AC Winding Analysis using Winding Function Approach

Transcription

1 AC Winding Analysis using Winding Function Aroach Gojko Joksimović Deartment of Electrical Engineering University of Montenegro, 0000 Podgorica Montenegro Abstract One of the crucial arts of every electrical machines course is the introduction to the basic elements of alternating current (AC) windings design. After introductory art regarding definition of the full-itch coil magnetomotive force (), usually continues analysis based on fundamental harmonic. This aer resents winding function aroach to the AC winding analysis. Although this aroach is well known and intensively used in the scientific circles, esecially last few years, author s imression is that this aroach is not sufficiently resent in the education rocess. This aroach has at least two rincial advantages over the conventional one. Firstly, it takes into account exact waveform i.e. all of the sace harmonics simultaneously. Secondly, this aroach is mainly comuter oriented and on such a manner it romotes very imortant asect of electrical engineering education - an interdiscilinary aroach. All of the basic terms in the field as distribution factor, itch factor, synchronous seed could be and are derived in this aer, using winding function definition. Additionally, basic elements of Fourier s analysis using built-in MATLAB functions are described and its alications are illustrated. eywords Turns function, Winding function,, distribution factor, itch factor, Fourier analysis INTRODUCTION With the advent of low-cost ersonal comuters and various easily accessible software ackages, comuter-aided teaching tools have become an essential art of both classroom lectures and laboratory exeriments in electrical machinery education []-[5]. These tools have articular

2 imortance in those cases when it is not convenient to organize laboratory exercises as in the case of AC winding design and analysis. The fundamentals of Electric machines are taught at the University of Montenegro in two main courses. The first course is an introductory course which covers the fundamental rinciles of electromechanical energy conversion and transformers. The second one deals with rotational electrical machines in deth. In the first course, one of the crucial arts is AC winding design and analysis, one of the most difficult toics in electric machines for students to understand. This is why a secial attention must be given to this toic with extensive use of grahics and animation. Presented aroach to this comlex matter at the Deartment of Electrical Engineering at University of Montenegro was carried out last two years. After learning the materials in the traditional manner, by resentation of slides and animation of rotating wave, the students take active articiation in the modeling of the concrete winding in the MATLAB environment. During the classroom lectures students have a chance to exlore and look at real winding machine, too, Fig. The quality of this way of active articiation in the educational rocess describes a high ercentage of success at the mid-term test: almost 86%. Fig. Stator AC winding in a real induction machine The described aroach facilitates students understanding of a very imortant concet in electric machinery: satial harmonics of. It makes a good basis for distinguishing these from the others, later introduced time harmonics of voltage or current. The described aroach has the

3 following additional good side: it rovides an animation of the rotating wave. Sectral analysis of rotating wave at different instants of time assures students of a constant harmonic content of the wave regardless of its different waveforms in different instants of time. WINDING FUNCTION DEFINITION AC windings in electrical machines basically have a twofold urose. In electrical generators AC windings are lace where electromotive force (emf) should be induced. In electrical motors, rimarily AC windings goal is to roduce rotating wave. In any case, AC windings have to be designed on such a manner that induced EMF in the AC windings or roduced rotating wave consists redominantly of the sace-fundamental sinusoidal comonent. eeing in mind that AC windings are a comlex subject having a books dedicated to it, [6], [7], and that are even today interesting from different oint of views, [8]-[], we will treat here first its basics. Let s look, for the beginning, the simlest form of the stator hase winding. It is a single concentrated coil with N n turns. Positions of coil s sides along stator circumference are described by mechanical angles θ and θ, Fig. A θ A θ C 3 4 θ Fig. Turns function definition 3

