Analysis and Calculating on Stator Slot Leakage Inductance of Six-phase

Transcription

1 Analysis and Calculating on Stator Slot eakage Inductance of Six-phase AC achines Department of Control Engineering, aval Aeronautical and Astronautical University, Yantai Abstract There are two sets of windings embedded in the same stator slots for six-phase machines. And mutual leakage flux coupling exists at the two windings. Calculating of slot leakage inductance depends mainly on slot layers, pitch, distributing between the two sets winding. The distributing winding configurations of the two windings usually include dual-layer windings, three-layer windings, four-layer. In this paper the analytic calculations of slot leakage inductance of six-phase AC machines with different winding structure are presented. the relevant formulas of slot self-leakage inductance, mutual leakage inductance of different pitch for two sets stator-winding with different layers are derived. Keywords: Six-Phase AC achine; Slot eakage Inductance; Windings Distributing. Introduction In recent years, the research and application of AC motor have been paid more and more attention, specially for six-phase machines[-]. The structure and distribution of the six-phase motor which is usually composed by dual stator-winding is more complex than that of the traditional three-phase motor. The accurate calculation of the winding leakage inductance is very important to the six-phase motor design and system controlling. The slot leakage inductance is the main component of stator winding leakage reactance and dual different stator-winding structural has large effect on calculating of the slot leakage inductance. For the traditional three-phase motor, winding design and slot leakage inductance calculating are very mature, however, for the six-phase motor, the two windings coexist on the same slot, it has the mutual leakage inductance flux coupling[-5].the calculating of slot leakage inductance is associated with the slot layers, winding pitch and two sets of stator-winding angle, which is associated with the winding structure, distribution of the winding in the slot. Especially, the slot mutual leakage inductance calculating has many differences with that of the traditional three phase motor, the formula of the existing literature can not be applied directly and need further research. This paper uses analytic method to research slot leakage inductance calculating of six-phase motor in the form of different winding distribution, the relation of mutual leakage inductance, self leakage inductance and pitch with same turns dual stator-winding with dual layer and the formula of slot leakage inductance with three layers and four layers are derived.. Definition of leakage inductance for six-phase AC machines The two sets of winding for six-phase induction motor are abc and xyz, respectively. In order to meet the motor control, it gets d-q equivalent circuit through Clark transforming, as Figure. The mutual leakage inductance matrix between stator-winding abc and xyz is equivalent to[]: lax lay laz l laz lax lay () lay laz lax International Journal of Advancements in Computing Technology(IJACT) Volume5,umber,February 0 doi:0.5/ijact.vol5.issue.57 90

2 u q i q R R i q u q l l j d j d m lr j( ) Rr r i qr d r uq r u d i d i d R R l l j q j q m lr j( ) Rr r i dr qr u d u dr Figure. The dq equivalent circuit for six-phase induction machine Transformed into d-q coordinates, mutual leakage inductance between d-d axis (or q-q axis) and mutual leakage inductance between d-q respectively are: cos cos( ) cos( ) (a) lax lay laz sin sin( ) sin( ) (b) lax lay laz ote that the value of the is positive number with xyz lagged of abc. If abc lagged of xyz, then: cos cos( ) cos( ) (a) Where, and x,y,z winding; lax lay laz sin sin( ) sin( ) (b) lax lay laz lax, lay, laz is respectively mutual leakage inductance between A phase winding, is respectively equivalent mutual leakage inductance of the two sets stator winding between d-d axis (or q-q axis) or between d-q axis; m is excited mutual inductance between the stator and rotor; l, l is two sets of stator-winding leakage inductance; lr is rotor winding self leakage inductance. Considering the proportion of slot leakage inductance is the largest among all the leakage inductances; therefore, the following discussion is the calculating of slot leakage inductance.. Calculating of slot leakage inductance. The relation between slot leakage inductance and pitch In order to reduce some high-order harmonic, induction motor design generally uses the dual layer winding, this would allow the current in some upper and lower layer winding not belong to the same phase, the slot leakage inductance can be divided into the self leakage inductance of the upper layer and self leakage inductance of lower layer, the mutual leakage inductance between the top and bottom layer[5]. That is ls lt lb K( p) () Where, lt, lb, is respectively self leakage inductance and mutual leakage inductance between the upper and lower winding side. K( p ) is mutual inductance coupling coefficient with pitch p. Example as A phase winding, leakage flux of A phase can be represented through the coupling between top and bottom layer of each phase current. That is la ( Kaa( p) ia Kab( p) ib Kac( p) ic (5) Kax( p) ix Kay ( p) iy Kaz( p) iz ) It can find the slot self-inductance coupling coefficient of A phase winding and mutual leakage inductance between A phase winding respectively with the xyz winding, putting the result into formula 9

