Acta Polytechnica doi:10.14311/AP.2019.59.0153 Acta Polytechnica 59(2):153–161, 2019 © Czech Technical University in Prague, 2019 available online at http://ojs.cvut.cz/ojs/index.php/ap GENERAL CALCULATION OF WINDING FACTOR FOR MULTI-PHASE/-LAYER ELECTRICAL MACHINES IRRESPECTIVE OF POLES NUMBER Daoud Ouamaraa, b, ∗, Frédéric Dubasa, Sid Ali Randic, Mohamed Nadjib Benallalb, Christophe Espanetd a Univ. Bourgogne Franche-Comté, Département ENERGIE, FEMTO-ST, CNRS, F90000 Belfort, France b Khemis Miliana University, Laboratory of Energy & Smart System LESI, Ain Defla, Algeria c Renault S.A., 78280 Guyancourt, France d Univ. Bourgogne Franche-Comté, 90000 Belfort, France ∗ corresponding author: daoud.ouamara@gmail.com Abstract. In this paper, a method to calculate the winding factor by only considering stator parameters without the rotor ones is developed. This is interesting because it allows the separation of the stator and rotor design, unlike the existing methods in the literature. A general method based on the matrix representation of a winding is presented. This approach requires the knowledge of four parameters : i) slots number, ii) phases number, iii) layers number, and iv) single-phase spatial distribution. A new feature of the multi-layer windings is introduced, it is called false-zero windings, which is divided into two categories: i) α-windings (i.e., odd false-zero windings), and ii) β-windings (i.e., even false-zero windings). The windings having no false-zero are categorized as γ-windings. The calculations are applied for single and multi-phase/-layer windings. The results of the comparison are satisfactory. The code used for the calculation is given in Appendix. Keywords: False-zero windings, matrix representation, multi-phase/-layer, winding factor. 1. Introduction 1.1. Context Rotational force generating devices, such as motors, electric generators and sirens, are generally composed of two components called stator and rotor, in which a winding is inserted in one or both sets. The winding is one of the most important and critical elements in such machines. Several types of windings are devel- oped and studied to achieve the desired performances for a given application, viz.: full-/short-/long-pitch winding, concentrated around teeth winding, . . . , dis- tributed winding, etc. Recently, Ouamara et al. (2018) [1] published an overview on the winding design and developed a research tool called ANFRACTUS Tool 1.0 allowing an automatic generation of all windings in multi-phase/-layer electrical machines by using the matrix representation. The way the coils are distributed in the stator and/or rotor slots directly affects the winding factor of each spatial harmonic and, therefore, the electro- magnetic performances of electrical machines (e.g., the self-/mutual-inductances, the back electromotive force, the electromagnetic torque, unbalanced mag- netic forces, . . . , the permanent-magnet eddy-current losses, the efficiency). The winding factor is then de- fined as the ratio of flux linked by an actual winding to flux that would have been linked by a full-pitch winding with the same number of turns [2]. The wind- ing is spatially distributed in the stator and/or rotor slots, whereby the flux penetrating the coils does not pass through all the coils simultaneously but with a certain phase shift. Consequently, the back electro- motive force of the winding is not directly calculated with the number of turns, but the winding factor corresponding to each harmonic must be taken into account [3]. In order to compensate the low torque of an electrical machine having a low winding factor, it is necessary to supply it with a higher current or using more turns [4]. In the literature, different methods for calculating the winding factor have been proposed. By using the Star of slots method [5], the computation is per- formed for fractional-slot three phase synchronous machines with single and two layers [6]-[8] and for multi-phase machines [9]. The electromotive force phasor vector is used to calculate the winding factor of concentrated multi-layer windings as explained in [10]. Using analytical expressions, the winding factor can be calculated from the distribution factor and the pitch factor [11]-[13]. In [14], the equations for a spe- cific winding layout have been applied for multi-phase permanent-magnet synchronous machines having all teeth wound concentrated windings. 