4 Assuming uniform air-ga length i.e. neglecting existence of stator and rotor slots, as well as assuming infinitely ermeable iron, Amere s circuital law, in common shae, C H dl J ds () S get much simler form, ( θ) g H ( ) g i n( θ) H 0 () where H(θ) is radial comonent of magnetic field intensity in the air ga at osition described by angle θ, g is air ga length, i is coil current and n(θ) is turns function, []. From Fig it is obvious that by changing the osition θ of the closed loo side (this closed loo takes art in the Amere s circuital law), turns function could be defined on following manner: n ( θ) N 0 n θ < θ θ for other θ (3) By emloying the second crucial law of magnetism, Gauss law, S B ds 0 (4) and analyzing only electromagnetic rocesses in the air ga, follows, µ 0 H ds 0 (5) S or, l π 0 0 H rdθdz 0 (6) Assuming that H does not deend on the axial coordinate, above exression is equal to, π 0 H dθ 0 (7) what could also be written as: π π 0 H dθ 0 (8) 4

5 From () and (8) follows, π π 0 i H ( 0) + n( θ) dθ 0 (9) g ( 0) + i ( θ) θ 0 π π H n g d (0) 0 i H ( 0 ) n( θ) () g where n ( θ) is the mean value of the turns function. From () and () follows: H [ ] i ( ) g n( θ) n( θ) θ () As H(θ) g is er definition, last exression could be written as, [ n( θ) n( θ) ] i N( θ) i (3) where, ( θ) n( θ) n( θ) N (4) is winding function, [3], [4]. Obviously, winding function is nothing else but er unit current. In further text terms winding function and er unit current will be used interchangeable. Winding function could be alternatively written as, N ( θ) Nn N αm π θ < θ θ αm π for rest of θ n (5) where α m is the coil itch in mechanical radians, α m θ θ. CONCENTRATED FULL-PITCH COIL For a full-itch coil, in a two ole machine (), for adoted system of reference, Fig 3, turns and winding functions are, 5

6 π 3π Nn < θ na ( θ) (6) 0 for other θ Nn π 3π < θ N A ( θ) (7) Nn for other θ Fig 3 show lacement of coil along the stator circumference, turns function rofile, winding function rofile together with two the most significant sace-harmonics as well as coefficients of the Fourier s exansion of winding function. A A C 3 4 θ a) b) Fig. 3. a) Cross section view, b) Turns function, winding function along with two the most significant higher saceharmonics and Fourier exansion of winding function for a full-itch coil in a two ole machine (). Coil has 0 turns. Obviously, only odd higher sace-harmonics are resent in Fourier series of winding function. The fundamental sace-harmonic amlitude is 6.37 turns, Fig b, i.e. 4/π higher than the winding function amlitude, so fundamental wave amlitude is, A max 4 N θ (8) π n ( ) I 6

7 where I is the alternating current rms value. Higher sace-harmonic of order ν has ν times smaller amlitude than the fundamental one: A max 4 Nn ν ( θ) I (9) π ν Turns and winding function for full itch coil as well as Fourier exansion of winding function given on Fig 3 could be easily defined using following few rows written in MATLAB : H04; dx*i/h; nn0; % number of samles in *i % discretization ste % number of turns in a coil % na turns function of hase (coil) A i; for teta0:dx:*i-dx, if((teta>i/)&(teta<3*i/)), na(i)nn; else na(i)0; end; ii+; end; clear i % Na winding function of hase (coil) A Nana-mean(na); % Fourier exansion F_series*abs(fft(Na)/H); stem(f_series(:30), filled ) It should have in mind following two facts: ) in order to obtain exact discrete Fourier transform, number of samles in π must be ower of two, i.e. H n where n is an integer; ) first element in Fourier exansion is coefficient equal to the mean value of the winding function and this element is not shown on Fig 3. Fig 4 shows cross section view of the machine as well as turns and winding function for a full-itch coil winding in a four ole machine,. Two full-itched coils (in a four ole machine full-itch coil has a itch α m π/ rad) are series connected. Total turns and winding function of hase 7

