 (τ ps+k)( +i) for 0 [(τ ps +k) ]L τ ps τ g (18) ( +i)] (τ ps 0.5 ) The ondition of [(τ ps +k)-( +i)] (τ ps 0.5 ) is always valid for m > and equivalent to Condition (16). Equation (1) and (18) an be ombined to form M ppik =[ 1 1 for 1 i q (τ τ ps +k)( +i) ]L g (19) q and 1 k q where τ q = for +i > τ ps +k (0) τ ps for +i τ ps +k E. Pole-to-pole phase mutual indutane for double layer windings The total pole-to-pole phase mutual indutane is the sum of the oil mutual indutanes between two adjaent poles: q q M pp = M ppik (1) Substituting (19) into (1), it yields M pp =[ q q q (τps +k)(τ +i) ] L τ g for all q () 1 i q, 1 k q Equation () is in a onise format and very easily used in any omputer program. In pratie, q is about 5 for small or middle power level mahines, τ ps is equal to mq, and is not greater than τ p. Therefore, this equation is not hard to be evaluated by hand. F. Phase indutane It is interesting to note that, the short pith windings are not suitable for the single-layer windings, and that full pith windings ( =τ ps and hene K τ = 1) must be used. The number of the oil turns N oil must be equal to the ondutor number per slot N. For a single-layer mahine with P poles and C iruits in parallel, given P/C iruits in series and using (11), the total indutane of eah iruit is L kt = P C [ q q(q 1) q(q 1) 6 6(τ ps ) ] N µr µ o Lτ p () H PM + µ r Again given C iruits in parallel, the overall mahine phase indutane with the Single Layer (SL) windings is found as L ph = µ rµ o LPN τ p C ( q H PM + µ r q(q 1) 6 q(q (4) 1) ) for SL 6(τ ps ) For the double-layer winding ase, the number of the oil turns N oil is equal to one half of the ondutor number per slot N, the pole indutane per phase takes the form of L pole = 1 4 ( q q(q 1) q(q 1) 6 6(τ ps ) ) N µr µ o L K τ τ p (5) H PM + µ r In addition, the mutual indutane between any two adjaent poles exists and takes the form of (). Given P pole indutanes, P mutuals between any two adjaent poles, and C iruits in parallel, the overall indutane of eah iruit for the double layer ase is L kt = P C (L pole+m pp ) (6) It an be extended to L kt = P 4C [ q q(q 1) q(q 1) 6 6(τ ps ) q q (τps +k)(τ +i) ] N (7) µr µ o L K τ τ p τ q H PM + µ r Given C iruits in parallel, the PM mahine phase indutane with the Double Layer (DL) windings is found as L ph = µ rµ o K τ LPN C [ q q(q 1) q(q 1) 1 1(τ ps ) 1 q q (τps +k)(τ +i) τ p ] for DL τ q H PM + µ r (8)

6 For the full-pith windings ( =τ ps =τ q ), (8) an be redued to L ph = µ q q rµ o L PN C [ q q(q 1) 1 6 k i ] (9) Noting that τ p H PM + µ r q(q 1) 1 = 6 q q k i (0) (9) is redued to the identity of (4). This result means that the phase indutane of the single-layer full pith windings is exatly same as that of the double-layer full pith windings if all the other onditions are kept same. This is easily understood sine the flux distributions are same for the both ases. This implies that (4) an be applied to the doublelayer winding if the effet of the short pith is negleted when alulating the pole-to-pole mutual indutane. For onentrated full-pith windings (q=1, =τ ps ), substituting q=1 and =τ ps =mq into (4), this expression redues to L ph = µ rµ o LPN τ p for onentrated 4C (1) H PM + µ r full pith windings It is worth noting that the air gap indutane is relatively small beause of the low relative permeability and large length of the PM with respet to the air gap. III. APPLICATION TO INDUCTION MACHINES AND COMPARISON WITH THE WINDING FUNCTION Given the differene between PM mahines and indution mahines, (4), (8) and (1) an be modified to apply to indution mahines as L ph = µ oτ p LPN C L ph = µ oτ p K τ LPN C [ q q(q 1) 1 q q (τps +k)( +i) [ q q(q 1) ] for SL IM () τ q q(q 1) 1(τ ps ) ]for DL IM 1 () L ph = µ oτ p LPN 4C for onentrated full pith IM (4) Equation () through (4) ould be rewritten into the forms L ph = µ oπ LR is N C [ q q(q 1) ] for SL