1.2. Objective of this Paper The purpose of this work is to establish a general and simple method for calculating the winding factor 153 http://dx.doi.org/10.14311/AP.2019.59.0153 http://ojs.cvut.cz/ojs/index.php/ap D. Ouamara, F. Dubas, S.A. Randi et al. Acta Polytechnica regardless of the number of slots, phases and layers. Only, the knowledge of the spatial distribution of single-phase coils in slots is sufficient. To the best of the author’s knowledge, the fact that the number of pole pairs is not taken into account for calculating the winding factor is not reported in the literature. This allows to design the windings without constraints on the number of pole pairs. The terms used to describe the method and notion of false-zero windings are described in Section 2, where the difference between α-windings and β-windings are explained. In Section 3, γ-windings are introduced, and also, the equations used to calculate the winding factor for each case are explained in detail with some examples. Section 4 deals with the multi-layer wind- ings in order to validate the calculations outlined in this article. Some specific windings in [15]-[17] have been used and compared with the results obtained by the developed method. Finally, the algorithm writ- ten in Matlab allowing the calculation of the winding factor is given in Appendix. 2. Notions of the Method 2.1. Matrix Representation Instead of using the usual diagrams, the stator and/or rotor windings distribution can be represented by a connection matrix [Mw] linking the m-phases (i.e., the matrix rows) to the stator and/or rotor slots (i.e., the matrix columns). In [18], this concept was used to design fractional-slot windings. A table was used for the coil arrangement in [19]. In order to represent multi-layer windings, [1] has subdivised the number of columns of [Mw] by the number of layers. This new connecting matrix [Cw] (of dimensions m × Qs · Ly whose m, Qs and Ly are respectively the number of phases, slots and layers) is defined by: 1 2 · · · Qs [Cw] =   [ C1,1wk ] [ C1,2wk ] · · · [ C1,Qswk ] [ C2,1wk ] [ C2,2wk ] · · · [ C2,Qswk ] ... ... ... ... ... ...[ Cm,1wk ] [ Cm,2wk ] · · · [ Cm,Qswk ]   1 2 ... ... m (1) The elements [ Ci,uwk ] (where i = 1, . . . ,m; u = 1, . . . ,Qs and k = 1, . . . ,Ly) are equal to: • 0 if no conductor of phase i in the uth slots and kth layers; • +1 if forward conductor of phase i in the uth slots and kth layers; • –1 if return conductor of phase i in the uth slots and kth layers. The false-zero windings are visible thanks to the use of [Cw] (see Section 2.2). Nevertheless, the windings’ distribution will be represented by: [Dw] = 1 Ly × [Cw] (2) For example, Figure 1 represents [Dw] for a stator of an electrical machine with Qs = 6, m = 3 and Ly = 2. Throughout this study, only the first row will be considered (i.e., the first phase) of [Dw], viz., [Vw] =[ D1,uwk ] (of dimension 1 × Qs · Ly). However, one of the other phases can also be chosen since the phases are balanced. For the last example, [Vw] is given by: [Vw] = 1 2 [ 0 1 0 0 −1 0 . . . . . . 0 −1 0 0 1 0 ] (3) The vector [Vw] may be separated into Ly sub- vectors [V ′w] (of dimension 1 ×Qs) according to: [V ′w]k = [Vw]{k + (u− 1) ·Ly)} ∀k (4) For the last example, [V ′w] of each layer is: [V ′w]1 = [Vw]{1, 3, 5, 7, 9, 11} (5a) [V ′w]2 = [Vw]{2, 4, 6, 8, 10, 12} (5b) [V ′w]1 = 1 2 [ 0 0 −1 0 0 1 ] (6a) [V ′w]2 = 1 2 [ 1 0 0 −1 0 0 ] (6b) Figure 2 shows the sub-vectors [V ′w]1 and [V ′ w]2 ex- tracted from the winding in Figure 1. 