8 winding is simly the sum of the turns and winding functions of every single coil in the winding. By other words, suerosition is allowed because infinitely ermeable iron was assumed. Now, the fundamental harmonic is second order harmonic (), third harmonic is sixth order harmonic (3) etc. The amlitude of the fundamental wave now is, A max 4 N θ (0) π A ( ) I where N A is the total number of turns in the hase winding. The amlitude of the higher sace-harmonic of order ν is: A max 4 N A ν ( θ) I () π ν A A A C 3 4 A θ a) b) Fig. 4. a) Cross section view, b) Turns function, winding function along with two the most significant higher-sace harmonics and Fourier exansion of winding function of two series connected full-itch coils (A -A and A -A ) in four ole machine (). Every single coil has 0 turns. DISTRIBUTED FULL-PITCH COIL PHASE WINDING Fourier analysis showed that the air ga roduced by a full-itch coil(s) consists of fundamental sace-harmonic comonent as well as a series of odd higher-order harmonic 8

9 comonents. In the design of the AC windings, serious efforts are made to distribute the coils making u the windings so as to minimize the higher-order harmonic comonents and to roduce an air-ga wave which consists redominantly of the sace-fundamental sinusoidal comonent. On this way, additionally, machine is better utilized. Therefore, in ractice, AC winding of induction machine is always designed as a distributed winding. Distributed hase winding of a three-hase machine occuies one third of stator slots along the circumference of the machine. Fig 5 shows such one hase winding in a two ole machine which stator has S8 slots. It means that one hase winding occuy six slots, three under one and three under other ole. A 3 A A A A A 3 θ a) b) Fig. 5. a) Cross section view, b) Turns function, winding function along with the two most significant higher-sace harmonics and the Fourier exansion of distributed winding consists of three series connected full-itch coils (A -A, A -A and A 3 -A 3 ) in two ole () machine. Every single coil has N n 0 turns, N A 30. Turns and winding function of hase A winding could be easily obtained by summing turns and winding functions of individual coils. As it is clear from Fig 5, resultant rofile of distributed full-itch winding has traezoid like shae what is much closer to the sinusoid than before, Figs 3 and 4. However, amlitude of the resultant winding function is not three times as greater than before, but somewhat smaller as it could be easily concluded observing the coefficients of the 9

10 Fourier series. The reason for that is in the sace dislacement of individual coils. The winding distribution factor d just define this ratio: er definition, distribution factor is ratio of the resultant sace-harmonic amlitude and sum of the sace-harmonics amlitudes of the constituent coils (ratio of vector and algebraic sum of constituent coil s s): dν ' A A max ν + Amax ν ' A A max ν + Amax ν... + ' n A A max ν coil max ν n n () For the fundamental sace harmonic, ν, from Fourier exansions, Figs 3 and 5: A max d 3 coil max 0.96 Similarly, for the fifth sace harmonic, ν5: A max 5 d 5 3 coil max 5 0. These results are in good correlation with results obtained from the well known analytical exression for distribution factor, [5]-[8], α π sin νq sin ν m d ν (3) α π q sin ν q sin ν mq where m is number of hases, q is number of slots er ole er hase and α is an electrical angle, α α m π/s. For analyzed hase winding, d d π π sin sin 3 6 π π 3 sin 3 sin π 5π sin 5 sin 3 6 π 5π 3 sin 5 3 sin

11 C C C 3 Having in mind definition of distribution factor, resultant distributed hase winding saceharmonic amlitude could be defined as: Amax ν 4 N θ (4) π ν A dν ( ) I THREE PHASE FULL-PITCH (SINGLE-LAYER) WINDING Three-hase machine has a three distributed windings on the stator. Phase windings are identical but shifted in sace for π/3 mechanical degrees. One hase winding in a three-hase machine occuies one third of number of stator slots under one ole, Fig 6. C 3 C B C B A 3 B 3 A A A A B 3 A 3 B B θ a) b) Fig. 6. a) Cross section view, b) Winding functions of hase windings. Every hase winding consists of three series connected full-itch coils. Two ole machine,. Every single coil has 0 turns. Winding functions i.e. s er unit current for such machine is given on Fig 6. In order to obtain the resultant wave, care must be taken about instantaneous value of hase currents. As it is well known, through the three-hase windings flows hase shifted currents. At one instant of time, say t 5ms, Fig 7, hase current i a has a value i a A while other two hase currents are