3 () can obtain, through the d-q coordinate transformation, then obtain slot self-inductance coefficient of A phase K () l l lt lb aa In order to improve the control performances of the six-phase machines, it need to minimize the mutual leakage inductance between each phase winding. Therefore, the relation of top and bottom layer mutual leakage inductance and pitch are discussed when two sets of winding angle respectively is 0, 0, 0 and 90.. Calculating slot leakage inductance of dual layer stator-winding ()When two sets of stator-winding angle is 0, calculating slot leakage inductance coupling coefficient between the top and bottom layer. The distribution of dual stator-winding abc and xyz is shown as Table. Table a -z b -x c -y x -c y -a z b Two sets of stator-winding spatial distribution is 0 phase-belt angle, the structure can not be a short pitch, only whole pitch winding can be used, that is p, and therefore: Kaa( p) 0, K ( ) 0 ab p, Kac( p) 0, Kax( p), Kay ( p) 0, Kaz ( p) 0,from formula(5), ()and(), it can respectively be attained ( p) ax( p)cos0 (7a) ax ( p)sin0 (7b) l l lt lb (8) ( ) When dual stator-winding angle is 0, calculating slot leakage inductance coupling coefficient between top and bottom layer. Twelve phase-belts are divided equality inside each pairs pole along stator slot inner circle, the width of phase-belt is 0 electrical angle, so it is the spatial distribution 0 phase-belt winding. When pitch p, winding distribution is positive phase sequence and xyz winding ahead of abc winding 0 electrical angle shown as Table. Table a x -c -z b y -a -x c z -b -y a x -c -z b y -a -x c z -b -y When pitch p 5, winding distribution shown as Table. Table a x -c -z b y -a -x c z -b -y x -c -z b y -a -x c z -b -y a The linear relation between inductance coupling coefficient and pitch is 5 p, then K ( p) p 0, K ( p) 0, K ( p) 0, K ( p) p, K ( p) p, aa Kaz( p) 0, From formula(5),()and(), there are ab ac ax ay 9

4 ( p) K ( p) (9a) ax ( ) 0 p (9b) ( p) ( p) K K (0) l l lt lb aa ax Inductance coupling coefficients of else pitch, shown as Table. Table 5 p k k k idq l = l p p-0 0 -p+ 0 k 5 p p p p 0 p 0 p-5 p- 0 k T + B +K aa - T + B +K aa - 0 -p T + B 0 -p T + B 0 p- p- 0 k p- 0 -p 0 k T + B +K aa - T + B +K aa - Where, slot top and bottom layer mutual leakage inductance coupling coefficient is defined as: Kaa Kab Kac Kax Kay Kaz Kba Kbb Kbc Kbx Kby K bz Kca Kcb Kcc Kcx Kcy Kcz K () Kxa Kxb Kxc Kxx Kxy Kxz Kya Kyb Kyc Kyx Kyy K yz Kza Kzb Kzc Kzx Kzy Kzz Combined with (), that is: k k k k k 0 k k k 0 k k k k k k 0 k K () k 0 k k k k k k 0 k k k 0 k k k k k When xyz winding lagged of abc winding 0, the slot mutual leakage inductance coupling coefficient between top slot layer and bottom layer slot is k k k k 0 k k k k k k 0 k k k 0 k k () K k k 0 k k k 0 k k k k k k 0 k k k k Where, the expressions of k k, and are the same as Table. ()When two sets of stator-winding angle is 0, calculating slot leakage inductance coupling coefficient between top layer and bottom layer. When stator winding is 0 phase-belt angle, xyz winding ahead of abc winding and p, the relation of winding is as Table 5. 9

5 Table 5 a x b y c z -y -c -z -a -x -b When p, the relation of winding is as Table. Table a x b y c z -c -z -a -x -b -y When xyz winding ahead of abc, the slot mutual leakage inductance coupling coefficient between top layer and bottom layer are shown as Table 7. Table 7 Pitchp k k k k idq l = l p - 0 p 0 p- 0 T + B +K aa - k TB p 0 p p 0 k TB 0 T + B +K aa - 0p p- 0 p 0 k TB 0 T + B +K aa - The mutual leakage inductance coupling coefficient of slot top-layer and bottom layer is k k k k k k k k k k k k k k k k k k () K k k k k k k k k k k k k k k k k k k When xyz winding lagged of abc winding 0, the slot mutual leakage inductance coupling coefficient between top layer and bottom layer is k k k k k k k k k k k k k k k k k k (5) K k k k k k k k k k k k k k k k k k k The expressions of k k, and are same as Table 7. ( ) When dual stator-winding angle is 90, calculating slot leakage inductance coupling coefficient. Analogizing the same method with 0 electrical angle, the slot mutual leakage inductance coupling coefficient between top layer and bottom layer can be introduced when xyz winding ahead of abc winding 90. 9