2.2. Notion of False-Zero 2.2.1. Definition Figure 3 shows an example of a one phase distribu- tion for a false-zero winding with Qs = 9, m = 3 and Ly = 2. Using the connection matrix [Mw], the first phase of this winding can be represented by: [ M1,uw ] = 1 2 [ 1 −1 0 0 −1 0 1 0 0 ] (7) Figure 1. Two-layer winding (Qs = 6, m = 3 and Ly = 2) and its connection matrix. 154 vol. 59 no. 2/2019 General Calculation of Winding Factor (a) (b) Figure 2. Two-layer winding (Qs = 6, m = 3 and Ly = 2): (a) First sub-vector extracted, and (b) Sec- ond sub-vector extracted. Figure 3. Example of a one phase distribution for a false-zero winding with Qs = 9, m = 3 and Ly = 2. The sixth column (i.e., the sixth slot) is filled with a zero, this indicates the absence of the first phase in this slot. Indeed, the sixth slot is filled by the first phase in the opposite direction (2-layer winding) as given by: [Vw] = 1 2   1 0 0 −1 0 0 0 . . .. . . 0 −1 0 –1 1 0 . . . . . . 1 0 0 0 0   (8) This case is called false-zero. Noting that a single layer winding can not have a false-zero (the slot is fully occupied by the phase). By introducing this new concept, multi-layer windings having false-zero will be subdivided into two categories: i) α-windings (i.e., odd false-zero windings), and ii) β-windings (i.e., even false-zero windings). The windings having no false-zero are categorized in γ-windings. 2.2.2. α-windings (i.e., odd false-zero windings) A false-zero winding is odd, if forward (+1) and return (–1) conductor numbers are unequal, viz.,∑ u [V ′w]1 (u) 6= 0 (9) By applying (4) on the vector (8) representing the winding in Figure 3, the following vector is obtained: [V ′w]1 = 1 2 [ 1 0 0 0 −1 −1 0 0 0 ] (10) The sum of [V ′w]1 elements is nonzero, and thus this winding is an α-winding (i.e., odd false-zero winding). 2.2.3. β-windings (i.e., even false-zero windings) A false-zero winding is even, if forward (+1) and return (−1) conductor numbers are equal, viz.,∑ u [V ′w]1 (u) = 0 (11) The winding given in Figure 4 is represented by: [Vw] = 1 2   1 –1 0 −1 0 0 0 . . .. . . 0 0 0 −1 0 –1 . . . . . . 1 0 1 0 0 0 . . . . . . 0 0 0 1 0   (12) By applying (4) to (12), the sub-vector is obtained: [V ′w]1 = 1 2 [ 1 0 0 0 0 −1 . . . . . . −1 0 0 0 0 1 ] (13) The sum of [V ′w]1 elements is zero and thus this wind- ing is a β-winding (i.e., even false-zero winding). 3. Winding Factor Calculation From the usual calculation of the winding factor based on the ditribution and pitch factors [2] and formulas given in [20], an adaptation on [Vw] is made. A generalized equation was obtained regardless of the number of slots, phases and layers. For the sake of clarity of the winding factor solution for each case, Figure 4. An example of a one phase distribution for a β-winding with Qs = 12, m = 3 and Ly = 2. 155 D. Ouamara, F. Dubas, S.A. Randi et al. Acta Polytechnica the following notations are adopted throughout the paper n = u⊗ [O]1,Ly (14) s = 1, . . . ,Qs ·Ly (15) where ⊗ represents the Kronecker product, and [O]1,Ly the ones matrix of dimensions 1 ×Ly. The flowchart of the method is presented in Fig- ure 5, showing steps of the calculation and the used equations. First, the vector [Vw] of the phase spatial distribution is tested if it is a false-zero winding. Then, the studied winding is categorized and the adequate equation is applied. 3.1. γ-Windings 3.1.1. Description For all single layer windings and multi-layer wind- ings with no false-zero, the winding factor is given by: ξγwh = m Qs × ∣∣∣∣∣∑ s [Vw](s) ×e−j 2π Qs ×n(s)×h ∣∣∣∣∣ (16) where j = √ −1, and h is the spatial harmonic orders. 