12 i b i c 0.5A. Resultant wave could be obtained by multilying hase winding functions with adequate instant values of the hase currents, ( t ms) + N i ( t 5ms) + N i ( t ms) S _ t 5ms N a ia 5 b b c c 5 (5) i.e. S _ t 5ms N a 0.5 Nb 0. 5 Nc (6) Similarly, for any other instant of time, say, t 0ms, ( t ms) + N i ( t 0ms) + N i ( t ms) S _ t 0ms N a ia 0 b b c c 0 (7) i.e. 3 ( N N ) S _ t 0ms b c (8) because i a (t 0ms)0, i b (t 0ms) 3/ and i c (t 0ms) 3/, Fig 7. Fig. 7. Three-hase, 50Hz, unit currents. Phase current rms value is / 0.707A. The resultant rofiles at these two instant of time are shown on Fig 8. Obviously, ositions of the maximal values of resultant s are shifted in sace, i.e. resultant wave is not fixed in sace: actually it is rotating wave. Rotating seed could be easily found from the following considerations: at t 5ms maximal value of the rotating is at osition described by the axes of hase winding A, i.e. θ t At t 0ms, maximal value of the rotating is at osition θ t Rotating seed, i.e. synchronous seed, is:

13 π [ rad] n angle π π 4 rad rad rad rev πf f 60 f time t [ ] [ ] t T T s s T s s min min (9) s More generally, in case of machine with ole airs, synchronous seed is times smaller: πf rad 60 f rev n s s min (30) Hence, resultant wave rotate with synchronous seed, having different waveforms in every different instant of time. However, regardless of shae of the rofile at different instants of time, Fourier s exansion of the is always the same. Sectral content of the resented waveforms is also shown on Fig 8. Fig. 8. Resultant rotating wave in two different instants of time, t 5ms and t 0ms and Fourier exansion of wave. Two ole machine. 3

14 For difference from the hase windings, resultant rotating wave does not contain any odd sace harmonic which is integer multily of three. By other words, all sace-harmonics belongs to the following series, ν 6 k + (3) where k0,±,±, From (3) follows that symmetrical three hase winding sulied from symmetrical three-hase voltage suly contains fundamental ν, fifth ν 5 th, seventh 7 th, eleventh th etc. sace harmonics. Minus sign means inverse rotating wave. Synchronous seed of νth sace-harmonic is ν times smaller than the synchronous seed of the fundamental one, πf rad 60 f rev n s ν ν s ν min (3) As it one can conclude from the Fig 8, noteworthy higher sace harmonics are 5 th and 7 th, so called hase belt harmonics which are direct consequence of traezoidal shae of the hase winding. However, the most significant higher sace harmonics are, so called, slot harmonics, of order S/±. In analyzed case it is 7 th and 9 th sace harmonics. They are direct consequence of the discrete nature of the winding i.e. conductor lacement in the slots. These harmonics on the best manner fills the gas in the stewise shae of resultant wave. By comarison the amlitudes of the sace-harmonics from Fig 5 and Fig 8, it is easy to conclude that the rotating sace-harmonic amlitude is.5 times higher than amlitude of the hase winding sace harmonics: S max ν N A dν 3 N A dν hasemax ν ( θ) I I (33) π ν π ν For analyzed winding, 3 N Adν S max I Aturns π ν π π More generally, for a m-hase symmetrical AC winding, 4