IM (5) N L ph = µ oπk τ LR is C [ q 1) q(q q(q 1) 1 1(τ ps ) q q (τps +k)(τ +i) ] for DL IM τ q 1 (6) L ph = µ C for onentrated full pith IM (7) Equation (7) and (5) are both the speial ases of (6) for the full pith windings and the onentrated full pith windings, respetively. The three equations above and the well-known indutane alulation method the winding funtion share the same advantage that all the harmonis are already inluded. However, an integral is needed in the winding funtion method, whih is not very straightforward and diffiult to be used in omputer programs. On the other hand, (6) is a losed-form analytial equation and suitable for omputer programs, a neessary tool to design mahines nowadays. In other words, (6) ould be taken as the integral result of the winding funtion approah for the general ase, in whih the windings with the oil pith of slots are uniformly distributed in q slots for eah phase eah pole. There are a total m phases, P poles, and C parallel iruits. To alulate the indutane of a speial ase, the winding funtion might be more onvenient while (6) is suitable for mahine designers, who usually use omputer programs to design mahines and ompare the indutanes of several different designs. In order to show that the equations derived here are equivalent to the winding funtion approah, onsider an example used in ourse ECE 711 at the University of Wisonsin-Madison: a total of N ondutors are uniformly distributed along the stator inner surfae of a phase pole mahine with the air gap. Assume N is large enough that the steps in the winding funtion aused by individual ondutors an be approximated as a smooth urve. The winding funtion approah gives the phase indutane is L ph = 7 µ o π LR is N H (8) 648 In this example, q is equal to infinity so that q 1=q, m=, =mq=q, and C=1. Substituting these onditions into (5), it is redued to L ph = µ oπ LR is 7q N H (9) 18 Given N=N qmp=6n q, the two results above are found to be idential. Consider another more general example: a double layer winding with q=, m=, τ ps =mq=9, =7 (short pith), P=8, and C=1. Equation (6) an be easily evaluated to be L ph =.0556 µ (40) To find the phase indutane by the winding funtion, the winding funtion N(θ) should be drawn first. It is shown in Figure 4

7 N(θ) 1.5N 1.5N Phase A urrent CURRE NT Slot No. π π Fig. 4. Win ding funtion for the double layer windings with q=, m =, τ ps=mq=9, =7, P=8, C=1 4π 5π θ The phase A indutane as per the winding funtion theory will be given by π L ph = µ o LR is C 0 N (θ) dθ =.0556 µ (41) whih is the same as that given by (6). Table I summarized the results alulated using the winding funtion method and (6) for 4 different ases. The identity between the two approahes is demonstrated. TABLE I INDUCTANCES CALCULATED FROM THE WINDING FUNCTION AND q=, m=, τ ps =9, P=8 C=1 =7 C=1 =8 C= =7 C= =8 The winding funtion results.0556 µ.4444 µ µ µ Equation (6) results.0556 µ.4444 µ µ µ IV. CONCLUSIONS General losed-form analytial expressions of air-gap indutane for - or more -phase surfae-mounted permanent magnet mahines have been derived and applied to indution mahines. The assumption of sinusoidally wound phase windings is unneessary. The derived expressions are suitable for all single- or double-layer, onentrated or distributed windings. The expressions and the winding funtion method have been demonstrated to be idential and share the same advantage that all the harmonis are inherently inluded, while these expressions are straightforward analytial equations and an be easily used in omputer programs. REFERENCES [1] T. A. Lipo, Analysis of Synhronous Mahines, ECE 511 ourse notes, University of Wisonsin - Madison (00), Chapter 1. [] D. W. Novotny, T. A. Lipo, Vetor Control and Dynamis of AC Drives, Oxford University Press, 1996, Chapter. [] D. C. Hanselman, Brushless Permanent-Magnet Motor Design, New York, MGraw-Hill, 1994, pp