3.1.2. Application : Qs = 20, m = 5 and Ly = 2 • Step 1: The distribution of the first phase of an γ-winding is given in Figure 6.(a) and represented by: [Vw] = 1 2   0 1 −1 −1 1 0 0 . . . . . . 0 0 0 0 0 0 . . . . . . 0 0 0 0 0 0 . . . . . . 0 0 −1 1 1 −1 . . . . . . 0 0 0 0 0 0 . . . . . . 0 0 0 0 0 0 . . . . . . 0 0 0   (17) • Step 2: According to (16), the calculation of the winding factor ξγwh is given in Figure 6.(b). Figure 5. Flowchart of the methodology. (a) (b) Figure 6. γ-winding with Qs = 20, m = 5 and Ly = 2: (a) First phase distribution, and (b) Winding factor ξγwh. 3.2. β-Windings (i.e., Even False-Zero Windings) 3.2.1. Description The winding factor for this category is given by: ξβwh = Ly × ξ γ wh (18) 3.2.2. Application : Qs = 12, m = 3 and Ly = 2 • Step 1: The distribution of the first phase of a β-winding is given in Figure 7.(a) and represented by: [Vw] = 1 2   1 –1 0 −1 0 0 . . . . . . 0 0 0 0 −1 . . . . . . 0 –1 1 0 1 . . . . . . 0 0 0 0 0 . . . . . . 0 1 0   (19) • Step 2: The sub-vector extracted is: [V ′w]1 = 1 2 [ 1 0 0 0 0 −1 . . . . . . −1 0 0 0 0 1 ] (20) • Step 3: The forward (+1) and return (−1) conduc- tor numbers are equal, viz.,∑ u [V ′w]1 (u) = 0 (21) • Step 4: According to (18), the calculation of the winding factor ξβwh is given in Figure 7.(b). 156 vol. 59 no. 2/2019 General Calculation of Winding Factor (a) (b) Figure 7. β-winding with Qs = 12, m = 3 and Ly = 2: (a) First phase distribution, and (b) Winding factor ξβwh. 3.3. α-Windings (i.e., Odd False-Zero Windings) 3.3.1. Description The winding factor for this category is given by: ξαwh = c× ξ β wh (22) where c = Qs/λ with λ a correction coefficient defined by: λ = { tz+1, for tz < Qs < tz+1 tz, for tz = Qs (23) in which t =   3, for z = 1 6, for z = 2 12 · (z − 2) , for z > 2 (24) with z ∈ N∗. 3.3.2. Application : Qs = 15, m = 3 and Ly = 2 • Step 1: The spatial distribution of the first phase of an α-winding is given in Figure 8.(a) and repre- sented by: [Vw] = 1 2   1 0 0 −1 0 0 0 0 . . .. . . −1 0 0 1 0 0 1 . . . . . . 0 1 –1 0 −1 0 0 . . . . . . −1 0 0 1 0 0 0 0   (25) • Step 2: The sub-vector extracted is: [V ′w]1 = 1 2 [ 1 0 0 0 −1 0 0 1 . . . . . . 1 0 0 −1 0 0 0 ] (26) (a) (b) Figure 8. α-winding with Qs = 15, m = 3 and Ly = 2: (a) First phase distribution, and (b) Winding factor ξαwh. • Step 3: The forward (+1) and return (−1) conduc- tor numbers are unequal, viz.,∑ u [V ′w]1 (u) 6= 0 (27) • Step 4: The coefficient λ = 24 which gives c = 0.625. • Step 5: The calculation of the winding factor ξαwh is given by Figure 8.(b). 4. Multi-Layer Windings In order to validate the method described in this paper for the winding factor calculation with Ly > 2, specific wingings, in the literature, has been used and compared with the results obtained by the developed method. 4.1. Windings extracted in Qi et al. [15] The authors of this paper (Qi et al.) presented a new multi-layer winding design method based on the wind- ing function. The multi-layer winding is obtained by a superposition of several double-layer windings. The example given in [15] is an 18-slots/16-poles fractional- slot concentrated-windings permanent-magnet ma- chine. Three multi-layer winding designs are obtained using the proposed method. The winding arrange- ments are shown in Figure 9, viz., i) a double layer, ii) two four-layer windings (i.e., Type1 and Type2), and ii) a six-layer winding. 157 D. Ouamara, F. Dubas, S.A. Randi et al. Acta Polytechnica Figure 9. Different type multi-layer windings (for Qs = 9 and m = 3) [15]. (a) Double layer winding (b) Type1 four-layer winding (c) Type2 four-layer winding (d) Six-layer winding Figure 10. Winding factor calculated by Qi et al. [15] and method of this paper. The winding factor is calculated for each winding and the results are compared with those calculated by Qi et al. [15] as shown in Figure 10. Except for some harmonics for which the negative direction is taken, the amplitude is the same between the compared results for each winding. 