15 S max ν m m N A dν hasemax ν ( θ) I (34) π ν THREE PHASE SHORTED-PITCH COIL (DOUBLE-LAYER) WINDING An additional measure taken in order to further ugrade the rotating waveform closer to the sinusoid is use of the short-itch coils. However, in order to use short-itch coils, stator hase winding must be laced in two layers along the stator circumference, i.e. short-itch coil must have one its side laced in one slot in one, say, bottom layer but other side must be laced in other slot in to layer. On this way double-layer winding is obtained. Using short-itched coils, with adequate coil itch, some of the higher sace-harmonics in the resultant rotating waveform could be canceled out or significantly attenuated. Let s look at the following examle: Fig 9 shows double layer three-hase stator winding. Every single coil in the hase windings is a short itch coil. Shortening of the coil is one stator slot itch. Now, instead of three coils in one hase winding, Fig 6, six shorted-itch coils makes one hase winding. In order that comarison with the reviously analyzed case could be ossible, every single coil has to have one half of number of turns as before. Phase winding from Fig 9 one could observe on a following manner: the hase winding is organized in two layers, and every layer, observed individually, is layer with full-itch coils. But, two layers are shifted in sace for one stator slot itch shortening of the coil. Therefore, resultant hase, on the harmonic basis, could be seen as a vector sum of the layer s. Here is conveniently to introduce other winding factor, chord or itch factor, as ratio of vector and algebraic sum of layer s fundamental wave, [5]-[8], β β β cos + cos cos β cos (35) + where β is an electrical angle which describe the shortening of the coil itch. 5

16 θ Fig. 9. Double layer three-hase winding. Two ole machine,. Shortening of the coil is one stator slot itch. Shortening of the coil could be described on the following manner, too, y π β k π π (36) τ so, itch factor is, y π sin (37) τ or, for any sace-harmonic, y π ν sin ν (38) τ By taking into account just defined itch factor, amlitude of the rotating sace-harmonic, in distributed double-layer three-hase winding is, S max ν 3 νdνn hase 3 νn hase I I π ν π ν (39) where ν is winding factor which include distribution and itch factor, ν dν ν. In many electrical machines textbooks above exression is given in the following form, Neff I S max ν. 35 (40) ν 6

17 where N eff ν N hase is so called number of effective turns er hase. Rotating waveshae for two different instant of time and for winding from Fig 9 is given on Fig 0. Fourier s coefficients for the sace harmonics are also given. For one stator slot itch shortening, itch factor is: 8 π sin Distribution factor is as before, 0.96 d so, effective turns number is, N eff Amlitude of the fundamental sace harmonic is, (40), ( / ) 7. Aturns Neff I 8.36 S max ν and somewhat is smaller than in a case of full-itch windings, due to the itch factor. This result is in close agreement with results obtained numerically, from Fourier s exansion of the resultant wave, Fig 0. However, the main result of short-itch coils using is in the attenuation of 5 th and 7 th sace harmonics in the rotating wave. Namely, in order to cancel out 5 th harmonic from the rotating wave, following condition must be satisfied, y π y π κπ 5 sin y κτ (4) τ τ 5 where κ is an integer, κ0,,, and κ must be chosen so that coil itch y is first number smaller of the ole itch exressed in number of stator slots. For analyzed winding, τ 9 slots, so, the reasonable solution for y is for κ: κ 36 y κ

18 Fig. 0. Rotating wave at two different instants of time and corresonding Fourier s coefficients. Two ole machine. Every single coil has 5 turns. As coil itch must be an integer, ossible solutions are y7 or y8. Obviously, for any choice, 5 th sace harmonic could not be eliminated but for y7 it will be significantly reduced. Similarly, for canceling out 7 th sace harmonic, following condition must be satisfied: sin 7 y π 0 7 y π κπ y κτ 7 (4) τ τ 7 κ 54 y κ Again, y could be 7 or 8 and, again, 7 th sace harmonic could be only attenuated, esecially for y8, but not absolutely eliminated from the wave. If coils with 7 stator slot itch are chosen, 5 th harmonic will be much attenuated than 7 th. Inversely, if coils with 8 stator slot itch are chosen, 7 th sace harmonic will be much attenuated than 5 th. In analyzed case study, coils with 8 stator slot 8

