IOSR Jounal of Engineeing (IOSRJEN) ISSN (e): 5-31, ISSN (p): 78-8719 Vol. 5, Issue 4 (Apil. 15), V1 PP 16- www.iosjen.og Detemination of The Winding Inductances Of A Two-Phase Machine. * I.K. Onwuka, ** U. U. Uma * Michael Okpaa Univesity of Agicultue Umudike, ifeanyikayonwuka@gmail.com ** Akanu Ibiam Fedeal PolytechnicUnwanna, Afikpo. gbogbonna@yahoo.co.uk Abstact: - An accuate method fo detemining the winding inductances of a -phase Stato winding is heein pesented. A -phase stato winding is consideed. The pocess begins with undestanding the winding configuation, obtaining the winding function fom that, and consideing the ai-gap function. Compaison is made fo the inductances of the same stato windings consideing saliency and non-saliency. It is obseved that in the case of salient pole otos, the inductances of the stato winding depends on the oto position. On the othe hand, the inductances of the stato winding in the case of non-saliency is constant. Non-linea effects on the magnetizing inductances wee not consideed. Keywods: Magnetizing Inductance, Winding function, Hamonics, Ai-gap function. I. INTRODUCTION Electic machines have assumed an enviable place in the dive of development and civilization, wold ove. Moe than 95% of wold electicity ae poduced fom electic machines. Again, electic machines ae the wok hose of industies, and find extensive applications in seveal domestic, tanspot, health, and agicultual facilities. One popety of immense impotance to the electic machine is the winding inductances. Often times though, the stato windings develop fault, maybe due to high tempeatue, powe system tansients, o some othe factos [1]. Because these machines ae expensive appaatus, it is often the pactice to e-coil the windings and estoe the machine to use. Moeove, fo the pupose of eseach, o to achieve some othe machine pefomance, it may be equied to eplace the windings of a stato to achieve cetain design objectives. Hence, it is impotant that at such times, the enginee o the designe must be able to detemine the new inductances of the machine with espect to the new stato winding. Whethe fo design o maintenance puposes, detemination of winding inductances is of immense impotance, as this affects next to the entie behavio of the machine, hence, its accuate computation is of huge impotance.while the majo inteest in inductance detemination would be to detemine the self and mutual inductances of the windings involved, a unique detemination of the inductances associated with the leakage flux can only be calculated o appoximated fom design consideations []. Hence, the leakage inductance is not dealt with in this wok. Howeve, it is the magnetizing inductances that is the inteest of this wok. II. WINDING DESCRIPTION A double-laye, integal-slot winding with choded coils will be used. Taking 1 slots pe pole, the stato numbe of slots is 4. Each slot holds 18 conductos, which could be of eithe o both phases. Windings wee distibuted as nea sinusoidal as possible. Refeences hee include [, 3, 4]. Futhemoe, the windings of both phases ae identical but displaced fom each othe by an angle of 9 o. The winding clock diagam appeas in figue 1. The following can easily be obseved fom the winding clock diagam: Numbe of slots = 4; Slot angula pitch = 15 o ; Phase belt = 1 slots; Phase spead = 15 o ; phase shift = 9 o ; Numbe of poles =; Pole-pitch = 1 slots pe pole. III. ASSUMPTION Each winding section is aligned axially within the ai gap. This is to say that the wie is neithe slanted in the cicumfeential diection no tilted in the adial diection as it passes though the ai gap. Also, the MMF in the ion is neglected. 16 P a g e
Winding Function, Na (phi-s) Detemination Of The Winding Inductances Of A Two-Phase Machine 7 6 8 5 9 A5 B1 B1 A5 4 A4 B B A4 1 A3 3 A3 B4 B4 11 A B5 A B5 1 A1 A1 1 φ 13 A1 A1 4 14 A B5 15 A3 B4 16 A4 17 A5 B B B1 B1 18 19 A5 A4 B5 B4 A3 1 A Figue 1: Winding Clock Diagam IV. WINDING FUNCTION The Winding function method of analysis of the Stato winding is pesented hee. This is because of the wellknown fact that the Winding Function is the basis of calculating machine inductances [5]. Again, the powe quality of the machine is a function of the shape of the ai-gap MMF due to stato windings. Equation (1) epesents the elationship between the ai gap MMF, F s, the winding function, N s (ϕ s ), and the stato cuent, i s. Subscipt s epesents stato quantity. F s = N s φ s i s (1) Whee ϕ s is an angula displacement along the stato inne cicumfeence. The winding function (WF) methodology developed in [6] was employed in this analysis. The tuns function, n(ϕ s ) and the aveage tuns function, n ave ae given by: n φ s = No. oftunsinintegationpat () n ave = 1 π n φ π s dφ s (3) The winding function is then given by: N s φ s = n φ s n ave (4) Figue and 3 shows the actual winding function, in stai-case fom. 3 6 4 - -4-6 1 3 4 5 6 7 Stato cicumfeential position, phi-s in adian unit Figue : Actual Winding function of the a-phase winding 17 P a g e