4.2. Windings extracted in Alberti et al. [16] A general theory of fractional-slot multi-layer wind- ings has been presented by Alberti et al. [16]. By applying the Star of slots method, two 9-slots/8-poles four-layer windings (see Figures 11.(b) and 11.(c)) are obtained from a 9-slots/8-poles two-layer winding (see Figure 11.(a)). In [16], the winding factor for four-layer windings is computed starting from the one of the two-layer winding with a shift angle and based on the Star of slots method. Compared results of the calculation are reported in Figure 12, where the same amplitude of the winding factor is obtained for each harmonic in both methods of the calculation. (a) 2-layer (b) 4-layer i (c) 4-layer ii Figure 11. Winding layout of phase A (for Qs = 9 and m = 3) [16]. 158 vol. 59 no. 2/2019 General Calculation of Winding Factor (a) 2-layer (b) 4-layer i (c) 4-layer ii Figure 12. Winding factor calculated by Alberti et al. [16] and method of this paper. 4.3. Windings extracted in Cistelecan et al. [17] In [17], the winding diagram for a 12-slots/10-poles single layer layout is presented in Figure 13.(a). The double layer presented in Figure 13.(b) may be ob- tained from the single layer winding by doubling the first with a half number of turns per coil and shifting by 5 geometrical slots, as explained in [17]. By dou- Figure 13. Winding diagrams for 12-slots/10-poles (only one phase is represented) [17]: (a) Single layer, (b) Double layer, and (c) Four-layer winding. (a) Single layer winding (b) Double layer winding (c) Four-layer winding Figure 14. Winding factor calculated by Cistelecan et al. [17] and by the method of this paper. bling again the double-layer winding and shifting by another 5 geometrical slots, a four-layer winding may be obtained as shown in Figure 13.(c). The winding factor has been calculated using an analytical method based on Star of slots and compared with the calculation made in this paper, results are identical. A comparison of winding factors obtained from both methods is presented in Figure 14. A special 12-slots/10-poles three layer winding is presented in [17] and shown in Figure 15. This winding has an unequal number of turns. In the developed method, the number of turns is sup- posed to be equal. The small difference in the results of the calculation of the winding factor, as given in Figure 16, is due to the unequal number of turns not taken into consideration in this paper. Figure 15. Winding diagrams for 12-slots/10-poles : Special three layer [17]. 159 D. Ouamara, F. Dubas, S.A. Randi et al. Acta Polytechnica Figure 16. Winding factor calculated for the special three layer by Cistelecan et al. [17] and by the method of this paper. 5. Conclusion A general and simple method for the calculation of the winding factor of electrical machines has been presented. This calculation is based on the matrix representation of a winding. Knowing the distribution of phases as well as the number of slots/phases/layers, the winding factor’s calculation is possible for all types of balanced windings with equal turns. The method has been tested on the multi-phase windings (the three phase included) and for any number of layers. Unlike the methods existing in literature, the number of pole pairs is not necessary, which allows the seperation of the rotor and stator design. The results of the comparisons are satisfying. This paper gives a useful tool for users to estimate the performance of a winding, focusing only on the stator parameters (i.e., number of slots, phases and layers), without the constraints on the number of pole pairs at the rotor. Since this method is limited to windings of an equal number of turns, future work will include expanding its potential to allow the computation of windings with an unequal number of turns. The hybrid windings having unequal number of layers (e.g., winding with three-four layers) may also be considered in future studies. 