6 k k k 0 k k k k k k 0 k k k k k k 0 () K 0 k k k k k k 0 k k k k k k 0 k k k The expressions of k k, and are same as Table. Opposition symbol of k in the above formula, mutual leakage inductance coupling coefficient between the top layer and the bottom layer when xyz winding lagged abc winding 90 can be got. The summary of slot leakage inductance calculating is shown as below. 5 p Figure. Relation between mutual leakage inductance and pitch of different electrical angle The relation of two sets of stator-winding mutual leakage inductance with different angle and pitch p is curved in Figure, where, the solid line is the relation of slot mutual leakage inductance and pitch when the angle is 0 and 90, the dotted line is the angle is 0, the black point is the angle is 0. () When angle is 0, ( ) and q axis is coupling, ( ), that is, the slot mutual leakage inductance between d, and the whole pitch winding is not conducive to reduce some of the high harmonics. () When angle is 0, 0, that is, the slot mutual leakage inductance between d and q axis is decoupling, and (p ), pitch p must be closed to to minimize, which make the fundamental waveform amplitude is expended largely, which isn t practical enough. () When angle is 0 and90, 0, slot mutual leakage inductance between d and q axis is decoupling. ( p ), the pitch need be close to, then is minimum, it make certain high-order harmonic decrease, and the expense of the fundamental amplitude is very small. Therefore, these two methods are feasible.. Slot leakage inductance for three layers stator- winding The distribution of conductors in the slots of three layers stator-winding is as figure., where layer and layer are represented for winding, and the layer is winding. 95

7 Zq (- b) Z q b (a) Slot of three layers winding Z q (b)distribution of conductor per phase in slot Figure. Distribution of conductor per phase in slot of three layer winding For phase A, the total flux per pole distance with / is : According to: Where, I a, I b, I c s a 0 l ef Z q I a I b I a I c ( ) ( ) s 0 l ef Z q I a I a I a I a () ( ) s s 0 lef ( ) Zq( Ix Iy ) s s 0 lef () Z ( I y I q y ) a a a lab b lac c slax x slay y slaz z (7) I I I I I I (8) are currents of winding, I x, I y, I z are currents of winding.,,,,, are slot leakage permeance and mutual leakage permeance respectively. Z is the number of slots per phase and per pole, is pitch. s, s are turns in series per phase, q The slot leakage inductance of winding is: l ( l l l (b )) (9) lsa s 0 ef The mutual leakage inductance between winding and winding are as followings: ss slax slxa 0 lef ( ) ss slay slya 0 lef ( ( ) ) (0) slaz slza 0. Slot leakage inductance for four layers stator- winding Winding is embedded at the top two layers, and winding embedded at the two bottom layers, as figure. 9

8 Z ( ) q Z q Z q (a)slot of four layer winding (b)distribution of conductors per phase in slot Figure. Distribution of conductors per phase in slot of four layer winding Shown as fig.(a),,,,,,,,,,,,,,,, are slot leakage permeance and mutual leakage permeance respectively. where, the relations of mutual leakage permeance are[5] () For phase A, the total flux per pole distance with / is : s a 0 lef Zq I a Ib I a Ic ( ) ( ) s 0 lef Zq I a Ia I a I a ( ) ( ) s s 0 lef ( ) Zq( I z I z I y I y ) s s 0 lef ( ) Zq( I z I z Iy I y ) The slot leakage inductance of winding is: s 0 ef () l ( l l l (b )) () lsa Slot leakage inductance of winding are: l ( l l l (b )) () lsa s 0 ef The mutual leakage inductance between winding and winding are as followings: slax slxa 0 s (5) s slay slya 0 lef ( ( ) ( ) ) s s slaz slza 0 lef (( ) ( ) ). Conclusions This paper analyzes the calculating rules of six-phase machines leakage reactance and calculating characteristics of slot self leakage inductance and mutual leakage inductance which is the double-layer winding structure, three layers and four layer. The calculating of slot leakage inductance is focused on 97

9 analyzing the same dual stator-winding turns and different winding distribution. These formulas and conclusions will be benefit to the six-phase machines design and performance calculating. 5. References [] R.H.elson,P.C.Krause. Induction machine analysis for arbitrary displacement between multiple winding sets[j], IEEE Trans on PAS, vol.9,no.,pp.8-88,97. [] i, Xiaohua,Huang, Surong. An Analysis of the Impact of the Coil on the atural Frequency of the Permanent agnet Synchronous otor Stator Core[J], JCIT, Vol. 7, o., pp. 9 ~ 0, 0 [] Jun-qiang ian, Shun-yi Xie, Wang Jian, Ping Hu, onlinear odel of Permanent-agnet Synchronous otors[j], JDCTA, Vol., o. 7, pp. 9 ~, 0 [] Chen Sikun. Design of electrical machines[], China achine Press,China,00 [5] Wuxusheng, a Weiming. Calculation of stator winding leakage inductance of and -phase machines with AC and DC output[j], Transactions of China electro-technical society,vol.9,no.,pp.9-,