b-phase hamonic magnitude a-phase hamonic magnitude Winding Function, Nb (phi-s) Detemination Of The Winding Inductances Of A Two-Phase Machine 6 4 - -4-6 1 3 4 5 6 7 Stato cicumfeential position, phi-s in adian unit Figue 3:Actual Winding Function of the b-phase winding V. HARMONICS ANALYSIS The method of Fouie seies [7] was applied to obtain the vaious hamonics pesent in the winding functions expessed in Figues and 3. The Fouie seies of the winding function was pefomed and vaious plots obtained using the MATLAB tool [8]. The fundamental components of the winding functions fo the two stato windings ae: N as = N s cos φ s δ (5) N bs = N s sin φ s δ (6) Whee, δ is the phase shift, and subscipt s efes to stato vaiables. Figues 4 and 5 give a pictue of the hamonic contents in the winding function, up to the 5 th non-zeo hamonic (odd hamonics only). This is a eflection of the distotion that will be pesent in the stato voltage and cuent wavefom. It is put in bette pespective by figue 6. 6 5 4 3 1 1 3 4 5 6 a-phasesampled hamonics Figue 4:Hamonics of the a-phase winding function 6 5 4 3 1 1 3 4 5 6 b-phase sampled hamonics Figue 5:Hamonics of the b-phase winding function 18 P a g e
Winding Function, Na (phi-s) Detemination Of The Winding Inductances Of A Two-Phase Machine Figue 6 shows a close appoximation of the fundamental component of the Winding Function to the esultant Winding Function. Thus, the value of the fundamental component of the Winding Function can be used fo inductance calculations, with a little eo toleance. These values ae given fo the two phases as: N as = 55.854.139ad (7) N bs = 55.854 1.4399ad (8) 6 4 Resultant winding function funamental component a-phase - -4 b-phase -6 1 3 4 5 6 7 Stato cicumfeential position, phi-s in adian unit Figue 6:Compaison between the esultant winding function and the fundamental component of the function. Hence, the effective numbe of stato tuns will be taken as N s = 55.854. It is clea that the influence of winding design featues which have not been consideed in the analysis ae accounted fo in N s. Ai-gap Function The ai gap function equies definition. It will be helpful to descibe the ai gap of the machine ove an angula displacement of π, as shown in Figue 7a, whee g 1 = the minimum ai gap length g = the maximum ai gap length β = the pole ac/pole pitch atio The invese ai-gap function is also shown in figue 7b. See Appendix A fo paamete values. Fo the pupose of inductance computation, it is beneficial to obtain the invese ai gap function, athe than the ai gap function itself. The Fouie seies appoach can be employed to detemine the invese ai-gap function moe accuately. Figue 7a: Appoximate Ai-Gap function 19 P a g e
Detemination Of The Winding Inductances Of A Two-Phase Machine Figue 7b: Appoximate invese Ai Gap Function Anothe accuate appoach to detemine the invese ai-gap function is as follows: α 1 = 1 β β g 1 g (9) α = sinβπ 1 1 π g 1 g (1) Then g φ s = 1 (11) α 1 α cos(φ s θ ) Theefoe g 1 φ s = α 1 α cos(φ s θ ) (1) Whee θ descibes the oto position. Obviously, both figue 7 and equations (9) though (1) accounted fo machine saliency. Howeve, if a ound oto is consideed, then: g 1 = g (13) Substituting the paamete values of Appendix A into equations (9) though (1), the invese ai gap function will be obtained as follows: With saliency: g 1 1373.3 1977.8cos s Without Saliency: 3333. 3333 s 1 g s (15) VI. MAGNETIZING INDUCTANCE (NON-LINEAR EFFECTS NEGLECTED) The detemination of self-inductance equies the flux linking a winding due to its own cuent to be computed. In the case of mutual inductance, the flux linking one winding due to cuent flowing in anothe winding is equied [].Geneally, the diffeential flux passing though a diffeential volume of coss-sectional aea dφ l and length g is given by [6], dφ = FdP = μ l Fdφ g s (16) Integating (16), beaing in mind that F is a function of φ s : Φ = μ l π F φ g s dφ s (17) Whee P = Pemeance, = inne stato adius,l = axial length of the ai gap, and µ =Pemittivity of fee space. It is common knowledge that flux linkage is expessed as: λ = NΦ = il (18) Whee L is inductance and I is cuent. Equation (17) is the flux linking one tun of an N s tun winding. If it is ecalled that N s is now a function of φ s, and the invese of the ai-gap length a function of (φ s θ ), then fo a winding with N s (φ s ) effective numbe of tuns, the flux linkage of winding A due to cuent in winding B will be given as: π λ AB = μ l N as φ s F B φ s g 1 φ s θ dφ s (19) If equation (1) and (18) ae kept in mind, then (19) can be solved fo the mutual inductance between windings A and B: π L AB = μ l N as φ s N bs φ s g 1 φ s θ dφ s () P a g e (14)