6. APPENDIX clear all, close all, clc, Winding’s Parameters titreFig=’Winding’’s parameters’; titreProp={’Qs : Number of slots’;... ’m : Number of phases’;... ’Ly : Number of layers’;... ’Vw : Vector of distribution of phase A’}; num_lines=1; val_init={’’;’’;’’;’’}; dataCell=inputdlg(titreProp,titreFig,... num_lines,val_init,’on’); Qs=str2double(dataCell{1}); m=str2double(dataCell{2}); Ly=str2double(dataCell{3}); Vw=str2num(dataCell{4}); Begin of Calculation h=30; % Number of harmonics Ns=Qs*Ly; n=kron(1:Qs,ones(1,Ly)); % Vector n for h=1:h s=1:Ns; % Vector s fabs{h} = abs((sum(Vw(s).* exp((-1i*2*pi*n(s)*h)/Qs)))); end fabs = cell2mat(fabs); % Detect a false-zero winding x=1:2:Ns-1; vect=Vw(x).*Vw(x+1); % Detect if an Alpha-/Beta-Winding Vw_p=Vw(x); sum=sum(Vw_p); % Alpha-Winding if (Ly > 1) && (min(vect)<0) && (sum~=0) % Calculation of coefficient lambda t1=[3 6]; t2=12:12:240; t=[t1 t2]; for z=1:size(t,2); if Qs > t(z) && Qs < t(z+1) lambda=t(z+1); elseif Qs >= t(z) && Qs < t(z+1) lambda=t(z); end end % Winding factor of Alpha-Winding fw=((m*Ly)/lambda)*fabs; % Beta-Winding elseif (Ly > 1) && (min(vect)<0) && (sum==0) % Winding factor of Beta-Winding fw=((m*Ly)/Qs)*fabs; else % Gamma-Winding % Winding factor of Gamma-Winding fw=(m/Qs)*fabs; end idx= ( fw < 1e-10); fw(idx)=0.0000; Post-Processing Create figure with bars f=figure; axeFig=axes(’Parent’,f,’Position’, [.1 .19 0.8 .74],’XLim’,[0,5],’YLim’,[0,5],... ’XGrid’,’on’,’YGrid’,’on’,’XMinorGrid’,’off’); stem(1:h,fw(1:h),’b’,’Marker’,’none’, ... ’LineWidth’,10); set(gca,’YGrid’,’on’); xlabel(’Harmonics’,’FontName’, ’Garamond’,... ’FontSize’, 11); xlim([0 31]), set(gca,’XTick’, [0:2:30]); 160 vol. 59 no. 2/2019 General Calculation of Winding Factor ylabel(’Amplitude [-]’,’FontName’, ... ’Garamond’, ’FontSize’, 11 ); ylim([0 1]); title(’Winding Factor : \xi_{w}’, ... ’FontName’,’Garamond’,’Color’, ’red’, ... ’FontSize’, 18); % Create the uitable and display fw t = uitable(’Parent’, f,’Data’,fw, ... ’FontName’,’Garamond’ ,’FontSize’, 14, ... ’Position’,[30 10 1300 80], ... ’RearrangeableColumns’,’on’, ... ’ForegroundColor’, ... 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The European Physical Journal Applied Physics 50(3):31102, 2010. doi:10.1051/epjap/2010058. 161 http://dx.doi.org/10.14311/AP.2018.58.0037 http://dx.doi.org/10.1109/IEMDC.2003.1211284 http://dx.doi.org/10.1109/T-AIEE.1943.5058623 http://dx.doi.org/10.1109/TIA.2006.876070 http://dx.doi.org/10.1109/ICELMACH.2008.4799937 http://dx.doi.org/10.4304/jcp.8.3.725-732 http://dx.doi.org/10.1109/TMAG.2013.2279100 http://dx.doi.org/10.1049/iet-epa.2011.0292 http://dx.doi.org/10.1109/IEMDC.1999.769226 http://dx.doi.org/10.1049/iet-epa.2015.0092 http://dx.doi.org/10.1109/ITEC-AP.2014.6940615 http://dx.doi.org/10.1109/TIA.2013.2242031 http://dx.doi.org/10.1109/ECCE.2010.5618267 http://dx.doi.org/10.1007/BF01236235 http://dx.doi.org/10.1051/epjap/2010058 Acta Polytechnica 59(2):153–161, 2019 1 Introduction 1.1 Context 1.2 Objective of this Paper 2 Notions of the Method 2.1 Matrix Representation 2.2 Notion of False-Zero 2.2.1 Definition 2.2.2 -windings (i.e., odd false-zero windings) 2.2.3 -windings (i.e., even false-zero windings) 3 Winding Factor Calculation 3.1 -Windings 3.1.1 Description 3.1.2 Application : Qs=20, m=5 and Ly=2 3.2 -Windings (i.e., Even False-Zero Windings) 3.2.1 Description 3.2.2 Application : Qs=12, m=3 and Ly=2 3.3 -Windings (i.e., Odd False-Zero Windings) 3.3.1 Description 3.3.2 Application : Qs=15, m=3 and Ly=2 4 Multi-Layer Windings 4.1 Windings extracted in Qi et al. Qi2014 4.2 Windings extracted in Alberti et al. Alberti2013 4.3 Windings extracted in Cistelecan et al. Cistelecan2010 5 Conclusion 6 APPENDIX References