19 itch are used (one stator slot itch shortening), so 7 th sace harmonic will be much more attenuated than 5 th, as could be easily concluded by comarison of wave sectrum from Figs 8 and 0. Usually, coil itch is chosen in such a way to attenuate 5 th and 7 th sace harmonic simultaneously, what is fulfilled for the following coil itch, y κτ 6 κτ 3 (43) together with additional condition, that κ must be chosen on a such way that coil itch y is slightly smaller than the ole itch. Condition (43) could be alternatively defined as, y τ τ 6 5 τ 6 (44) However, from Figs 8 and 0 it should be observed that shortening of the coils have no any imact on the intensity of the slot harmonics! Coil, regardless of its itch, must begin in one slot and end in the other, and therefore the coil s sacing is an integral multile of the basic sacing causing slot harmonics in the first lace. CONCLUSIONS A winding function aroach for AC winding analysis is resented. This aroach has at least two rincial advantages over conventional one. Firstly, it takes into account all of the sace harmonics simultaneously, i.e. exact waveform. Secondly, this aroach is mainly comuter oriented and on such manner it romotes very imortant asect of electrical engineering education - an interdiscilinary aroach. All of the basic terms in the field as distribution factor, itch factor, synchronous seed are derived in this aer using winding function aroach. Additionally, basic elements of Fourier s analysis using built-in MATLAB functions are described and its alications are illustrated. This aroach could be easily imlemented for analysis of fractional slot windings as well as any form of asymmetrical windings. 9

20 REFERENCES [] A.F. Zobaa, T.A. Boghdady, Integration into undergraduate courses of transformer tests using MATLAB/Simulink, International Journal of Electrical Engineering Education, Vol. 44, No.4,.39-33, December 007. [] S. Ayasun, G. arbeyaz, DC Motor Seed Control Methods Using MATLAB/Simulink and Their Integration into Undergraduate Electric Machinery Courses, Comuter Alications in Engineering Education, Vol. 5, No.4, , 007. [3] S. Ayasun, C. Nwanka, Induction Motor Tests Using MATLAB/Simulink and Their Integration Into Undergraduate Electric Machinery Courses, IEEE Transactions on Education, Vol. 48, No.,.37-46, February 005. [4] Y.N. Anagreh, I.M. Al-Refae e, Teaching the self-excited induction generator using MATLAB, International Journal of Electrical Engineering Education, Vol. 40, No.,.55-65, January 003. [5] J. Cathey, Electric Machines: Analysis and Design Alying MATLAB, McGraw Hill, 00. [6] H. Sequenz, The windings of electrical machines, Vol. 3, A.C. Machines, Sringer Verlag, Vienna, 950 (in German). [7] M.M. Liwschitz-Garik, Winding of alternating current machines, van Nostrand Publications, 950. [8] A. Demenko, Descrition of electrical machine windings in the finite element sace, COMPEL, Vol. 7, No. 4,. 7-79, 008. [9] J. Steinbrink, Design and analysis of windings of electrical machines, International Symosium on Power Electronics, Electrical Drives, Automation and Motion, SPEEDAM, ,

21 [0] D. A. ocabas, A. F. Mergen, A novel method to reduce the effects of sace harmonics in alternating current machines, International Symosium on Industrial Electronics, ISIE, , 008. [] H. Xueliang, D. Qiang, H. Minqiang, A Novel Exact and Universal Aroach for Calculating the Differential Leakage Related to Harmonic Waves in AC Electric Motors, IEEE Transactions on Energy Conversion, Vol. 9, No.,. -6, March 004. [] N.L. Schmitz, D.W. Novotny, Introductory electromechanics, Ronald Press, 965. [3] R.B. Robinson, Inductance coefficient of rotating machines exressed in terms of winding sace harmonics, Proc. Inst. Elec. Eng, Vol., No. 4, , 964. [4] T.A. Lio, Theory and control of synchronous machines, ECE 5 lecture notes, University of Wisconsin-Madison [5] A.E.Fitzgerald, C. ingsley, S.D. Umans, Electric machinery, McGraw Hill Professional, 00 [6] I. Boldea, S.A. Nasar, The induction machine handbook, CRC Press, 00 [7] P.C.Sen, Princiles of electrical machines and ower electronics, Wiley, 996 [8] P.Cochran, Polyhase induction motors analysis, design and alication, CRC Press, 989