Detemination Of The Winding Inductances Of A Two-Phase Machine Fom equations (18) though (), it is clea that ecipocity holds since the ode of the two windings may be intechanged. Hence, L AB = L BA (1) The above equations ae also valid fo cases whee windings A and B ae one and the same. Hence the magnetizing inductances of windings A and B, espectively, ae given by: π L AA = μ l N as φ s g 1 φ s θ dφ s () π N bs φ s g 1 φ s θ L BB = μ l dφ s (3) With saliency: Substituting equations (5), (6) (neglecting δ), and (1) into equations (), (), and (3), and solving the same, the following esults will be obtained: L AA = μ lπn as α 1 α cosθ (4) L BB = μ lπn bs α 1 α cosθ (5) α L AB = μ lπn as N sinθ bs (6) Without Saliency: Consideing equations (9) and (1) when equation (13) is tue, it will be obseved that α 1 = 1 (7) g α = (8) Substituting equations (7) and (8) into equations (4), (5), and (6) esults to: L AB = (7) L AA = μ lπ N g as (8) L BB = μ lπ N g bs (9) Using the paamete values of Appendix A, Table 1 and figue 8 wee obtained. The Q- and D-axes Magnetizing Inductances Often times in the modeling and simulation of electic machines, it is convenient to eliminate all time-vaying inductances by a change of vaiables fom the stationay machine vaiables to the q-d vaiables of a convenient efeence fame. If the oto efeence fame is assumed, then the q-d magnetizing inductances can be estimated as follows: The tansfomation matix to be used is: T m = cosθ sinθ (3) sinθ cosθ Let L s = L AA L BA L AB L BB Then L mdq = T m L s T 1 m (3) L mdq = diag L md L mq (33) Equation (3) yields equation (33), whee L mq and L md ae espectively the q- and d-axes inductances equied, θ is an angula displacement of the oto. Discussion and Conclusion The leakage inductances wee not consideed in this wok. Nevetheless, the self-inductances of the windings will be the sum of thei leakage inductances and thei magnetizing inductances. Of couse, non-linea effects on the inductances ae not consideed at this stage. Table 1: The magnetizing inductances Magnetizing inductance With Saliency Without Saliency L AA.761.548 cos.1846 (31) L BB L AB L BA.761.548 cos.1846.548 sin.548 sin 1 P a g e
Magnetizing Inductance Detemination Of The Winding Inductances Of A Two-Phase Machine Stato and Ai-gap Dimensions =.6m l =.75m g 1 =.3m g = 5g 1 μ o = 4πe-7.14.1.1.8.6.4. -. LAA -.4 LBB LAB -.6 1 3 4 5 6 7 Roto angula displacement Figue 8: The vaiation of Stato inductances with oto angula positionin the case of saliency. It will be obseved that the magnetizing inductances ae a function of the oto position in the case of salient pole machines, while they ae constants in the case of ound oto machines. Moeove, consideing that the windings ae displaced in space 9 o fom each othe, it is clea that thei mutual inductance will be zeo in the case of ound oto machines, as also was obtained. Howeve, the saliency of the oto, in the case of salient pole otos, leads to some flux linkage between the windings, esulting to a non-zeo mutual inductance. The pocess applied in this wok can be applied as well to unbalanced windings and to 3-phase windings to obtain accuate values of the magnetizing inductances of any stato winding. REFERENCES [1] S. B. Ibahim. A Review of Failue Causes and Condition Monitoing Techniques fo Rotating Electic Machines Past, Pesent, and Futue, Poceedings of the Maiden National Engineeing Confeence of CEET, Michael Okpaa Univesity of Agicultue Umudike, May 7-9, 14. [] Paul C. Kause, Oleg Wasynczuk, Scott D. Sudho_. Analysis of Electic Machiney, IEEE Pess, New Yok, 1995. [3] Mittle, V. N., and Mittal A., Design of Electical Machines, Standad Publishes Distibutos, Delhi, 6. [4] Juha Pyhonen, Tapani Jokinen, Valeia Habovcova, Design of Rotating Electic Machines, John Wiley and Sons Ltd, 8. [5] P. S. Bimbha, Electical Machiney, Khana Publishes, Delhi, Seventh Edition, 8. [6] Obe, E. S. Nnadi, D. B and Eke, J. Inductances and Ai-gap Flux density of a Synchonous Reluctance Machine using the Actual Machine Geomety, NSE Technical Tans. Vol. 44. No. 4, pp.49-63, 9. [7] Ewin Keyszig. Advanced Engineeing Mathematics, John Wiley and Sons, Inc, New Yok, 1979. [8] MATLAB. Mathwoks 7 : Natic, Pennsylvania, USA. P a g e